JAIST Repository
https://dspace.jaist.ac.jp/

Title

ゲームエンターテイメントと報酬の計算手法の評価

Author(s)高, 娜穎CitationIssue Date

2024-03

Type

Thesis or Dissertation

Text version

ETD

URL

http://hdl.handle.net/10119/19064

RightsDescription

Supervisor: 飯田 弘之, 先端科学技術研究科, 博士

Japan Advanced Institute of Science and Technology

Doctoral Dissertation

Computational Measures of Game Entertainment and Reward

GAO Naying

Supervisor: Hiroyuki Iida

Graduate School of Advanced Science and Technology
Japan Advanced Institute of Science and Technology
Information Science

March 2024

Supervised by
Prof.Dr.Hiroyuki Iida

Reviewed by
Prof.Dr.Kokolo Ikeda
Prof.Dr.Kiyoaki ShiraiProf.Dr.Jin Yoshimura
Prof.Dr.Masnizah Mohd

Abstract
The Game Refinement (GR) theory, widely employed for assessing the enjoyment andintricacy of games, is herein examined through the lens of audience perception. This
theory delves into the cognitive information processes occurring in our minds while engaging in strategy games, in adherence to the principles set forth by Newton’s laws. Its
central focus is on the uncertainty surrounding game outcomes, employing abstraction
to dissect game components and assess the progression of game information based onachievements and game duration.
The quantification of this progression leads to the measurement of informational
acceleration, akin to Newton’s second law (F = ma), wherein informational acceleration correlates with mental force and excitement. Changes in acceleration significantly
impact game dynamics, especially in high-stress scenarios, and can be rectified using“jerk” values derived from acceleration readings. In this context, a comprehensive exploration of various physical factors influencing psychological experiences in gaming is
conducted.
This study prominently emphasizes the intersection of physical and psychologicalelements by depicting the progression of game information within our minds. By establishing reasonable informational jerk values associated with the success rate andinformational acceleration of the game process, the development of game refinement
theory enhances the overall player experience.
The concept of “jerk”, signifying a sudden change in acceleration, is a fascinating
phenomenon experienced by humans and possesses practical applications in mechanicaliand engineering domains, such as elevator rides. The paper forges a connection between
the logistical model of game progression and “jerk” measurements, particularly in gameswith incomplete information, typified as AD values (representing addiction and thepropensity for repeated gameplay).
To investigate the implications of AD values concerning GR metrics and its extension, the motion in mind model, several popular card games, including Wakeng,
Doudizhu, Winner, Big Two, and Tien Len, are utilized. Self-playing simulations conducted by artificial intelligence (AI) agents validate these findings with empirical data,
yielding valuable insights into game design and gameplay experiences.
Players of varying skill levels may encounter and endure diverse reward frequencies denoted as N and jerk values during the progression of game information. Thus,
an exploration into the interplay between the player’s performance level (k), reward
frequency (N ), and jerk values is undertaken.
The inherent game risk, often represented as “m” is a well-acknowledged factor.
To overcome these challenges, players must skillfully devise strategies. Exceptionallyskilled players can navigate uncertainties with ease, characterized by a game velocity
represented as v⃗0 = 1 − m.
However, players possess differing abilities, resulting in varying levels of proficiency
in dealing with uncertainties. Prior research employs “k” as a measure of performance,
where higher values indicate lower performance. For imperfect players, the game velocity can be calculated as v⃗k = (1 − km)v⃗0 = (1 − km)(1 − m).
In the realm of reinforcement learning, “N ” often signifies the “number of steps”
or “time steps”, representing the interval for acquiring rewards. In games with perfect
information, players possess complete knowledge of moves, while imperfect information
games entail partial insights or educated guesses. In such instances, “N ” can symbolize
the number of actions considered during decision-making, akin to the Monte Carlo Tree
Search algorithm.
In this study, the Game Refinement (GR) theory is explored from the perspectiveii

of the audience. The theory focuses on the cognitive processes that occur while playing strategy games, emphasizing the uncertainty of game outcomes and how it affects
player experience. It introduces the concept of “jerk” to measure sudden changes inacceleration in games. The study investigates the relationship between “AD” values,
representing addiction and repeat gameplay, and GR metrics in various card games.
Additionally, it delves into the interplay of player performance (k), reward frequency(N ), and unpredictability (AD) in gaming. This research provides valuable insights
into game design and gameplay experiences.
Keyword: Game refinement theory;Motion in mind ;Perfect information game;Imperfect
information game;Card game;Jerk ;Addiction;iii

Acknowledgments
First and foremost, I want to express my deep gratitude to my esteemed supervisor,
Professor Hiroyuki Iida. As a distinguished figure in the field of Artificial Intelligence,
he not only possesses a vast reservoir of expertise and research acumen but also showcases remarkable versatility across various knowledge domains. Despite his demandingrole as the president at JAIST, he consistently finds time to mentor and guide us,
his students, in our research pursuits, even while traveling for business commitments
abroad. His genuine concern for our well-being is evident through his generous gesturesof hosting dinners and organizing recreational activities, often at his expense. Whenever he encounters us, he unfailingly inquires, “May I be of assistance to you?” Such isthe caliber of my supervisor – not just a paragon of research prowess, but a benevolent
and approachable mentor who treats all students as his proteges. It is my privilege andfortune to be under Professor Iida’s tutelage. In moments of adversity, his unwavering
support and encouragement are akin to a paternal figure, imparting not only scholarly
knowledge and research skills but also the virtues of character. Without his guidance,
my academic journey would have remained incomplete.
I must also extend my heartfelt gratitude to my co-supervisor, Professor Kokolo
Ikeda. His expertise and unwavering commitment to academic rigor have been invaluable in my academic journey. Professor Ikeda has provided me with a multitude of
insightful questions and perspectives on my research topic, thereby enriching my intellectual horizons and significantly improving the quality of my research work. I am
profoundly thankful for his meticulous guidance in revising my thesis.
iv

Furthermore, my minor research supervisor, Professor Shirai Kiyoaki, deserves mygratitude for his dedicated mentorship during my master’s and doctoral studies. His
conscientious demeanor and wealth of experience have greatly contributed to my academic growth, enabling me to acquire specialized knowledge in hardware and broaden
the scope of my research perspectives.
I would like to extend my heartfelt appreciation to the esteemed members of mydissertation committee, Prof. Dr. Jin Yoshimura and Prof. Dr. Masnizah Mohd. Their
unwavering kindness and dedication to my academic journey have been invaluable.
Throughout my research, they have not only offered me highly tailored advice andguidance but have also provided me with an abundance of warm-hearted support. Theircommitment to my success has made a profound difference in my academic pursuits,
and I am truly grateful for their mentorship and encouragement.
I am profoundly grateful to my beloved parents, whose sacrifices and unwavering
support have served as the bedrock of my academic journey. Their unwavering understanding and encouragement have been pivotal to my accomplishments. My mother’snurturing care, reminiscent of a devoted guardian, has been my steadfast source ofstrength. She celebrates my moments of joy and triumph, providing boundless comfort
and solace during periods of sorrow, discouragement, and uncertainty. Through her, I
have cultivated a profound sense of gratitude and happiness.
While my father may be stern and demanding, he unfailingly provides nothing but
the best for me. He embodies both strictness and benevolence in his role as a father.
To my dear parents, I extend my heartfelt thanks and wishes for their continued healthand happiness.
Additionally, I express gratitude to my older brother, with whom I shared thejourney of growing up. We mutually supported and encouraged each other, creating abond that has been instrumental in our development.
Lastly, I wish to express my gratitude to all the colleagues in our Iida laboratory, whohave become an extended family to me. I extend my appreciation for the invaluablev

assistance, cherished memories, and joyful moments you have provided. Under thesagacious leadership of Dr.Akmal, our senior colleague, we have cultivated a cultureof mutual support and benevolence, following the teachings of our esteemed mentor,
Professor Iida. Regardless of time or place, we stand united, and together, we are an
extraordinary team.

vi

Contents
Abstracti

Acknowledgments

iv

1 Introduction

1

1.1

Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Problem Statement and Research Questions . . . . . . . . . . . . . . .

2

1.3

Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52 Related Works

8

2.1

Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2Game Refinement Theory and Motion in Mind Concept . . . . . . . . .

8

2.2.1

Motion in mind model . . . . . . . . . . . . . . . . . . . . . . .

10

2.2.2

Jerk and comfort in mind . . . . . . . . . . . . . . . . . . . . .

13

2.3

Card Games as Testbed in This Study . . . . . . . . . . . . . . . . . .

15

2.4

Perfect Information Game and Imperfect Information Game . . . . . .

17

2.4.1

Perfect information game . . . . . . . . . . . . . . . . . . . . . .

17

2.4.2

Imperfect information game . . . . . . . . . . . . . . . . . . . .

18

Reward Frequency in Reinforcement Learning . . . . . . . . . . . . . .

19

2.5.1

Influence of reward frequency in gaming . . . . . . . . . . . . .

19

Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20

2.5

2.6

vii

3 A Computational Game Experience Analysis via Game Refinement
Theory22

3.1

Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

22

3.2

Measurement of Play in Games . . . . . . . . . . . . . . . . . . . . . .

25

3.2.1

Player Experiences in Games . . . . . . . . . . . . . . . . . . .

25

3.3

Player Psychology in Games . . . . . . . . . . . . . . . . . . . . . . . .

27

3.4

Game Refinement Theory and Its Development . . . . . . . . . . . . .

29

3.4.1

Game Refinement Theory and Its Development . . . . . . . . .

29

3.4.2

Jerk and Comfort in Mind . . . . . . . . . . . . . . . . . . . . .

30

Gamified Experience From Metaphysical Perspectives . . . . . . . . . .

34

3.5.1

Basketball . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

3.5.2

Soccer board . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

Physics and Psychophysiology Processes in Games . . . . . . . . . . . .

40

3.6.1

Interaction Dynamic . . . . . . . . . . . . . . . . . . . . . . . .

41

3.6.2

Game Playing Experience and Flow Theory . . . . . . . . . . .

43

3.6.3

Conceptual Basis of Motion in Mind . . . . . . . . . . . . . . .

47

3.6.4

Limitations and Future Works . . . . . . . . . . . . . . . . . . .

52

Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

53

3.5

3.6

3.7

4 Implications of Jerk’s On The Measure of Game’s Entertainment: Discovering Potentially Addictive Games

55

4.1

Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

4.2

Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

4.3

Analysis of Card Games . . . . . . . . . . . . . . . . . . . . . . . . . .

61

4.3.1

Suits irrelevant card games: Wakeng and Doudizhu . . . . . . .

62

4.3.2

Suits relevant card games: Big Two, Winner, and Tien Len . . .

65

Proposed Computational Models . . . . . . . . . . . . . . . . . . . . .

66

4.4.1

66

4.4

Game progress model . . . . . . . . . . . . . . . . . . . . . . . .

viii

4.5

4.6

4.7

4.4.2

Motion in mind model . . . . . . . . . . . . . . . . . . . . . . .

69

4.4.3

Experimental design . . . . . . . . . . . . . . . . . . . . . . . .

72

Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

74

4.5.1

Result analysis of DouDiZhu . . . . . . . . . . . . . . . . . . . .

74

4.5.2

Result analysis of Wakeng . . . . . . . . . . . . . . . . . . . . .

77

4.5.3

Result analysis of suits-relevant card games . . . . . . . . . . .

77

4.5.4

Result analysis of fixed AI levels . . . . . . . . . . . . . . . . . .

79

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.6.1

Comparison on different game complexities . . . . . . . . . . . .

81

4.6.2

GR and AD relative to addictive situation . . . . . . . . . . . .

84

Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

89

5 The Impact of Performance Degree on Players: Exploring the Dynamics of Player Engagement and Enjoyment in Game Process

92

5.1

Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

92

5.2

Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

93

5.2.1

The meaning of playing performance . . . . . . . . . . . . . . .

94

5.2.2

The acceleration ak of different kinds of players . . . . . . . . .

95

5.2.3

The balance of perfect player and imperfect player from the perspective of potential energy . . . . . . . . . . . . . . . . . . . .

95

Exploring optimal rounds for distinguishing real strength . . . .

97

Performance Degree (k): In-Depth Analysis . . . . . . . . . . . . . . .

100

5.3.1

The risk rate m and performance degree k . . . . . . . . . . . .

100

5.3.2

The explanation of the correspondence system of m and the max-

5.2.4
5.3

imum acceptable performance degree k . . . . . . . . . . . . . .
5.3.3

100

Comparison of motion in mind measures based on performance
degree k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ix

103

5.4

Dynamic Interaction between Performance Level k, Reward Frequency
N , and AD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

107

5.4.1

Performance level k and reward frequency N . . . . . . . . . . .

107

5.4.2

Performance level k and AD . . . . . . . . . . . . . . . . . . . .

108

5.4.3

Reward frequency N and AD . . . . . . . . . . . . . . . . . . .

109

5.4.4

The difference between perfect information games and imperfect
information games . . . . . . . . . . . . . . . . . . . . . . . . .

115

5.5

The influence of ratio ϕ (GR/AD) . . . . . . . . . . . . . . . . . . . . .

116

5.6

Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

6 Conclusion

126x

List of Figures
2-1 An illustration of move selection model based on skill and chance . . .

10

2-2 Objective and subjective reinforcement when k = 3 . . . . . . . . . . .

13

2-3 The cross point between the line with velocity v, curve with accelerationa and curve with jerk j. t1 ; t2 and t3 represent the bound for effort,
achievement, and discomfort, respectively. . . . . . . . . . . . . . . . .

15

3-1 The cross point between the line with velocity v, curve with accelerationa and curve with jerk j. t1 ; t2 and t3 represent the bound for effort,
achievement, and discomfort, respectively. . . . . . . . . . . . . . . . .

32

3-2 The cross point between the curves of the velocity v, acceleration a, andjerk j, where such a cross point describes the comfortable moment of
the basketball game. After the cross point, it can be observed that withenough training and skill, achieving rewards becomes easy. However, thefeeling of discomfort will also be higher due to boredom and insufficient
challenge. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38

3-3 Using board game to play soccer . . . . . . . . . . . . . . . . . . . . . .

39

3-4 Using a game tree model to visualize the scoring process . . . . . . . .

41

3-5 Challenge vs. Skill, illustrating the ”flow” region Source: English Wikipedia.
https://en.wikipedia.org/wiki/File:Challenge vs skill.jpg . . .

45

3-6 The description of game process using ∆ scores . . . . . . . . . . . . .

45

xi

3-7 The description of Flow theory using the game-playing process to associate the context of the expected experience when playing, based on
self’s (ability) and opponent’s (challenge) score. . . . . . . . . . . . .

46

3-8 The analysis of motions in mind based on dynamical scores gap . . . .

49

3-9 An example of the dynamic interactions and game-playing experience of
one game process based on its association with the Flow theory . . . .

52

4-1 Objective and subjective reinforcement when k = 3 . . . . . . . . . . .

72

4-2 The tendency of GR and AD based on the ability level of sophisticatedcard games. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

76

4-3 The tendency of GR and AD based on the ability level of classical card
games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

79

4-4 The tendency of GR and AD value of different complexity game . . . .

81

4-5 The relations between GR and AD . . . . . . . . . . . . . . . . . . . .

85

4-6 The relations between reward frequency (N ) and game length (D) . . .

86

4-7 The relations between reward frequency (N ) and AD . . . . . . . . . .

87

4-8 The crosspoint between fairness (y), reinforcement (v), entertainment(GR), and unpredictability (AD) . . . . . . . . . . . . . . . . . . . . .

88

5-1 Game progression velocity as a function of risk rate m . . . . . . . . . .

94

5-2 Motion in Mind Measures for k = 3 . . . . . . . . . . . . . . . . . . . .

96

5-3 Total and each round solved game uncertainty . . . . . . . . . . . . . .

99

5-4 Possible relation between performance degree and risk rate . . . . . . .

101

5-5 The comparison of energy in mind based on the player performance level k103
5-6 Motion in mind measure compared based on the performance degree ktransition from 3 to 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . .

105

5-7 Comparison of momentum in mind based on the player performance level k105
5-8 The comparison of subjective momentum in mind based on the player
performance level k . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii

106

5-9 Performance Level k and Reward Frequency N . . . . . . . . . . . . . .

108

5-10 The relations between k and AD . . . . . . . . . . . . . . . . . . . . .

109

5-11 The relationship between N and AD in the sports domain . . . . . . .

110

5-12 The relationship between N and AD in the board games domain . . . .

111

5-13 The relationship between N and AD in the card games domain . . . .

113

5-14 The relations between N , k and AD. . . . . . . . . . . . . . . . . . .

113

5-15 Gamification, Game and Competition . . . . . . . . . . . . . . . . . . .

122

xiii

List of Tables
2.1

Measures of game refinement for board games . . . . . . . . . . . . . .

14

3.1

Measures of game refinement for popular board games, adopted from [1]

32

3.2

Contextual correspondence between game information progress, Newton
dynamics, and their link . . . . . . . . . . . . . . . . . . . . . . . . . .

35

3.3

Contextual link between physics, games, and psychology . . . . . . . .

36

3.4

Quintessential two-sided time-limited shooting game - basketball (adopted
from [2] and basketball reference website⋆ ) . . . . . . . . . . . . . . . .

37

3.5

Measures of game refinement for soccer game according to [3] . . . . . .

39

3.6

Links between board game to soccer, where D is total shots, B is average
feasible options, B1 is average promising options (i.e., n is assumed as√
ideal options n = B1 ), GR is the informational acceleration, and AD
is the informational jerk. . . . . . . . . . . . . . . . . . . . . . . . . . .

3.7

The Correspondence between Game Context and Non-Game Context
using S (Scores) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.8

39

44

Analogical translation between motion in minds, its game-playing implications, and its psycho-physiological context . . . . . . . . . . . . . . .

47

3.9

Dynamical emotions and the corresponding description . . . . . . . . .

51

4.1

Comparison of card games considered in this study . . . . . . . . . . .

62

4.2

The card types of Wakeng . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.3

The card types of Doudizhu . . . . . . . . . . . . . . . . . . . . . . . .

64

xiv

4.4

Measures of game refinement for board games . . . . . . . . . . . . . .

69

4.5

The experiment design of Wakeng and Doudizhu . . . . . . . . . . . . .

74

4.6

Measures of game refinement for classical DouDiZhu . . . . . . . . . . .

75

4.7

Results of different levels of AI with different DouDiZhu game settings

75

4.8

The analysis of Wakeng based on different level AI based on the settingof (3, 1, 2)(20,16,16) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.9

77

The analysis of several card games based on different level AI and setting 78

4.10 Closeness to reasonable zone of GR and AD given by different levels of
players of different game complexity . . . . . . . . . . . . . . . . . . . .

80

4.11 The entertainment aspects of different GR and AD value expression . .

80

4.13 The comparison of suit-relevant card games based on a standard setting

82

4.12 The comparison of suit irrelevant based on a standard setting . . . . .

82

4.14 Summary of motions in mind measures of different games . . . . . . . .

83

4.15 Possible corresponding games . . . . . . . . . . . . . . . . . . . . . . .

89

4.16 Possible principle of game element . . . . . . . . . . . . . . . . . . . . .

89

5.1

Relations between m and k. . . . . . . . . . . . . . . . . . . . . . . .

102

5.2

The relationship between N and AD in the sport games domain . . . .

110

5.3

The relationship between N and AD in the board games domain . . . .

111

5.4

The relationship between N and AD in the card games domain . . . .

112

5.5

Correlation between player performance and game characteristics . . .

114

5.6

Comparison between Perfect Information Games and Imperfect Information Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

5.7

Measures of game refinement for board games [4] . . . . . . . . . . . .

117

5.8

Comparison of Card Games using GR and AD values [5] . . . . . . . .

117

5.9

Comparison of Basketball and Soccer using GR and AD values [6] . . .

118

5.10 Comparison of Hotels using GR and AD values [7] . . . . . . . . . . . .

119

5.11 Comparison of Languages in Duolingo using GR and AD values [8] . .

120

xv

5.12 Game types compared using ϕ values . . . . . . . . . . . . . . . . . . .

xvi

121

Chapter 1
Introduction
1.1

Chapter Introduction

Game Refinement Theory, originally proposed by Professor Hiroyuki Iida in 2004, isa widely recognized framework that utilizes the game progress model to analyze theuncertainty of game outcomes [4]. It introduces the concept of the “GR value” asa standard measure for assessing the entertainment value of games, with a desirable
range typically falling within GR ∈ [0.07, 0.08]. Furthermore, this theory has foundapplications in diverse domains such as business evaluation [9], educational assessment[10], and even in examining the evolution of popular board games [4] and [11].
In recent years, Game Refinement Theory has evolved and expanded its scope toconsider the analogy between in-game actions and physical movements, shedding lighton the connection between these actions and players’ psychological experiences. This
research aims to investigate how different ranges of in-game movements can affect players’ comfort and immersion. Achieving a state of comfort during gameplay is crucial
because it can lead players to enter a state akin to “flow” where they are deeply engaged
and motivated to continue playing and start the next round.

1

1.2

Problem Statement and Research Questions

In the world of gaming, players and their opponents come together in carefully orchestrated conflicts for the sake of entertainment, all governed by a set of rules and leading
to quantifiable outcomes. Traditionally, these games have hinged on crucial elements
such as difficulty levels and reward systems.
Drawing from previous research on game refinement theory, we have gained insight
into the information science aspect of games. Games can be seen as a journey from astate of uncertainty to one of certainty, usually signaled by achieving a goal or determining a winner. The duration of the game, in terms of time, plays a pivotal role inshaping the gaming experience and the excitement surrounding the outcomes. Games
that are excessively short or overly long tend to be seen as unreliable or tiresome.
This research delves into the realm of quantifiable measures, borrowing from theworld of physical analogies, to understand the inner workings of our cognitive processes
while engaging in distinct gaming scenarios. It seeks to gauge the level of uncertainty
and unpredictability that players encounter, all while maintaining a broader perspectiveon the logic of these processes. The ultimate goal of this study is to create a formal
model that can be applied to enhance the design and analysis of future games. A
key contribution of this research lies in its transformation of temporal aspects, such as
game duration, into contextual forms, such as search space and information uncertainty.
These contextual forms are leveraged to characterize the psycho-physiological processes
that underlie our interaction with information progression during gameplay.
Moving forward, the discussion explores the concept of “jerk” within the context ofacceleration and its multifaceted applications as a parameter for comfort and safety invarious domains. This segues into the examination of game refinement theory, with its
central focus on the rate of change of information speed, known as the GR value, withingames. This theory provides insights into how game progress unfolds and how players
make choices, with a specific emphasis on understanding uncertainty and success rates.

2

The narrative then shifts its attention to card games, particularly those characterized by incomplete information. These games are known for their brevity, replayability,
reliance on chance, and strategic depth. They have enjoyed a surge in popularity inthe digital era, challenging players to navigate complex game states, remember theiropponents’ moves, and make probabilistic assessments.
The core objective of this study is to explore the addictive nature of these card
games. Here, “addictive” refers to the intense enthusiasm and motivation that players
exhibit when repeatedly engaging in these games, rather than indicating a psychologicalor substance-related addiction. Through simulations, the research investigates how
different dynamics of game progress (GR) and unpredictability (AD) values impact
player experiences. It also examines how varying levels of AI skill and player emotional
responses, such as engagement and surprise, manifest in different rule scenarios. The
ultimate aim is to identify potential metrics related to these emotional experiences, allgrounded in the context of card games.
Furthermore, this study delves into a comparison of the psychological states andimmersion levels of players with varying skill levels when confronted with games featuring different reward frequencies. It seeks to understand how reward frequency becomesa controllable or enriching factor as players refine their skills and explore various facetsof gameplay. This comprehensive examination uncovers the interconnectedness of theseelements, highlighting how they mutually influence and continually enhance the gaming
experience.
Consequently, this thesis aims to address the following research questions:• Apart from acceleration, what role do other physical quantities play during gameplay?
• How does motion in mind processing of movement function when players train or
explore games of varying complexity?
• For players with different skill levels, how do players perceive the game when faced3

with different reward frequencies?

1.3

Research Objectives

The second to fourth chapters are dedicated to addressing three specific objectives
outlined in the introduction, providing comprehensive answers and explanations forthe research questions posed, the research objectives are introduced, highlighting theneed to expand the game refinement theory and investigate the role of various physicalquantities in the gaming process.
Chapter 2 focuses on the first objective, aiming to extend the existing game refinement theory. The game process is developed, starting with the first derivative of
acceleration, jerk. The expanded AD value is defined concerning the GR value. Concepts and calculation methods of AD are presented, along with an exploration of how
speed, acceleration, and jerk influence the game progression and subsequently impact
the player. Visualization techniques are employed to illustrate the game’s progress.
The second objective is addressed in Chapter 3, which delves into the implicationsof jerk on the measure of game entertainment. The game information process is established, and solutions for each physical quantity are identified. Card games serve as a
medium to study player abilities in the face of varying game complexities, observing
changes in the AD value (jerk) and discerning its impact on the gaming experience. The
conjecture that jerk measures unpredictability and surprise in a physical environmentis explored and validated using card game data. Findings reveal that strong players
exhibit smaller AD values for complex games, demonstrating adaptability and control
over the gaming environment.
The third objective is explored in Chapter 4, focusing on the interplay between
game design (specifically, reward frequency), player abilities, and AD (motion in mind).
The relationship between individuals and the overall gaming experience is analyzed.
Insightful correlations between game design elements, player capabilities, and the AD

4value are uncovered, providing a comprehensive understanding of how these factors
collectively contribute to the gaming experience.
In our study, we aim to delve into the impact of physical quantities beyond acceleration on the gameplay experience. We will investigate the cognitive processingof in-game movements and how they influence players’ performance and engagement,
especially in the context of games with varying levels of complexity. The research concludes by summarizing the key findings and contributions of each chapter, emphasizing
the broader implications of the study for game refinement theory, player experience,and game design.

1.4

Thesis StructureThis thesis comprises six main chapters, given in the following structure.
• Chapter 1: Introduction
The primary aim of this chapter is to provide an overarching view of the research, encompassing key definitions, the interrelationships among the core keywords within the study, and a concise historical context of the relevant domain.
This chapter is instrumental in elucidating the central problem that the researchendeavors to address. Additionally, the Introduction chapter encompasses the articulation of research questions, along with the objectives and significance of thestudy. Finally, this chapter concludes with an outline of the dissertation’s structure.
• Chapter 2: Related Works
This chapter reviews the theoretical background relevant to the present study andprovides an overview of the latest research in the field. The first section of thischapter briefly introduces the subsequent literature review, followed by several
5

sections that delve into necessarily related works, such as Game refinement theory,
motion in mind, and the game models that will be employed in the subsequent
analysis and so on.
• Chapter 3: A Computational Game Experience Analysis

via Game Refinement Theory.
In this chapter, dynamic variation and process visualization are utilized throughinteractive games involving basketball and soccer to extend and elaborate upon
the game refinement (GR) theory. The exploration begins with acceleration andextends to jerk, allowing players to not only experience forces (represented as attractiveness within the game) but also perceive changes in those forces (variations
in gravitational attraction). Here, the focus is not solely on the mean value of acceleration but also on its fluctuations, emphasizing the importance of understanding
both aspects.
• Chapter 4: Implications of Jerk’s on the Measure of Game’sEntertainment: Discovering Potentially Addictive GamesIn this chapter, we primarily use card games as a medium to compare playersof different skill levels horizontally and vertically. When players face games ofvarying complexities, they have the opportunity to enhance their skills or explore
different strategies during training. We delve into the players’ distinct perceptions
throughout this process and examine the relationship between physical attributes
in the information process and their psychological perceptions.
• Chapter 5: The Impact of Performance Degree on Players:

Exploring the Dynamics of Player Engagement and Enjoyment in Game Process6In this chapter, we use player skill level as a baseline to explore games of varying
difficulty levels. When facing both incomplete information games and complete
information games, different games exhibit varying reward frequencies, leading todifferences in player immersion and the addictive nature of repeated play. We
examine the relationships between player skill level (k), reward frequency (N ),and the degree of addiction (AD) experienced by players.
• Chapter 6: Conclusion
The last chapter is the conclusion of the dissertation. It concludes the whole
dissertation relative to the main aim and objectives of the dissertation. Somepotential future works are also outlined.

7

Chapter 2
Related Works
2.1

Chapter Introduction

This chapter reviews the theoretical background related to this research. It introduces
the latest research results in this field, which include the influence of game refinement
theory and its new perspective, the impact of jerk during the game process, motion inmind, the methods of data collection we used, and so on.

2.2

Game Refinement Theory and Motion in Mind
Concept

The game progress model of game uncertainty is based on early work by [4]. It has
been previously applied to measuring the design sophistication in domains of business[9], and education [10], and acts as a tool for exploring the evolution of popular board
games [4] [11]. The GR values for most popular games are located in a reasonablezone of GR ∈ [0.07, 0.08]. From the player’s viewpoint, the information on the game
result is an increasing function of time (the number of moves in board games) t. Here,the information on the game result is defined as the amount of solved uncertainty (or8information obtained) x(t), as given by (2.1). The parameter n (where 1 ≤ n ∈ N ) is
the number of possible options and x(0) = 0 and x(T ) = 1.

x′ (t) =

n
x(t)
t

(2.1)

x(T ) stands for the normalized amount of solved uncertainty. Note that 0 ≤ t ≤ T ,0 ≤ x(t) ≤ 1. Equation (2.1) implies that the rate of increase in the solved informationx′ (t) is proportional to x(t) and inversely proportional to t. Solving (2.1), (2.2) isobtained.
 ntx(t) =
T(2.2)

It is assumed that the solved information x(t) is twice derivable at t ∈ [0, T ]. The
second derivative of (2.2) indicates the accelerated velocity of the solved uncertainty
along the game progress, which is given by (2.3).

x′′ (t) =n (n − 1) n−2
t
|t=T
Tn

=

n (n − 1)
T2

(2.3)

Accelerated velocity implies the difference in the rate of acquired information during
the game’s progress. Then, the acceleration motion or free-fall motion in mind, a isgiven by (2.4). In the domain of board games, a is approximated as (2.5), where√B(n = B) and D stand for the average number of possible moves and game length,
respectively.

a=1x(t) = at2
2

(2.4)

n (n − 1)
B
≈ 2
2
T
D

(2.5)

Figure 2-1 illustrates a model of move candidate selection based on skill and chance.
This illustration shows that skillful players would consider a set of fewer plausible
9

candidates (say b) among all possible moves (say B) to find a move to play, and that
there is a core part of its original game with branching factor B. The core part is astochastic game with a smaller branching factor, b since it is assumed that each amongb candidates may be equally selected.

Figure 2-1: An illustration of move selection model based on skill and chance

2.2.1

Motion in mind model

In operant conditioning, a variable-ratio schedule is a schedule of reinforcement wherea response is reinforced after an unpredictable number of responses [12] [13]. This typeof schedule creates a steady, high rate of reaction. Stochastic games such as gambling
and lottery games are typical examples of a reward based on a variable-ratio schedule.
Mind sports [6] games such as chess and Go were also essentially stochastic games
while applying the move selection model [1]. This situation implies that a reward ofvariable-ratio reinforcement schedule characterizes a game [14].
Therefore, the reward function of a game can be characterized by defining the reinforcement schedule’s variable rate (denoted as V R(N )). Then, velocity v (win rate)
and mass m (win hardness) of the motion in mind model are given by (4.11), where0 ≤ v ≤ 1 and 0 ≤ m ≤ 1. As such, N was used to measure the frequency of getting
rewards, where the player can get a reward in a total average of N steps [15]. Let v0 be
the reward function over various masses for the perfect player as given by (4.12), whichcorresponds to the objectivity of play. Note that there is a distinctive computation of
the v for the board and scoring games [1]. The success rate is defined as v = G
for scorT
ing games (such as basketball, soccer, etc.), where G and T are the average successful

10

Bscore and the total shoot attempts. Meanwhile, the success rate is defined as v = 2D

in a board game (i.e., Chess, Shogi, etc.), where B is the average branching factor, andD is the average game length.
1
N

and m = 1 − v, where 1 ≤ N ∈ R(2.6)

m + v0 = 1, where 0 ≤ m ≤ 1 and 0 ≤ v0 ≤ 1

(2.7)

v=

The notion of energy conservation had been proposed by [16] as a potential measureof engagement, where the formulation of momentum in the game (p⃗1 ) and potential
energy in mind (Ep ) are given by (4.13) and (4.14), respectively. Then, the momentum
in mind (p⃗2 ) can be derived based on the conservation of energy in mind, given by(4.15), which is associated with the measure of player’s engagement, given by (4.16).

p⃗1 = mv

(2.8)

Ep = 2mv 2(2.9)

Ep = p⃗1 + p⃗2(2.10)

p⃗2 = Ep − p⃗1 = 2m3 − 3m2 + m(2.11)

Applying (4.16) while assuming p⃗2 = mv2 , the subjective velocity v2 is given by(4.17). Let vk (m) be a reward function over various m for a player with ability parameterk. Then, the relation is generalized as vk using a parameter (say k where 0 ≤ k ∈ R) thatis the nature of the game under consideration, as shown in (4.18). The ability parameterk stands for players’ strength in the competitive game context or error tolerance in11the social or non-competitive context. For example, there is no error tolerance forthe perfect player v0 . Note that objectivity and subjectivity perspective enables us
to deepen the understanding of engagement and addictive mechanisms in games [16].
Thus, the objective velocity (v0 ) and subjective velocity (vk ) were determined.
v2 = 2m2 − 3m + 1 = (1 − 2m)(1 − m)

(2.12)

vk = (1 − km) v0 , where 0 ≤ k ∈ R(2.13)The notion of potential energy in mind was originally discussed by [1] and its formula
is given by (4.14). The notion of velocity is derived from the reinforcement schedule
V R(N ) with frequency N , so we call objective reinforcement (E0 ) for the potential
energy in mind of the perfect player (v0 ). Otherwise, we call subjective reinforcement(Ek ) for the potential energy in the mind of other players (vk ). A game would produce
its potential energy in the field of play (hence we call it potential energy of play) by
which people would feel engagement or reinforcement.
In behavioral psychology, the term “reinforcement” refers to an enhancement of
behavior. This term was used as a positive interpretation, i.e., greater reinforcement
gives people a more substantial interest in staying in the event under consideration. In
the game context, reinforcement depends on the player’s ability. The potential energy ofplay (Ek ) is given by, Ek = 2mvk2 which is denoted as subjective reinforcement. For theperfect player or game theoretical reward (k = 0), denoted as objective reinforcement
E0 .
Figure 4-1 illustrates the objective and subjective reinforcement when k = 3. The
reward function (vk ) represents a player’s model or his/her sense of value. When assuming k > 3, vk < 0 holds at m = 13 where the objective reinforcement is maximized.
This situation implies the learning context’s most comfortable (peak of E0 ). Therefore,
it is highly expected to have k ≤ 3. Furthermore, a board game like Go (m = 0.42) is12

still not yet solved; thus, 2.38 < k is expected to hold.

motion in mind measures

1
v0
vk
E0
Ek

0.8
0.6
0.4
0.2
0
−0.2
0

0.1

0.2

0.3

0.4

0.5 0.6
mass m

0.7

0.8

0.9

1

Figure 2-2: Objective and subjective reinforcement when k = 3

2.2.2

Jerk and comfort in mind

Two processes with the same GR-value at the end of the game information progress
may have different instantaneous GR-value tendencies. For example, two basketballteams have the same successful shootings and total shot attempts. So their GR values
are the same when the game is finished. However, each team felt different tendencies
to get scores. The team with a stable scoring process is predictable, and vice versa. As
such, each team’s force was not only felt but also the change of the force. In physics,
acceleration can be felt in motion and the feeling of jerk [17].
The third derivative of (2.2) indicates the change of accelerated velocity (or jerk[17]) of the solved uncertainty along the game progress [18], which is given by (2.14).
Hence, the motion with constant jerk j is given by (2.15), where it is approximate in
the domain of board games as (2.16).

13

x′′′ (t) =n(n − 1)(n − 2) n−3n(n − 1)(n − 2)t|t=T =nT
T3
1
x(t) = jt36

j=(2.14)(2.15)

n(n − 1)(n − 2)
B
≈3 3
3
T
D

(2.16)

Table 2.1 shows the measures of game refinement for board games. For sophisticatedboard games such as Chess, Shogi, and Go, it is assumed that there exists a reasonablezone for the acceleration (a) and jerk (j), which is between 0.07−0.08, and 0.045−0.06,
respectively.
Table 2.1: Measures of game refinement for board games

Chess
Shogi
Go

B

D

35
80
250

80
115
208

a

√
3

0.074
0.078
0.076

0.059
0.054
0.044

√

j

We show, in Figure 2-3, an illustration of solved information over time, with ourinterpretation. The cross-point between the acceleration curve and jerk curve is thepoint where the maximum amount of achievement is greater than the discomfort (t1 ),after t1 , the discomfort will be larger than achievement. The cross-point between velocity and jerk is the point where effort is greater than the discomfort (t2 ), and thecross-point between velocity and acceleration is the point where effort is more excellent than achievement(t3 ). The cross-point interval ensures a reasonable zone for gamelength [19].

14

300

t2

t3

200
t1

Solved uncertainty y(t)

250

150
100
50
0

0

20

40

60

80 100 120 140 160 180 200 220 240
time t (game progress)

v

a

j

“zone” area

Figure 2-3: The cross point between the line with velocity v, curve with accelerationa and curve with jerk j. t1 ; t2 and t3 represent the bound for effort, achievement, anddiscomfort, respectively.

2.3

Card Games as Testbed in This Study

Card games are a diverse category of recreational activities that utilize a standard deckof playing cards or specialized cards designed for a particular game. These games havebeen enjoyed for centuries and are popular worldwide due to their simplicity, strategic
depth, and social nature. Here are some key points to understand about card games:
1. Variety: Card games encompass a wide range of options, from classic games like
Poker, Bridge, and Solitaire to modern and innovative titles. They can be broadly
categorized into several types, including trick-taking games, shedding games, and15

matching games.
2. Deck of Cards: Most card games use a standard deck of 52 playing cards,
which is divided into four suits: hearts, diamonds, clubs, and spades. Each suitcontains thirteen ranks, from Ace to King. However, some card games may require
additional decks or unique cards with specific designs.
3. Objective: The objectives of card games can vary significantly. Common objectives include achieving a certain hand or pattern of cards, accumulating points,
or being the first to get rid of all cards. The specific goal depends on the rules of
the game being played.
4. Strategy: Many card games involve a strategic element, where players makedecisions based on the cards they hold and the actions of other players. Strategic
thinking, planning, and adaptability are often key factors in winning card games.
5. Popularity: Card games have stood the test of time and remain popular acrossgenerations. They are played in homes, casinos, and online platforms, catering toa wide range of players.
6. Rule Variations: Card games often have multiple variations and house rules.
Players may adapt the rules to suit their preferences or regional traditions.
7. Educational Benefits: Card games can help develop various cognitive skills
such as memory, concentration, and strategic thinking. They are also used for
educational purposes in mathematics and probability lessons.
Whether you’re looking for a casual pastime or a competitive challenge, card games
offer a rich and diverse world of entertainment for players of all ages and skill levels.
Shedding games are a category of card games with the primary objective of quickly
getting rid of all the cards in hand. This is typically achieved by matching the cards
played to previously played cards or by following specific card patterns. Players aim to16

empty their hands as rapidly as possible while adhering to the established rules. Oneof the most widely played and representative shedding card games in our research is“DouDizhu”.

2.4

Perfect Information Game and Imperfect Information Game

2.4.1

Perfect information game

Perfect information games are a category of games where all players have complete
knowledge of the game state at all times.
1. Full Knowledge: Players have access to all relevant information, including thecurrent state of the game, the available options, and the history of moves madeby all players.
2. No Hidden Information: There are no hidden cards, pieces, or elements that
players are unaware of. Everything is open and visible to all participants.
3. Examples: Classic examples of perfect information games include Chess and
Tic-Tac-Toe, where both players have complete knowledge of the board and all
the pieces on it. In these games, success is determined solely by strategy and skill.
4. Solved Games: Some perfect information games have been fully solved, meaningthat optimal strategies have been determined, and the outcome of the game is
known with perfect play. Chess, for example, has not been fully solved, but
certain endgames have been.

172.4.2Imperfect information game

Imperfect information games, on the other hand, are games in which players do nothave complete knowledge of the game state. In these games:
1. Hidden Information: Players lack complete information about the state of
the game. This can include hidden cards in card games, hidden movements of
opponents in board games, or uncertainty about the environment in video games.
2. Bluffing and Uncertainty: Imperfect information games often involve elementsof bluffing, psychology, and decision-making under uncertainty. Players must
make choices based on limited information and try to anticipate their opponent’s
actions.
3. Examples: Poker is a classic example of an imperfect information game. In
Poker, players have hidden cards (hole cards), and they must use betting andpsychological tactics to make the best possible decisions.
4. Complexity: Imperfect information games can be more complex and challenging
due to the added element of uncertainty. Success in these games requires not only
strategic thinking but also the ability to read and manipulate opponents.
5. AI Challenges: Creating artificial intelligence (AI) agents to excel in imperfect
information games, like Poker, has been a significant challenge in the field of AI
research.
In summary, perfect information games involve complete and open knowledge ofthe game state. In contrast, imperfect information games introduce elements of hidden
information and uncertainty, requiring players to make decisions based on incomplete
data and psychological tactics. Each type of game offers unique challenges and opportunities for strategy and skill development.

18

2.5

Reward Frequency in Reinforcement Learning

Reinforcement Learning is a subfield of machine learning where an agent learns tomake decisions by interacting with an environment. The agent takes actions in theenvironment to maximize a cumulative reward signal over time.
Reward frequency is a critical aspect of reinforcement learning and refers to how
often an agent receives feedback or rewards from the environment. It plays a significant
role in shaping the learning process and affecting the agent’s behavior.

2.5.1

Influence of reward frequency in gaming

In a gaming context, the concept of reward frequency takes on specific meanings depending on the type of game being played.
Scoring games
In scoring games, reward frequency is a measure of how many attempts or actions aplayer needs to make to successfully score points or achieve a desired outcome. For
example: In basketball, reward frequency might represent the number of shots a player
needs to take before successfully making a basket.
This type of game typically focuses on players achieving points by completing specifictasks, earning scores, or reaching particular objectives. The primary goal in scoring
games is usually to accumulate as many points as possible, with the score often serving
as a measure of a player’s success. Classic examples of scoring games include brickbreaker games, shooting games, and racing games.
In these cases, a higher reward frequency may suggest that success is challenging
to achieve, requiring multiple attempts, while a lower reward frequency indicates thatsuccess is easier to attain.

19

Choosing game
In board games, reward frequency refers to the number of available options or choices aplayer has when making decisions related to a specific location or position on the gameboard.
Choosing games emphasizes players making decisions during gameplay that impact
the progress and outcomes of the game. Players often need to balance different options
to select the most suitable strategy for achieving success in the game. These games
may include strategy games, role-playing games (RPGs), and adventure games.
In these scenarios, a higher reward frequency means that the player has more strategic options and flexibility when making decisions related to a particular position on thegame board. Conversely, a lower reward frequency implies limited choices and potentially more straightforward gameplay.

2.6

Chapter Conclusion

This chapter thoroughly examines several facets of gaming, game theory, card games,
and reinforcement learning. These areas of study collectively contribute to our comprehension of the mechanics behind gameplay, the decision-making processes involved,
and the profound impact of rewards on player behavior.
Our exploration commenced with an in-depth look at the diverse universe of card
games. We highlighted their rich variety, the standard deck of cards used, and the overarching objectives that players strive to attain. Card games, whether they are classics
or contemporary creations, offer a wellspring of entertainment, strategic thinking, andopportunities for social interaction that cater to players of all backgrounds.
Subsequently, we delved into the distinctions between perfect and imperfect information games. Perfect information games, exemplified by venerable classics like
Chess and Tic-Tac-Toe, revolve around complete knowledge and transparent information, where success is primarily determined by strategic prowess and skill. In contrast,
20

imperfect information games, such as Poker, introduce elements of hidden information
and unpredictability, wherein psychology and decision-making become paramount.
Expanding our horizons even further, we ventured into the domain of reinforcement
learning. In this subfield of machine learning, agents acquire the art of decision-making
through iterative interactions with their environment, with the ultimate goal of maximizing cumulative rewards over time. The concept of reward frequency emerged as apivotal factor that shapes the learning process and molds player behavior in a plethoraof gaming contexts.
In summary, our comprehensive exploration of gaming, game theory, card games,
and reinforcement learning casts light on the multifaceted landscape of decision-making,strategic thinking, and the role of rewards within the gaming world. These insights
offer invaluable perspectives for both gaming enthusiasts and researchers in fields suchas artificial intelligence and psychology. As our journey into the dynamics of gamesand decision-making continues, we gain a deeper appreciation for the intricacies andnuances that render gaming a captivating and ever-evolving realm.

21Chapter 3
A Computational Game Experience
Analysis via Game Refinement
TheoryThis chapter is based on the integration, update, and abridgment of the following publication:• N. Gao, Y. Gao, Z. Zhang, M. N. A. Khalid, and H. Iida, “A computational
game experience analysis via game refinement theory, Telematics and Informatics
Reports”, in Telematics and Informatics Reports, vol. 9, pp. 100039, 2023.

3.1

Chapter Introduction

Activity continuum was previously associated with the notion of spontaneity and freedom [20], providing the foundation of various game categorization and taxonomy [1, 21].
Games may be roughly described as a system in which one or more players engage in22manufactured conflicts for the sake of amusement or enjoyment, which are regulated
by established rules and result in a definite, quantifiable objective or outcome. The
primary elements of the game category were traditionally key difficulties and gaming
mechanisms [21].
“Huizinga” [22] described play as an important activity in creating civilizations
and cultivating culture, which had evolved to eventually impact people’s values [23]. A
game would move from an uncertain to a certain state from the viewpoint of information
sciences (when the goal was reached or the winner was determined). The size of thetime (or rather, length of the play) played a role in describing the gaming experienceand the outcome anticipation (too short or too lengthy were judged to be unreliable ortiresome, respectively) [1].
Players will have varied psychological sensations depending on the level of the game’s
information progression. When a player has the upper hand, they feel at ease since theirrisks are decreased, and they can anticipate winning. They will experience anxiousness
as a result of being on the losing end. As a result, there should be a variety of conditions(e.g., success rate) to balance situations of advantage (relaxation) and disadvantage(anxiety), in which both the player and the opponent are engaged and comfortable(connected with the ‘attractiveness’ area). As a result, various success rates (speed orvelocity) give an objective perspective and sense of the game throughout, which might
be valuable and depicted based on our feelings.
Each game’s progress is meant to give a range of utilitarian goals that indicate theamount of effort, success, and suffering (unpredictability) associated with the game’s
outcomes. Quantitative measures may be used to explain the utility of game information
processes, and the mechanisms of game information processes can be discovered. The
game refinement (GR) theory has been studied as a measure of complexity based on
the “principle of seesaw” game, where the amount of option variants remained constantthroughout the games [4]. Since then, GR has been used to determine the sophisticationand magnitude of expected attractiveness in both games [11, 24, 25] and non-game
23

contexts [9, 10]. According to this research, the second derivative of game information
progress may be used to evaluate the attraction of F = ma in our minds in the samemanner that velocity can be used to estimate the game’s pace.
The evolution of game information is analogically represented with physical quantities in this study, where the average acceleration (a derivative of the speed of solving
uncertainty in games; associated with attractiveness) and jerk (a derivative of expectedattractiveness in games; associated with unpredictability) can be measured. Basketballand soccer were chosen as the case study for such an analysis. Furthermore, the research looked at alternative representations of game information progress using gametree structure and value functions to simulate the most exciting and appealing situations
during gameplay. By connecting to the well-known Flow theory, this visualization gave
realistic estimations of the underlying play experience of the game-playing processes.
The quantities measured by physical analogies relevant to our mind’s motions are
explored concerning distinct process mechanisms connected with different playing circumstances. Furthermore, the degree of solved uncertainty and unpredictability isdetermined, as well as the extended perspective’s logic. This research attempts to formalize the game-playing experience into a formal model that may be used to improvethe design and analysis of future games. The study’s primary contribution is the transformation of temporal forms (e.g., game duration) into contextual forms (e.g., search
space and information uncertainty), in which the interaction of information progress isutilized to characterize psychophysiology processes.
The organization of this chapter is as follows. Section 3.2 introduces quite a little literature from different perspectives of game-playing experience measures, such asenergy expenditure, perceived behavioral control, and social violence. Section 3.4 provides an overview of the GR theory and its recent developments. Section 3.5 provides a
few example applications of the GR theory to scoring and choosing games. Section 3.6
expands the game refinement theory using physical quantities for two-sided dynamics,
scoring dynamics, and game process visualization. Finally, Section 3.7 concludes the24chapter.

3.2

Measurement of Play in GamesIn this section, a brief review of the related works on player experiences that wereobserved in a game setting was outlined and described in Section 3.2.1. Then, a brief
review of the previous works on player psychology was also described in the subsequent
Section 3.3.

3.2.1

Player Experiences in Games

Activity continuum exists by extending play to games, and then into sports, characterized by decreasing spontaneity and freedom into regularization, orientation, habituation, and institutionalization (play-game-sport continuum) [20]. Such a continuum
grounded the competitive activities into equal chances (agôn), unequal chances (alea),pretense (mimicry), and pursuit of giddiness (ilinx) as the classification system of games.
Nevertheless, it is apparent that play, like games and sports, represents a specific kindof experience [20].
Several game-playing experience measures have been studied, such as energy expenditure, perceived behavioral control, and social violence. Based on the behaviorof experienced “Dance Dance Revolution” (DDR) players and inexperienced players,
it was found that different skillful players have different energy expenditures. Participants with greater playing experience can work at higher intensities, promoting greater
energy expenditure [26]. Data collected using a structured questionnaire from 1584 university students in Malaysia with different backgrounds shows that perceived behavioral
control has the most substantial influence on actual use. Other variables found to influence actual usage include perceived enjoyment, subjective norms, attitude, perceived
enjoyment, and flow experience [27].
In the previous research about assessing the potential impact of playing video games,
25

particularly violent games, the Game Engagement Questionnaire (GEQ) was developed
using classical and Rasch rating scale models. The GEQ provides a psychometricallyrobust measure of levels of engagement specifically elicited while playing video games;
thus, showing promise for future research examining risk and protective factors fornegative game impact [28, 29]. Meanwhile, another study was conducted by [30] to potentially influence and evaluate quality of experience (QoE) in gaming service providers
and provide expert evaluation in a passive viewing-and-listening scenario or interactive
online/cloud gaming scenario; have roughly classified into three classes of influence factors: (a) human, (b) system, and (c) context. The author also highlighted that such
QoE measurements would be invaluable in adjustable game input/output and mechanics, provide better realism, and quantify interaction degradation.
Some researchers investigate whether individual difference influences the characteristic experience of game playing [31]. The results suggest that physically aggressive
people tend to engage in a more aggressive playing style after controlling the differencesin gender and previous gaming experience. Implications of these findings and directions
for future studies are discussed. Another study combines several 3-dimensional spatial
technologies to enhance players’ gaming experience [32], where the result showed thatthe play experience of different player levels and performance could be optimized, especially in games with 3-dimensional tasks.
From these perspectives, when engaged in any game experience, players’ behavior (i.e., energy consumption) can measure skill level and violent tendencies. Thus,
these studies showed the prospect of visualizing and measuring quite a few abstract
factors during the game-playing experience. From another perspective, gravitational
acceleration (like 4g and 5g) is essential in the context of the roller coaster ride toleave an impactful experience in people’s hearts, and minds [1]. In essence, this study
is interested in providing an alternative measure of game-playing experience by formalizing information progress of games via the game refinement theory to describe
psycho-physiological processes. Recently, topics such as autonomous driving and gam26

ification have become more and more popular. Vehicular automation involves the useof mechatronics, artificial intelligence, and multi-agent systems to assist the operator
of a vehicle (car, aircraft, watercraft, or otherwise). These features and the vehicles
employing them may be labeled as intelligent or smart vehicles or transportation. A vehicle using automation for difficult tasks, especially navigation, to ease but not entirely
replace human input, may be referred to as semi-autonomous [33], whereas a vehicle
relying solely on automation is called robotic or autonomous [34].

3.3

Player Psychology in GamesIn this section, I discussed Player Psychology in Game, and how research by Johannes
demonstrates the effectiveness of digital biomarkers in gaming for assessing social anxiety, with behaviors such as player accuracy and movement patterns indicating thiscondition. Additionally, Tng discovered that avatar identification is mediating in internet gaming disorder (IGD). Dang’s analysis of temporal gaming patterns aimed tocomprehend learner motivation, while Sevcenko revealed that in-game metrics can reliably forecast cognitive load. Furthermore, Harris established a connection between
trait competitiveness and problematic online gaming.
A study conducted by [35] explored game-based digital bio-markers’ effectiveness forsocial anxiety assessment, embedded in a gaming task, based on a player’s interaction
with and around a non-player character. The results of our study show that social
anxiety manifests in several behaviors in-game, similar to avoidance behaviors from thephysical world, useful as digital biomarkers. In particular, social anxiety was reflected
in player accuracy and the movement paths (i.e., safety behaviors where the player
stays farther away from non-player characters). Furthermore, this study presents theinfluence of customized avatars and camera perspectives on how much social anxiety
is expressed in a gaming task. Meanwhile, [36] presents a study that examines theassociation of determinants, such as motivation of gaming and identity of in-game27

avatar, relative to the internet gaming disorder (IGD) among multiplayer online battle
arena (MOBA) gamers. The study found the underlying determinants of IGD whereavatar identification was found as a significant mediator, although such psychologicalmotivations of gaming reflect unique meanings and consequences. For instance, youth
must determine a more positive self-concept to strengthen greater self-acceptance andvoluntary integration of the ‘fatigue’ system by game developers to discourage excessive
time spent on online gaming indirectly.
Quitting and procrastination are evidence of exercising self-regulation in learners
and provide the potential to understand their motivation by identifying and analyzing
less desirable behaviors in learning. Such temporal patterns in gaming behaviors havebeen analyzed by [37] on an intelligent tutoring system in detecting motivation. The
study’s findings suggest that gaming behaviors indicate partial cognitive engagement
and session-level influences on learning motivation. In addition, self-regulated behavior
theories might infer temporal fluctuations in the motivations (such as ego-depletionand task-switching). Such information could be capitalized for responsive cognition
and motivational dynamics. Another study conducted by [38] found that simple ingame metrics that describe the behavior and performance in an emergency simulation
game can reliably assess and predict cognitive load based on a theory-driven approach.
The result of the study indicates that increased difficulty causes significantly higher
subjective ratings of cognitive load, accompanied by significantly poorer performance.
Moreover, the early stages of gameplay are more informative and predictive for later
gameplay outcomes than an aggregated score accumulated over a more extended time.
The development of problem video gaming was prominent in high-risk online games(i.e., massively multiplayer online first-person shooter games) where direct competitionis the key motivator for participating in such a game. [39] had examined the relationshipbetween three facets of trait competitiveness and problem online video gaming, where
the study found that competitiveness motivates online gaming, suggesting that inherent
dominant competitive tendencies may increase the risk of problematic participation in28

competition-centered online video games. Meanwhile, [40] conducted an exploratory
analysis of the role of skill versus chance in determining outcomes in skill-based gaming
machines (SGMs1 ) to assess and measures player understanding of gaming activities anderroneous beliefs of gambling. The study suggested that SGMs appeal to gamblers andnew cohorts, especially younger players. At the same time, the use and understandingof SGMs are likely to change over time as players adapt to the presence of these newgambling activities and potentially lead to greater illusions of control in individuals,
overconfidence in understanding how outcomes are determined, and greater gamblingproblem severity.

3.4

Game Refinement Theory and Its Development

In this section, the GR theory is revisited, where its original formulation from thegame progress model of outcome uncertainty is described in Section 3.4.1. Then, a
new perspective from the third derivative of the game progress model is introduced anddescribed in Section 3.4.2.

3.4.1

Game Refinement Theory and Its Development

The game progress model of game uncertainty is based on early work by [4]. It has
been previously applied to measure the design sophistication in domains of business[9], and education [10], and act as a tool for exploring the evolution of popular board
games [4, 11]. The GR values for most popular games are located in a reasonablezone of GR ∈ [0.07, 0.08]. From the player’s viewpoint, the information on the game
result is an increasing function of time (the number of moves in board games) t. Here,the information on the game result is defined as the amount of solved uncertainty (orinformation obtained) x(t), as given by (5.1). The parameter n (where 1 ≤ n ∈ N ) is1

Gaming machine with a skill-element within the random mechanisms of electronic gaming machines (EGMs)

29

the number of feasible options and x(0) = 0 and x(T ) = 1.

x′ (t) =

n
x(t)
t

(3.1)

x(T ) stands for the normalized amount of solved uncertainty. Note that 0 ≤ t ≤ T ,0 ≤ x(t) ≤ 1. (5.1) implies that the rate of increase in the solved information x′ (t) is
proportional to x(t) and inverse proportional to t. Solving (5.1), (5.2) is obtained.
 ntx(t) =
T(3.2)

It is assumed that the solved information x(t) is twice derivable at t ∈ [0, T ]. The
second derivative of (5.2) indicates the accelerated velocity of the solved uncertainty
along with the game progress, which is given by (5.3).

x′′ (t) =n (n − 1) n−2
t
|t=T
Tn

=

n (n − 1)
T2

(3.3)

Accelerated velocity refers to the difference in the rate at which information isgathered as the game progresses. Then (5.4)gives the acceleration motion or free-fallmotion in mind, a. a is estimated in the domain of board games as (3.5), where B andD denote the average number of possible moves and game length, respectively.

a=

3.4.2

1
x(t) = at2
2

(3.4)

n (n − 1)
B
≈ 2
2
T
D

(3.5)

Jerk and Comfort in Mind

Two processes with the same GR value at the end of the game information progress
may have different instantaneous GR value tendencies. For example, two basketballteams have the same total shots and total attempts. So their GR values are the same30

when the game is finished. However, each team felt different tendencies to get scores.
The team with a stable scoring process is predictable, and vice versa. As such, force(uncertainty) was not only felt but also the change of the force (unpredictability). In
physics, acceleration can be felt in motion and the feeling of jerk [17].
The change in accelerated velocity (or jerk [17])of the solved uncertainty along withthe game progress [18], which is supplied by (3.6), is shown by the third derivative of(5.2). As a result, (3.7)gives the motion with continuous jerk j, which is approximatedin the domain of board games by (3.8). In this study, j represents the change in informational acceleration (the magnitude of thrills, the tendency of getting attractiveness,and related to game addiction), which is linked to an addictive-like event due to thetendency of motivation retention [18]; thus, denoted as AD value.

x′′′ (t) =n(n − 1)(n − 2) n−3n(n − 1)(n − 2)t|t=T =
n
T
T3

(3.6)

1
x(t) = jt3
6

(3.7)

3a
B
≈3 3
t
D

(3.8)

j=

Table 3.1 shows the measures of game refinement for board games. For sophisticatedboard games such as Chess, Shogi, and Go, it is assumed that there exists a reasonablezone for the acceleration (a) and jerk (j), which is between 0.07−0.08, and 0.045−0.06,
respectively. The cross point (Figure 3-1) between acceleration and jerk t1 is the point
where the maximum amount for achievement is greater than the discomfort (t1 ). Thiscondition is the first reasonable chance to stop the game; otherwise, the discomfort
will be higher than achievement. Moreover, before the cross point between accelerationand jerk, players are considered beginners in the game, where lots of effort is vested instudying and training, and the achievement is much lower than the effort spent; thus,
it is too short to stop the game.
31

Table 3.1: Measures of game refinement for popular board games, adopted from [1]Board games

B

D

Chess
Shogi
Go

35
80
250

80
115
208

√

a = GR

√
3

0.074
0.078
0.076

j = AD
0.059
0.054
0.044

B:
√ Average branching factors; D: Average game length;a = GR: informational acceleration (thrill);√3
j = AD: informational jerk (surprise);

300

t3

t2

200
t1

Solved uncertainty y(t)

250

150
100
50
0

0

20

40

60

80 100 120 140 160 180 200 220 240
time t (game progress)

v

a

j

“zone” area

Figure 3-1: The cross point between the line with velocity v, curve with accelerationa and curve with jerk j. t1 ; t2 and t3 represent the bound for effort, achievement, anddiscomfort, respectively.

32

However, after t1 , the discomfort or unpredictability becomes larger than achievement but still lower than effort, so t1 is the first chance to stop the game. Otherwise, thediscomfort will be higher than the achievement. The cross point between velocity andjerk (t2 ) is where effort is greater than the discomfort or unpredictability, and it is thesecond reasonable chance to stop the game. After t2 , the discomfort is higher than theeffort, inducing the feeling of discomfort and unpredictability. Furthermore, the crosspoint between velocity and acceleration is when the effort is greater than achievement(t3 ), making it the last chance to stop the game.
Quite a few levels of risk in game-playing are the motives behind the desire andconcentration to face the challenge, which is the point of entertainment. After t3 ,the achievement will be higher than the effort, similar to the feeling of “reap without
sowing.” This situation is where the player had achieved mastery and possessed theability to play the game expertly; thus, making the game bland. The interval of thecross points constitutes the reasonable zone for a reasonable game length. Moreover,players need enough steps to show their abilities to make the game fair enough; ifgame steps are too less, players do not have enough chance to show their skill, so gamesteps should be longer than t1 , So t1 ≤ t ≤ t3 can make sure game’s fairness andentertainment.
A game is a constant task that invariably contains quite a few risks, where rewards
are not expected at every step. As such, the best effort is devoted to playing thegame. When the achievement is higher than the effort devoted, it makes us feel “reapwithout sowing” and “gain without pain.” As such, effort with quite a few degreesof achievement makes the game progress a comfortable experience. However, ensuring
a comfortable experience means that discomfort should be less than achievement andeffort.

33

3.5

Gamified Experience From Metaphysical Perspectives

The reasonable GR (acceleration) and AD (jerk) measures of the game process ofthe sophisticated and popular board games such as Chess, Shogi, and Go have beenpreviously established (Table 3.1). However, such measures were dependent on thetemporal dimensions of game-playing, which makes generalizing the model of the playprocess challenging when dealing with problems with little to infinite time continuum.
Therefore, considering the different dimensions of information progression in the gameis essential.
The success rate in the game process is associated with the speed of what a player
can gain, which is regarded as the physical domain’s velocity. The total attempts(steps) are associated with the time or total costs in the physical domain, such as fiveminutes or steps. The total successful shoots in the game process are associated withthe outcome gained, which is regarded as displacement in the physical domain. Hence,the correspondence of those physical quantities can be utilized to estimate the metaphysical game process in our minds. As total attempts and total shoots constitute thegame process model, the physical model can express such progress while investigating
the different interplay between various metaphysical properties. Hence, the analogical correspondence of the Newtonian dynamics and the game information progress isbroadly determined as summarized in Table 3.2.
In this study, if the velocity of the process is directly proportional to the speed ofgetting a reward, the players invariably take n times for a single shoot. As such, thev = n1 , implies that there is no uncertainty during the process. The effort during thegame process is described in (3.9). However, in the real world, the process is constantly
subjected to uncertainty. Then, a continually changing velocity process is described asv = at, which produces (3.10).

34

Table 3.2: Contextual correspondence between game information progress, Newton
dynamics, and their link⋆⋆

Scoring games

Newton dynamics

Link

y(t)
t
v

Total goals
Total shoot attempts
Total score rate

Displacement
Time
Velocity

Total gain
Total cost
Process speed: Mathematical notation; v: uncertainty solving rate; t: game length; y(t): solved uncertainty;

y(t) = vt(3.9)

1y(t) = at2
2(3.10)

Furthermore, the process is also subjected to quite a few degrees of unpredictability.
For example, when taking the elevator [41], if the acceleration is constant, the humanbody is immediately subjected to an upward thrust of m(a + g) when the elevator
starts, which causes discomfort to the passengers. Hence, the designers invariably set
the acceleration to a gradient, giving the human body a period of adaptation, avoiding
discomfort. By introducing a = jt in the process, (3.11) is obtained.
1y(t) = jt3
6(3.11)

The cross-linking between three domains (physics, games, and psychophysiology)
was established by bridging the physics concepts to their psycho-physiological meaning, determined based on the cross point of various psycho-physiology quantities (Section 3.4.2). The ‘velocity’ is the speed to get the reward associated with the effort devoted. The ‘acceleration’ is the thrills of the obtained rewards associated with achievement due to changes in the effort. Finally, the ‘jerk’ is the degree of discomfort associated with unpredictability due to changes in the expected achievement. The analogy
35

Table 3.3: Contextual link between physics, games, and psychology
Physics

Games

Psychology

Velocity (v)Solved uncertainty Information gain (effort)
Acceleration (a) Thrill magnitude
Reward gain (achievement)
Jerk (j)Level of surprise
Unpredictability (discomfort)of such links is given in Table 3.3.

3.5.1

Basketball

A quintessential two-sided time-limited sports game is basketball. The shot attempts
and successful shots, determine the game information progress by promoting a similar
idea from board games to sports games. In the previous research, the analysis of gameinformation progress focuses on the data of one team. However, focusing on the entire
basketball process, the game information progress needs to be created by the attempts
and goals of the two teams.
If the game process progresses at a constant speed, then the game process is certain,
which is given as in (3.12) where G and T stand for the average number of goals andshoot attempts, respectively. In the domain of sports games, if the process is uncertain,
then the informational acceleration is given as in (3.13). If the process is unpredictable,√
the informational jerk is given as in (3.14). Solving a = 2G
and j = 6G
, then a = 0.073
T2
T3√
and 3 j = 0.046 were obtained, respectively.
GtT(3.12)

1y(t) = at2
2(3.13)

1y(t) = jt3
6(3.14)

y(t) =

36

This situation implies that if acceleration is reasonable (GR ∈ [0.07, 0.08]), thegame is enjoyed with an appropriate degree of uncertainty and thrills. This condition
corresponds with the attractiveness “felt” during the game process, which is related tothe measure of force (F = ma). Meanwhile, jerk is a part of the underlying componentsthat are essential in our daily life, where it generates phenomena such as vibrato inmusic and the elevator system with progressive acceleration. Quite a few times, slow
and gentle change of force is better perceived when the amount of jerk is reasonable(AD ∈ [0.045, 0.06]). A game with tolerable perturbation from unpredictability anddiscomfort during the game process would remain enjoyable. It can be observed from
Table 3.4 that the GR (acceleration) and the AD (jerk) values of the basketball are
0.073 and 0.046, respectively. Hence, basketball has a reasonable amount of thrills andunpredictability, which makes the game characterized as a sophisticated scoring game.
Table 3.4: Quintessential two-sided time-limited shooting game - basketball (adopted
from [2] and basketball reference website⋆ )Basketball
†
⋆

G†

T‡

GR

AD

72.76

164.02

0.073

0.046

: Successful goals; ‡ : Total shots; GR: Informational acceleration; AD: Informational jerk;: https://www.basketball-reference.com/leagues/NBA stats per game.html;

The correspondence between physical quantities in the game and psychological feelings enables the visual representation of such processes to be mapped to its effort,
achievement, and unpredictability, as given in Figure 3-2. The velocity of the gameis linear based on effort. With the change in velocity, its acceleration describes thegame’s achievement levels. With the change of acceleration, unpredictability can be
expressed through the linear jerk. Thus, the game progress satisfies the relationshipsof unpredictability ≤ achievement ≤ ef f ort. This relationship provides an importantjustification that shows a clear separation between different progressions relative to thenumber of actions that can be made in the game setting, which corresponds to the37expected psychological feelings of a player.

Solved uncertainty y(t)80
y1 (t) = vt
y2 (t) = 12 at2y3 (t) = 16 jt3

60

40

20

0

0

20

40

60 80 100 120 140 160 180 200
time t (game progress)

Figure 3-2: The cross point between the curves of the velocity v, acceleration a, andjerk j, where such a cross point describes the comfortable moment of the basketballgame. After the cross point, it can be observed that with enough training and skill,achieving rewards becomes easy. However, the feeling of discomfort will also be higher
due to boredom and insufficient challenge.

3.5.2

Soccer board

Quite a few times, the difficulty for every game is different, where there exist quite
a few games with low points and successful score rates being much less than 12 , e.g.,soccer [42]. In soccer, it is hard to get a goal, and the game becomes very exciting
when a team gets a single goal. The average number of goals is 2.64 per match, andthe promising shots (shots on target) are 7.86 [43]. As such, the previous study has
calculated the game refinement measure for soccer [3], given as in Table 3.5.
However, quite a few countries have quite a few board games to play soccer, suchas “Paper Soccer” in Poland [44] and “Philosopher’s Football” in the USA [45]. Thus,
it is reasonable to apply the board game model to soccer based on such a context. A
quintessential soccer board game called “Zuqiuqi” in Chinese and “soccer board” in38

Table 3.5: Measures of game refinement for soccer game according to [3]Goals

Shots on target

Total shoots

GR

2.64

7.86

22

0.073

English [46], is given in Figure 3-3.

Figure 3-3: Using board game to play soccer
22
The average feasible options (or feasible time) (B) to get a single goal is 2.64
= 8.33.

Meanwhile, the average promising options (average feasible time over the shots on
22target) B1 for soccer is 7.86
= 2.8 (Table 3.6). Hence, the promising options B1 are√√√approximate as the root feasible options B ( B = B1 , 8.33 ≈ 2.8).

Table 3.6: Links between board game to soccer, where D is total shots, B is average
feasible√ options, B1 is average promising options (i.e., n is assumed as ideal optionsn = B1 ), GR is the informational acceleration, and AD is the informational jerk.
D

B

B1

n

GR

AD

22

8.33

2.8

1.67

0.076

0.09

In the soccer case, the best physical strength and the ideal state of a player correspond to the strongest AI in the board game cases, we can see the promising options√
are approximated to the root square of feasible locations( B = B1 ), so the ideal value√of n = B1 = 1.67 is considered. As such, an ideal state for the players is 1.67 times
39for one shoot. By associating with soccer board games, a new expression of accelerationand jerk for soccer was obtained and given as in (3.15) and (3.16), respectively.
n (n − 1)
B1
≈ 2
2
T
D

(3.15)

n(n − 1)(n − 2)
B1
≈
3
T3
D3

(3.16)a=j=

Every game has a different success rate where its velocity differs. Consequently,
a different design of unpredictability to balance the success rate and retain players’interest and loyalty is required. According to the soccer board results, players cannot
quickly and frequently achieve the reward because the velocity is too low. Although the√informational acceleration falls in the zone values ( a ∈ [0.07, 0.08]), it does not sufficiently describe soccer’s attractiveness. Therefore, having a higher jerk value increases
the level of unpredictability, which offsets the rewarding difficulty in soccer.
If the velocity (successful score rate) is small, a high jerk value helps to generate
an unpredictable (or surprise) experience, which improves the entertainment outlook.
While rewards are not easily obtained, the game can still be enjoyed (watched). On
the other hand, if the successful score rate is too high and the process is perceived as
too easy, a low jerk value would stabilize players’ enjoyment and comfort.

3.6

Physics and Psychophysiology Processes in GamesIn this section, the application of physics quantities relative to its psychophysiology expectation based on flow theory, in the context of games, is investigated. In Section 3.6.1,
the dynamics of an adversarial setting (two players/teams) are explored based on theinteraction of velocity, uncertainty, and forces. Then, the relationship between scoring
dynamics in the game and flow theory in psychology is established (Section 3.6.2). Finally, the interplay between the various physical quantities leading to the conception40

of the motions in mind is discussed in Section 3.6.3.

3.6.1

Interaction Dynamic

Generally, the game process can be expressed using a game tree structure. For example,
Figure 3-4 depicts the game process of the players in such a way that the first player
spends three attempts on the first shoot, two attempts for the second shoot, and a single
attempt for the third one. Then, the player’s corresponding success rates (velocity) in
every state can be determined, where the acceleration and jerk can be derived to express
different game-playing processes at each moment.

Initstate

lose

win

lose

lose win
winlose win
Figure 3-4: Using a game tree model to visualize the scoring process

The successful score rate at any moment is the velocity of the game to obtain the
reward. For two side players (or teams), p1 and p2 denote the first player (or team)and second player, respectively, and their velocity (or successful score rate) is v1 andv2 , respectively. Therefore, it can be conjectured that if v1 = v2 = 21 , the level of
attractiveness felt by both parties would be maximized. This situation reflects the bestscoring effort given by both sides to gain a reward (or score), where both sides have an
equal chance.
From another perspective, the scoring process can be regarded as a quintessential
41

binary variable, where the player’s success rate and loss rate have a distribution of[v, 1 − v]. In the context of information theory, entropy can be used to measure theuncertainty in the random variables in each state [47]. Then, the uncertainty of statebased on such distribution is given as (3.17), where the uncertainty is maximum (H = 1)
when v = 21 . Therefore, if both the success rate and loss rate are 12 , then the uncertainty
is highest. This situation describes that the game’s outcome is uncertain (the highest
uncertainty), which again implies that both sides have an equal chance.

H = −v log(v) − (1 − v) log(1 − v)

(3.17)

Solving the cross point between (3.9) and (3.10), then the information acceleration
expression is obtained as a = 2vt . Then, v can be aggregated based on each player byconsidering the interactions between two players (teams), in which the resulting expression is given as (3.18). In the physics domain, there are two quintessential interactive
forces: universal gravitational force and electromagnetic force, which are expressed as
2
Fu = G MR1 M[48] and Fe = k qR1 q22 [49], respectively. Here M and q are the weight of2

two masses and the quantity of electric charge, respectively. For the attractiveness in
the game process, these are the interactive forces in mind. Relative to the interactive
forces, the constant G and k in the game-playing context are associated with players’
ability and the successful score rates of players (given by v1 and v2 ). Hence, the analogyof such interactive force in the game is given by (3.19).

a1 =

2v1
,
t

2(1 − v1 )t

(3.18)

4v1 (1 − v1 )t2

(3.19)a2 =F = Ka1 a2 = K

Based on the premise of v1 + v2 = 1 in two-sided game, only when v1 = v2 = 12 ,the force between each other becomes highest, which is K
. If one side is much stronger
t2
than another side, their abilities and success rate are either 13 or 23 , respectively, while42their interactive force will be less than K(or 8K). The notion of mass is associatedt2
9t2
with the weights of the objects (how heavy it is to hold the object) and the difficulty ofchanging its state (difficulty of putting stress on an object). Mass in the game context
is prevalent based on the latter, which refers to the difficulty of moving. This situation
corresponds to the challenge of the game (a measure of risk rate). Hence, m1 = 1 − v1
and m2 = 1 − v2 refers to the mass of first and second players (or team), respectively.
The analogy of universal gravity in the game context can be described as given by(3.20). When the risk rates of both sides approach 12 , the universal gravity in the gamecontext is highest.

Fu = G

m1 m2(1 − v1 )(1 − v2 )v2 (1 − v2 )
=G
=G
2
2
R
R
R2

(3.20)

In essence, interaction dynamics in games can be analyzed when it is assumed thateach side has similar abilities, implying successful score or risk rate is both approaching1. In such a condition, the interactive attractiveness between the two players becomes2

the highest. It is easy to understand that the winning possibilities will also be the same
when the ability is well-matched with both sides in a contest, where the interactive
force in our minds is highest when both sides tried their best, they will feel indulged
like flow in a non-game context.

3.6.2

Game Playing Experience and Flow Theory

It was found that in two-sided games, when both players (or teams) abilities are similar,
the successful score rates are approximately balanced, and the interactive attractivenessbetween two-sided players becomes the highest. However, what happens when the
successful score rate is not 12 ? Based on such regards, differences between the two sides
are best described in conjunction with the Flow theory. Such condition is bridged via
the measures of scores and their differences to describe conditions in a game-playing
context (Table 3.7).

43

Table 3.7: The Correspondence between Game Context and Non-Game Context using
S (Scores)
Non-Game Context

Game Context

Notation

Skill
Challenge∆(Skill-Challenge)

Player’s Ability
S
Opponent’s Ability
Opponent’s S
The Distance between Abilities ∆S

S: Scores; ∆(S): Score Gap between 2 sides;

The Flow theory is a distinguished psychological theory that explains the feelingsof people acting on quite a few things, fully immersing or engaged, based on differences
between two dimensions: skills and challenges (Figure 3-5). The standard of measurement is the ∆(skill − challenge) [50]. When ∆(skill − challenge) = 0, both skill
and challenge are high, where the experience that happened is defined as the “flow.”
Meanwhile, ∆(skill − challenge) > 0 is where the experience people feel is related tocontrol and relaxation. Meanwhile, the ∆(skill − challenge) < 0 is related to arousal
and anxiety. Nevertheless, what are the challenges and skills in the game context?
If the skills are the abilities a player obtains, it can be linked to the success rate in
the game, and its opponent’s ability is the challenge they need to overcome. Hence, therisk rate is the expected challenge in the game context.
Different success rates are utilized to estimate the affected processes during thegame relative to the Flow theory. Since the scoring process is a dynamic mechanism
during the game progression, the velocity (v), acceleration (a), and jerk (j) can bevisualized. The sine function is used to describe the scoring process (v) starting from0:0 (Figure 3-6). Then, the derivative of v is given by the cosine function to describe thea that is associated with attractiveness. Subsequently, the derivative of a is the minus
sine function representing j, associated with the unpredictability or discomfort in thegame process. Consistent with previous research, [18] when the scores are similar, theabsolute value of acceleration is the highest (the greatest attraction).
Similarly, Figure 3-7 depicted the player scores to measure their abilities, and the44

Figure 3-5: Challenge vs. Skill, illustrating the ”flow” region Source: English
Wikipedia. https://en.wikipedia.org/wiki/File:Challenge vs skill.jpg

6:4
2:0
1:0

5:4

2:1
4:4

0:0

2:2
3:4
2:3
2:4
v

a

j

Figure 3-6: The description of game process using ∆ scores45

opponent’s scores correspond to the challenge the player needs to face. In the gameplaying process, when a player and his opponent’s abilities (scores) are similar andstrong, they will be fully concentrated and attentive to the game because such a moment
is the most uncertain, and both players can expect the greatest excitement. The moststable (predictable) situation means that players can expect the most comfort andcontrol. GR theory explains that when two side teams’ ∆scores = 0, the absolutevalue of acceleration is the highest because this condition is the most uncertain. In
such a setting, every feasible thing could happen, since similar skills were expectedfrom both sides.

Figure 3-7: The description of Flow theory using the game-playing process to associate
the context of the expected experience when playing, based on self’s (ability) andopponent’s (challenge) score
In addition, when the success rates are comparable, interactive attractiveness becomes the highest, implying a balance between ability and challenge in the game context(∆[skill − challenge] = 0). As such, players will experience the “flow” zone, which has
the highest absolute value of acceleration (the greatest thrills). As a result, the game46

outcome becomes unpredictable. At the same time, the j → 0 because it is hard tochange the highest uncertainty; players will not feel discomfort and unpredictability.
In essence, players will potentially feel both thrill and anxiety, blended or in parallel,
for the mere possibility of a situation change. Besides the ∆scores = 0 condition, different ability gaps correspond to psycho-physiological experiences of various non-gamecontexts.

3.6.3

Conceptual Basis of Motion in Mind

Different game-playing mechanisms were revealed in each state based on v, a, and j√during their game processes. Since, a = 2GandGR=a, then the GR value would
2
T
approach the sophisticated zone as the game progressed. In physics, mass is the factor
determining the difficulty of changing the state of motion when an object is stressed.
Thus, mass is a physical quantity that describes the inertia of an object, which describes
the “heaviness” and the difficulty of holding an object [51]. In games, the risk factor
is the reason that determines the difficulty of progression and; thus, the difficulty ofobtaining scores. Therefore, assuming the mass as the risk rate given by m = 1 − v, theanalogical translation between physics, games, and psycho-physiological context can be
determined, which is given in Table 3.8.
Table 3.8: Analogical translation between motion in minds, its game-playing implications, and its psycho-physiological context
Motion
v
m
p⃗1
E⃗p
p⃗2

Implication

psycho-physiological

Solved uncertainty
Risk uncertainty
Continuity tendency
Information expectationp⃗1 and E⃗p gap

Information gain (effort)
Uncertainty gain (challenge)
Concentration
Expectation“knowledge” gapv; velocity; m = 1 − v: mass; p⃗1 : game’s objective momentum; E⃗p : game’s potential energy; p⃗2 :game’s subjective momentum;

Concerning the relative motions in the mind, the analogy of momentum in mind (⃗p)
47

is adopted, which is given as (3.21). To identify the peak point, the first derivative of p⃗
is p⃗′ = 1 − 2v = 0, where v = 12 , then p⃗ = 41 holds. Initially, the probabilities of win andloss are both 12 , so the velocity is 12 and the risk rate m = 1 − v = 12 . In this situation,
the momentum p⃗ is highest.

p⃗ = m · v = (1 − v) · v(3.21)In physics concept, momentum p⃗ is the difficulty of stopping moving objects andthe tendency of moving objects to keep moving. Also, momentum is an instantaneous
variable, and it measures the momentary trend based on the current velocity. Concerning the game process, momentum in mind p⃗ describes players’ tendency to maintainthe game’s state and continue to focus on the game. It can be seen that their successrate is the same, and their tendency to keep playing the game is highest. Hence, the p⃗measures the players’ tendency to keep playing the game in different velocity situations[16].
Potential energy (E⃗p ) is the energy stored in a specific location. It is related to the
expectation in the specific state. In different winning rate states, there are different
energy and different expectations in our minds. The potential energy E⃗p = 2mv 2 in
the game is defined as the amount of the required game information a player needs inthe game process [1]. It explains the expectation of the player in finishing the game.
As a game progresses, the game’s potential energy reflects the amount of anticipation
the game may give to the player (degree of win comfort). More importantly, this helps
the game designer determine the game sophistication’s appropriate stability, given as(3.22).
E⃗p = 2mv 2(3.22)

Relative to the process in our mind, momentum p⃗1 = mv is the momentum of thegame’s motion, which is followed by the momentum of mind’s motion (p⃗2 = E⃗p − p⃗1 ),48

where p⃗2 is adopted as the measurement of the magnitude of engagement. In thecontext of psychophysiology, the gap between concentration and expectation can be
related to the epistemic rationality of determining the “knowledge gap,” in which theformer is related to the truth system (actual/observed information) and the latter isrelated to the belief system (predicted/expected information) from the classical theoryof knowledge [16, 52].

p⃗2 = 2mv 2 − mv

(3.23)

When both sides of the players’ success rate are similar, ∆S approaches 0, andmomentum p⃗ increases. This situation implies that the difficulty of stopping the gameis high, and the tendency to keep playing the game is high. Otherwise, when ∆S
increases, momentum p⃗ approaches 0 because the game’s uncertainty is reduced, andthe disadvantaged side already had no chance to win the game. Such game processes
are illustrated as in Figure 3-8.
2

⃗v
∆S
p⃗1
E⃗p
p⃗2

Motions in mind

1

0

−1

−2

0

1

2

3

4
Steps

5

6

7

8

Figure 3-8: The analysis of motions in mind based on dynamical scores gap

49

If ∆S approaches 0 or 1 when they evenly match each other, the player’s expectation
is high, and if the player comes to the disadvantaged side, and ∆S approaches -1,
lessen the player’s expectation. When the situation became apparent, ∆S is 2 or 2; expectation reduced to 0 because the winner is almost determined, so any work
cannot reverse the tide. To this end, p⃗2 represents the gap between expectation andconcentration; when the score difference is 1 or -1, the situation at this time is a situationthat can quickly change. A player quintessentially wants to anticipate the reverse ofsuch a situation, making the magnitude of engagement the highest.
Based on the conceptual basis of the motion in mind and flow theory, the professional basketball games’ dynamic process is analyzed. The process of obtaining in-game
scores can be associated with the player’s abilities. Meanwhile, the abilities of their opponent (opponent’s scores) can be regarded as a challenge. Hence, the instantaneous
psychophysiology measures, expected from the interaction dynamics of the two sides(player and their opponent) in the game-playing, can be visualized. For this purpose,actual data of NBA games between two teams were adopted, where the game between
the Milwaukee Bucks (ranked second) versus the New York Knicks (Ranked 26th) was
analyzed for their interaction dynamics.
The analysis of the game match between the two teams was summarized in Table 3.9.
Their score S is less than ten at the beginning of the process, and their score gap (∆S)
is smaller than one. Such a situation makes the prospect of the game unclear since
the score gap is not apparent; thus, associated with apathy. However, as the gameprogresses, S > 10 while ∆S were still smaller than one, implying that the game istrapped in a stalemate and white-hot stage; in this situation, the potential to reverse
the situation could happen in time. Flow conditions could occur due to the need to beattentive and indulged in such a moment.
As the game progresses further, their score gap gradually becomes larger (2 ≤|∆S| ≤ 4); when their scores are less than 10, it means that it is also the beginningof the process, the advantageous team would experience boredom while the other team50

Table 3.9: Dynamical emotions and the corresponding description
Emotions

Game Score

Skill:Challenge Ratio

ApathyFlow
Boredom
WorryControl
ArousalAnxiety
Relax|∆S| ≤ 1, S ≤ 10|∆S| ≤ 1, S > 10
2 ≤ |∆S| ≤ 4, ∆S is positive, S ≤ 10
2 ≤ |∆S| ≤ 4, ∆S is negative, S ≤ 10
2 ≤ |∆S| ≤ 4, ∆S is positive, S > 10
2 ≤ |∆S| ≤ 4, ∆S is negative, S > 10|∆S| ≥ 5, ∆S is negative|∆S| ≥ 5, ∆S is positive

Low:Low
High:High
Medium:Low
Low:Medium
High:Medium
Medium:High
Low:High
High:Low

experience worried. As the game progresses gradually, this condition implies that theadvantageous team understands this process and experiences the notion of control, andthe other team experiences arousal. Also, when two side teams’ ∆S > 5, the condition
becomes too difficult to reverse, in which the team in an advantageous position would
undergo a relaxing experience while the other team in a disadvantaged position would
experience anxiety. Such condition can be validated from the observation made fromthe recorded game recap of the final matches between Milwaukee Bucks versus the
New York Knicks (i.e., Game Recap: Bucks 112, Knicks 97, see https://www.you
tube.com/watch?v=8VduEjEsNO8; Game Recap: Bucks 112, Knicks 100, see https://www.youtube.com/watch?v=sKra9m6YqUo; and Game Recap: Knicks 113, Bucks 98,
see https://www.youtube.com/watch?v=2dlearLKJEk).
The listing of such dynamic processes for the winner’s side is depicted in Figure 3-9 toconsider the composition of such experiences. It can be observed that the advantageous(or stronger) team would experience mainly relaxation and control during the gameprogression due to their skill being higher than the challenge they faced (opponent’s
ability). Therefore, considering these interactions and dynamics of the game-playing
experience would be beneficial in designing a personalized entertainment, specialized
monitoring, or training program (e.g., medical rehabilitation) and leveraging the flow
experience in a specific activity or task.

51

Relax44 %24 %Control

8% 7%

1%
4%
5%
7%

Anxiety
Boredom
ApathyFlow

ArousalWorry

Figure 3-9: An example of the dynamic interactions and game-playing experience of
one game process based on its association with the Flow theory

3.6.4

Limitations and Future Works

The limitations of the study are as follows. The proposed model of interaction dynamic and game-playing experience involves mapping the game processes as psychophysiological measures. However, such a measure would be a reliable measure when
the game-playing scores of the game are obtained by actual human players (such as
the NBA score data). However, this measure would not provide a psycho-physiologicalmeasure of players when the data is obtained using AI agents and requires further
justification of human players in those contexts.
In addition, validating the collective emotion experienced by the players in a teambased setting remains a challenging feat in the research community. Moreover, justification was needed from the theoretical studies in psychology, empirical evidence on theinfluence of games on human perceptions, the importance of personality characteristics,
and the oscillating psychological states in game-playing. Those areas would be fertile
grounds for future prospective studies to verify and validate this study.
Future works can also investigate the influences of information dynamics, such as
52

jerk value, on different player strengths, varying game complexity and rules, and differences between simple classical board games and sophisticated modern video games.
Another aspect of potential future study involves developing a framework of playerdriven game-playing experience by manipulating surprising events via such information
dynamic (e.g., jerk value).

3.7

Chapter Conclusion

Our research linked game progression to emotions through game refinement theory anda physical model, in games, effort, achievement, and unpredictability are aligned withvelocity, acceleration, and jerk. Higher jerk values in games increase surprise and makeplayers addictive and repetitive.
The construction of a game information model is rooted in a game tree, striving for acomprehensive visualization and quantitative representation of intricate game processes.
Concurrently, it integrates pertinent theories such as game refinement theory and flow
theory, intending to thoroughly assess the entertainment value and complexity of games.
Psychological states in the game’s progression were elucidated by linking scoring
situations with the emotional states outlined in flow theory. This connection was revealed through the lens of game refinement theory and the analogy of a physical model.
Various game elements were correlated with the physical model to ascertain the mind’s
psychological conditions, where appropriate effort, achievement, and unpredictability
corresponded to the physical quantities of velocity, acceleration, and jerk, respectively.
The dynamics of diverse associated psychological states during gameplay were established, showcasing how sophisticated games leverage jerk value to heighten difficulty,
leading to players experiencing sudden rewards that enhance the element of surprise.
Additionally, players were anticipated to contend with interactive forces, illustratingthat when players’ abilities are similar, the expected “attractiveness” peaks, aligningwith the flow zone. Conversely, when players’ success score rates differ, distinct psy-53

chological aspects are associated with varying measures of game progression. A higher
success score rate than the opponent is linked with relaxation, while a lower score rateis associated with anxiety.
It was found that momentum (⃗p) is essential to measure the tendency of playing
the game during the progress when their scores are different. When linked to the gameprocess, p⃗ implies the tendency to retain focus. When ∆S becomes lessened, the gametends to be continued (peak p⃗ value). Otherwise, the game tends to stop due to gap
differences between the losing and winning teams (high ∆S).
Essentially, a novel perspective on game refinement theory emerged through theintroduction of the jerk value, offering a quantitative framework for game design andmechanistic measures of the player’s anticipated psychology. This viewpoint facilitates a more player-driven experience by pinpointing pertinent jerk values concerning
success rate and informational acceleration. Consequently, maintaining a stable gameprogression yields a comfortable and appealing gaming experience, while incorporating
relevant surprise and discomfort enhances the motivational and attractive aspects ofa more challenging game progression. Motions in mind for design, ensuring a playercentric experience balancing skill and challenge, integrating physics, psychology, anddesign for immersive gaming.

54

Chapter 4
Implications of Jerk’s On The
Measure of Game’s Entertainment:
Discovering Potentially Addictive
Games
This chapter is based on the integration, update, and abridgment of the following publication:• N. Gao, H. Chang, Z. Zhang, M. N. A. Khalid and H. Iida, “Implications of Jerk’s
On the Measure of Game’s Entertainment: Discovering Potentially Addictive
Games”, in IEEE Access, vol. 10, pp. 134362-134376, 2022.

55

4.1

Chapter Introduction

The time derivative of acceleration, termed jerk (or jolt), represents a physical phenomenon that is represented through a sudden change of acceleration (expressed in
m/s3 ), which humans can feel. It has been widely used as a standard (dis)comfort
parameter (e.g., roads, rails, vehicles, trains, elevators, amusement rides) and was applied in manufacturing, engineering, sports science, safety science, and more [53][54].
Other aspects of jerk were studied, focusing on minimizing the wear-and-tear effectof jerk-induced operations of machinery [55] and its impact on the structural-dynamic
applications (i.e., bio-mechanics and robotics) [56]. Nevertheless, jerk application fromthe domain of games was limited since its measurements were typically focused on
physical and external implications in games (c.f. [17][57]).
Research on game refinement (GR) theory focused on exploring the dimensional
implications of the game information progress [58][59][6], where the balance between
certainty and uncertainty in reaching a goal is considered the rate of change of information speed (or acceleration), denoted as GR value. Therefore, game progress and moveselection models were proposed from the game context using the game refinement theory and its new perspective, called the motion in mind model [1]. This model allowed
the uncertainty of the game progress to be analogously measured relative to a physicalmeasure associated with the success rate (v).
Subsequently, considering its first (acceleration, a) and second (jerk, j) derivatives
were associated with speculative psychology, such as engagement and unpredictability,
respectively. Therefore, a jerk in games is interchangeably regarded as an informationaljerk, leading to the association of unpredictability (or surprise), which is denoted asa AD value [60]. When comparing different games, AD value changes could provide
insights between classical and sophisticated games. As such, such a measure could
establish the concept of unpredictability and stability concerning jerk dimensions ingames. The higher the AD value of the game, the more unpredictable and full of56

surprises, and vice versa.
The card game is a typical imperfect information game. Players can only see theircards and the cards that have already been played. Therefore, they need to learn all theinformation about the game process. People indulge in this kind of game (incomplete
information game) since it can be characterized as short, repeatable rounds (replayability), chance-based, and strategy-oriented, making it among the most entertaining games
among adults and teenagers. Coupled with the digital platform and low-cost mobile
devices, the popularity of multiplayer online card games has been increasing where its
market value of about $4 billion in 2019 [61], and has positively grown its market sharein recent years [62]. With a unique blend of recreational and intellectual entertainment,
card games extensively test the ability of the player to navigate massive state space,recall other players’ moves, probabilistically estimate the missing information induced
from incomplete and asymmetric knowledge, and possibly even be addictive. Therefore,
this study adopted card games as the test environment suited for the interest of thisresearch.
There are many types of popular card games in China, and among the most popularwere the Doudizhu[25], 24 points (also called as 24 puzzle) [63], and Wakeng [64]. They
all have various playing styles and are widely accepted by different target audiences.
Wakeng and Doudizhu were two games that share rule similarities but differ in gameplay.
This gameplay includes one based on single cards, pair cards, triple cards, quadruple
cards, chain cards, and various extensions and variations. In addition, it is only related
to the number of points and has nothing to do with the suit. Wakeng’s gameplay
rules were conservative and straightforward, while Doudizhu, the succession version of
Wakeng, was more varied, changeable, and sophisticated.
This study investigates the impact of the rule design of Wakeng, Doudizhu, and othersimilar card games. As these card games were prevalent, it is unclear why these kindsof incomplete information games were associated with being addictive. The addictive
term used here refers to the game being repeatedly played and/or preferred from the57behavioral standpoint, which is not referring to a psychological or substance-related
addiction; instead, a “healthy excessive enthusiasms” that relate to motivation and froma non-pathological perspective [65]. As such, addictive term used in this study refers
to excessive enthusiasm and motivation to repeat and continue the behavior, which
statistically becomes a popular choice by most. As such, simulations were conducted
to imitate difficult situations faced by players beyond their ability, a condition thatmatches player ability, and a case that improves player’s intuitions to explore more
ways to increase the game’s playability; each represented by the different dynamics
of GR and AD values. Several computational experiments were adopted to achieve
such conditions. First, this study compares the game against other related card games
using the game refinement theory and its new perspective. Second, different AI levels
were adopted to simulate different skill levels of players, to understand the stability
and controllability of the game (horizontal view). Third, players’ affective modality(such as engagement and surprise) regarding the gameplay with different rules was also
investigated (vertical perspective). Finally, the potential measures of such affective
modality were identified based on the card games.

4.2

Previous work

The terminology of the “jerk” had been extensively provided by [57], which defined astandard term widely used; the first derivative of acceleration. However, interestingly,
there were alternative terms, such as “acceleration onset rate”, and “jerk-dot”, thatrefer to the same concept that describes the fourth derivative of position. Therefore,
for consistency and generality, the term “jerk” is used throughout this study to describe
the fourth position derivation relative to its alternative uses.
The jerk had been used to measure, control, or generate smooth trajectories, motion
profiles (horizontal or vertical or lateral or rolling), ride (dis)comfort criteria or parameters, human kinesiology and sport science, and shock spectrum analysis [57]. The

58

advancement of unmanned machinery (i.e., robotic excavators) has led to the investigation of optimizing jerk for the joint trajectory planning problem of hydraulic robotics
excavators. The study found that stable and smooth motion with accurate trajectory
can be achieved while mitigating equipment damage (structural and wear) and prolonging its service life [66][67]. Furthermore, understanding and studies on jerk had
been increasing for earthquake engineering had been conducted for the improvement of
the structural dynamics such as structural control, isolation systems, and dampers areapplied in mitigating jerk [68].
Previous research also focuses on the influence of acceleration and higher derivativesof motion relative to the classical mechanics of rigid bodies [17] and acoustics [69][70];often associated with stability and vibration. However, such research works focus on
the physical phenomenon where the influence of jerk had been investigated from thephysical activities and other bodies of implications or systems. Adopting the analogyof classical mechanics and motions, the information in a game can be modeled as
the canonical representation of location, where information can undergo displacement(progress), velocity (success rate), and acceleration (attractiveness). Then, the law of
motion in the game is formalized by considering the derivatives of the game progressmodel, where the link between game refinement theory and flow theory is determined;leading to a more natural yet pleasurable human experience based on the sophisticatedgame’s underlying mechanism [1].
Based on human experience modeling in the game, scarce works had been focusedon it, but rather on developing artificial intelligence (AI) agents or systems that could
efficiently maximize the winning rate in the card game. For instance, the Big Two
AI (Big 2 AI) framework had been proposed by [71] to predict multiple movementsof multiple opponents and determine the card set with a high chance to win. The
known played-card information and unknown holding-card information are explored
to analyze the superiority of card sets. In addition, the weight values for different
card sets are customized based on the analyzed card superiority and game-playing
59data. Furthermore, historical data teaches frequent playing patterns with different
game features. Experimental results show that Big2AI outperforms existing AI and
can achieve the highest win rate (if a winning chance exists) or the least losing points(if there is no chance to win) against computer and human players.
From the perspective of incomplete information games, [72] proposes a complete
game framework for Doudizhu, fully considering the confrontation and cooperationin the Doudizhu game, considering a multirole modeling-based card-playing framework that includes role modeling, card-carrying, and decision-making strategies. Rolemodeling learns different roles and behaviors by using a convolutional neural network.
Cards carrying can calculate reasonable rules using an evaluation algorithm. Finally,
decision-making is for implementing different card strategies for different player roles
while discussing some key factors affecting them. Experimental results showed that thiscard-playing framework makes playing decisions like human beings, learn, collaborate,
and reason when facing incomplete information game problems. Meanwhile, the complexity and attractiveness of Doudizhu as a multiplayer incomplete information gamewere investigated by analyzing the combinations of player numbers, player roles, andscoring systems [25]. AI players of different levels were constructed for self-game simulation. Game refinement measures were used to evaluate and identify the best settings
for the Doudizhu game and clarify its popularity.
Previous works focused on taking the physical measure of “jerk” in smoothing andcomfort of physical operations. Meanwhile, such a measure is still a nascent area inthe development of games [60], where it was essentially dominated by the developmentof AI. Although game concepts have been adopted in various problem domains, understanding the nature of game playing’s underlying mechanism has been limited. This
study investigates such a mechanism by adopting a logistic model of game outcome uncertainty, where a measure of game refinement has been derived. It was applied to the
popular board and sports games where the implications of its sophisticated design had
been investigated [2],[58][59][6]. It also has defined the harmonic balance between skill
60and chance, where theoretical analysis using empirical data confirms the effectivenessof the proposed model.

4.3

Analysis of Card Games

Some of the prominent characteristics of defining card games were their suites, thenumber of players, starting conditions, outplay mechanisms, and card decks. A suitis one of the groups into which the cards in a deck are classified as playing cards[73][74]. Typically, each card contains one of many pips (symbols) indicating which
suit it belongs to (♣: clubs; ♦: diamonds; ♥: hearts; ♠: spades); the suit may also bedenoted by the color written on the card. Except for face cards, the rank of each cardis determined by the number of pips on it. There is no order between the suits unlessestablished in the rules of a given card game (for instance, in Tien Len, the rankingof cards based on numerical and suit: 2♠, A♥, A♦, A♣, A♠, K♥), so ranking reveals
which cards within a suit are better, higher, or more valuable than others [74]. Thereis exactly one card of each rank in each suit in a single deck. In addition, a deck may
contain one-of-a-kind cards that do not belong to any suit, known as jokers.
General characteristics of the considered card games were outlined and compared inTable 4.1. The studied card games were considered climbing games, where the player
can simply pass if they cannot or do not wish to beat the previous play [74]. Among thefive-card games considered, only two were suits irrelevant. Note that the characteristics
outlined were the typical conditions or rules associated with the game characteristics.
In addition, these card games were suited for evaluating this study due to the uniquecharacteristics associated with the card games. Firstly, card games were incomplete
information games, where some information about the game is hidden throughout thegame-playing progression. Secondly, card games involve some magnitude of the affective
modalities (engagement and surprise), where the change of information throughout thegame progression impacts the play experience. Therefore, the variants of card games

61

were adopted where the impact of the rule design and their combinations with the skill
level of the players were explored.
Table 4.1: Comparison of card games considered in this study

Characteristic

Wakeng Doudizhu

WinnerBig Two

Tien Len

Suit Types
Typical Player Numbers
Standard Card Deck
Initial Cards
Start Condition

N/A
Three
54
16
4♥

Suits
Four
52
13
Trick card⋆

Poker Hands
Four
53
13
Lowest card (3♦)

ReverseFour
52
13
Lowest card (3♠)

N/A
Three
54
17
Role Bidding†

N/A: Suit irrelevant card games, where players can play the game with all the suits erased fromthe cards; ⋆ : trick card determined by the dealer, placed face up in the pile of cards he/she deals;†: role bidding involving bidding for the landlord or peasant-based on self-declared stakes (lowto high stakes, 1–3); Poker hands: Each hand belongs to a category determined by the patterns
formed by its cards [75]. A hand in a higher-ranking category consistently ranks higher than a
hand in a lower-ranking category. A hand is ranked within its category using the ranks of its cards.
Individual cards are ranked from highest to lowest: A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, and 2. Suits
are not ranked, so hands that differ by suit are of equal rank [76].

4.3.1

Suits irrelevant card games: Wakeng and Doudizhu

Wakeng
The meaning of Wakeng is “dig the pit” there may be treasures and gold in the pit,
but there also may exist a crisis in the hole, so sometimes Wakeng implies “trap” inChinese. Usually, the game is played by three players, and everyone keeps 16 cards
first. Meanwhile, four cards are hidden in the “pit”, and the priority player can decideto dig or not; those four cards may be a bonus but sometimes a punishment for the
digger. It is a card game popular in Northwest China and a basic classical poker game.
The player who keeps “4 of Hearts” has the priority to be the digger to get four more
hidden cards and start the first round.
Moreover, if the “4 of Hearts” was hidden, the priority player changed to be theplayer who keeps “5 of Hearts”. And so on. When the priority player does not want
62

Table 4.2: The card types of Wakeng
Card Type

Description

ExamplesLS

HL

singleone card
43pair
2 same cards
44
33
triple3 same cards
444
333
quadruple
4 same cards
4444
3333
⋆
chain
≥3 single
456
4567890JQK†⋆
paired≥3 pairs
4455664567890JQK*2
triple†≥3 triples⋆
444555666
4567890JQK*3†⋆
quadruple≥3 quadruples 444455556666 4567890JQK*4†⋆: chain; : consecutive; LS: lowest & short rank;
HL: highest & longest rank;
the hidden card, other players can choose to dig the four more hidden cards. Afterthe hidden cards are decided to keep by someone, the rest of the players will become ateam automatically and against the digger, and the player who plays all the cards out
first wins. There are only 8 card types, as Table 4.2 shows, and every player can playaccording to these 8 card types. Also, the next player only can play the cards with thesame type and more robust than the card of the previous hand.
There are 52 cards in total (two jokers are not included). Except for the four cards
covered in the pit, the remaining 48 cards are equally divided among each player, 16
cards each. The player who holds the “4 of hearts” decides whether the four hidden
cards at the pit are required. If not, the priority is transferred to the next player. If no
players want it, then the priority holder has to pick it. When none of the three players
hold the card “4 of hearts”, the priority will be transferred to the player who holds
the card “5 of hearts”. So, the player who holds the card “5 of hearts” will prioritize
getting four hidden cards or not. This process could repeat until the card “8 of hearts”.
At the same time, the person holding the card “4 of hearts” first plays the card;
he has the priority to play and set the card type. After that, he can play any of the63Table 4.3: The card types of Doudizhu
Card

Description

Type

ExamplesLS

HL

singlestraight
pair
wings

one card
3
JOKER≥5 single⋆
34567 34567890JQKA
2 same33
22≥3 pairs⋆
334455 34567890JQKA*2
3 same333
222
triplet3 same + a kiker
3334
222A
3 same + pair kiker 33344 222AA
plane≥2 triplets
333444 QQQKKKAAA88990022
bomb
4 of a kind
3333
2222
rocket
JOKER and joker⋆: consecutive; LS: lowest & short rank; HL: highest & longest rank;available card types, and the next player needs to draw the corresponding card type,which must be stronger than the previous player’s card. The process repeats until itcannot be continued, constituting one round. If a player’s card cannot be defeated byany other player and the round is stopped, then this player has the priority to playcards and can play any of the eight card types to start a new round. He and his teamwill win if someone plays out all the cards in his hand.
DouDiZhu
DouDiZhu also called ‘Fighting Landlord,’ ‘Landlords,’ or ‘2 against 1,’ is China’s mostpopular computer card game. In 2017, the download amount of DouDiZhu network
clients reached 1.13 billion [77]. Although the categories of cards are more diverse, thecategories of cards of the Doudizhu game, as given in Table 4.3, and its rules are moresophisticated but still similar [78]. As a result, Doudizhu is more popular and has abroader audience than Wakeng.
The classic DouDiZhu game features three players, two of whom, referred to as “thepeasants”, must collaborate against another, referred to as “the landlord”. The game64

usually lasts between one and three minutes long [77][78]. This circumstance allowed
individuals to play the game anytime and from any location. If the peasants’ hands areplayed first, they win; otherwise, the landlord wins. Profits and losses are shared among
the peasants in the game, whereas the landlord carries himself alone, indicating thatthe game is a zero-sum game satisfying Nash Equilibrium. As in most card games, thestarting hand of DouDiZhu has the most considerable influence on the game’s outcome.
The rules of DouDiZhu are simple, but the two most essential parts of winning thegame demand strategies and expertise.

4.3.2

Suits relevant card games: Big Two, Winner, and Tien
Len

Winner
The Winner shared similarities to the DouDiZhu. The main differences between Winnerand classic DouDiZhu are found in two aspects. Firstly, the number of players where theWinner can play at least two players, usually not more than eight, using two decks ofcards when there are more than four players. Secondly, each player of the Winner fights
on their own. The Winner plays out all the hands, while other players continue until
there is just one player, and they score by sequence ranks to play out cards. Thirdly,
hands in Winner have a different weight of the suit, which ranked from biggest to
lowest (♠ > ♥ > ♣ > ♦). Suit discrimination gives the Winner many card combination
possibilities [79].
Big Two
Big two (also known as deuces, Capsa, pusoy dos, dai di, and various other names)
is a card game of Chinese origin. It is similar to the games of the Winner, Daifugō,President, Crazy Eights, Cheat, and other shedding games [80]. The game is very
popular in East Asia and Southeast Asia. It is played casually and as a gambling game,65

usually with two to four players, where the entire deck is dealt out in either case (orsometimes with only 13 cards per player if there are fewer than four players). The gameinvolves using poker hands, where the hand belongs to a category determined by thepatterns formed by its cards [75]. The game’s objective is to be the first to get rid of
all the cards in hand by constantly one-upping opponents with cards played singly orin certain poker combinations (pair cards, three cards, or five cards) [81].
Tien Len
Tien Len, also known as Vietnamese cards, Thirteen, Poison, or Killer 13, is a sheddingtype card game popular in Vietnam [82]. It is derived from the Chinese card game
Winner, which uses a specially printed deck of cards, and Big Two. Considered thenational card game of Vietnam, the game is intended and best for four players. For
the first hand, the dealer is picked randomly; for subsequent hands, the loser of theprevious hand deals. Cards ranked from high to low: 2 A K Q J 10 9 8 7 6 5 4 3.
Within the numerical ranking, suits rank (from high to low): Hearts ♥ – Diamonds ♦– Clubs ♣ – Spades ♠. So for example (from higher to lower): 2♠ A♥ A♦ A♣ A♠ K♥.
Thus, 2♥ is the highest card and 3♠ the lowest. Players, in turn, discard a single or
combination of cards to a central face-up pile. The objective is to avoid being the last
player to hold any cards.

4.4

Proposed Computational Models

4.4.1

Game progress model

The proposed logistic model of game uncertainty is based on the early work by [4]. Fromthe player’s viewpoint, the information on the game result is an increasing function oftime (the number of moves in board games) t. Here, the information on the game result
is defined as the amount of solved uncertainty (or information obtained) x(t), as given66

by (4.1). The parameter n (where 1 ≤ n ∈ N ) is the number of possible options andx(0) = 0 and x(T ) = 1. x(T ) is the normalized amount of solved uncertainty, where0 ≤ t ≤ T and 0 ≤ x(t) ≤ 1. Equation 4.1 implied that the rate of increase in thesolved information x′ (t) is proportional to x(t) and inversely proportional to t. Solving(4.1), (4.2) is obtained.
n
x(t)
t

(4.1)

 n
t
x(t) =T(4.2)

x′ (t) =It is assumed that the solved information x(t) is twice derivable at t ∈ [0, T ]. The
second derivative of (4.2) indicates the accelerated velocity of the solved uncertainty
along with the game progress, given by (4.3). The acceleration of velocity implies thedifference in the rate of acquired information during the game’s progress. The motion
with constant acceleration a or free-fall motion in mind is given by (4.4). In the domainof board games, the acceleration a is approximated as (4.5), where B and D represent
the average number of possible moves and game length, respectively.

x′′ (t) =n (n − 1) n−2
t
|t=T
Tn

a=

=

n (n − 1)
T2

(4.3)

1
x(t) = at2
2

(4.4)

n (n − 1)
B
≈ 2
2
T
D

(4.5)

Two processes with the same GR-value at the end of the game information progress
may have different instantaneous GR-value tendencies. For example, two basketballteams have the same successful shootings and total shot attempts. So their GR values
are the same when the game is finished. However, each team felt different tendencies
67

to get scores. A team with a stable scoring process is predictable, and vice versa. As
such, each team’s force was not only felt but also the change of the force. In physics,
acceleration can be felt in motion, and the feeling of jerk [17].
The third derivative of (4.2) indicates the change of accelerated velocity (or jerk [17])of the solved uncertainty along the game progress [18], which is given by (4.6). Because
acceleration is always associated with the uncertainty of the game process, we can measure the attractiveness F = ma concerning Newton’s second law, which measures theacceleration process of attractiveness. Then, the first derivative of acceleration, called“jerk”, would be associated with the measures of the tendency of getting attractiveness(or unpredictability), which could be speculated with the addiction mechanism. Hence,
the motion with constant jerk j is given by (4.7), where it is approximate in the domainof board games as (4.8).

x′′′ (t) =n(n − 1)(n − 2) n−3n(n − 1)(n − 2)t|t=T =nT
T3
1
x(t) = jt36j=n(n − 1)(n − 2)
B
≈3 3
3
T
D

(4.6)

(4.7)

(4.8)

Table 4.4 shows the measures of game refinement for board games. For sophisticatedboard games such as Chess, Shogi, and Go, it is assumed that there exists a reasonablezone for the acceleration (a) and jerk (j), which is between 0.07−0.08, and 0.045−0.06,
respectively. Given the relations described by (4.9) and (4.10), then Table 4.4 can beobtained, which shows the reasonable acceleration and jerk zone of games that arederivative of popular, sophisticated board games.

GR =

√

√
B
a=
D

68

(4.9)

√
3
p
3B
3
AD = j =D(4.10)

Table 4.4: Measures of game refinement for board games

Chess
Shogi
Go

4.4.2

B

D

35
80
250

80
115
208

a

√
3

0.074
0.078
0.076

0.059
0.054
0.044

√

j

Motion in mind model

In operant conditioning, a variable-ratio schedule is a schedule of reinforcement wherea response is reinforced after an unpredictable number of responses [12, 13]. This typeof schedule creates a steady, high rate of reaction. Stochastic games such as gambling
and lottery games are typical examples of a reward based on a variable-ratio schedule.
Mind sports games such as chess and Go were also essentially stochastic games while
applying the move selection model [1]. This situation implies that a reward of variableratio reinforcement schedule characterizes a game [14].
Therefore, the reward function of a game can be characterized by defining the reinforcement schedule’s variable rate (denoted as V R(N )). Then, velocity v (win rate)
and mass m (win hardness) of the motion in mind model are given by (4.11), where0 ≤ v ≤ 1 and 0 ≤ m ≤ 1. As such, N was used to measure the frequency of getting
rewards, where the player can get a reward in a total average of N steps [15]. Let v0 be
the reward function over various masses for the perfect player as given by (4.12), whichcorresponds to the objectivity of play. Note that there is a distinctive computation of
the v for the board and scoring games [1]. The success rate is defined as v = G
for scorT
ing games (such as basketball, soccer, etc.), where G and T are the average successfulBscore and the total scores. Meanwhile, the success rate is defined as v = 2D
in a board

69

game (i.e., Chess, Shogi, etc.), where B is the average branching factor, and D is theaverage game length.
1
N

and m = 1 − v, where 1 ≤ N ∈ R(4.11)

m + v0 = 1, where 0 ≤ m ≤ 1 and 0 ≤ v0 ≤ 1(4.12)v=

The concept of energy conservation was introduced by [16] as a prospective metricfor engagement. The formulation of momentum in the game (p⃗1 ) and potential energy
in the mind (Ep ) is provided by Equations (4.13) and (4.14), respectively. Subsequently,
the momentum in the mind (p⃗2 ) can be deduced through the conservation of energy
in the mind, as expressed in Equation (4.15). This derived momentum in the mind isclosely linked to the quantification of a player’s engagement, as defined by Equation(4.16).

p⃗1 = mv

(4.13)Ep = 2mv 2(4.14)Ep = p⃗1 + p⃗2(4.15)

p⃗2 = Ep − p⃗1 = 2m3 − 3m2 + m(4.16)

Applying (4.16) while assuming p⃗2 = mv2 , the subjective velocity v2 is given by(4.17). Let vk (m) be a reward function over various m for a player with ability parameterk. Then, the relation is generalized as vk using a parameter (say k where 0 ≤ k ∈ R) thatis the nature of the game under consideration, as shown in (4.18). The ability parameter70

k stands for players’ strength in the competitive game context or error tolerance inthe social or non-competitive context. For example, there is no error tolerance forthe perfect player v0 . Note that objectivity and subjectivity perspective enables us
to deepen the understanding of engagement and addictive mechanisms in games [16].
Thus, the objective velocity (v0 ) and subjective velocity (vk ) were determined.
v2 = 2m2 − 3m + 1 = (1 − 2m)(1 − m)

(4.17)

vk = (1 − km) v0 , where 0 ≤ k ∈ R(4.18)The notion of potential energy in mind was originally discussed by [1] and its formula
is given by (4.14). The notion of velocity is derived from the reinforcement schedule
V R(N ) with frequency N , so we call objective reinforcement (E0 ) for the potential
energy in mind of the perfect player (v0 ). Otherwise, we call subjective reinforcement(Ek ) for the potential energy in the mind of other players (vk ). A game would produce
its potential energy in the field of play (hence we call it potential energy of play) by
which people would feel engagement or reinforcement.
In behavioral psychology, the term “reinforcement” refers to an enhancement of
behavior. This term was used as a positive interpretation, i.e., greater reinforcement
gives people a more substantial interest in staying in the event under consideration. In
the game context, reinforcement depends on the player’s ability. The potential energy ofplay (Ek ) is given by, Ek = 2mvk2 which is denoted as subjective reinforcement. For theperfect player or game theoretical reward (k = 0), denoted as objective reinforcement
E0 . Figure 4-1 illustrates the objective and subjective reinforcement when k = 3.
The reward function (vk ) represents a player’s model or his/her sense of value. Whenassuming k > 3, vk < 0 holds at m = 13 where the objective reinforcement is maximized.
This situation implies the learning context’s most comfortable (peak of E0 ). Therefore,
it is highly expected to have k ≤ 3. Furthermore, Go (m = 0.42) is still not yet solved;

71

thus, 2.38 < k is expected to hold.

motion in mind measures

1

v0
vk
E0
Ek

0.8
0.6
0.4
0.2
0
−0.2
0

0.1

0.2

0.3

0.4

0.5 0.6
mass m

0.7

0.8

0.9

1

Figure 4-1: Objective and subjective reinforcement when k = 3

4.4.3

Experimental design

The experiment is based on several logical settings to mimic the different experiencesof players. Different logical rules were set by listing all possible actions, and a pruningalgorithm was utilized to find the optimal solution based on the different weights (Table 4.5). The first experiment focuses on fixing the game but is played with different
AI levels (horizontal perspective): weak, fair, and strong. As such, analysis of playerswith different skill levels was simulated.
Subsequently, the second experiment was conducted on the same player (fixed AI
level) who played games with varying levels of complexity (vertical perspective). Suchanalyses were simulated from the different rules represented by the increasing complexity of the card games, from classical ones to more sophisticated games (i.e., between
Wakeng and Doudizhu).

72

Algorithm 1 Card Game Algorithm
1: pass count l ← 0
2: pass count p ← 0
3: is end ← false
4: while not is end do
5:
if is played then6:last played ← 0
7:
else
8:
pass count leq1
9:
end if10:
for i ← 0 to 2 do
11:check end(card groups[i])12:player action policy(card group, i + 1, last player, his, valid moves)
13:
if is played then14:
last played ← 0 ELSE
15:pass count l1
16:
end if17:
end for18:
current player ← id
19:
opponent ← 3 − current player {Find the opponent player}
20:
if player id == last player then21:
valid cards, length ← list greater cards(, card group)
22:
else
23:
if check weight(his[−1][0], his[−1][1]) ≤ thresholdandlength > 1 then24:
valid cards, length ← list greater cards(his[−1][1], card group)25:
end if26:
end if27:
played cards ← play rules(valid cards)
28:
remove cards(valid cards, card group)29:
if length == 1 then30:
return false31:
end if
32: end while=073

Table 4.5: The experiment design of Wakeng and Doudizhu
AI level

Description

Weak

The priority players play the longest weakest cards first,
and other players play their cards as much as possible
based on the cards of the previous player.

Fair

Besides the basic logic of weak AI, two players otherthan the digger cooperate. If one of them plays a cardwith a large difference in weight to defeat the digger’scard, he chooses not to defeat his teammate.

Strong

Add a new restriction to the previous logic, play any
cards but can not cause two more solo cards. If a player
can defeat the previous player, but he played this hand,
it will cause two more solo cards, then he will give up
playing cards.

4.5

Computational Results

4.5.1

Result analysis of DouDiZhu

The classic Doudizhu is the most popular version in China, played by three players.
Valid cards include a single card, pair card, straight card, triple card, double-wing card,plane card, bomb card, rocket card, and 300 cards, which kickers can play. DouDiZhu
game has a unique score system. The basic score is set at 3 points in this study,
doubling the score when the player played a bomb or rocket [78]. For example, thelandlord obtains a total of 24 points while the peasant loses 12 points if there arethree bombs; thus, the landlord wins the game. The conducted simulation considers areasonable level of DouDiZhu AI, which corresponds to the average level of a humanplayer (Table 4.6).
The result of the conducted simulation considers a reasonable level of DouDiZhu AI,
which corresponds to the average level of a human player. As the GR = 0.0715, it islocated in the sophistication zone (GR ∈ [0.07, 0.08]). So, most people could enjoy the74Table 4.6: Measures of game refinement for classical DouDiZhu
B

D

GR

AD

8.197 40.032 0.0715 0.0726
B: average possible options;
D: total steps;
Doudizhu game. Moreover, the AD = 0.0726 is higher than the reasonable zone value(AD ∈ [0.045, 0.060]). This situation indicates that the Doudizhu game contains lotsof unpredictability and instability. Because of Doudizhu’s vibrant and diverse rules,
players often feel surprised and unpredictable. Subsequently, the results of different
levels of AI with different settings were analyzed (Table 4.7).
Table 4.7: Results of different levels of AI with different DouDiZhu game settings
Setting†

AI levelB(3, 0, 3)(18, 18, 18)

fairstrong

7.344
6.657

45.418 0.060 0.062
53.212 0.048 0.051

weak(3, 1, 2)(20, 17, 17) fairstrong

7.667
8.198
6.613

38.486 0.072 0.074
40.032 0.072 0.073
53.212 0.048 0.051

weak(3, 1, 2)(18, 18, 18) fairstrong

7.232
7.644
6.021

38.964 0.069 0.072
40.696 0.068 0.070
53.075 0.046 0.049

weak(3, 1, 2)(22, 16, 16) fairstrong

9.312
10.02
8.037

37.605 0.081 0.081
39.070 0.081 0.080
50.929 0.056 0.057

weak(3, 1, 2)(24, 15, 15) fairstrong13.438 36.520 0.100 0.094
14.373 37.810 0.100 0.093
11.761 48.026 0.071 0.068

†

D

GR

AD

: the first parenthesis represents the player’s configuration (number ofplayers, landlord number, peasant number); the second parenthesis represents the respective player’s initial hands (landlord, peasant 1, peasant2); B: average possible options; D: total steps;75Two observations can be made from the analysis of the result of the sophisticatedcard game Doudizhu (illustrated as Figure 4-2). Firstly, weak players’ ability is insufficient to adapt to the game’s difficulty. At this time, the GR value is higher than
the reasonable range. As a result, the game becomes difficult for the player, where theAD value is higher than the reasonable range, which makes the game too varied andunpredictable. After learning and training, players can master and enjoy the game, and
the GR value and AD value become lower and close to the reasonable zone. Secondly, if
the game is not balanced and fair enough for the players (for example, if the differencein the number of cards is too large), the uncertainty and unpredictability of the game
become too high. For players, the AD value has a more significant impact at this time,
which they will indulge in surprises.

Figure 4-2: The tendency of GR and AD based on the ability level of sophisticatedcard games76

4.5.2

Result analysis of Wakeng

Wakeng was explored and analyzed using different levels of AI to simulate different
players with varying skill levels (Table 4.8). Based on the table, it can be observedthat the GR value increases as the player’s skill increases (canonical to players of weakskill to strong skill). This condition implies that when players explore the variation of
the classical gameplay and develop more ways to play, they will feel more attracted toplaying the game.
Table 4.8: The analysis of Wakeng based on different level AI based on the setting of(3, 1, 2)(20,16,16)
AI level

B

D

GR

AD

weak
fair
strong4.10 49.72 0.041 0.046
4.69 54.15 0.040 0.045
4.12 37.48 0.054 0.062

B: average possible options; D: total steps;

However, classical games do not have much variety where the rules are relatively
fixed and straightforward, making the unpredictability very low (AD ∈ [0.045, 0.060]).
Nevertheless, players can play more cards after learning and training, improving thegames’ unpredictability (as indicated by the increasing GR and AD values). However,the change between the different AI levels was insignificant, implying that even thoughmastery of the game is achieved, players’ entertainment value and controllability will
not change significantly.

4.5.3

Result analysis of suits-relevant card games

It can be observed that the stability and controllability measured by the AD value basedon different AI levels and levels of sophistication between Doudizhu and Wakeng wererelated to the points of cards; suits are irrelevant. Subsequently, similar suit-relevant
77card games, such as Winner, Big Two, and Tien Len, were explored further. From thedata analysis of Table 4.9, a similar observation can be found from these suit-relevant
card games, verifying the previous findings. For players with a high skill (simulated bystrong AI), or the game was advanced and sophisticated enough, the GR and AD values
increased and reached a reasonable zone leading to a higher entertainment experienceand higher unpredictability.
Table 4.9: The analysis of several card games based on different level AI and setting
Games

AI level

B

D

GR

weak

3.22

63.29

0.028 0.034

fair

3.33

65.67

0.028

0.033

strong

3.68

44.71

0.043

0.050

weak

3.43

61.10

0.030 0.036

fair

4.54

64.10

0.033

0.037

strong

3.88

42.27

0.047

0.055

weak

3.95

56.94

0.035 0.040

Tien Le fair

5.09

56.93

0.040

0.044

3.72

43.70

0.044

0.051

Big two

WinnerstrongAD

The influence of suits usually demands the game to be played by more players.
Typically, the number of cards in each player’s hand will be reduced, slightly impacting
the balance it has on the gameplay. Collected data showed that GR and AD values
increase to the reasonable zone when the player becomes stronger. For the classicalgames, the GR and AD values of strong players are higher than the weak, implying
players need to train to explore more gameplay to make the relatively bland gameprocess more entertaining (Figure 4-3).
78

Figure 4-3: The tendency of GR and AD based on the ability level of classical card
games

4.5.4

Result analysis of fixed AI levels

Based on the standard game settings, Doudizhu of parameters of (3, 1, 2)(20, 17, 17)
and Wakeng of parameters of (3, 1, 2)(20, 16, 16) have the AD value between 0.051 to
0.074 and 0.04 to 0.045, respectively. It can be observed that the AD value of Wakeng
is lower than Doudizhu because its gameplay was conservative, with fewer options,and thus much more stable than Doudizhu. Therefore, the rules of Wakeng are morepredictable, associated with the low value of AD. This finding is aligned with previous
studies that associate low AD values with simple and plain games [18]. Thus, therelationship between player level and game complexity, based on GR and AD values,
was described in Table 4.10.
The GR and AD values decrease when players become stronger for high-complexity
games like Doudizhu. When the player starts to play, the ability is not enough toadapt to the game’s difficulty. Currently, the GR and AD values are higher than thereasonable range. The game is challenging and unpredictable for the player. However,players can master and enjoy the game after learning and training. For simple games,
The GR and AD values increase when players become stronger. When the player starts

79

Table 4.10: Closeness to reasonable zone of GR and AD given by different levels of
players of different game complexity
Player Level

Classical

Advanced

Weak<>
Normal≈≈Skillful==<: less than reasonable zone of GR/AD;>: more than reasonable zone of GR/AD;≈: approximately close to reasonable zone of GR/AD;=: within the reasonable zone of GR/AD;
to play, they will feel simple and predictable. Both GR and AD are smaller than areasonable range. However, different gameplay can be developed and explored after
training, so the GR and AD values will slowly increase, and players will slowly build
interest and unexpectedness.
Table 4.11: The entertainment aspects of different GR and AD value expression
GR & AD

Description< zoneskill > sophistication, too easy, need to continue to explore theevolution in the game rules to make the game more interesting= zonePerfect match> zoneskill < sophistication, too difficult, need to improve their skills tomaster the game

Table 4.11 summarizes the expression of different GR and AD values from the gameentertainment perspectives. If the game is challenging for the player, the player keeps
training on the process, where they can become proficient and skillful, improving theirskills. The process is slow, yet mastering the game would gradually match the game,and the player can feel the joy of conquering. On the other hand, if the game is easy forthe player, the player keeps developing and exploring other gameplay, which they findvariable ways to play. The process includes exploring more game possibilities to make80

the process simpler, more exciting, and more variable. As such, GR and AD values
tendencies relative to the training time influence the game’s difficulty and mastery(Figure 4-4).

Figure 4-4: The tendency of GR and AD value of different complexity game

4.6

Discussion

4.6.1

Comparison on different game complexities

The suit-relevant and suit-irrelevant card games were further explored from the perspective of the motion in mind model, in comparison to the game progress model (GRand AD values), in which the results were presented in Table 4.12 and Table 4.13.
For suit-irrelevant games, the velocity of Doudizhu is higher than Wakeng, And forsuit-relevant games, the velocity of Tien Len is higher than Big Two. This condition
showed that the speed of a sophisticated game is higher than a classical and simple
game. Then, the momentum of the game’s motion (p⃗1 ) described the players’ attention
and concentration and the tendency to keep playing games. It can be observed thatthe momentum (p⃗1 ) of Doudizhu is higher than Wakeng, while Tien Len is higher than
Big Two. These situations implied that players tend to continue the sophisticated andadvanced games and have a greater tendency to play advanced games.
81

Table 4.13: The comparison of suit-relevant card games based on a standard setting

Setting

AI level B

D

GR AD

v

m

p⃗1

E⃗p

p⃗2

a2

strongTien Len [(4, 2, 2)(13, 13, 13, 13)] fairweak

3.721 43.700 0.044 0.051 0.043 0.957 0.041 0.003 -0.037 0.039
5.090 56.930 0.040 0.044 0.045 0.955 0.043 0.004 -0.039 0.041
3.950 56.931 0.035 0.040 0.035 0.965 0.033 0.002 -0.031 0.032

strongfair
weak

3.680 44.705 0.043 0.050 0.041 0.959 0.039 0.003 -0.036 0.038
3.330 65.670 0.028 0.033 0.025 0.975 0.025 0.001 -0.023 0.024
3.220 63.290 0.028 0.034 0.025 0.975 0.025 0.001 -0.024 0.024

Big two[(4, 2, 2)(13, 13, 13, 13)]

Table 4.12: The comparison of suit irrelevant based on a standard setting
Setting

D

GR

AD

v

m

p⃗1

E⃗p

AI level

B

p⃗2

a2

strong6.61 53.21 0.048 0.051 0.062 0.938 0.058 0.004 -0.055 0.751

DDZ [(3, 1, 2)(20, 17, 17)] fair

8.20 40.03 0.072 0.073 0.102 0.898 0.092 0.009 -0.082 0.590

weak

7.67 38.49 0.072 0.074 0.100 0.900 0.090 0.009 -0.081 0.602

strong4.12 37.48 0.054 0.062 0.055 0.945 0.052 0.006 -0.046 0.049

fair

4.69 54.15 0.040 0.045 0.043 0.957 0.041 0.004 -0.038 0.040

weak

4.10 49.72 0.041 0.046 0.041 0.959 0.040 0.003 -0.036 0.038

WK [(3, 1, 2)(20, 16, 16)]

Also, the potential energy (E⃗p ) of Doudizhu is higher than Wakeng, and Tien Len
is higher than Big Two. These conditions imply that the player’s expectation of finishing the advanced game is higher than a classical game [83]. It can be observed that
Doudizhu and Tien Len have higher AD values and higher (E⃗p ) values. This likely canbe speculated as the unpredictability is great, and the players keep the expectation high.
Moreover, advanced games’ absolute value of p⃗2 is higher than a classical one, whichmeans it has more engagement to continue. The result (Table 4.12 and Table 4.13) also
revealed that advanced games have high velocity, AD value, p⃗1 , E⃗p , and p⃗2 compared82

to the classical games. This condition implies that the sophisticated game has moreunpredictability and expectation, thus, better engagement. Relative to the AD value,such a condition can also be induced, which means that higher unpredictability andsurprise can make us feel more concentrated, engaged in continuing the game, and havea greater expectation of the game process.
A summary of motion in mind measures for different games was given in Table 4.14.
It can be observed that the more advanced and complex the game, the higher thevelocity. The velocity of DouDiZhu is the highest, and the Big Two is the lowest, whichmeans that the speed of information progression in a sophisticated game is higher thanin a classical game, where the information contained in each step requires multiple
options of consideration. Correspondingly, the higher the difficulty and complexityof the game, the lower the mass, and the easier it is to progress. The higher the
momentum, the higher the tendency to continue, and the higher the potential energy,
the more expectation the players have.
Table 4.14: Summary of motions in mind measures of different games
Card games

Rank

m

v

DoudDiZhu
WakengTien Len
WinnerBig Two

1
2
3
4
5

0.898
0.954
0.959
0.964
0.970

0.010
0.046
0.041
0.035
0.030

p⃗1

⃗
Ep

N

0.092 0.0187 9.8
0.044 0.0040 21.7
0.039 0.0030 24.4
0.034 0.0024 28.2
0.029 0.0020 33.3

Moreover, in card games, if N > 20, the game is too random, and players feel too
easy, forcing players to improve their ability to explore more gameplay to reduce theimpact of a surprise and make the process more enjoyable. The GR and AD values
would increase to the reasonable zone. In contrast, if N < 15, the game is too static,
where limited N is required to acquire the skill to distinguish the optimal choice. Such
conditions challenge players to become more skillful in adapting to the game rules andfeel enjoyable; thus, the GR and AD values decrease to the reasonable zone.
83Some games have perfect information (such as Chess, tic-tac-toe, Checkers, Go, andso on), where each player can see all the pieces on the board at all times [84]. For
these kinds of games, the game length (denoted as D) is enormous, while the optional
moves/positions to be made (denoted as B) at each step are small, resulting in a low-riskrate with a long game duration and a minimal number of rounds. Therefore, players will
be more cautious because they know all the current and possible future information todetermine the winning state. Moreover, given enough rounds (or unlimited resources),the game mimics the skill demonstrated by the strong player, where only a limited
number of rounds is enough to distinguish the winners.
However, some games are characterized by their aspect of the play hidden from
opponents, such as the cards in poker and bridge, which were typical examples of gameswith imperfect information [85]. For card games, the D is small, and B is high, making
the risk rate high, short game duration, and a considerable number of rounds. Thiscondition implies that in imperfect information games, players are more based on luck(or likelihood) for each hand because most information is hidden. Furthermore, eachround does not have enough game steps to demonstrate the skill to identify winners,
which demands more rounds to determine the real winner. So for imperfect information
games, players need many rounds to distinguish winners.

4.6.2

GR and AD relative to addictive situation

The impact of GR and AD values were further explored where the cross point of
GR = AD is determined, which could be related to the balance between entertainment/excitement (associated with the gravity of play) and surprise/fluctuation (related
to the jerk of play). If GR = AD, people should feel exceedingly engaged and addicted.
This condition is caused by not only the player feeling entertained due to the GR sophisticated zone, but the high AD value is associated with the feeling of surprise andunpredictability. As such, the play is likely to be repeated to maintain fairness, with
speculation of winning the next round.
84

Revisiting (4.9) and (4.10), solving GR = AD would have B = 9. Then, considering GR = AD from the perspective of reward frequency N would involve resolution
involving (4.11); thus, leading to (4.19) which results in N = 9. When N = 9, there
is a cross point between GR and AD, where the velocity of the game is v = 91 andm = 89 = 0.89 (Figure 4-5). At the same time, when B = 9 and GR ∈ [0.07, 0.08], thenD ≈ 40. That is the reason (B, D) = (9, 40) would be the candidates for an addictive
game. This condition would explain why the setting of Doudizhu of (B, D) = (8, 40)
is the most popular and addictive game for common players among many similar card
games.
sAD =3

3 B 2 2D··
2 D4 B

s


=

3

3
· GR4 · N
2


(4.19)·10−2
GR = AD
N =3
N =5
N =9

8

AD

6

4

2

1

2

3

4

5
GR

6

7

8

9

·10−2

Figure 4-5: The relations between GR and AD
Based on the given N and GR, solving them can help in discerning the reasonablegame length (D), given by the (4.20). It can be observed that the D is shorter (Figure 4.6.2) when AD and N get higher (Figure 4.6.2). This situation implies that for85

a fast-paced game, the rate of getting the reward is shorter, high predictability, full of
surprise, and encourages players to play the game repeatedly, thus, making it addictive.

D=

2v
2
=
a
N · GR2
GR = 0.070
GR = 0.075
GR = 0.080

250Game Length (D)(4.20)

200

150

100

50
2

3

4
5
6
7
Reward Frequency (N )89Figure 4-6: The relations between reward frequency (N ) and game length (D)Based on results and previous studies, the GR value could provide the balanceof the game length; if it is too short, the game is unfair, and players do not haveenough steps to show their ability. Otherwise, it is too boring. The cross point between
velocity and acceleration shows the balance between fairness and entertainment. In
that sense, the winner can be determined within reasonable steps. Meanwhile, basedon the value of AD, it provides the balance of the game rounds. The higher jerk valuemeans unpredictability; it causes shorter steps (the game process relies on chance, andplayers have insufficient steps to show skill in the stochastic game). Otherwise, thegame steps are longer. This situation can be defined based on y1 = vt, y2 = 21 at2 , andy3 = 61 jt3 where the relation AD and GR were given by AD3 = j and GR2 = a. When86

·10−28GR = 0.070
GR = 0.075
GR = 0.080

AD

7

6

5

4
2

3

4
5
6
7
Reward Frequency (N )89Figure 4-7: The relations between reward frequency (N ) and AD
AD=GR, the cross point between acceleration and jerk can balance the unpredictability
and entertainment; in that sense, the winner can be determined with good rounds(Figure 4-8).
In addition, the reward frequency (N ) differences between perfect and imperfect
information games are observed, where the N of the former is lower than the latter. Thiscondition implies that imperfect information games have random choices compared toperfect information games, so the entire process requires careful choosing. For example,
if the game is stochastic, there is a high possibility of surprises, the game steps are too
short, and the jerk value is high. This situation can be found in a card game, where
many rounds were played to balance the impact of surprises. On the other hand, if thegame is too static, while the player can fully demonstrate the ability, longer steps areneeded to complete the game. This situation causes the jerk value to be low (like aboard game), where the winner can be distinguished under limited rounds.
A low jerk value can balance the deterministic and stochastic of the game features.

87y1 = vty2 = 12 at2
y3 = 16 jt3

Game outcome (y)

80

60

40

20

0

0

10

20

30

40 50 60 70
Game length (t)

80

90

100

Figure 4-8: The crosspoint between fairness (y), reinforcement (v), entertainment (GR),and unpredictability (AD)Otherwise, a high jerk value would be attractive to the players where the steps arelimited, full of surprises, and engaging. The data from card games could suggest an
equivalence relation between game refinement theory and motion in mind to determine
the nature of addictive games. Given the condition of GR ∈ [0.07, 0.08] (zone values)and AD = GR, it is where the N = 9 was obtained, which suggests the nature ofstrongly addictive games when m = 0.9 (N = 9) under the assumption that k = 2.5.
As such, Table 4.15highlights the possible corresponding popular games.
An extension of the game progress model leads to the four possible major principlesof play. These principles were defined relative to the measures of the game progressmodel, based on both GR and AD values, which were given in Table 4.16. GR value
implication was derived from the principles of (1) to (3). The game length should not be
too small/large; otherwise, unfair/boring. It was found that addictive entertainment isderived from all principles. Games that satisfy principles (3) and (4) would be exceedingly engaged and should be repeatedly played to meet fairness relative to principles

88

Table 4.15: Possible corresponding games
Game type

N

v

m

AD

2

Perfect information

1.5
2
3
5

/3
1
/2
1
/3
1
/5

1

/3
/2
2
/3
4
/5

0.041 237
Go
0.046 178 Basketball
0.052 119
Shogi
0.062 71
Chess

Imperfect information

8
9

1

7

0.072
0.075

/8
1
/9

1

8

/8
/9

t

44
40

Games

Mahjong
Doudizhu(1) and (2). A certain type of game with a larger mass (for instance, m > 0.9) can be
implemented on a computer and thus can be easily played repeatedly.
Table 4.16: Possible principle of game element
Game Element

In-game Description

Game length (t)Velocity (v)Acceleration (a)Jerk (j)Cost of getting reward
Fairness
Frequency of getting reward
Reinforce of play
Uncertainty of getting reward
Attractiveness of play
Unpredictability of getting reward Surprise of play

4.7

In-game Implication

Chapter Conclusion

This study explored the implications of motion in mind measure, in particular jerk value,
to measure a game’s entertainment aspects and identify what constitutes an addictive
game. Several card games with different complexities and rule designs were investigatedand computationally analyzed using varying AI levels to represent different skill levelsof the player. By simulating the performance of players with different skill levels incard games of various difficulty levels, we can conduct both horizontal and verticalcomparisons to delve into the interactive relationship between players and games. This
allows us to explore the balance between player abilities and game complexity. The
89

differences between the two dimensions of games were observed: (1) deterministic andstochastic games; and (2) simple and sophisticated games.
For sophisticated games, such as board games, the GR value is 0.07-0.08 and the AD
value is 0.045-0.06, ensuring entertainment and predictability (stability, controllability).
For games with imperfect information, such as poker and mahjong, the AD value issimilar to the GR value. The AD value is higher than that of the perfect information
games, which means that the game is highly unpredictable and there are surprises in
the process. The balance between the AD value and the GR value makes the gameaddictive. The process is unparalleled and players prefer to play repetitively.
For deterministic games (i.e., board games), the possible options (N ) are small; they
are deterministic and stable, ranging from 1.5 ≤ N ≤ 5, where the GR and AD value
are located in their sophistication zones (GR ∈ [0.07, 0.08] and AD ∈ [0.045, 0.06]) andGR > AD. Longer steps make the process more deterministic and less surprising. For
stochastic games (i.e., card games), the possible options (N ) are large, ranging from5 ≤ N ≤ 10, but limited in terms of game length while being more unpredictableand full of surprise (AD > GR), while solving GR = AD allowed the N value to be
determined. When N = 9, there is a cross point between GR and AD curves (seeFigure 4-5), where B = 9 and D = 40. This condition is the most addictive andentertaining case, which explains the reason for card game popularity and addiction(for instance, the Doudizhu game setting of (B, D) = (8, 40)).
Analysis of the AD value relative to the GR value showed that it could measure
the stability and predictability of game players in complete and incomplete information games. Analyzing GR and AD provides insights into player entertainment andunpredictability. Complex games start above zone values, posing initial challenges,
while simpler games begin below, becoming more engaging over time. For instance,high-complexity games (like Doudizhu) typically have GR and AD values higher thanthe zone values, implying that the game difficulty and unpredictability were not easily
mastered by the player by just having a better ability. Nevertheless, players can become90more skillful in mastering and enjoying the game after learning and training. Meanwhile, for simpler games (like Wakeng, Big Two, Winner, and Tien Len), the player
could quickly get bored since the GR and AD values are smaller than the zone values.
However, different gameplay can be developed after exploring and training, so the GR
and AD values will slowly increase, and players will feel interested and unexpected.
Furthermore, the value of the potential energy (E⃗p ), the momentum in the game(p⃗1 ), and in mind (p⃗2 ) of a complex game is higher than the simple game, implying a highlevel of player’s tendency, expectation, and engagement to play sophisticated games.
This condition can also be related to the AD value, where higher unpredictability
can be observed. While GR and AD focused on balancing the game sophistication
corresponding to the game length and winning rate, while relative to the E⃗p , p⃗1 , andp⃗2 , provided the indication of the game stability based on the player’s tendency andexpectation. Therefore, the principle of play can be implied, providing insights into thegame’s elementary components.
Nevertheless, the study has several limitations. Firstly, the collected data were basedon the established assumption that the difference between the levels of AI indicates theplayer’s skill level. The gap between amateur-level and expert-level players may behuge in some game types and insignificant in others; thus, caution is advised, and thefindings are interpreted with a grain of salt. Secondly, the findings were the applicationof the proposed computational model in the domain of card games. The implicationin complex real-time continuous games (such as video games) is currently unknown,
which is open for future studies. Finally, the study’s theoretical basis was based on
the culmination of works built on analogical relations between classical physics andmechanical game-playing schemes. Games based purely on sensors or gestures, psychophysiological measures or signals, and another mode of interfaces (augmented or virtual
reality technology) would be an exciting area to explore in the future.

91

Chapter 5
The Impact of Performance Degree
on Players: Exploring the Dynamics
of Player Engagement andEnjoyment in Game Process
5.1

Chapter Introduction

In the gaming domain, it is common to acknowledge the presence of inherent risks,
represented by the variable “m”. Players are entrusted with the challenge of formulating
strategic solutions to overcome these obstacles. Proficient players who have mastered
the game can effortlessly progress, resulting in a game progression ideal velocity denotedas v⃗0 = 1 − m [86].
However, the gaming landscape reveals a diverse range of player abilities, leading
to varying competencies in managing uncertainties and advancing in the game. Prior
research introduces the variable “k” to evaluate player performance, with higher values
indicating diminished performance. For players with imperfections, the game progres-92

sion velocity can be computed as v⃗k = (1 − km)v⃗0 = (1 − km)(1 − m). In this regard, we
will discuss the comparison between perfect players and imperfect players in the fieldof motion in mind during the game, and explore how the psychological changes evolveas players continue to improve and become proficient players.
Performance level “k” is important on the player side, and reward frequency “N ”
also plays a big role on the game side. In the context of reinforcement learning, “N ”
often signifies the “number of steps” or “time steps” representing the timeframe for
earning rewards. In games with perfect information, players enjoy complete access tomoves influencing the game’s outcome or the count of potential positions that can beeliminated. Conversely, in games involving imperfect information, players may possess only partial insights into other players or engage in informed speculation about
strategies and inclinations. In such scenarios, “N ” may denote the number of options
or actions considered during decision-making. For instance, in the Monte Carlo Tree
Search algorithm, “N ” frequently signifies the number of simulations or playouts conducted from a specific game state, so the velocity also can be described as v⃗0 = N1 .
As mentioned earlier, the concept of “AD” emphasizes the inherent unpredictability of games. This study aims to delve into the intricate relationship between theperformance level represented by “k” and the reward frequency represented by “N ”, aswell as the unpredictability manifested by “AD” in the gaming domain. These threefundamental elements are closely intertwined, forming a cohesive whole that provides
valuable insights for analyzing both the player’s and the game’s perspectives. This offers valuable perspectives for game designers, optimizers, product managers, and others
involved in the gaming industry.

5.2

Related Works

This section begins by examining the disparity between perfect players and imperfectplayers from the standpoint of physical quantities such as speed, acceleration, and93

potential energy. Additionally, it investigates the conditions under which non-perfectplayers can emulate sensations akin to those in an ideal state. Furthermore, it explores
the relationship between the speed of the game and the optimal number of turns.

5.2.1

The meaning of playing performance

In different player performances, discernible distinctions emerge in the velocities ofgame progression. As illustrated in Figure 5-1, players with lower performance levels
demonstrate a diminished capacity to overcome uncertainties, even when faced with anequivalent risk rate, as compared to their higher-performing counterparts[1].
1
v⃗0
v⃗1
v⃗2
v⃗3

reward winning rate

0.8
0.6
0.4
0.2
0
−0.2
0

0.2

0.4
0.6
risk rate m

0.8

1

Figure 5-1: Game progression velocity as a function of risk rate mUpon solving the inequality vk = (1 − km)(1 − m) ≥ 0, we determine the criticalthreshold for the admissible risk rate, corresponding to a player’s given performancelevel. This threshold is precisely defined as mx = k1 .
This clarifies that a player’s velocity maintains a positive trajectory within a range
spanning from 0 to k1 , as long as the risk rate remains within this defined limit. Nevertheless, if the risk rate surpasses this critical threshold, the capacity to resolve uncer94

tainties diminishes significantly. Players may then encounter stagnation in their gameprogress when employing strategies based on their performance level.

5.2.2

The acceleration ak of different kinds of players

From the previous discussion, we have the derivation of vk .

v⃗k = (1 − km)v⃗0(5.1)

Correspondingly, we can also derive the acceleration ak
ak = v⃗k ′ = (1 − km)v⃗0 ′ = (1 − km)a0(5.2)

The acceleration ak consistently stays below a0 , and proficient players may experience greater entertainment than less skilled players.

5.2.3

The balance of perfect player and imperfect player fromthe perspective of potential energy“Subjective expectation”, E0 can be translated as“personal expectation[87]” or “individual expectation”. This term is typically used to indicate an individual’s subjective
estimation or anticipation of a certain outcome or event, rather than being based on
objective data or probabilities.
“Objective expectation”, Ek can be translated as “factual expectation”. This term
usually denotes the expected value based on objective facts, probabilities, or data,
without consideration of subjective viewpoints [16]. By solving the intersection between perfect and imperfect players, E0 represents the objective expectation (Ideal
reinforcement), while Ek denotes the subjective expectation (Players side).

E0 = Ek95(5.3)

So we have2mv0 2 = 2mvk 2(5.4)

In this scenario, both perfect and imperfect players participate on equal footing,
implying that regular players can share the same motivation and expectations. This
inclusivity ensures the game’s appeal to a broad audience of ordinary players. Solving(5.4), we have
k=

2
m

(5.5)

As m always falls within the range of m = [0 − 1], if k is greater than 2, anintersection point exists between E0 and Ek .
Assume k = 3 as depicted in Figure 5-2. In high-risk games, such as classicalboard games (i.e., m = 23 ), the skill level of ordinary individuals is farther from that ofmasters (k = 3). However, ordinary players have the advantage of extended thinking

Motion in Mind Measurestime, making this skill gap acceptable.
⃗v0
⃗v3
E0
E3
∆3

0.5

0

−0.5

0

0.2

0.4
0.6
Risk Rate m

0.8

1

Figure 5-2: Motion in Mind Measures for k = 3
When the risk rate hovers around m = 13 , player concentration is at its peak. As
the risk rate increases beyond m = 13 , the disparity between E0 and Ek diminishes,96contributing to a heightened sense of comfort among players. This trend persists until
m = 23 , where, E0 aligns with Ek , resulting in maximum player engagement. Hence,m = 32 is the most suitable and reasonable risk rate for ordinary players.
If the risk rate surpasses m = 32 , the challenges become too formidable for bothoriginal and perfect players, leading to decreased engagement. However, even at thispoint, actual players may still exhibit higher engagement levels compared to perfectplayers.

5.2.4

Exploring optimal rounds for distinguishing real strength

In prior research, game outcomes have been traditionally characterized using averagevelocity as a representation of the effort required. The equation for the game outcome
in an ideal case y0 (v = 1) is expressed as follows:

y0 = t(5.6)

For other cases with a speed of v = N1 , the equation becomes:
y1 = vt =

1
t
N

(5.7)

However, since the speed of resolving uncertainty is not constant, considering average acceleration as a measure of the game outcome becomes relevant. Acceleration,in this context, is relative to attraction, and mental attraction is closely linked withachievement. The equation for y2 is presented as follows:1y2 = a0 t2 where a0 = GR2
2

(5.8)

√

Here, a0 is given by a0 ≈ DB2 , hence GR = DB , in the domain of board games withthe average branching factor B and game length D [4].
Subsequently, we can determine the intersection point (say t02 ) between ideal effort

97

and achievement by solving y0 = y2 , leading to:1
t02 = a0 2 t02 2
2

(5.9)

The initial sensible option for terminating the game is t = t02 , representing the minimum game duration. If this duration is exceeded, the achievement becomes excessively
high, the attainment comes too effortlessly, the game appears too simple and smooth,and players may lose motivation to overcome challenges, leading the game into a dull
and unengaging stage.

t02 =

2
(= T )
a0(5.10)In this phase, the resolved uncertainty associated with y02 can be quantified. If
players decide to conclude the game at this point, it is reasonable for a single round to
determine the winner, ensuring a relatively fair outcome. However, if the objective is to
identify the genuinely stronger side, a single round may not suffice. Repeating multiple
rounds becomes necessary to mitigate the influence of randomness and establish a moreaccurate assessment of skill.
y02 = T

(5.11)At the same time, we can solve the cross point (say t12 ) between y1 and y2 .
2v
T
=
a0
N

(5.12)

1T
t12 = 2NN

(5.13)

t12 =Hence, y12 can be described as,y12 =Therefore, multiple rounds are required to encompass the entire spectrum of uncertainty and effectively discern the genuinely stronger side.

98

Figure 5-3: Total and each round solved game uncertainty

Rounds =

y02
= N2
y12

(5.14)

Corroborating with real-life observations, in the case of perfect information games
where the process is openly transparent and there are fewer available options, coupledwith a lower frequency of rewards N , the number of rounds tends to be lower. Simultaneously, each round takes longer than in imperfect information games; every move ineach round requires careful consideration, making each round relatively “stable” and“fair”.
For imperfect information games, the game process is unpredictable, necessitating
the determination of actions amidst numerous possibilities. The frequency of rewards
is higher, leading to an increased number of rounds. However, each round’s duration
is often shorter in such cases, enabling a swift transition to the next round. Repeatediterations balance the potential “randomness” and “bias” in each round. These surprises
contribute to an addictive, almost irresistible feeling, akin to an addictive experience.

99

5.3

Performance Degree (k): In-Depth Analysis

In this section, our primary focus is on exploring the connection between playing performance (k) and the suitable complexity of the game. We delve into the specific evolutionof “motion in mind” as players enhance their skills through training and exploration.

5.3.1

The risk rate m and performance degree k

There exists a negative correlation between the risk rate m and the performance degree
k. In games with higher stochastic characteristics, featuring a high-risk rate and shorter
game steps, chance assumes a more significant role. This leads to a reduced gap between
ordinary and expert players, resulting in a lower value of k. Conversely, in games withmore statistical characteristics, marked by a low-risk rate and a greater emphasis on
skill over chance (as indicated by a higher GR), the gap between ordinary and expert
players widens, yielding a higher value of k.
In most common games, a balance is struck between stochastic and statistical elements, with the risk rate typically falling within the range of 13 to 32 . For classic games,
a player’s performance level can be quantified using a k value, which typically ranges
from 3 to 6.
mx =

5.3.2

1
k

(5.15)The explanation of the correspondence system of m andthe maximum acceptable performance degree k

Certainly, here are some general guidelines for the correspondence between the risk ratem and the maximum acceptable performance degree k.
High-risk games (m > 32 ): In high-risk games where the risk rate m exceeds 23 , the recommended maximum acceptable player performance level k
typically falls within the range of 1 to 1.5. These games have a significant
100

10

performance degree k

k
8
6
4
2
0

0

0.1

0.2

0.3

0.4 0.5 0.6
risk rate m

0.7

0.8

0.9

Figure 5-4: Possible relation between performance degree and risk rateelement of luck, so players need to have a higher level of skill to navigate
the elevated risk and strive for GR cater control over the game.
Balanced-risk games ( 13 ≤ m ≤ 23 ): Games with a balanced risk rate typically see a maximum acceptable player performance level k ranging from
1.5 to 3. These games strike a harmonious balance between skill and luck,allowing both casual and professional players to participate in a fair andchallenging gaming experience.
Low-risk games (m < 31 ): For low-risk games with a risk rate m less than 13 ,it’s advisable to have a maximum acceptable player performance level k in
the range of 3 to 10. These games primarily emphasize skill, and luck plays
a diminished role in determining the game’s outcome.
It is important to keep in mind that these guidelines are flexible, and variations may
arise based on the specific game and player preferences. Moreover, individual players
may possess varying skill levels and luck, influencing their perception of the maximum101acceptable playing performance level.
Table 5.1: Relations between m and k
m

N

k

0.1

1.11

10

0.2

1.25

5

0.3

1.43

3.3

0.4-0.6

1.7-2.5

1.7-2.5

0.7-0.9

3.3-101.1-1.4

Explanation
Typically, the game is designed for young children andis relatively easy, making it accessible even to those withlimited skills.
Geared towards elementary school students, this gameis relatively easy, allowing children with varying abilities
to enjoy it.
Targeted at older elementary school students, as theymature and learn, the skill gap among students narrows.
The intended players are typical adults, and the risk
rate in most two-player games is around 21 . When bothplayers have similar abilities, and the game is fair, thewin rate should be roughly equal, offering enjoyment toa broad range of players. However, the scope of players
who can enjoy the game is limited.
High-risk games primarily rely on randomness rather
than skill. To relish these games, strong skills are necessary, further restricting the range of player levels.

Players with lower gaming skills often lean towards games with lower difficulty levels(higher reward frequency). In such scenarios, the jerk value of the game is reduced,
resulting in a more predictable and straightforward gaming experience. These players
enjoy a higher rate of success and derive GR cater satisfaction from their achievements.
Conversely, players with advanced gaming skills tend to favor games with higher difficulty levels (lower reward frequency). Although these games are associated with higher
jerk values, they also offer GR cater challenges. Players with high gaming skills relishthe process of identifying optimal choices from a plethora of options and experience anenhanced sense of accomplishment.
In the case of random games, characterized by larger values of N and increased
randomness and unpredictability, gameplay tends to be more intense. However, highly
skilled players often opt for games with elevated difficulty levels. On the other hand,102statistical games, featuring smaller N values and GR cater certainty, typically offer lessintense gameplay. These games find diverse applications in fields such as business andeducation, where statistical analysis and decision models are frequently utilized.

5.3.3

Comparison of motion in mind measures based on performance degree k

In prior studies examining the connection between motion and mindset, it was discovered that the concept of energy, denoted as Ep , can serve as a metric for measuring
a player’s expectation [88]. Notably, players with different skill levels exhibit distinct
energy levels, and when k surpasses 2, a crossover point emerges between perfect andimperfect players. At this juncture, imperfect players can harbor similar expectations
to those of perfect players.
Ep = 2mv 2(5.16)Ep , k = 0
Ep , k = 2
Ep , k = 2.5
Ep , k = 3

energy in mind

0.25
0.2
0.15
0.1
5 · 10−2
0

0.1

0.2

0.3

0.4

0.5 0.6
m

0.7

0.8

0.9

1

Figure 5-5: The comparison of energy in mind based on the player performance level k
In the gaming domain, when k = 3 and m = 23 , an intriguing equilibrium arises with103

Ek = E0 , delicately balanced. Exploring the evolution of players transitioning from theintroductory stage when k = 3 to the realm of heightened skill, we unveil a fascinating
phenomenon: at k = 2.5 and m = 54 , once again, the convergence of Ek = E0 unfolds.
This prompts profound contemplation. From k = 3 to k = 2.5, as a player’s skill
advances, the crossover point shifts to the right. This suggests that players with higher
skill levels will achieve the same expectations but at a higher risk rate. It implies thatskillful players can confront more substantial challenges.
This signifies that for players endowed with relatively refined expertise, the allureof more challenging games seems to strike a chord akin to the expectations harboredby perfect players. As players transition from k = 3 to an elevated tier of skill, theirperformance ascends. As k is attenuated to 2.5, while the risk factor escalates to m = 45 ,the anticipated outcomes of the game are reminiscent of those of an ideal player. In
essence, even when confronted with more formidable challenges, adept players manage
to evoke the anticipation reminiscent of an idyllic scenario. This intimates that forskillful players, engagement with games of heightened complexity can elicit expectations
on par with the archetype of the perfect player.
Building upon prior research findings, it has been established that when k = 3, there
exists a corresponding value of m = 32 , resulting in E0 = Ek . This convergence point
signifies that players can experience expectations similar to those of perfect players.
Moreover, in the domain of incomplete-information games like card games [5], ourinvestigation reveals that when N falls within the range of 8 to 9 and then m spansfrom 78 to 98 , with k ranging between 2.25 and 2.3, the equilibrium condition GR = AD
is attained. This balance point suggests that players’ enjoyment and addiction levels
are in equilibrium. Consequently, in this chapter, we will delve into the physical andpsychological changes in our minds as players undergo continuous training, observing
the shift in k from 3 to 2.
In addition to potential energy Ep [89], we utilized the momentum concept to assess
the concentration level among players. It becomes apparent that momentum varies for104

Figure 5-6: Motion in mind measure compared based on the performance degree ktransition from 3 to 2
players with different performance levels. In instances where performance is subpar,
the corresponding momentum diminishes. This suggests that players with weaker skills
display reduced focus and concentration on the game.
p1 = mv(5.17)

Energy in Mind

0.2
0.1
0
−0.1

p1 , k = 0
p1 , k = 2
p1 , k = 2.5
p1 , k = 3

−0.2
−0.3

0.1

0.2

0.3

0.4

0.5 0.6
m

0.7

0.8

0.9

1

Figure 5-7: Comparison of momentum in mind based on the player performance levelk
The subjective momentum represents the engagement of players, and the subjective
momentum of players with different performances is also different, but performance105

level k and subjective momentum p2 are not linear relationships.

p2 = Ep − p1

p2 , k = 0
p2 , k = 2
p2 , k = 2.5
p2 , k = 3

0.3
energy in mind

(5.18)

0.2
0.1
0
−0.1
0.1

0.2

0.3

0.4

0.5 0.6
m

0.7

0.8

0.9

1

Figure 5-8: The comparison of subjective momentum in mind based on the player
performance level k
It is evident that through continuous training and performance enhancement, transitioning from k = 3 to k = 2, stronger players can overcome the bottleneck at a lower
risk rate, showcasing their readiness to face challenges and achieve higher levels of participation than perfect players. Additionally, we observe a progressive increase in the
maximum value of p2 . For players with higher skill levels, both the peak and averagevalues of p2 are elevated. Essentially, this signifies that players with greater capabilities
experience higher average and peak engagement levels, setting them apart from thosewith weaker skills.
This trend underscores the intricate interplay between players’ skill levels, perceived
risk, and subjective momentum. It reveals how these factors collectively shape the degree of players’ immersion and commitment to the gaming experience, shedding light on
the nuanced dynamics of player engagement. We have explored the intricate relation106

ship among player competence, risk rate, and subjective momentum, offering valuable
insights into understanding player engagement across diverse contexts. This contributes
to the enhancement of game design and user experience, enabling us to better cater tothe needs and expectations of various types of players.

5.4

Dynamic Interaction between Performance Levelk, Reward Frequency N , and ADThis section delves into the interplay among three vital elements: k, N , and AD, fromboth player and game perspectives.
The symbiotic relationship between player performance and game rules is evident,
as they mutually complement and influence each other, fostering a dynamic interplay.
Importantly, players of varying skill levels can align their expertise (k) with a game’s
unpredictability (AD) or the number of choices (N ) through dedicated training andexploration. This adaptability empowers players to tailor their skills and strategies,
fostering a personalized gaming experience that achieves a harmonious balance between
enjoyment and challenge.

5.4.1

Performance level k and reward frequency N

Building on our earlier discussion, novice players are typically inclined towards simpler
games characterized by higher speed of getting rewards, resulting in a more predictableand gratifying gaming experience (GR). On the flip side, proficient players gravitate
towards demanding games with lower velocity, relishing the complexity and the senseof achievement they provide.
In random games with larger values of N and higher unpredictability, gameplay
reaches heightened intensity, attracting highly skilled players. Conversely, statisticalgames with smaller N values and increased predictability are less intense and are107frequently employed in business and educational settings for statistical analysis anddecision-making, showcasing their applicability beyond the realm of gaming.

k=

1
N
=
m
N −1

(5.19)

3

N (Reward Frequency)

k
2.5

2

1.5

1

1

2

3

4
5
6
7
8
k (Performance Degree)

9

10

11

Figure 5-9: Performance Level k and Reward Frequency N

5.4.2

Performance level k and AD

Building upon prior research on AD through card game analysis [5], it becomes evident
that in simpler games, where the player’s skill surpasses the game’s difficulty, the AD
value starts small, and players continually explore ways to enhance the game’s enjoyment. Conversely, in complex games, where the player’s skill falls short of coping withthe game’s difficulty, the AD value is excessively high, rendering the game uncontrollable. In such cases, players persistently engage in training to improve themselves andenhance their skills to bring the AD value within a reasonable range.

108

Figure 5-10: The relations between k and AD
Players with varying levels of skill can indeed achieve a better match between theirskill level (k) and the level of unpredictability (AD) or the number of choices (N ) in
the game through continuous training and exploration. By refining their skills andadapting their strategies, players can tailor their gaming experience to a level that isboth enjoyable and appropriately challenging for their abilities.

5.4.3

Reward frequency N and AD

In reinforcement learning, N often represents the “number of steps” or “time steps”
indicating that a reward unit is obtained from every N step.
From a gaming perspective, N can be interpreted as the reward frequency, considering game expertise and motion in mind. The G − T (Goal-Total steps) model implies
receiving a reward once every N operation. For example, in basketball, we may makeN attempts before successfully scoring a basket, and each successful basket results ina reward [6].

109Table 5.2: The relationship between N and AD in the sport games domain
Games

N

D

AD

basketball

2.25

178

0.046

soccer

8.33

22.0

0.090

·10−2

Value

N

AD

8

8

6

6

4

4

2

2

0
basketballsoccer 0
Sport Games

Figure 5-11: The relationship between N and AD in the sports domain
It is evident that in sports games [6], there is a positive correlation between thenumber of options (N ) and the level of unpredictability (AD). In other words, as
the number of available options increases, the uncertainty and unpredictability of thegame also increase. This is because more options imply more variables and possibilities,
making it more challenging to predict the outcome of the game.
In the B − D(Branch factor - Depth) model, it is inferred that we select the best
option from N choices. This concept is similar to a game tree, where we consider N
possible positions and make our final decision. For example, in a chess game, we may
evaluate N potential moves and select the one with the highest expected outcome as
our final decision [6].
110

Table 5.3: The relationship between N and AD in the board games domain
Games
Go
shogi
chess

N (B)
1.66
2.88
4.57

D
237.0
119.0
71.0

AD
0.041
0.052
0.062

Similarly, for board games [90], there is also a positive correlation between thenumber of options (N ) and the level of unpredictability (AD). In other words, as thenumber of available options increases in a board game, the level of unpredictability
also tends to increase. This is because a higher number of options for GR introduces
increased complexity and potential outcomes, making it more challenging to anticipate
the final result of the game.
·10−2

Value

N

AD

6

6

4

4

2

2

0

Go

Shogi

Chess

0

Board Games
Figure 5-12: The relationship between N and AD in the board games domain
In general, for perfect information games, where all game information is available toplayers, the more options there are to choose from, the higher the level of unpredictability. This is because we need to narrow down the range of possibilities from numerous
available options to make the best decision.
In the case of imperfect information games like card games [5], where factors such111

as suits, rules, and the game state are not entirely known, and there is significant
uncertainty, your assumption aligns with the concept of a game tree. In this context,
N can be considered similar to the concept of branching.
In a game tree model, N would represent the number of possible actions or moves
available at a given game state. Each action or move leads to a different branch in thegame tree, representing a different possible outcome or game state. By exploring thebranches and considering N different options, the player can make informed decisions
based on the available information and their strategic understanding of the game.
Considering the uncertainty and imperfect information in card games, the player
can use the available information to assess the potential outcomes associated with eachbranch and make decisions accordingly. The N branches represent the different choices
or actions available to the player at a particular game state, and the player would aim
to select the most favorable or strategic option based on their analysis of the game tree.
It’s worth noting that the implementation and specific details of modeling card
games using game trees or similar approaches can vary depending on the specific gameand the chosen algorithm or framework. However, your hypothesis of N being analogous to the concept of branching in a game tree model aligns well with the context ofimperfect information games like card games.
Table 5.4: The relationship between N and AD in the card games domain
Games

N (B)

D

AD

Big Two

3.33

65

0.033

Winner

4.54

64

0.037

WK

4.69

54

0.045

Tien

5.09

57

0.044

DDZ

8.20

40

0.073

112

·10−2
8

8

Value

N

AD

6

6

4

4

2

2

0

Big two Winner Wakeng Tien Doudizhu

0

Card Games
Figure 5-13: The relationship between N and AD in the card games domain
Additionally, in the realm of card games[5], it’s noteworthy that there exists a positive correlation between the number of options (N ) and the degree of unpredictability(AD). This implies that as the number of options (N ) increases in card games, so does
the level of unpredictability, adding an intriguing layer of complexity to the gaming
experience.

Figure 5-14: The relations between N , k and AD
The three important elements(k, N , and AD) on the player side and the game side
are related in pairs.
113

Table 5.5: Correlation between player performance and game characteristics
Variables

Relationship
We observe a clear correlation between players’ skill levels (k) and gamecharacteristics, particularly in game selection and experience. Novice players (with higher k) tend to prefer simpler games with higher speed (resulting
in lower reward frequency N ) for a more predictable and enjoyable gamingkNexperience. In contrast, experienced players gravitate towards more challenging games with lower reward frequencies, appreciating the complexity
and sense of achievement they offer. Thus, k and N exhibit a negative relationship.
Furthermore, our analysis of card games indicates that as players train
or explore different complex games, their performance level (k) tends tokADconverge towards that of an ideal player, leading to a decrease or increasein the average deviation (AD) within a reasonable range.
Additionally, upon examining various game types, we discovered a positiveNAD
correlation between reward frequency (N ) and average deviation (AD).

In random games with higher N values and AD cater to unpredictability, the gaming
experience becomes more intense, attracting highly skilled players. Conversely, statistical games with smaller N values and higher predictability are utilized in business andeducational contexts for statistical analysis and decision-making.
Player performance and game rules complement each other, promote each other,
continuously iterate, and influence each other. Crucially, players with varying skill levels can align their expertise (k) with a game’s unpredictability (AD) or the number ofchoices (N ) through training and exploration. Adapting their skills and strategies enables players to customize their gaming experience, striking a suitable balance between
enjoyment and challenge.

114

5.4.4The difference between perfect information games andimperfect information games

In general, when comparing perfect information games [91] with imperfect information
games, the latter typically present a greater number of choices and more intricate
scenarios. Perfect information games are renowned for their explicit information-sharing
characteristics, where players possess complete and clear information throughout thegame. In contrast, imperfect information games involve a higher degree of uncertainty
and concealed information, requiring players to make decisions in the absence of perfect
information.
In imperfect information games [92], players need to consider the potential actionsof others and hidden information while also making decisions based on probabilities
and conjectures. In such scenarios, there may exist a broader range of strategies andvariations, as different hidden information and varying opponent behaviors can impact aplayer’s optimal decision-making process. Compared to perfect information games, thefeatures of imperfect information games contribute to a more diverse strategic landscapeand a more complex decision-making process, as players must balance risks and benefits
in the face of incomplete information

115

Table 5.6: Comparison between Perfect Information Games and Imperfect Information
Games
Type

Perfect Information Games

Imperfect Information Games

Players have perfect information Players may lack perfect informaGame
about the game state and all past tion, leading to uncertainty andCharacteristics
player actions.

hidden information.

Chess, Go, many strategic board

Card games, games with hidden

games

information

ExamplesBased on limited information,
Based on perfect information forPlayer Decisions

probabilities, strategies, and poscalculations and decisions
sible opponent actions
Requires decisions based on spec-

Information

Every move in the game can beulation and possible opponent ac-

Impactcalculated and predicted.
tions

Possible choices(N )Prone to fewer choices and com-

Prone to more choices and com-

plexities

plexities

Lower

HigherUnpredictability
(AD)

5.5

The influence of ratio ϕ (GR/AD)In previous studies, we investigated the balance between game refinement (GR) andaddictiveness (AD), recognizing that a harmonious interplay between these factors results in an engaging player experience characterized by entertainment, repetition, andimmersion in the game’s surprises and delights. This chapter delves into the implications of imbalanced GR and AD values, examining how different ratios convey unique116

meanings and influence the overall gaming experience.
Here, we employ the ratio of GR (Game Refinement) value to AD (Addictiveness)value, denoted as ϕ, as a metric for assessing the relationship between entertainment
value and the level of addictiveness. Through this analysis, we explore the impact ofenjoyment and addiction (repeatability) across various game genres. This approach
offers insights into the underlying motivations guiding game designers’ decisions andfacilitates a discussion on the varying importance of entertainment and habituation ingame design.
Table 5.7: Measures of game refinement for board games [4]
Games

B

D

GR

AD

ϕ

Chess

35

80

0.074

0.059

1.25

Shogi

80

115

0.078

0.054

1.44

Go

250

208

0.076

0.044

1.73

In the realm of board games, the ϕ value typically resides in the [1.25-1.75] range.
This range suggests that board games offer an enjoyable and pleasurable experience,coupled with a degree of stability and statistical influence. Engaging in board games
demands a certain level of strategic thinking and decision-making.
Table 5.8: Comparison of Card Games using GR and AD values [5]
Games

B

D

GR

AD

ϕ

DoudDiZhu

8.2

40.03

0.072

0.073

0.99

Wakeng

4.69

54.15

0.040

0.045

0.89

TienLen

5.09

56.93

0.040

0.044

0.91

Winner

4.54

64.1

0.033

0.037

0.89

BigTwo

3.33

65.67

0.028

0.033

0.85

117

In the realm of card games, the ϕ value typically falls within the [0.85-1] range,
signifying that card games tend to display more unpredictability than board games andare marked by a higher level of instability. Engaging in card games not only necessitatesskillful play but also demands the ability to adapt to unforeseen situations.
Table 5.9: Comparison of Basketball and Soccer using GR and AD values [6]
Games

B

D

GR

AD

ϕ

Basketball72.76

164.02

0.073

0.046

1.59

Soccer

2.8

22

0.076

0.09

0.84

In the realm of sports games, the ϕ value commonly falls within the approximate
range of [0.85-1.5]. This range underscores the importance of entertainment impact
over unpredictability, suggesting that the enjoyment derived from sports games is a key
factor. The implication is that entertainment value plays a more pivotal role than unpredictability in sports games. These games often showcase higher statistical measures,reflecting a greater reliance on skills rather than chance-based elements, as indicatedby the GR parameter.
The concept of gamification has gained widespread popularity in recent years andis applied across various facets of life, including education [93] and business [94]. In
the following sections, we will delve into typical applications of gamification in botheducational and business contexts.

118

Table 5.10: Comparison of Hotels using GR and AD values [7]
Hotel

B

D

GR

AD

ϕ

IHG

3

75

0.023

0.028

0.833

Hilton

5

60

0.037

0.041

0.907

Starwood

3

50

0.035

0.042

0.833

Marriott

6

75

0.033

0.035

0.935

Hyatt

5

60

0.037

0.041

0.907

This table compares different hotel brands based on the ratio of GR values to
AD (Addictive Dynamics) values, denoted as ϕ. This ratio assesses the relationshipbetween entertainment value and addictiveness level, providing insights into the appeal
and gaming aspects of various brands.
Similar to card games, when the ϕ value is less than 1, it signifies that entertainment
value and addictiveness are significant. Due to the AD value surpassing the GR value,
repetition plays a crucial role at this point. Under the influence of gamification concepts,
there is a tendency for players to consistently choose this hotel, fostering loyalty andhabituation.
However, what sets it apart is that in this scenario, both the GR and AD values are
relatively small. This implies that the amusement and addictive qualities of commercial
activities are relatively low. They merely apply gamification concepts, lacking theequivalent enjoyment and thrill players might experience in traditional games.

119Table 5.11: Comparison of Languages in Duolingo using GR and AD values [8]Language

B

D

GR

AD

ϕ

English55.619291.958

0.026

0.019

1.35

Spanish64.142319.571

0.025

0.018

1.39

French72.222346.333

0.025

0.017

1.41

German89

381.25

0.025

0.017

1.47

Portuguese

68

379

0.022

0.016

1.40

Italian

66

385

0.021

0.015

1.39

In the domain of educational games, the ϕ value typically hovers around 1.5. This
indicates that entertainment value holds a significant role within the realm of educational gaming, catering to the importance of the GR parameter. The overall gameplay
tends to be more stable and transparent, characterized by a clear and open process.
However, what distinguishes it in this scenario is that both the GR and AD values
are relatively small. This suggests that the entertainment and addictive elements ofeducational games are somewhat modest. While these games often incorporate gamification concepts, unlike traditional games, players tend to perceive platforms such as
Duolingo primarily as learning tools rather than entertainment platforms.

120

Table 5.12: Game types compared using ϕ values
Game TypesGR

ϕBusiness Games

0.033

0.8-0.9

ImpactExplanation

Repetitive (Builds Habit-forming games often

Card Games

0.072

0.85-1.0

Loyalty)found in daily activities

Enjoyable

and Games associated with ran-

Addictive (Create

domness and exhibit addic-

Harmonize)tive tendencies

Enjoyable

andGames connected to physi-

Sports Games

0.074

0.85-1.6

Exercise (Promotes
cal activities and fitness
Health)Games designed for learning

Educational

Learning (Encour0.024

and expected skill enhance-

1.35-1.5

Gamesages Improvement)
ment
Enjoyable

andGames involving competi-

Board Games

0.076

1.25-1.75

Stable (Cultivatestion with stable dynamics
Competition)

Different values of ϕ represent the boundaries between games and various activities.
Through observation, we can discern that when ϕ is larger and both GR values andAD values are higher, as seen in board games, it marks the intersection between games
and competition. Conversely, when ϕ is smaller and both GR values and AD values
are lower, as evident in business games, it signifies the boundary between games andhabits.
In the case of games that combine elements of competition, such as board games,
they exhibit higher ϕ values along with elevated GR and AD values. This indicatesthat these games not only offer entertainment and enjoyment but also possess a certain
121

Figure 5-15: Gamification, Game and Competitionlevel of addictiveness, possibly tied to the competitive nature and outcomes. These
games ignite players’ competitive spirit, urging them to strive for better performance,
all while maintaining their entertainment value.
On the other hand, examples of business games showcase lower ϕ values, accompanied by lower GR and AD values. This highlights that these games emphasize
cultivating habits and daily activities rather than pursuing high entertainment or addictiveness. While these games may incorporate gamification elements to engage users
in specific business activities like purchases or earning points, players generally view
them as routine tasks rather than traditional games.
In gamification scenarios, a lower ϕ value indicates a greater emphasis on the AD
value. In contrast to entertainment, designers aim for the development of habits, encouraging repetition and fostering increased loyalty.
In game-to-competition scenarios, an increase in the phi value suggests a greater
significance of the GR value. This implies that the game becomes progressively moreentertaining, standard and stable, less surprising, and unpredictable. The overall pro122

cess, blending entertainment and stability, aids in identifying winners and showcasing
strength.
In conclusion, ϕ values serve as a significant metric for gauging game characteristics,
aiding us in better understanding the distinctions in terms of entertainment, addictiveness, and impact among different game types, as well as the varying positions they holdin players’ perspectives.

5.6

Chapter Conclusion

In this section, we primarily explored the relationship between player skill level (k),reward frequency (N ), and unpredictability (AD).
Firstly, in terms of player skill level (k), it’s evident that players at different skill
levels have varying speeds in handling game uncertainties, with less-skilled players solving uncertainties at a slower pace than highly-skilled players. Consequently, the rateof information processing in the game also differs, with less-skilled players experiencing a lower acceleration than highly-skilled players. This implies that highly skilledplayers derive more enjoyment from the game than average players. Moreover, we analyzed player performance when they can achieve similar expectations as perfect players,
denoted as k = 3. Nevertheless, the equilibrium point, where GR equals AD, is approximately at k = 2.3. As a result, we examined how various motions in mind change
as players transition from k = 3 to k = 2, and how they enhance their performance toattain a heightened gaming perception.
To ensure that player speed is not catering than 0, there exists a negative correlation
between player skill level (k) and game risk rate (m). In games with high randomness,high-risk rates, and shorter game steps, the role of chance is more prominent, resulting
in a smaller gap between average players and professional players, with lower values ofk. Conversely, in games with stronger statistical elements, lower risk rates, and GR
catering to emphasize skill rather than chance, the gap between average players and123professional players is larger, leading to higher values of k.
This suggests that less experienced players select games with higher reward frequencies and simplicity, aiming for a more predictable and satisfying gaming experience.
Conversely, skilled players prefer games with lower reward frequencies to cater to challenges, appreciating the complexity and sense of accomplishment they provide. The
relationship between k and N displays a negative correlation.
Expanding from the analysis of card games, it’s apparent that in simpler games,
players’ skills surpass the game’s difficulty, resulting in initially low AD values, as players continuously explore methods to enhance the enjoyment of the game. Conversely,
in more complex games, players’ skills may not match the game’s difficulty, leading toexcessively high AD values, making the game unmanageable. In such cases, players
need persistent training to improve themselves and their skills, maintaining AD values within a reasonable range. Players with different skill levels can indeed achieve a
better match between their skill level (k) and the level of unpredictability (AD) or thenumber of choices (N ) in the game through continuous training and exploration. By
refining their skills and adapting their strategies, players can customize their gaming
experience to strike a balance between enjoyment and an appropriate level of challenge.
Simultaneously, game designers and promoters can gather user data, and tailor and promote games that align with their preferences, achieving a mutually beneficial two-way
matching for a win-win outcome.
Additionally, our analysis encompassed various types of games, including card games,business games, educational games, and board games, to explore the relationship between N and AD, which also exhibited a positive correlation.
Moreover, we expanded our analysis beyond AD to include the ratio ϕ of GR to
AD, offering additional insights into diverse game types and their respective boundaries. Various ϕ values represent the distinctions between games and various activities.
Through observation, we noted that when ϕ is larger and both GR and AD values arehigher, as seen in board games, it signifies the intersection between games and competi124

tion. In contrast, when ϕ is smaller and both GR and AD values are lower, as observedin business games, it denotes the boundary between games and habits. Using board
games as an example, which integrate competitive elements, they displayed higher ϕvalues along with increased GR and AD values. This indicates that these games not
only offer entertainment and enjoyment but also possess a certain level of addictiveness,possibly related to their competitive nature and outcomes. These games ignite players’
competitive spirit, motivating them to strive for better performance while maintaining
their entertainment value. On the other hand, examples of business games showcased
lower ϕ values, accompanied by lower GR and AD values. This highlights that thesegames emphasize cultivating habits and daily activities rather than pursuing high entertainment or addictiveness. While these games may incorporate gamification elements
to engage users in specific business activities like purchases or earning points, players
generally view them as routine tasks rather than traditional games.
In conclusion, ϕ values are a significant metric for assessing game characteristics,
assisting game developers and optimizers in adjusting and improving games to cater todifferent player preferences and levels.

125

Chapter 6
Conclusion
This research has significantly advanced our understanding of game design and player
behavior through a comprehensive investigation into various aspects of gaming. The
examination of three critical areas in this study has yielded valuable insights capableof guiding game designers and developers. Ultimately, these findings contribute to thecreation of more captivating and well-balanced game experiences for players.
Our principal contribution revolves around the creation of an extensive game information model based on game trees. We innovatively introduced the concept of “jerk”,
rooted in game refinement theory, and explored the integration of flow theory within
the gaming context. Introducing the ∆score, effectively bridging the divide between
skill and challenge outside a game setting, added depth to our exploration. By incorporating dynamical velocity, acceleration, and jerk, we probed the intricate links between
motion and emotions, particularly concerning flow theory. This model not only provides
a comprehensive visualization of intricate game processes but also quantifies them. By
synthesizing established theories such as game refinement theory and flow theory, ourmodel emerges as a valuable instrument for evaluating games, taking into account boththeir entertainment value and complexity. This comprehensive approach provides developers with the insights necessary to make informed decisions during the design andoptimization of games, ultimately elevating the overall gaming experience.
126After introducing the concept of jerk and the game dynamics model, our expanded
study explored the motion in mind measure, particularly the jerk value, to assess gameentertainment and addiction potential. We conducted computational analyses on various card games using AI to represent different player skill levels and game complexities.
By simulating players with varying skills in card games of different difficulties, we
performed horizontal and vertical comparisons, investigating the interaction between
players and games. This exploration aimed to find a balance between player abilities andgame complexity in two dimensions: weak vs strong players and plain vs sophisticatedgames. For deterministic games, like board games, options (N ) are limited (1.5 ≤ N ≤5), leading to deterministic and stable gameplay with GR >AD. Longer steps increase
determinism. Stochastic games, such as card games, offer more options (5 ≤ N ≤ 10),
resulting in unpredictable and surprising gameplay with AD >GR. Solving GR =AD identifies the most addictive and entertaining scenario, occurring at N = 9 (e.g.,
Doudizhu with parameters B = 9 and D = 40).Analyzing the relations between AD and
GR reveals the ability to measure player stability and predictability in games. Highcomplexity games challenge players, while simpler games may become more engaging
after exploration and training. Additionally, we observed that complex games havehigher potential energy (Ep ), game momentum (p1 ), and mental momentum (p2 ) values,
indicating greater player engagement, expectations, and unpredictability. This also
correlates with higher AD values. These insights provide a deeper understanding ofgame dynamics.
Finally, we explored the relationship between player skill level (k), reward frequency(N ), and unpredictability (AD), from the player side and game side including reward
frequency and reward surprises.
Firstly, varying player skill levels (k) impact the handling of game uncertainties.
Less-skilled players solve uncertainties more slowly than highly-skilled ones, resulting
in slower information processing and less enjoyment for the former. Analyzing player
performance at k=3, we found the equilibrium point of GR = AD around k=2.3.
127

We examined how player transitions from k=3 to k=2 affect their performance andgaming experience enhancement. A negative correlation exists between player skill
level (k) and game risk rate (m). Higher randomness, risk rates, and shorter gamesteps diminish the gap between average and professional players, yielding lower k values.
Conversely, games emphasizing skill over chance lead to larger k values, indicating apreference for greater challenges.Less experienced players gravitate towards games withhigher reward frequencies and simplicity for predictable enjoyment. Skilled players favor
lower reward frequencies, relishing complex challenges. The k-N relationship displaysa negative correlation. Analysis revealed that in simpler games, player skills surpass
game difficulty, resulting in initially low AD values. In more complex games, skills may
not match game difficulty, resulting in excessively high AD values. Continuous training
and skill improvement help maintain AD values in a reasonable range, allowing players
to balance enjoyment and challenge.
We expanded our analysis to various game types, finding a positive correlation between N and AD. Additionally, we introduced the GR-AD ratio (ϕ) to distinguishbetween games and other activities. Larger ϕ values, as seen in competitive board
games, indicate increased addictiveness due to competition. Smaller ϕ values, as observed in business games, suggest a focus on routine activities over entertainment. In
summary, ϕ values are crucial for evaluating game characteristics and assisting developers in catering to diverse player preferences and levels.
In conclusion, our comprehensive exploration has shed light on the intricate nuancesof player psychology and gameplay dynamics. The integration of Flow theory, gamerefinement theory, and physical analogies has offered valuable insights into how psychological states evolve during gameplay. Moreover, our examination of various game
dimensions, ranging from complexity to unpredictability, has deepened our understanding of what makes games engaging and even addictive. This holistic perspective equips
us with a richer understanding of player experiences in diverse gaming scenarios, paving
the way for more player-centric and enjoyable game design in the future.
128

Bibliography[1] Hiroyuki Iida and Mohd Nor Akmal Khalid. Using games to study law of motions
in mind. IEEE Access, 8:138701–138709, 2020.
[2] Shuo Xiong, Long Zuo, Rachaya Chiewvanichakorn, and Hiroyuki Iida. Quantifying
engagement of various games. In The 19th Game Programming Workshop 2014.
Information Processing Society of Japan, 2014.
[3] Shuo Xiong and Hiroyuki Iida. Attractiveness of real time strategy games. In The
2014 2nd International Conference on Systems and Informatics (ICSAI 2014),
pages 271–276. IEEE, 2014.
[4] Hiroyuki Iida, Kazutoshi Takahara, Jun Nagashima, Yoichiro Kajihara, andTsuyoshi Hashimoto. An application of game-refinement theory to mah jong.
In Matthias Rauterberg, editor, Entertainment Computing – ICEC 2004, pages
333–338, Berlin, Heidelberg, 2004. Springer Berlin Heidelberg.
[5] Naying Gao, Hengyuan Chang, Zeliang Zhang, Mohd Nor Akmal Khalid, andHiroyuki Iida. Implications of jerk’s on the measure of game’s entertainment:
Discovering potentially addictive games. IEEE Access, 10:134362–134376, 2022.
[6] Arie Pratama Sutiono, Rido Ramadan, Peetikorn Jarukasetporn, Junki Takeuchi,
Ayu Purwarianti, and Hiroyuki Iida. A mathematical model of game refinement
and its applications to sports games. 2015.

129

[7] Long Zuo, Nazhif Rizani, Hiroyuki Iida, and Shuo Xiong. An analysis of points
system of hotel loyalty program based on the return on investment. In 2018 International Conference on Electrical Engineering and Computer Science (ICECOS),
pages 357–360. IEEE, 2018.
[8] Duy Huynh, Long Zuo, and Hiroyuki Iida. Analyzing gamification of “duolingo”
with focus on its course structure. In Games and Learning Alliance: 5th International Conference, GALA 2016, Utrecht, The Netherlands, December 5–7, 2016,
Proceedings 5, pages 268–277. Springer, 2016.
[9] Long Zuo, Shuo Xiong, ZHI-CHAO Wang, and Hiroyuki Iida. Evaluation of hotel loyalty program with game refinement theory and analytic hierarchy process.
DEStech Transactions on Computer Science and Engineering, (ceic), 2018.
[10] Duy Huynh, Long Zuo, and Hiroyuki Iida. An assessment of game elements inlanguage-learning platform duolingo. In 2018 4th International Conference on
Computer and Information Sciences (ICCOINS), pages 1–4. IEEE, 2018.
[11] Wu Yicong, Htun Pa Pa Aung, Mohd Nor Akmal Khalid, and Hiroyuki Iida. Evolution of games towards the discovery of noble uncertainty. In 2019 International
Conference on Advanced Information Technologies (ICAIT), pages 72–77. IEEE,
2019.
[12] Burrhus Frederic Skinner. ’superstition’in the pigeon. Journal of experimental
psychology, 38(2):168, 1948.
[13] Richard J. E. James, Claire O’Malley, and Richard J. Tunney. Understanding
the psychology of mobile gambling: A behavioural synthesis. British Journal of
Psychology, 108(3):608–625, 2017.
[14] Kang Xiaohan, Mohd Nor Akmal Khalid, and Hiroyuki Iida. Player satisfaction
model and its implication to cultural change. IEEE Access, 8:184375–184382, 2020.
130

[15] Janis Zuters. Realizing undelayed n-step td prediction with neural networks. In
Melecon 2010-2010 15th IEEE Mediterranean Electrotechnical Conference, pages
102–106. IEEE, 2010.
[16] Mohd Nor Akmal Khalid and Hiroyuki Iida. Objectivity and subjectivity in games:
Understanding engagement and addiction mechanism. IEEE Access, 9:65187–
65205, 2021.
[17] David Eager, Ann-Marie Pendrill, and Nina Reistad. Beyond velocity and acceleration: jerk, snap and higher derivatives. European Journal of Physics, 37(6):065008,
2016.
[18] Gao Naying, Anggina Primanita, Mohd Nor Akmal Khalid, and IIDA Hiroyuki.
A key factor to maintain engagement: Case study using ‘login system’. In Sriwijaya International Conference on Information Technology and Its Applications(SICONIAN 2019), pages 492–497. Atlantis Press, 2020.
[19] Gao Naying, Gao Yuexian, Mohd Nor Akmal Khalid, and Hiroyuki Iida. A computational game experience analysis via game refinement theory. Telematics andInformatics Reports, 9:100039, 2023.
[20] Deborah P Vossen. The nature and classification of games. Avante, 10(1), 2004.
[21] Gabriel Alves Mendes Vasiljevic and Leonardo Cunha de Miranda. Brain–computerinterface games based on consumer-grade eeg devices: A systematic literature review. International Journal of Human–Computer Interaction, 36(2):105–142, 2020.
[22] Johan Huizinga. Homo ludens: A study of the play-element in our culture. Routledge & Kegan Paul, 1949.
[23] Kang Xiaohan, Mohd Nor Akmal Khalid, and Hiroyuki Iida. Player satisfaction
model and its implication to cultural change. IEEE Access, 8:184375–184382, 2020.

131

[24] Liu Xinyue, Fu Luping, Mohd Nor Akmal Khalid, and Hiroyuki Iida. Finding
appropriate settings for fairness and engagement of a newly designed game: A casestudy using japanese crossword game ‘myogo renju’. Entertainment Computing,
34:100358, 2020.
[25] Yuexian Gao, Wanxiang Li, Yuhao Xiao, Mohd Nor Akmal Khalid, and Hiroyuki
Iida. Nature of attractive multiplayer games: Case study on china’s most popularcard game—doudizhu. Information, 11(3):141, 2020.
[26] Katie Sell, Tia Lillie, and Julie Taylor. Energy expenditure during physicallyinteractive video game playing in male college students with different playing experience. Journal of American College Health, 56(5):505–512, 2008.
[27] Ahmed Ibrahim Alzahrani, Imran Mahmud, Thurasamy Ramayah, Osama Alfarraj, and Nasser Alalwan. Extending the theory of planned behavior (tpb) to
explain online game playing among malaysian undergraduate students. Telematics
and Informatics, 34(4):239–251, 2017.
[28] Jeanne H Brockmyer, Christine M Fox, Kathleen A Curtiss, Evan McBroom, Kimberly M Burkhart, and Jacquelyn N Pidruzny. The development of the game engagement questionnaire: A measure of engagement in video game-playing. Journalof experimental social psychology, 45(4):624–634, 2009.
[29] Christine M Fox and Jeanne H Brockmyer. The development of the game engagement questionnaire: A measure of engagement in video game playing: Responseto reviews. Interacting with Computers, 25(4):290–293, 2013.
[30] Sebastodes Moller, Steven Schmidt, and Saman Zadtootaghaj. New itu-t standards
for gaming qoe evaluation and management. In 2018 Tenth International Conference on Quality of Multimedia Experience (QoMEX), pages 1–6. IEEE, 2018.

132

[31] Wei Peng, Ming Liu, and Yi Mou. Do aggressive people play violent computer
games in a more aggressive way? individual difference and idiosyncratic gameplaying experience. CyberPsychology & Behavior, 11(2):157–161, 2008.
[32] Arun Kulshreshth, Kevin Pfeil, and Joseph J LaViola. Enhancing the gaming
experience using 3d spatial user interface technologies. IEEE computer graphics
and applications, 37(3):16–23, 2017.
[33] Junyan Hu, Parijat Bhowmick, Inmo Jang, Farshad Arvin, and Alexander Lanzon.
A decentralized cluster formation containment framework for multirobot systems.
IEEE Transactions on Robotics, 37(6):1936–1955, 2021.
[34] Junyan Hu, Parijat Bhowmick, and Alexander Lanzon. Group coordinated control of networked mobile robots with applications to object transportation. IEEE
Transactions on Vehicular Technology, 70(8):8269–8274, 2021.
[35] Martin Johannes Dechant, Julian Frommel, and Regan Mandryk. Assessing social
anxiety through digital biomarkers embedded in a gaming task. In Proceedings ofthe 2021 CHI Conference on Human Factors in Computing Systems, pages 1–15,
2021.
[36] Soo Ting T’ng and Kee Pau. Identification of avatar mediates the associations
between motivations of gaming and internet gaming disorder among the malaysian
youth. International Journal of Mental Health and Addiction, 19(4):1346–1361,
2021.
[37] Steven C Dang and Kenneth R Koedinger. The ebb and flow of student engagement: Measuring motivation through temporal pattern of self-regulation. International Educational Data Mining Society, 2020.
[38] Natalia Sevcenko, Manuel Ninaus, Franz Wortha, Korbinian Moeller, and Peter133

Gerjets. Measuring cognitive load using in-game metrics of a serious simulation
game. Frontiers in Psychology, 12:906, 2021.
[39] Nick Harris, Kayla B Hollett, and Joshua Remedios. Facets of competitiveness as
predictors of problem video gaming among players of massively multiplayer online
first-person shooter games. Current Psychology, pages 1–10, 2020.
[40] Sally M Gainsbury, Kahlil S Philander, and Georgia Grattan. Skill gamblingmachines and electronic gaming machines: participation, erroneous beliefs, andunderstanding of outcomes. International Gambling Studies, 20(3):500–514, 2020.
[41] Donivan L Hall, Richard C Loshbough, and Gerald D Robaszkiewicz. Jerk, acceleration, and velocity limited position pattern generator for an elevator system,
August 4 1970. US Patent 3,523,232.
[42] timmurphy.org. Uefa champion league. https://www.uefa.com/uefachampionsleague/,
2021. Accessed on Jan, 2021.
[43] sports.ndtv.com. 2018 fifa world cup stats. https://sports.ndtv.com/2018-fifaworld-cup/stats/league-statsdetail, 2018. Accessed on Jan, 2021.
[44] Wikipedia. papersoccer. https://en.wikipedia.org/wiki/Paper soccer. papersoccer.
[45] Elwyn R Berlekamp, John H Conway, and Richard K Guy. Winning ways for yourmathematical plays, volume 4. AK Peters/CRC Press, 2004.
[46] Zhang Chunsheng. soccerboard. https://bit.ly/3gcoOme. soccerboard.
[47] Robert M Gray. Entropy and information theory. Springer Science & BusinessMedia, 2011.

134

[48] Genyun Sun, Qinhuo Liu, Qiang Liu, Changyuan Ji, and Xiaowen Li. A novel
approach for edge detection based on the theory of universal gravity. Pattern
Recognition, 40(10):2766–2775, 2007.
[49] MI Antonoyiannakis and JB Pendry. Electromagnetic forces in photonic crystals.
Physical Review B, 60(4):2363, 1999.
[50] Jeanne Nakamura and Mihaly Csikszentmihalyi. Flow theory and research. Handbook of positive psychology, pages 195–206, 2009.
[51] Alessandro Bacchetta, Giuseppe Bozzi, Martin Lambertsen, Fulvio Piacenza,
Julius Steiglechner, and Werner Vogelsang. Difficulties in the description of drellyan processes at moderate invariant mass and high transverse momentum. PhysicalReview D, 100(1):014018, 2019.
[52] Jonathan L Kvanvig. The value of knowledge and the pursuit of understanding.
Cambridge University Press, 2003.
[53] Lichen Zhang, Mohsen Mutasem Diraneyya, JuHyeong Ryu, Carl T Haas, andEihab Abdel-Rahman. Assessment of jerk as a method of physical fatigue detection. In International Design Engineering Technical Conferences and Computers
and Information in Engineering Conference, volume 51739, page V01BT02A010.
American Society of Mechanical Engineers, 2018.
[54] Vladimir Vukobratović and Sergio Ruggieri. Jerk in earthquake engineering: Stateof-the-art. Buildings, 12(8):1123, 2022.
[55] Yunyue Zhang, Zhiyi Sun, Qianlai Sun, Yin Wang, Xiaosong Li, and Jiangtao
Yang. Time-jerk optimal trajectory planning of hydraulic robotic excavator. Advances in Mechanical Engineering, 13(7):16878140211034611, 2021.
[56] Wolfgang Steiner. On the impact of the jerk on structural dynamics. PAMM,
21(1):e202100202, 2021.
135

[57] Hasti Hayati, David Eager, Ann-Marie Pendrill, and Hans Alberg. Jerk within thecontext of science and engineering—a systematic review. Vibration, 3(4):371–409,
2020.
[58] Shuo Xiong, Long Zuo, and H Iida. Quantifying engagement of electronic sports
game. Advances in Social and Behavioral Sciences, 5:37–42, 2014.
[59] Nathan Nossal.

Expansion of game refinement theory into continuous move-

ment games with consideration on functional brain measurement. PhD thesis,
Ph.D.Thesis, Japan Advanced Institution of Science and Technology, 2015.
[60] Gao Naying, Anggina Primanita, Mohd Nor Akmal Khalid, and IIDA Hiroyuki.
A key factor to maintain engagement: Case study using ‘login system’. In Sriwijaya International Conference on Information Technology and Its Applications(SICONIAN 2019), pages 492–497. Atlantis Press, 2020.
[61] Sharanya Eswaran, Vikram Vimal, Deepanshi Seth, and Tridib Mukherjee. Gaim:
Game action information mining framework for multiplayer online card games(rummy as case study). In Pacific-Asia Conference on Knowledge Discovery andData Mining, pages 435–448. Springer, 2020.
[62] Sanjay Agrawal, Mridul Sachdeva, and Tridib Mukherjee. Targeted upskilling
framework based on player mistake context in online skill gaming platforms. In
The First International Conference on AI-ML-Systems, pages 1–7, 2021.
[63] Meita Fitrianawati, Zulhaj Aliansyah, Nur Robiah Nofikusumawati Peni,
Imam Wahyudi Farid, and Lukman Hakim. Monte carlo method at the 24 gameand its application for mathematics education. Journal of Honai Math, 5(2):83–94,
2022.
[64] Wakeng. Retrieved 05 September 2022.

136

[65] Marloes LC Spekman, Elly A Konijn, Peter HMP Roelofsma, and Mark D Griffiths.
Gaming addiction, definition and measurement: A large-scale empirical study.
Computers in Human Behavior, 29(6):2150–2155, 2013.
[66] Min Su Lee, Young Il Shin, Seung Joon Choi, Han Byul Kang, and Ki Yong
Cho. Development of a machine control technology and productivity evaluationfor excavator. Journal of Drive and Control, 17(1):37–43, 2020.
[67] Yunyue Zhang, Zhiyi Sun, Qianlai Sun, Yin Wang, Xiaosong Li, and Jiangtao
Yang. Time-jerk optimal trajectory planning of hydraulic robotic excavator. Advances in Mechanical Engineering, 13(7):16878140211034611, 2021.
[68] Vladimir Vukobratović and Sergio Ruggieri. Jerk in earthquake engineering: Stateof-the-art. Buildings, 12(8):1123, 2022.
[69] Steven H Schot. Jerk: the time rate of change of acceleration. American Journalof Physics, 46(11):1090–1094, 1978.
[70] Jianyi Wang, Qingwei Jia, and Youyi Wang. Hard disk drive seeking acoustics
reduction by a new two-degree-of-freedom seeking controller. IEEE transactions
on magnetics, 46(6):1587–1590, 2010.
[71] Lien-Wu Chen and Yiou-Rwong Lu. Challenging artificial intelligence with multiopponent and multimovement prediction for the card game big2. IEEE Access,
10:40661–40676, 2022.
[72] Shuqin Li, Saisai Li, Hengyang Cao, Kun Meng, and Meng Ding. Study on thestrategy of playing doudizhu game based on multirole modeling. Complexity, 2020,2020.
[73] David Parlett. The Oxford guide to card games. Oxford University Press, USA,
1990.

137

[74] www.pagat.com. Games classified by type of cards or tiles used. https://www.pagat.com/class/\#equipment. Retrieved 05 September 2022.
[75] Lou Krieger. Poker player’s bible: Raise your game from beginner to winner, 2004.
[76] www.wsop.com. Hand ranking. https://www.wsop.com/poker-hands/. Retrieved
05 September 2022.
[77] Yuexian Gao, Wanxiang Li, Mohd Nor Akmal Khalid, and Hiroyuki Iida. Quantifying attractiveness of incomplete-information multi-player game: case study using
doudizhu. In Computational Science and Technology, pages 301–310. Springer,
2020.
[78] en.wikipedia.org. Doudizhu. https://en.wikipedia.org/wiki/Dou dizhu. Retrieved 05 September 2022.
[79] en.wikipedia.org. Winner (card game). Retrieved 05 September 2022.
[80] en.wikipedia.org. Bigtwo. https://en.wikipedia.org/wiki/Big two. Retrieved
05 September 2022.
[81] Jude Elliot Tan. How to play dai di (big two). https://www.timeout.com/singa
pore/things-to-do/how-to-play-taiti-big-two. Dai Di (Big Two).
[82] en.wikipedia.org. Tien len. https://en.wikipedia.org/wiki/Ti\%E1\%BA\%BF
n l\%C3\%AAn. Retrieved 05 September 2022.
[83] Paul G Mezey. Potential energy hypersurfaces. 1987.
[84] Jan Mycielski. Games with perfect information. Handbook of game theory witheconomic applications, 1:41–70, 1992.
[85] Tuomas Sandholm. Solving imperfect-information games. Science, 347(6218):122–
123, 2015.
138

[86] Hiroyuki Iida and Mohd Nor Akmal Khalid. Using games to study law of motions
in mind. IEEE Access, 8:138701–138709, 2020.
[87] Norman T Feather. Expectations and actions: Expectancy-value models in psychology. Routledge, 2021.
[88] Hiroyuki Iida and Mohd Nor Akmal Khalid. Using games to study law of motions
in mind. IEEE Access, 8:138701–138709, 2020.
[89] Kang Xiaohan, Mohd Nor Akmal Khalid, and Hiroyuki Iida. Player satisfaction
model and its implication to cultural change. IEEE Access, 8:184375–184382, 2020.
[90] Hiroyuki Iida, Kazutoshi Takahara, Jun Nagashima, Yoichiro Kajihara, andTsuyoshi Hashimoto. An application of game-refinement theory to mah jong. In
International Conference on Entertainment Computing, pages 333–338. Springer,
2004.
[91] Robert W Rosenthal. Games of perfect information, predatory pricing and thechain-store paradox. Journal of Economic theory, 25(1):92–100, 1981.
[92] Matthew L Ginsberg. Gib: Imperfect information in a computationally challenging
game. Journal of Artificial Intelligence Research, 14:303–358, 2001.
[93] Ilaria Caponetto, Jeffrey Earp, and Michela Ott. Gamification and education: A
literature review. In European Conference on Games Based Learning, volume 1,
page 50. Academic Conferences International Limited, 2014.
[94] Dennis L Kappen and Lennart E Nacke. The kaleidoscope of effective gamification:
deconstructing gamification in business applications. In Proceedings of the firstinternational conference on gameful design, research, and applications, pages 119–122, 2013.

139

Publications

Journal papers[1] Naying Gao, Hengyuan Chang, Zeliang Zhang, Mohd Nor Akmal Khalid, and Hiroyuki Iida. Implications of jerk’s on the measure of game’s entertainment: Discovering
potentially addictive games. IEEE Access, 10: 134362-–134376, 2022.
[2] Naying Gao, Yuexian Gao, Khalid, Mohd Nor Akmal Khalid, and Hiroyuki Iida. A
Computational Game Experience Analysis via Game Refinement Theory. Telematics
and Informatics Reports, Volume 9, March 2023, 100039.
[3] Zhang Zeliang, Gao Naying, Li Siqi, Mohd Nor Akmal Khalid, and Hiroyuki Iida.
Action Games Evolution Analysis: A Case Study Using the God of War Series. IEEE
Access, vol. 10, pp. 123697–123710, 2022.
[4] Ben Yang, Jinghan Wu, Aoran Sun, Naying Gao, Xuetao Zhang. Robust landmark
graph-based clustering for high-dimensional data. Neurocomputing, 496, 72–84, 2022.
[5] Liu Chang, Huang Shunqi, Naying Gao, Mohd Nor Akmal Khalid, and Hiroyuki Iida.
A solver of single-agent stochastic puzzle: A case study with minesweeper. KnowledgeBased Systems, 246 (2022) 1086.
[6] Gao Yuexian, Liu Chang, Gao Naying, Mohd Nor Akmal Khalid, and Hiroyuki Iida.
Nature of arcade games. Entertainment Computing, Volume 41, 2022, 100469
140

International Conferences[7] Gao Yue Xian, Gao, Naying , Khalid, Mohd Nor Akmal and Iida, Hiroyuki. Finding Flow in Training Activities by Exploring Single-Agent Arcade Game Information
Dynamics[C]. 19th International Conference on Entertainment Computing (ICEC).
Springer International Publishing, 2020: 126-133.
[8] Gao Naying, A. Primanita, Mohd Nor Akmal Khalid, and Hiroyuki Iida. A key
factor to maintain engagement: Case study using ‘login system’. Sriwijaya International
Conference on Information Technology and Its Applications (SICONIAN 2019). 2020,
pp. 492–497.
[9] Gao Naying, Mohd Nor Akmal Khalid, and Hiroyuki Iida. E-Proceedings of The
5th International Multi-Conference on Artificial Intelligence Technology (MCAIT 2021).
Analysis of Professional Basketball League via Motion in Mind, Online, 5 August 2021.
[10] Naying Gao, Hengyuan Chang, Mohd Nor Akmal Khalid and Hiroyuki Iida, The
1st Artificial Intelligence and Entertainment Science Workshop (AIES2021). Influenceof Jerk on Gaming Engagement: A Case Study Using Card Games, Online, November
2nd, 2021.
[11] Naying Gao, Hengyuan Chang, Mohd Nor Akmal Khalid Hiroyuki Iida, The
3rd ASEAN Workshop on Information Science and Technology (AWIST 2020).Using
Motions in Mind to Analyze Process Stability Measured by GR Value and AD Value,
Online, 1 December 2020.

141