Discussion Paper No.365
Unawareness of Actions and Myopic Discovery Process
in Simultaneous-Move Games with UnawarenessYoshihiko Tada
Chuo University
Economic Reserch Institute,
Associate Researcher

February 2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Unawareness of Actions and Myopic Discovery
Process in Simultaneous-Move Games with
Unawareness ∗Yoshihiko Tada†‡
February 18, 2022

Abstract
This study discusses the convergence to a solution in simultaneousmove games with unawareness. Since games with unawareness assume
players’ unawareness of actions, in some plays, players might revise theirsubjective views when they observe opponents’ play, which they were unaware of. Recently, a model of discovery processes was proposed to analyze
the revision process of subjective views. Pioneering work in discovery processes shows that the revision process that converges to subjective views
is not adopted by all players. However, previous studies do not show thatthe players’ play converges to a particular solution. This paper introduces the closedness under rational behavior (CURB) notion, that is, a
set-valued solution concept, to simultaneous-move games with unawareness, and shows that the play of myopic players, who best respond to theprevious play, converges to the generalized CURB set. Moreover, we propose that CURB block games, which consist only of a realizable CURB
set, represent the economy of knowledge of games’ structures.
JEL classification: C70; C72; D80; D83
Keywords: Game Theory; Unawareness; Discovery of Actions; Growing
Awareness; Adaptive Play; Closedness under Rational Behavior∗ This article is based on the author’s master’s thesis, Tada (2018), and two earlier versions, Tada (2020) and Tada (2021). The earlier versions of this article were presented at the2018 Japanese Economic Association Autumn Meeting at Gakushuin University, September
8–9, 2018, Tokyo, and at the 2021 Japanese Economic Association Spring Meeting at Kwansei Gakuin University, May 15–16, virtual. The author thanks Norimasa Kobayashi, Yasuo
Sasaki, Masakazu Fukuzumi, Toichiro Asada, and Hirokazu Takizawa for their insightful comments on this study.
† Graduate School of Economics, Chuo University, 742-1, Higashinakano, Hachioji-shi,
Tokyo 192-0393, Japan‡ yoshihiko.tada.4@gmail.com

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1

Introduction

This study introduces closedness under rational behavior (CURB) notions tosimultaneous-move games with unawareness, models a myopic discovery process
in which every player best responds to the immediately preceding plays, andshows that myopic best responses converge to realizable CURB sets in any
myopic discovery process. Games with an unawareness model lack conceptionof the games’ situations, for example, the players’ set, the actions’ set, and state
spaces. This means that certain players may not be able to perceive parts ofthe sets of players, actions, and states.1
Since players may be unaware of actions, some of them might revise theirview when they observe opponents’ play, which they were previously unaware of.
Schipper (2021) discusses such revision process, namely, the discovery process,whereby players observe actions that they might have previously been unawareof, and revise their subjective views by adding the new actions to their subjective
views.
A discovery process is diﬀerent from a learning process. Learning is an
update process of probability distributions. However, as shown by Schipper(2013), unawareness of an event is not the same as assigning the event to zeroprobability. Any event that the agents are unaware of is not included in thesubjective state space. Hence, the agents cannot assign any probability to theevent. If such an event occurs, their subjective state space must be expanded.
In games with unawareness, the set of actions must also be increased when
actions that players were unaware of are played. Therefore, learning cannot beapplied to games with unawareness. Discovery processes are alternative modelsof learning processes that were proposed to avoid this very issue.
Schipper (2021) shows that when all players implement rationalizable strategies in each stage of the game, any rationalizable discovery process converges
to form larger subjective views that do not need further revision2 . However,
he does not show that players’ play converges toward a particular solution, although he does show that such subjective views possess the same rationalizableself-confirming equilibrium, that is, such revision processes do not converge tothe equilibrium.
This paper tries to show that, through the discovery process in simultaneousmove games with unawareness, all the players’ play converges to a particularsolution. Our model assumes that every player best responds to the opponents’
immediately preceding plays in each stage of the game. We shall call the play a“myopic play. A myopic play is a type of Markov play. Unlike rationalizable
plays, myopic plays may not be rationalizable. However, in the real world, mostpeople are not rational and do not mutually believe in each other’s rationality. Therefore, they do not play based on rationalizable strategies. Hence, anassumption of myopic play is suitable for real world analysis.
1 Pioneering works about games with unawareness include Feinberg (2021), C̆opic̆ Galeotti(2006), Ozbay (2007), Heifetz et al. (2013b), Halpern and Rêgo (2014), Rêgo and Halpern(2012), and Grant and Quiggin (2013). For a historical survey, see Schipper (2014).
2 Note that larger subjective games might not be common.

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We consider the solution concepts in games with unawareness before we examine a convergence to a solution concept in the myopic discovery processes.
There are two main notions of solution concepts in studies on games with unawareness. The first is the equilibrium notion, which has been used in Feinberg(2021), C̆opic̆ Galeotti (2006), Ozbay (2007), Halpern and Rêgo (2014), Rêgoand Halpern (2012), Grant and Quiggin (2013), and Meier and Schipper (2014).
An equilibrium is interpreted as an equilibrium in beliefs. However, unlike standard game models, games with unawareness assume unawareness of actions.
Therefore, some equilibria may not be steady state equilibria, which means thatno rational agent deviates from the equilibrium because every equilibrium includes some action that some player is unaware of. To avoid this issue, several
previous studies provide a steady state notion refined equilibrium, for example,
cognitive stability notion (Sasaki, 2017) and self-confirming notion (Schipper,2021).
The second notion is the set-wise notion, more specifically, the rationalizability notion, used by Heifetz et al. (2013b, 2021), Perea (2018), and Guarino(2020). This notion is appropriate for game situations where beliefs have notyet been formed. In addition, since the rationalizable action set is supported by
all mixed-strategy equilibria, the diﬃculty of disproving the probability distribution in mixed action equilibria can be eliminated.
However, both the steady state and rationalizability notions have issues.
Learning processes are not appropriate for games with unawareness because
there could be a growing awareness of actions in the subjective game of eachplayer in each stage of the game. Therefore, we cannot explain belief formation
for steady state equilibria. Moreover, if the equilibrium is a mixed strategy,
players might not be able to disconfirm the distribution of such a strategy. A
rationalizability notion avoids these issues. However, the set of rationalizableactions is much larger than the support for any given equilibrium. Moreover,although the notion might be appropriate for one-shot games or games that
players face at first, it is diﬃcult to discuss the steady state using rationalizability notions.
To avoid these issues, this study introduces the notion of CURB to simultaneousmove games with unawareness. The CURB notion, proposed by Basu andWeibull (1991), is a refinement of the strict Nash equilibria and rationalizability. A CURB notion can be interpreted as a steady state, as done by Myerson
and Weibull (2015), and is a set-wise notion. Furthermore, any CURB set is asubset of the rationalizable action set. Hence, CURB notions seem appropriate
for games with unawareness.
Certain actions of certain players may not belong to a player’s available
choices, because the agent is unaware of the action. Hence, she or he cannot
play the action. In other words, there exist some actions that do not belong to
the subjective view. In such a scenario, a CURB set of the objective view is
not included in the subjective view. Therefore, we must generalize the CURB
notion to simultaneous-move games with unawareness. This paper proposes a
generalized CURB set, named a “realizable CURB set.
By using a model of myopic discovery processes and realizable CURB no3

tions, we can show that, in any myopic discovery process all the players’ playconverges to a realizable CURB set. More precisely, a subset of play supports
converges to a realizable CURB set. Our discovery process converges to bothlarger subjective views and a particular solution. In some realizable CURB setsthat players’ play converges to, actions outside the CURB set are redundantbecause rational players do not deviate from the CURB set. Therefore, delineating such actions is not a problem. We can construct a game without suchactions, and name the game without the redundant actions as the “CURB block
game. A block is a Cartesian product of non-empty subsets of players’ actions,and a block game is a game constructed by a block. In a simultaneous-movegame with unawareness, a myopic discovery process converging to the realizable
CURB set might not be a CURB in the objective game. Hence, in some CURB
sets, a player who can observe a CURB set of the objective game, might thinkthat the opponent is irrational. By contrast, in a CURB block game, the realizable CURB set must be CURB. Therefore, we can interpret constructing the
CURB block game as economy of knowledge of a game’s structure, and players’
cognitive costs for excluding cognitive dissonance. Moreover, we can say that amyopic discovery process proposes a candidate for games in which each player
economizes knowledge about actions.
In constructing models of unawareness, we used type-based approaches.3
Although our type-based model seems similar to Harsanyi (1967), games withunawareness diﬀer from incomplete information games, specifically, standard
Bayesian games. In standard Bayesian models, a player’s unawareness of actions is represented with extremely low pay-oﬀs. In formulation, the rational
participant does not perform the actions. However, if participants are irrationalin models with an assumption, they might perform the actions. It means that
our representation is a contradiction. In contrast to Bayesian games, unawareness of actions in games with unawareness is represented by using a (semi-)
sub-lattice of game structures. The models with unawareness provide diﬀerent
subjective games or views of the games to each player, and assume that eachplayer is unaware of opponents’ subjective games. Thus, they cannot perform
actions outside the action set in their own subjective games.
The rest of this paper is organized as follows. Section 2 provides preliminaries. Section 3 develops a generalization of CURB, and Section 4 formulates
a discovery process, called the myopic discovery process. Section 5 shows therelationships with a cognitively stable, generalized Nash equilibrium. Section
6 discusses a block game, and economy of knowledge of a game’s structure.
Finally, Section 7 provides the discussion and the conclusion.
3 Type-based models were first introduced to games with unawareness by Meier and Schipper (2014). Another pioneering model was proposed by Perea (2018). However, both themodels have diﬀerent definitions. Meier and Schipper (2014) assume that players’ types are
directly associated with views of the games, whereas Perea (2018) believes that although players’ types are associated with beliefs about views of the games, the types cannot be associatedwith the views themselves. While our formulation is similar to Perea (2018), our assumptions
diﬀer. Perea (2018) does not assume fixed belief hierarchies, but rather, probabilistic beliefs.
By contrast, we assume belief hierarchies. Our formulation is a simplified version of Kobayashi
et al. (2021) although their models are epistemic, and they assume probabilistic beliefs.

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2

Preliminaries

2.1

Simultaneous-Move Games with UnawarenessThis section provides a definition of simultaneous-move games with unawarenessthat are type-based models, a generalized Nash equilibrium, and rationalizability. Let G = (I, A, u) be a standard finite simultaneous-move game. I is a finite
set of players, and I−i = I \ {i}. A = ×i∈I Ai , where Ai is the non-empty finite
set of i’s actions, and each element on the set is ai ∈ Ai . Let A−i = ×j∈I−i Aj .
u = (ui )i∈I , where ui : A → R is i’s utility function. Denote i’s mixed action on
Ai by mi ∈ M (Ai ), where M (Ai ) is the set of i’s mixed actions, and a mixed
action profile on A by m = (mi )i∈I ∈ M (A) = ×i∈I M (Ai ). We denote i’s
expected utility for m ∈ M (A) by Eui (m).
First, we define simultaneous-move games with unawareness.4 For any standard simultaneous-move game G, let V = ×i∈I (2Ai \ {∅}) be the set of possible
views of G. That is, the set of a Cartesian product of a non-empty action subset. Like most previous works, this study assumes that the set of players iscommonly known, and that each player’s utility for each action profile is thesame among all the possible views. Let v ∈ V be a (possible) view or a block,5and Avi be the set of i’s actions in v = ×j∈I Avj . Let Av−i = ×j∈I−i Avj . Here,when a player i is given v, i is aware of a ∈ v, and unaware of a ∈ A \ v. For′
any v, v ′ ∈ V , v is contained in v ′ if Avi is a subset of Avi for any i ∈ I, that′
is, Avi ∪⊆ Avi ∪. Let M (Avi ) = {mi ∈ M (Ai )|Σai ∈Avi mi (ai ) = 1}. Given any′
δ, δ ∈ v∈V X∈2I \{∅} M (×i∈X Avi ), δ ≡ δ ′ means that δ and δ ′ have the same
supports and probabilities. Therefore, we can say that δ and δ ′ are equivalent.
Let Γ = (G, (Ti )i∈I , (vi )i∈I , (bi )i∈I ) be a simultaneous-move game with unawareness as follows: for each i ∈ I,
• Ti is a finite and non-empty set of i’s type, one of which is their actual
type t∗i .
• vi : Ti → V is i’s view function.
• bi : Ti → T−i is i’s belief function, where T−i = ×j∈I\{i} Tj . If bi (ti ) =(tj )j∈I\{i} , then for each j ∈ I \ {i}, vj (tj ) must be contained in vi (ti ).
Simply put, we do not assume probabilistic beliefs.
Let us call G an objective game (in Γ). An objective game can be interpreted
as the “true game” in Γ.6 i’s type ti describes their view about the game, and4 Our definition is essentially the same as Kobayashi et al. (2021). Note that Kobayashi etal. (2021) allow probabilistic beliefs, whereas our model assumes fixed beliefs. Our definitions
are similar to that of Perea (2018). Note that there are two major diﬀerences. First, the Perea’s(2018) model does not fix belief hierarchies on views. We assume that the “actual type” of theplayers is given. Second, Perea (2018) deals with probabilistic beliefs on awareness, whereas
our players always have point beliefs on their opponents’ awareness, as is often assumed in
the literature on games with unawareness.
5 A block is a Cartesian product of non-empty subsets of actions.
6 The term “objective game” was used by Halpern and Rêgo (2014). Feinberg (2021) refers

5belief about the opponents’ types. At ti , vi (ti ) = v means that i is aware ofv, and unaware of A \ v, while bi (ti ) = (tj )j∈I\{i} means that at ti , i believes
that the others’ types are (tj )j∈I\{i} , and that each j’s view is vj (tj ). Given(i, ti ) ∈ I × Ti , we denote a sequence ti1 , ti2 , . . . , tih , . . . , where ti1 = ti , andfor any h ≥ 2, tih = bih−1 (tih−1 )(ih ). We say that ti leads to tj if, and only if,
there exists
ti1 , . . . , tih such that ti1 = ti and tih = tj . Here, we∪ a subsequence∪
suppose i∈I Ti = i∈I {t∗ih }h≥1;t∗i =t∗i .
hv (t∗ )The set of each player’s actual play Ai i i may be a proper subset of i’sv (t∗ )
full action set Ai . In such a scenario, they cannot play ai ∈ Ai \ Ai i i . In
other words, a player’s realized actions exclude the non-realized actions. Letv (t∗ )
×i∈I Ai i i be the realizable action set. Some players may not perceive therealizable action set.

2.2

Solution Concepts

In this subsection, we consider two solution concepts as equilibrium notions(Feinberg, 2021; Rêgo and Halpern, 2012; Halpern and Rêgo, 2014; Grant andQuiggin, 2013; Meier and Schipper, 2014) and set-wise notions (Heifetz et al.,2013, 2021; Perea, 2018).
Let us define the generalized strategies. For any i ∈ I, let si : Ti →∪vi (ti )v (t )) with si (ti ) ∈ M (Ai i i ) for all ti ∈ Ti . Then, given ti ,ti ∈Ti M (Aiv (t )si (ti ) ∈ M (Ai i i ) is i’s local action at ti . We denote i’s generalized strategy by si = (si (ti ))ti ∈Ti , and a generalized strategy profile by s = (si )i∈I . In
the generalized strategy profile s, each player i’s actual play is mi ∈ M (Ai )with mi ≡ si (t∗i ), and the profile is called the objective outcome induced froms. Let us define the belief system µ = (µi )i∈I , where for any (i, ti ) ∈ I × Ti ,vi (ti )µi (ti ) ∈ M (A−i), and µi = (µi (ti ))ti ∈Ti . This study allows correlated beliefs.
2.2.1

Equilibrium Notion

First, let us consider equilibrium notions. Halpern and Rêgo (2014) introduce anequilibrium notion in games with unawareness as a generalized Nash equilibrium– in a simultaneous-move game with unawareness Γ, s∗ is a generalized Nashequilibrium if there exists belief system µ such that for any (i, ti ) ∈ I × Ti ,1. s∗i (ti ) ∈ arg maxx∈M (Avi (ti ) ) Eui (x, µi (ti )), andi

2.

µi (ti ) ≡ (s∗j (bi (ti )(j)))j∈I−i .

As they point out, a generalized Nash equilibrium is best interpreted as “anequilibrium in beliefs” (Halpern and Rêgo, 2014: 50). However, as pointed
out by Schipper (2014), there exists some generalized Nash equilibrium, which
consists of wrong beliefs. In that case, each player would revise their own beliefs
to such a game as the “modeler’s normal-form game,” and Perea (2018) calls it the “basegame.” In this context, we follow the study by Halpern and Rêgo (2014).

6

about a game’s structure and the opponents’ play, and may not play the samegeneralized Nash equilibrium.
To avoid this scenario, previous literature has discussed two approaches:equilibrium restricted by steady state notions and rationalizability. In this subsection, we focus only on steady state equilibrium notions.
First, let us consider the generalized Nash equilibrium with stable belief
hierarchies, and the cognitively-stable generalized Nash equilibrium proposedby Sasaki (2017):• The generalized Nash equilibrium s∗ has stable belief hierarchies if belief
system µ satisfies that for any (i, ti ) ∈ I × Ti , µi (ti ) ≡ (s∗j (t∗j ))j∈I−i .
• The generalized Nash equilibrium s∗ is cognitively stable if for any (i, ti ) ∈I × Ti , s∗i (ti ) ≡ s∗i (ti∗ ).
The former notion means that every player’s belief about the opponents’ plays
is correct, while the latter notion means that each player’s decision making
regarding their arbitrary type is equivalent to their actual play.
Sasaki (2017) expresses the following:
Remark 1. Generalized strategy profile s∗ is a generalized Nash equilibriumwith stable belief hierarchies if, and only if, s∗ is a cognitively-stable generalized
Nash equilibrium.
Such a generalized Nash equilibrium is called a cognitively-stable generalized
Nash equilibrium in this paper. A cognitively-stable generalized Nash equilibrium has the following property.
Remark 2. For any simultaneous-move game with unawareness Γ, let s∗ be acognitively-stable generalized Nash equilibrium. Then, the objective outcomem∗ ≡ (si∗ (t∗i ))i∈I is a Nash equilibrium on the realizable action set.
Proof. Suppose that s∗ is a cognitively-stable generalized Nash equilibrium, thatis, for any (i, ti ) ∈ I × Ti , s∗i (ti ) ≡ m∗i . Then, m∗ ≡ m ∈ M (vj (tj )) for anyv (t∗ )
(j, tj ) ∈ I × Tj . Assume that m′ ∈ M (×i∈I Ai i i ) with m′ ≡ m∗ is not a Nashequilibrium on the realizable action set. In other words, there exists i ∈ I suchthatm∗i ≡ m′i ̸∈ arg max ∗ Eui (x, m′−i ).
v (t )
ix∈Ai iThen, for any (i, ti ) ∈ I × Ti ,m∗i ≡ m′i ≡ m′′i ̸∈ arg max Eui (x, µi (ti )),v (ti )x∈Ai iwhere µi ≡ m′−i ≡ m∗−i . However, since s∗ is a cognitively-stable generalized
Nash equilibrium, this is a contradiction. Therefore, m′ is a Nash equilibriumon the realizable action set.

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Next, let us consider a self-confirming equilibrium proposed by Fudenberg
and Levine (1993). Schipper (2021) generalizes a rationalizable self-confirming
equilibrium to include extensive-form games with unawareness. Kobayashi et al.
(2021) focus on only simultaneous-move games with unawareness, and discuss arationalizable self-confirming equilibrium by using epistemic models. We discuss
the k-self-confirming equilibrium, which means that all the players in the k-thorder mutually believe that all the players’ beliefs are correct. The following
definition of self-confirming equilibria is based on Kobayashi et al. (2021). s∗ isa k-self-confirming equilibrium if there exists belief system µ such that for any
h = 1, . . . , k + 1 and ih ∈ I, where ti1 = t∗i ,1. s∗ih ∈ arg maxvix∈M (Ai h

(ti )
h )

Eui (x, µi (tih )), and2. µih ≡ (s∗ih+1 (tih+1 ))ih+1 ∈I−ih .
2.2.2

Set-Wise Notion

Next, we consider the set-wise notions. Rationalizability is the only set-wisenotion discussed in previous literature on games with unawareness.
Hence, this subsection discusses only rationalizability.7 Rationalizability is
proposed by Bernheim (1984) and Pearce (1984). Heifetz et al. (2013) generalize Pearce’s extensive-form rationalizability to games with unawareness. This
notion is one of the approaches to avoid the issue of equilibrium notions in general. This study shows that rationalizability in simultaneous-move games withunawareness, based on Kobayashi et al. (2021), is as follows: In a simultaneousmove game with unawareness Γ, for any t ∈ ×i∈I Ti , let
for all i ∈ I :




∪
vj (bi (ti )(j))′v
0); andS (t) = α ∈ v∈V ×i∈I M (Ai ) (1)α−i ≡ α−i ∈ ×j∈I−i M (Aj′(2)αi ≡ si (ti ) arg maxx∈M (Avi (ti ) ) Eui (x, α−i)i, and define S n (t) for n = 1, 2, . . . inductively by∩
S n (t) = S n−1 (t)S n−1 (ti , bi (ti )).
i∈I∩

Then, S(t) = n=0 S n (t) is the rationalizable mixed ∪action set at t. Here, let
us define the rationalizable pure action set as R(t) = α∈S(t) supp(α).
Kobayashi et al. (2021) propose a k-rationalizable self-confirming equilibrium as follows: In a simultaneous-move game with unawareness Γ, s∗ is a
k-rationalizable self-confirming equilibrium if it is a k-self-confirming equilibrium, and (s∗i (t∗i ))i∈I ≡ α ∈ S((ti )i∈I ), where for any i ∈ I, there exist j ∈ I
and h = 1, . . . , k + 1 such that ti = tjh , where tj1 = t∗j . As shown by Kobayashi
et al. (2021), in the k-rationalizable self-confirming equilibrium, the k-th ordermutual beliefs of rationality are correct. Thus, some player’s k + 1-th orderbelief might not be correct.
7 Section 3 proposes a CURB notion as another set-wise notion.

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3

Closedness under Rational Behavior

Before modeling a discovery process, we consider the strategy subsets closed
under rational behavior (CURB sets), as proposed by Basu and Weibull (1991).
In this section, we generalize a CURB notion for a simultaneous-move gameswith unawareness. Previous literature provides two approaches to avoid thesteady state notion from not being applied to equilibrium notions, in general, ingames with unawareness: a steady state equilibrium notion and a rationalizable
notion. However, both notions have issues.
1. As pointed out by Schipper (2014, 2021) and Heifetz et al. (2021), wecannot apply a learning process to formulate beliefs about the opponents’plays in games with unawareness. Moreover, a steady state equilibriummight have mixed actions. Thus, if such a steady state equilibrium is
unique and mixed, players might not be able to disconfirm a distributionof mixed actions.
2. Since rationalizable actions are support every mixed equilibrium, the solution is too large. Although rationalizable notions might be appropriate forone-shot games or games that players face at first, it is diﬃcult to discuss
the steady state using rationalizability.
To avoid the above issues, we use the other notion, CURB, which has characterizations of both steady state equilibrium notions and rationalizable notions.
A CURB notion refines a strict Nash equilibrium and rationalizability. This
notion avoids the diﬃculty of disconfirming mixed actions as rationalizability,and has a steady state notion as a steady state equilibrium. Hence, the CURB
notion seems appropriate.
Although Basu and Weibull (1991) first define a CURB notion on a standard
game, this paper defines it on each view. Given any standard simultaneous-movegame G, any possible view v̂ ∈ V and mixed action profile m ∈ M (v̂). Let
βiv̂ (m−i ) = {ai ∈ Ai |ai ∈ supp(mi ) be such that mi ∈ arg max Eui (x, m−i )}x∈M (Av̂
i)be the set of i’s pure-action best responses to their belief about m−i ∈ M (Av̂−i ).
For any v ⊆ v̂, let∪
βiv̂ (Av−i ) =βiv̂ (m−i )′v
m−i ∈M (Av̂−i ):m−i ≡m−i ∈M (A−i )be the set of i’s optimal actions under beliefs in M (v), and let β v̂ (v) = ×i∈I βiv̂ (Av−i ).
Then, CURB is defined as follows.
Definition 1. Give a standard simultaneous-move game G and v̂ ∈ V . C ⊆ v̂is a CURB set on v̂ if β v̂ (C) ⊆ C. C is a minimal CURB set on v̂ if C is CURB
on v̂, and every proper subset of C is not CURB on v̂.

9Basu and Weibull (1991) show that every standard game has a minimal
CURB set.
Remark 3. Given any standard game, every possible view has a minimal CURB
set.
In standard games, only one CURB set is necessary for the full action set.
However, since a given possible view for each player may not be consistent
with the full action set in games with unawareness, realized CURB sets are
diﬀerent for standard games, and games with unawareness. Hence, we must
distinguish CURB notions between the two models. Let us discuss the CURB
notion under unawareness. We define a CURB set on the realizable action set,called a realizable CURB set.
Definition 2. Given a simultaneous-move game with unawareness Γ, let v ∗ =v (t∗ )
×i∈I Ai i i be the realizable action set. C ∈ V is a realizable CURB set if∗
C ⊆ v ∗ and β v (C) ⊆ C. C is a minimal realizable CURB set if it is CURB on∗v , and every proper subset of C is not CURB on v ∗ .
Realizable CURB notions have the following property.
Lemma 1. Every simultaneous-move game with unawareness Γ has a minimal
realizable CURB set; it is non-empty.
Proof. Let us construct a game G′ = (N, A′ , u′ ) such that the following assumptions hold.
• N is common in Γ.
v (t∗ )• A′ = ×i∈I Ai i i .
• For any i ∈ I, u′i : A′ → R such that ui (a) = u′i (a) for any a ∈ A′ .
Following Basu and Weibull (1991), there must be a (minimal) CURB set C ⊆A′ in G. In other words, there exists a set of each player’s pure-action best
response, β ′ (C), such that β ′ (C) ⊆ C in G. Since β ′ (C) is defined on A′ =v (t∗ )
×i∈I Ai i i , C is a minimal realizable CURB set.
Given Γ, some realizable CURB set C ∈ V may be C ⊆ vi (ti ) for any(i, ti ) ∈ I × Ti . However, the set is not CURB in vi (ti ) at some ti . Givena realizable CURB set, we distinguish between a case wherein the realizable
CURB set is CURB in every vi (ti ) for any (i, ti ) ∈ I × Ti , and one where it is
not as follows.
Definition 3. In a simultaneous-move game with unawareness Γ, C ∈ V is
a common realizable CURB set if for any (i, ti ) ∈ I × Ti , C is a (minimal)
realizable CURB set and C ⊆ vi (ti ). C is a common (minimal ) CURB set if it∗
C 8
is a common (minimal) realizable CURB set, and for any i, βiv (AC−i ) ⊆ Ai .
8 In one of the previous versions of this paper, Tada (2020), notes that a common CURB
set means only a CURB set on the full action set. In contrast, this paper generalizes thisnotion by focusing on a realizable action set.

10

From the definition 3, it is obvious that for any (i, ti ) ∈ I × Ti , ti is not i’s
actual type, and C is CURB on vi (ti ).
Common CURB notions have the following property.
Proposition 1. Any simultaneous-move game with unawareness possessing acommon CURB set has a cognitively-stable generalized Nash equilibrium.
Proof. Assume that any simultaneous-move game with unawareness has a common CURB set C ∈ V . That is, for any (i, ti ) ∈ I × Ti , C is CURB on vi (ti ).
Then, following Basu and Weibull (1991), there exists a Nash equilibrium onvi (ti ), m∗ ∈ M (vi (ti )), satisfying m∗ ≡ m′ ∈ M (C). Suppose that m′ is not av (t )Nash equilibrium on v. In other words, there exists (i, ai ) ∈ I × Ai i i , suchthat Eui (ai , m′−i ) > Eui (m′ ). However, since C is a common CURB set, thisis a contradiction. Therefore, m′ is a Nash equilibrium on C. Thus, since(i, ti ) ∈ I × Ti is arbitrary, m∗ ≡ m′ is a Nash equilibrium on vi (ti ). Thus, s∗with s∗i (ti ) ≡ m∗i for any i ∈ I and ti ∈ Ti is a cognitively-stable generalized
Nash equilibrium.
Proposition 1 suggests one of the conditions for the existence of a cognitivelystable generalized Nash equilibrium in any game with unawareness. The contraposition is that if there does not exist any cognitively-stable generalized Nashequilibrium, then there does not exist any common CURB set. This meansthat if some players cannot perceive any CURB set in the realizable action set,
then the players are surprised about a realized play because the player’s belief
about the opponents’ plays is wrong. Our proposition suggests a condition for
all players’ stable plays, that is, rational players do not deviate from a specificplay.9The following corollary is obvious from the above proof of proposition 1: .
Corollary 1. Given any simultaneous-move game with unawareness, a common
CURB set includes the support of some cognitively-stable generalized Nashequilibrium.
Remark 4. Given any simultaneous-move game with unawareness Γ, any common realizable CURB set is a support for the objective outcome, induced froma 0-rationalizable self-confirming equilibrium. Given the actual type profile,
every common CURB set is included in the rationalizable action set, and the
maximum CURB set is equal to the rationalizable action set.
Example 1. Let us consider that two players, Alice (A) and Bob (B), face thefollowing objective game:
A/B
a1
v =
a2
a3
O

b1
3, 3
5, 0
0, 0

b2
0, 5
1, 1
0, 0

9 I thank Masakazu Fukuzumi for this suggestion.

11

b3
0, 0
0, 0
2, 2

.

Here, if Alice is unaware of her own action a2 , then, her view is as follows:
vA =

A/B
a1
aa

b1
3, 3
0, 0

b2
0, 5
0, 0

b3
0, 0
2, 2

,

If Bob is unaware of his own action b2 , then, his view is as follows:

v

B

A/B
a1
=
a2
a3

b1
3, 3
5, 0
0, 0

b3
0, 0
0, 0
2, 2

.

Let us suppose that Alice believes that Bob’s view is the same as hers, thatis, they both believe that they hold the same view v A , while Bob believes that
Alice’s view is the same as his, that is, they both believe that they hold thesame view v B .
Here, we formulate this game (with unawareness) Γ = (v O , (TA , TB ), (vA , vB ), (bA , bB ))as follows:
TA = {t∗A , tA }, and TB = {t∗B , tB };∗) = tB ;given t∗A , vA (t∗A ) = vA , and bA (tAgiven tA , vA (tA ) = vA , and bA (tA ) = t∗B ;given t∗B , vB (t∗B ) = vB , and bB (t∗B ) = tA ; andgiven tB , vB (tB ) = vB , and bB (tB ) = t∗A .
Since Alice’s realizable actions are a1 and a3 , and Bob’s realizable actions
are b1 and b3 , the realizable action set is the following table:
vR =

A/B
a1
a3

b1
3, 3
0, 0

b3
0, 0
2, 2

.

First, let us consider generalized (pure) Nash equilibria in the game. Thereexist four generalized pure Nash equilibria:
s1 = ([sA (t∗A ) = a1 , sA (tA ) = a2 ], [sB (t∗B ) = b1 , sB (tB ) = b2 ]);s2 = ([sA (t∗A ) = a1 , sA (tA ) = a3 ], [sB (t∗B ) = b3 , sB (tB ) = b2 ]);s3 = ([sA (t∗A ) = a3 , sA (tA ) = a2 ], [sB (t∗B ) = b1 , sB (tB ) = b3 ]);, ands4 = ([sA (t∗A ) = a3 , sA (tA ) = a3 ], [sB (t∗B ) = b3 , sB (tB ) = b3 ]).
Then, the cognitively-stable generalized Nash equilibrium is only s4 .
Second, let us consider self-confirming (pure) equilibria in the game. Thereexist a 0-self-confirming equilibrium, and ∞-self-confirming equilibrium. The
0-self-confirming equilibrium is onlys5 = ([sA (t∗A ) = a1 , sA (tA ) = a1 ], [sB (t∗B ) = b1 , sB (tB ) = b1 ]).

12Note that the 0-self-confirming equilibrium is not the k-self-confirming equilibrium (k > 1) because in k + 1, Alice’s a1 does not respond best to Bob’s b1 , andBob’s b1 does not respond best to Alice’s a1 .
In contrast, the ∞-self-confirming equilibrium is only s4 . When comparing
a cognitively-stable generalized Nash equilibrium and ∞-self-confirming equilibrium, both the equilibria are the same.10
Third, let us consider rationalizability. Given three tuples t1 = (t∗A , t∗B ),t2 = (t∗A , tB ), and t3 = (tA , tB ), the pure rationalizable actions are as follows:
R(t1 ) = {a1 , a3 } × {b2 , b3 };R(t2 ) = {a2 , a3 } × {b1 , b3 }; andR(t3 ) = {a1 , a3 } × {b1 , b3 }.
Then, it is obvious that 0-self-confirming equilibrium s5 is a 0-rationalizableself-confirming equilibrium, and that ∞-self-confirming equilibrium s4 is a ∞rationalizable self-confirming equilibrium. Kobayashi et al. (2021) show that the∞-rationalizable self-confirming equilibrium is a cognitively-stable generalized
Nash equilibrium, and vice versa.
Finally, we consider a (realizable) CURB notion. There exist three CURB
sets on the realizable action set, that is, three realizable CURB sets:
C 1 = {a1 } × {b1 };C 2 = {a3 } × {b3 }; andC 3 = {a1 , a3 } × {b1 , b3 }.
Here, C 3 is a maximum CURB set. Since, C 1 , C 2 , C 3 ⊆ v A and C 1 , C 2 , C 3 ⊆v B , every realizable CURB set is a common realizable CURB set. Moreover,C 2 is the only unique common CURB set because the common CURB set is
CURB on v A and v B .
Let us compare the CURB notion with the other notions. First, we compare
the CURB notion and equilibrium notion. By proposition 1, any common CURB
set includes a support for the objective outcome, induced from cognitively-stable
generalized Nash equilibrium. Since the ∞-rationalizable self-confirming equilibrium and cognitively-stable generalized Nash equilibrium are the same, C 2
includes a support for the objective outcome (a3 , b3 ), induced from a cognitivelystable generalized Nash equilibrium and ∞-rationalizable self-confirming equilibrium, s4 . In contrast, a common realizable CURB set C 1 , that is not acommon CURB set, is a support for the objective outcome (a1 , b1 ), induced
from 0-rationalizable self-confirming equilibrium s5 . In C 1 or s5 , each player isrational but their 1st order belief is irrational play11 . Moreover, they might becertain about the opponent’s irrationality.
10 This finding is similar to that of Kobayashi et al. (2021). Kobayashi et al. (2021) showthat any cognitively-stable generalized Nash equilibrium is a ∞-rationalizable self-confirming
equilibrium, and vice versa.
11 Rational play means that players maximize their utilities.

13

Next, let us consider a relationship with rationalizability. Given a type t1 ,R(t1 ) = C 3 . As shown by Basu and Weibull (1991), a rationalizable action set
is equivalent to a maximum CURB set. In contrast, minimal CURB sets, C 1
and C 2 , are subsets of the rationalizable action set, that is, any minimal CURB
set is a refined notion of rationalizability. □As shown in the above example, a realizable CURB notion is related to othersolution concepts, that is, the CURB notion has similar characterizations as theother notions.

4

Myopic Discovery ProcessStandard game models study a convergence to a minimal CURB set by using alearning model or an adaptation model, for example, Hurkens (1995) and Young(1998). However, in games with unawareness, some players might not notice theopponents’ actions. Since the players must revise their belief about the gamestructure, and the opponents’ views, a learning model cannot be applied togames with unawareness, as pointed out by Heifetz et al. (2021). To avoid thisissue between games with unawareness and learning models, Schipper (2021)
proposes a model of discovery processes.
A discovery process represents an update process by which each player revises
their own belief about the game’s structure, and the opponents’ play. Schipper(2021) is the first to introduce a discovery process in extensive-form gameswith unawareness, which is based on Heifetz et al. (2013). This study models a
discovery process in simultaneous-move games with unawareness based on Perea(2018). Although our definition, at first glance, may seem diﬀerent from thatof Schipper (2021), both are essentially the same.
Definition 4. Γ′ = (G, (Ti′ )i∈I , (vi′ )i∈I , (b′i )i∈I ) is a discovered game with s =(si )i∈I in Γ = (G, (Ti )i∈I , (vi )i∈I , (bi )i∈I ) if for any (i, ti ) ∈ I × Ti , there exists
t′i ∈ Ti′ such that
1. vi′ (t′i ) = ×j∈I [Aj i i ∪ supp(sj (t∗j ))], where tj∗ is j’s actual type in Γ; andv (t )

2. for any (j, tj ) ∈ I−i × Tj satisfying bi (ti )(j) = tj , there exists tj′ such thatb′i (t′i )(j) = t′j and vj′ (t′j ) = ×k∈I [Akj j ∪ supp(sk (t∗k ))], where t∗k is k’s
actual type in Γ.
v (t )

Note that some Γ, Γ′ may be T ̸⊆ T ′ and T ′ ̸⊆ T , or T ∩ T ′ = ∅.
Example 2. Consider the following objective game played by Colin (C) andDavid (D):
C/D
c1
v =
c2
c3
0

d1
3, 3
5, 0
-1, 0
14

d2
0, 5
1, 1
0, 1

d3
0, -1
1, 0
2, 2

;

and two possible views as follows:

v1 =

C/D
c1
c3

d2
0, 5
0, 1

d3
0, -1
2, 2

and v 2 =

C/D
c1
c2

d1
3, 3
5, 0

d3
0, -1
1, 0

.

Let us formulate the game with unawareness Γ = (v 0 , (TC , TD ), (vC , vD ), (bC , bD ))as follows:
TC = {t∗C , tC }, and TD = {t∗D , tD };given t∗C , vC (t∗C ) = vC , and bC (t∗C ) = tD ;given tC , vC (tC ) = vC , and bC (tC ) = t∗D ;given t∗D , vD (t∗D ) = vD , and bD (t∗D ) = tC ; andgiven tD , vD (tD ) = vD , and bD (tD ) = t∗C .
Here, suppose that Colin and David play a generalized strategy profile:s = ([sC (t∗C ) = c1 , sC (tC ) = c2 )], [sD (t∗D ) = d2 , sD (tD ) = d3 ]).
The objective outcome is (c1 , d2 ) induced by s.
Let Γ′ be the discovered game with s in Γ. Then, each player’s type set in
′∗ ′
′
′
′
Γ is TC′ = {t′∗C , tC }, and TD = {tD , tD }, where′∗b′C (tC) = t′D ;b′C (t′C ) = t′∗D;′b′D (t′∗D ) = tC ;b′D (t′D ) = t′∗
C;

′
′
′
v̂ 1 = vC(t′∗C ) = vD (tD ) =

C/D
c1
c3

d2
0, 5
0, 1

d3
0, -1
2, 2

;

and
′
′
v̂ 2 = vC(t′C ) = vD
(t′∗
D) =

C/D
c1
c2

□
A discovery process is defined as follows.
15

d1
3, 3
5, 0

d2
0, 5
1, 1

d3
0, -1
1, 0

.

Definition 5. A discovery process P = (⟨Γ1 , s0 ⟩, ⟨Γ2 , s1 ⟩, . . . , ⟨Γλ , sλ−1 ⟩, . . . ),is defined as follows:• for any λ, Γλ = (G, (Tiλ )i∈I , (viλ )i∈I , (bλi )i∈I ),• when λ = 0 and s0 = ϕ, while for any λ ≥ 1, sλ is a played generalized
strategy profile in Γλ , and• for any λ ≥ 2, Γλ is a discovered game with sλ−1 in Γλ−1 .
Let us call Γ1 the initial game with unawareness (in P ).
By definition 4, definition 5 implicitly assumes perfect recall. If we exclude
the assumption, some player may forget some action at λ even if they are awareof the action at λ − 1.
This study assumes that every player implements a pure action. Meanwhile,
standard game models might assume that every player implements and observesa mixed action. In contrast, in games with unawareness, it does not seemappropriate that every player implements, and observes a mixed action because
under unawareness, players cannot observe the frequency of their opponents’actions at each stage of the game during the course of any discovery process.12
Schipper (2021) models rationalizable discovery processes. We formulate
rationalizable discovery processes based on Perea (2018) as follows.
Definition 6. A discovery process P = (⟨Γ1 , s0 ⟩, ⟨Γ2 , s1 ⟩, . . . , ⟨Γλ , sλ−1 ⟩, . . . )is a rationalizable discovery process if for any λ and t ∈ ×i∈I Ti , (sλi (ti ))i∈I ∈S λ (t), where S λ (t) is the rationalizable mixed action set on Γλ .
In a discovery process, cautious players might carefully revise their beliefs
about the game, the opponents’ plays and rationalities, and pay-oﬀ uncertainty.
However, in the real world, The agents must pay a higher cost for revising such
beliefs, and implementing rationalizable strategies. If players are myopic, they
do not pay a high cost for revising their beliefs. This section explains the myopic
discovery process in which every player responds best to the opponents’ previous
plays.
First, we define a strategy of myopic play in a discovered game as follows.
Definition 7. Let Γ′ be a discovered game from Γ. s′ is a myopic best responsein Γ if there exists the belief system, µ′ = (µ′i )i∈I such that for any (i, t′i ) ∈ I×Ti′ ,1. s′i (t′i ) ∈ arg max

v ′ (t′ )i )x∈M (Ai i

Eui (x, µ′i (t′i )); and2. µ′i (t′i ) ≡ (s∗j (t∗j ))j∈I−i , where s∗ is played in Γ, and for any j ∈ I−i , t∗j ∈ Tj
is j’s actual type in Γ.
Next, we provide a myopic discovery process.
12 I thank an anonymous referee for pointing this out.

16

Definition 8. Any discovery process P = (⟨Γ1 , s0 ⟩, ⟨Γ2 , s1 ⟩, . . . , ⟨Γλ , sλ−1 ⟩, . . . ),is a myopic discovery process if for any λ ≥ 2, sλ is a myopic best response at
λ.
Some important questions arise here. Does any myopic discovery process
converge to a discovered game in which the players cannot revise each other’sviews? What solution concept does the discovered game have? The following
theorem answers the above questions.
Theorem 1. Given any simultaneous-move game with unawareness Γ, every
myopic discovery process, P , converges to a discovered game, possessing a common realizable CURB set. Thus, a subset of the supports of all the agents’
myopic best responses converges to common realizable CURB set.
Proof. Since we consider a myopic discovery process, it is necessary to focus
only on the realizable action set. For any objective outcome in the initialv (t∗ )
game m ∈ M (×i∈I Ai i i ), let β λ (m) be an objective outcome induced bya myopic best response on the realizable action set, and it is defined as follows: β 0 (m) = supp(m), β 1 (m) = β ′ ◦ β 0 (m), β 2 (m) = β ′ ◦ β 1 (m), . . . ,β λ (m) = β ′ ◦ β λ−1 (m), . . . . Suppose that for any CURB set on the realizablev (t∗ )
action set C ⊆ ×i∈I Ai i i , and natural number λ, β λ (m) ̸⊆ C. As pointed out
by Basu and Weibull (1991), since the set of the rationalizable strategy profilev (t∗ )v (t∗ )
on ×i∈I Ai i i , R ⊆ ×i∈I Ai i i , is CURB,13 β λ (m) ̸⊆ R for any λ, this is acontradiction. Therefore, there exists a realizable CURB set, C, and natural
number, n, such that β n (m) ⊆ C. Suppose that there exists λ ≥ n, satisfying
β λ (m) ̸⊆ C. In other words, β ′ ◦· · ·◦β ′ ◦β n (m) ̸⊆ C. However, since C is CURB,′
that is, β(v ′ ) ⊆ C for any v ′ ⊆ C with ∅ ̸= Avi ⊆ AC
i , this is a contradiction.
λλTherefore, β (m) ⊆ C for any λ ≥ n. Since, β (m) supports an objective outcome induced by a myopic best response at λ, and C is a realizable CURB set,
the support for the objective outcome is included in the realizable CURB set.
By definition 4, since CURB, C, is common, C is a common realizable CURB
set.
It is known that many intuitively appealing adjustment processes eventually
settle down in a minimal CURB set (cf., Hurkens, 1995; Young, 1998). Theorem
1 adds to the previous literature, highlighting the importance of the CURB
set. However, the process therein converges to a general CURB set, And, notnecessarily, a “minimal” one, such as in Hurkens (1995) and Young (1998).
Example 2 (Continued.) Let Γ be an initial game, that is, a game at λ = 1.
Then, the realizable action set is as follows:
vR =

C/D
c1
c3

d1
3, 3
-1, 0

d3
0, -1
2, 2

.

13 Specifically, R is a maximum tight CURB set. An action profile set, C ∈ V , is a tight
CURB set if β(C) = C.

17

v R has three CURB sets, C 1 = {c1 } × {d1 }, C 2 = {c3 } × {d3 }, and C 3 ={c1 , c3 } × {d1 , d3 }. Here, C 2 is a mutual CURB set.
In Γ, there exist two generalized Nash equilibria:
s1 = ([sC (t∗C ) = c1 , sC (tC ) = c2 ], [sD (t∗D ) = d1 , sD (tD ) = d2 ]); ands2 = ([sC (t∗C ) = c3 , sC (tC ) = c2 ], [sD (t∗D ) = d3 , sD (tD ) = d3 ]).
Obviously, both equilibria are cognitively unstable.
First, let us focus on the former equilibrium, s1 . The objective outcome is(c1 , d1 ). Since Colin is unaware of d1 , he is surprised and revises his view asfollows:
C/D
c1
c3

′

v1 =

d1
3, 3
-1, 0

d2
0, 5
0, 1

d3
0, -1
2, 2

.

′
′
′
Then, at λ = 2, the discovered game Γ′ = (G, (TC′ , TD), (vC, vD), (b′C , b′D )),
where
2∗ 2
′
2
TC′ = {t2∗C , tC }, and TD = {tD , tD };
′

′
1
′
2∗
2
vC
(t2∗
C ) = v , and bC (tC ) = tD ;′(t2C ) = v 2 , and b′C (t2C ) = t2∗
vC
D;
2
2∗
2
′
′
(t2∗
vD
D ) = v , and bD (tD ) = tC ; and′′vD(t2D ) = v 1 , and b′D (t2D ) = t2∗
C .

At λ = 2, when they play myopic best response, the generalized strategy profile
is
2 2
2
2∗
2
2
s21 = ([s2C (t2∗C ) = c1 , sC (tC ) = c1 ], [sD (tD ) = d1 , sD (tD ) = d1 ]).

Both players do not discover the opponents’ actual plays. Hence, the nextstage game, at λ = 3, is the same as Γ′ . In Γ′ , the play s21 is not a generalized
Nash equilibrium, and the objective outcome is (c1 , d1 ). The support of theobjective outcome, {c1 } × {d1 }, is a subset of a realizable CURB set, C 1 .
Next, let us focus on the latter generalized Nash equilibrium s2 . The objective outcome is (c3 , d3 ). Since David is unaware of c3 , he is surprised andrevises his view as follows:
C/D
c1
v2 =
c2
c3
′

d1
3, 3
5, 0
-1, 0

18

d3
0, -1
1, 0
2, 2

.

′′
′′
′′
Then, at λ = 2′ , Γ′′ = (G, (TC′′ , TD), (vC, vD), (b′′C , b′′D )), where
′

′

′

′

′′
TC′′ = {t2C ∗ , t2C }, and TD= {t2D∗ , t2D };′′′′′ 2 ∗vC(tC ) = v 1 , and b′′C (t2C ∗ ) = t2D ;
′

′

′

′

′′ 2
vC
(tC ) = v 2 , and b′′C (t2C ) = t2D∗ ;
′

′

′

′

′′ 2 ∗vD(tD ) = v 2 , and b′′D (t2D∗ ) = t2C ; and
′

′

′

′′ 2
vD
(tD ) = v 1 , and b′′D (t2D ) = t2C ∗ .

At λ = 2′ , when they play myopic best response, the generalized strategy profile
is
′

′

′

′

′

′

′

′

′

s22 = ([s2C (t2C ∗ ) = c3 , s2C (t2C ) = c3 ], [s2D (t2D∗ ) = d3 , s2D (t2D ) = d3 ]).
Both players do not discover the opponents’ actual play. Hence, the nextstage game, at λ = 3′ , is the same as Γ′′ . In Γ′′ , the play, the generalized strategy
profile is a cognitively-stable generalized Nash equilibrium, and a support of theobjective outcome, {c3 } × {d3 }, is a subset of a common CURB set, C 2 in Γ′′ .
□5Relationships with Cognitively-Stable Generalized Nash Equilibria

The previous section focused only on CURB notions. However, steady state
notions are not only CURB notions, but also equilibrium notions.14 Below, we
consider the convergence to discovered games in a discovery process, possessing
a steady state equilibrium, by associating it with CURB notions.
First, we provide a mutual CURB notion that each player’s actual view has
the same CURB set.
Definition 9. In any simultaneous-move game with unawareness Γ, C ∈ V isa mutual CURB set if for any i ∈ I, C is a non-empty CURB set in vi (t∗i ).
A mutual CURB notion has the following property.
Lemma 2. Every mutual CURB set is a realizable CURB set.
Proof. Given any mutual CURB set, C ∈ V , C ⊆ vi (t∗i ) for any i ∈ I. Supposethat C is not a realizable CURB set, that is, there exists some i, such that
14 Kobayashi et al. (2021) consider a rationalizable self-confirming equilibrium. They try toapply steady state notions to rationalizability by associating it with equilibrium notions.

19C
βi∗ (AC−i ) ̸⊆ Ai in the realizable action set. Since the realizable action set isv (t∗ )v (t∗ )

C
defined by ×i∈I Ai i i , βi i i (AC−i ) ̸⊆ Ai . This contradicts that C is a mutual
CURB set. Hence, C is a realizable CURB set.

Lemma 3. In a simultaneous-move game with unawareness, if a mutual CURB
set is present in every view, then there exists a common CURB set.
Proof. Suppose that a mutual CURB set, C, is present in every view in asimultaneous-move game with unawareness. Suppose that for some (i, ti ) ∈I × Ti , C is not CURB in vi (ti ). Since for some j ∈ I vi (ti ) ⊆ vj (t∗j ), where t∗j is
j’s actual type and t∗j leads to ti , C is not CURB in vi (t∗i ). This is a contradiction. Therefore, the mutual CURB set is the CURB in every view in the game.
Then, by lemma ??, since the mutual CURB set is a realizable CURB set, theset is a common CURB set.
When relating mutual CURB notions with steady state equilibrium notions,
we can show the condition for converging a discovered game possessing somesteady state equilibrium. Moreover, we can show the condition for converging agame, such that every equilibrium is a steady state equilibrium. This is provedin the following theorems.
Proposition 2. In any simultaneous-move game with unawareness, if there
exists a mutual CURB set, such that the CURB set is CURB in every view in
the game with unawareness, then there exists a cognitively-stable generalized
Nash equilibrium.
Proof. Suppose that some mutual CURB set is present in every view in asimultaneous-move game with unawareness. By lemma 3, the mutual CURB
set is a common CURB set. Then, by proposition 1, there exists a cognitivelystable generalized Nash equilibrium.
Theorem 2. Suppose a simultaneous-move game with unawareness, Γ, has amutual CURB set. Then, there exists a myopic discovery process converging toa discovered game possessing a cognitively-stable generalized Nash equilibrium.
Proof. Suppose that there exists a mutual CURB set, C, in Γ. By lemma 2,
C is a realizable CURB set. By theorem 1, there exists a myopic discovery
process, P , converging to C. Since the C is a common realizable CURB set bylemma 3, by proposition 2, there exists a cognitively-stable generalized Nashequilibrium.
Corollary 2. Suppose that every realizable CURB set is a mutual CURB set
in Γ. Then, every myopic discovery process converges to a discovered gamepossessing a cognitively-stable generalized Nash equilibrium.

20Note that the process considered in the present study starts from an arbitrary
generalized strategy profile. Our convergence result holds even if the starting
point is not, necessarily, a generalized Nash equilibrium.15
Next, let us consider a relationship with a Nash equilibrium in an objective
game. Sasaki (2017) discusses relationships between a cognitively-stable generalized Nash equilibrium and a Nash equilibrium in an objective game in any
simultaneous-move game with unawareness. The researcher shows the following
proposition.
Proposition 3. Given any simultaneous-move game with unawareness Γ, forv (t∗ )
any i ∈ I, if Ai i i = Ai , where t∗i is i’s actual type, then every cognitivelystable generalized Nash equilibrium induces an objective outcome to be a Nashequilibrium in an objective game G.
Proof. Given any simultaneous-move game with unawareness Γ, suppose thatv (t∗ )
for any i ∈ I, Ai i i = Ai , where t∗i is i’s actual type. Suppose that generalized
strategy profile, s∗ , is a cognitively-stable generalized Nash equilibrium. By
remark 2, the objective outcome induced from a cognitively-stable generalized
Nash equilibrium is a Nash equilibrium of the realizable action set. Since every
player is aware of their own actions in the objective game, the realizable action
set is equivalent to the action set of the objective game. Hence, since every Nashequilibrium of the realizable action set is a Nash equilibrium in the objective
game, the support of the objective outcome, induced by a cognitively-stable
generalized Nash equilibrium, is a Nash equilibrium in the objective game.
By theorem 1 and proposition 3, we show the following theorem.
Theorem 3. In any simultaneous-move game with unawareness Γ, for any i ∈ I,v (t∗ )
if Ai i i = Ai , where t∗i is i’s actual type, then any myopic discovery process
converges to a discovered game, such that any cognitively-stable generalized
Nash equilibrium induces an objective outcome to be a Nash equilibrium in an
objective game G.

6

A CURB Block Game and Economy of Cognitive Costs

Discovery processes represent convergence of all the agents’ subjective game,where players need not revise their subjective game. In other words, theseprocesses are aimed toward at searching for a game’s true structure. Schipper (2021) states that a rationalizable discovery process is a process where each
15 Tada (2018) discusses a revision process in which players play a generalized Nash equilibrium in each round, and conjectures that the process converges to a cognitively-stable
generalized Nash equilibrium, if there is any. However, the conjecture is wrong in assumingthat players play a generalized Nash equilibrium in each round. This study yields a result inthe same spirit as that, under another condition, in which players play myopic best responses.

21

player searches for their larger subjective game, and shows that every rationalizable discovery process converges to a discovered game possessing a rationalizableself-confirming equilibrium. Our myopic discovery process converges to a discovered game possessing a common realizable CURB set, and in the myopic
discovery process, supports of plays converge to the common realizable CURB
set.
In our model, as shown in Section 3, some myopic discovery processes do notconverge to a discovered game possessing a common CURB set. Some players
may be certain of the opponents’ irrationality. However, by using the block gamenotion (e.g., Myerson and Weibull, 2015) of a smaller game than each player’s
subjective game, players can reconstruct a block game possessing a common
CURB set from a discovered game that a myopic discovery process converges
to, and they can be certain about each other’s rationality.
Let us consider a case where a discovered game possesses a realizable CURB
set. When all the players implement a generalized strategy profile so that theobjective outcome is in the realizable CURB set, if they are rational, they do
not perform actions outside the realizable CURB set. Thus, all the actionsin the complement set of the realizable CURB set are redundant for them.
Therefore, each player excludes the actions in the complement set to economize
cognitive costs of the true structure of the game. If they economize the cognitive
costs, their subjective games are the smallest games in which the action set is acommon realizable CURB set. The following definition represents the “economyof knowledge” about a game’s structure.
Definition 10. Given any game with unawareness, Γ = (G, (Ti )i∈I , (vi )i∈I , (bi )i∈I ),and any common realizable CURB set, C ∈ V in Γ, Γ′ = (G, (Ti′ )i∈I , (vi′ )i∈I , (b′i )i∈I )is an economized game by C in Γ, if for any (i, ti ) ∈ I × Ti , there exists t′i ∈ Ti′
so that• vi′ (t′i ) = C; and• for any (j, tj ) ∈ I−i × Tj with bj (tj )(j) = tj , there exists t′j ∈ Tj′ so thatb′j (t′j )(j) = t′j , and vj′ (t′j ) = C.
Then, GC = (I, C, uC ) is called a realizable CURB block game with C, whereCC
uC = (ui )CI∈I , and ui : C → R so that for any a ∈ C, ui (a) = ui (a).
In the example 2, when Colin and David play s1 in the initial game, ands21 in the next stage of the game, since the objective outcome induced by s21 is1
1C1(c1 , d1 ), the realizable CURB block game with C 1 is GC = (I, C 1 , (uCC , uD )).
1
1
Thus, in the economized game, ΓC , all subjective games are GC .
The following remark is obvious.
Remark 5. Every economized game, Γ′ by C, in Γ has a cognitively-stable
generalized Nash equilibrium.
1

In ΓC in the example 2, there exists a unique generalized Nash equilibriumsuch that Colin and David play c1 and d1 in each subjective game, respectively.
22

Thus, by the definition of cognitive stability, the generalized Nash equilibrium
is cognitively stable.
When Γ is a discovered game that a myopic discovery process converges
to, every subjective game is a realizable CURB block game with a CURB set
such that supports of players’ actual actions converge in the process. Hence, arationalizable discovery process is a search process for larger subjective games,
whereas our myopic discovery process is a search process for common, smaller
subjective games, that is, realizable CURB block games.

7

Discussion

7.1

Adaptive PlayThis study considers myopic agents and myopic play. In the model, each player
responds best to the opponents’ strategies in the previous stage of the game.
However, a bounded agent may not be able to provide their best response to the
opponents’ strategies. Young (1993) provides adaptive play models that allow
participants to not provide best responses to previous plays. This subsection
discusses a generalization of the adaptive plays to simultaneous-move gameswith unawareness.
First, we provide a definition of adaptive plays in a discovered game.
Definition 11. Let Γ′ be a discovered game from Γ, and let ε > 0 be anerror rate such that ε is suﬃciently small. Generalized strategy profile, s′ , isan adaptive play in Γ′ , if for any (i, t′i ) ∈ I × Ti , with probability 1 − ε, player
i chooses a best response to i’s beliefs µ′i (t′i ) ≡ (s∗j (t∗j ))j∈I−i such that s∗ is a
generalized strategy profile played in Γ, and t∗j is j’s actual type in Γ; further,v ′ (t′ )with probability ε, i chooses an action in Ai i i at random.
We propose a discovery process with an adaptive play as follows, based on
the definition 11.
Definition 12. Any discovery process, P = (⟨Γ1 , s0 ⟩, ⟨Γ2 , s1 ⟩, . . . , ⟨Γλ , sλ−1 ⟩, . . . ),is an adaptive discovery process, if for any λ ≥ 2, sλ is an adaptive play profileat λ.
In a game without unawareness, Hurkens (1995) and Young (1998) use anadaptive play notion, and show a convergence to a minimal CURB set. Our
proof of the theorem 1 focuses on only the realizable action set. Additionally,
we can conjecture the following.
Conjecture 1. Given any simultaneous-move game with unawareness, in any
adaptive play, supports of the objective outcome, induced by adaptive plays,
converge to a common minimal realizable CURB set.
Although we omit a proof of this conjecture, we show an informal proof as
follows.

23

The informal proof. Given any simultaneous-move game with unawareness, andany adaptive discovery process.
1. It is necessary to focus on only the realizable action set as per the theorem
1.
2. Based on Hurkens (1995) and Young (1998), adaptive plays converge to aminimal CURB set of the realizable action set.
3. Then, the set is a common minimal realizable CURB set. □7.2

Growing AwarenessA study of discovery processes entails an analysis of growing awareness or updating awareness. Karni and Vierø(2013, 2017) discuss decision-making underunawareness, and propose a reverse Bayesian model. As pointed out by Schipper (2013), an agent who is unaware of an event is diﬀerent from an agent who
assigns probability zero to the event. This means that an unaware agent cannot assign a probability to an event that they are unaware of. Given such an
event, Karni and Vierø’s (2013, 2017) model discusses the methods to revise
such agents’ beliefs.
Galanis and Kotoronis (2021) provide generalizations of the results of Genakoplos and Polemarchakis (1982) and Ostrovsky (2012). They suppose that updating awareness is minimal, and a true state is never excluded. Traders eventuallyagree on the price of the security. Moreover, if the security is separable, tradersagree on the correct price and there is information aggregation.

7.3

Limitations

Our research has the following limitations.
1. In a game with unawareness, in a generalized Nash equilibrium or under arationalizable strategy, each player may be convinced that they are playing
a higher-order subjective game, or that the opponents are unaware of certain actions. However, in certain plays, each player may discover actions
that they were unaware of, which may confirm that the players’ subjective
game was wrong. Here, the question arises, why was the player convincedthat their higher-order subjective game was correct in the initial gamewith unawareness? In the example 2, there exist two cognitively-unstable
generalized Nash equilibria, s1 and s2 , in the initial game. This study
does not yield any appropriate answer to the question as to which equilibrium Colin and David play when they both implement a generalized Nashequilibrium play.
Our discovery process, and that of the previous works, explain how tobuild each player’s subjective game under unawareness, however, they do
24not explain how to do so in an initial game with unawareness. This issue
is an subject for future research on games with unawareness.
2. Each player pays attention to the opponents’ subjective games in the initial
game with unawareness, however, they do not pay attention to them ina discovered game. We do not have any appropriate answer to why eachplayer ceases to pay attention.
3. Models of discovery processes suppose that each player recognizes the opponents’ plays and actions that they were previously unaware of. However,
the assumptions may be too strict. For example, most children of preschool age would not be able to understand conversations among adults,
or, at least, cannot have the same conversations. In further research, we
aim to relax this assumption, and reconstruct the models of discovery
processes.

7.4

Related Literature

Unawareness in General
The first motivation of studies on unawareness is overcoming the No-Trade Theorem presented by Milgrom and Stokey (1982). Previous works about unawareness, that address this issue, had two approaches. One was a non-partitional
state space model, for example, Geankoplos (2021); and the other was an unawareness structure model, for example, Heifetz et al. (2006, 2013a), and Galanis (2013, 2018).
Interpretations about unawareness under the two approaches are diﬀerent.
The former corresponds to a lack of knowledge, that is, an agent does not know
an event, and does not know that he or she does not have that knowledge.
The other literature on this approach includes Samet (1990), Shin (1993), andEwerhart (2001). However, in (non-)partitional models, several assumptions
lead to trivial unawareness, that is, an agent cannot be unaware of any event; see
Modica and Rustichini (1994, 1999), Dekel et al. (1998), and Chen et al. (2012).
The latter model is proposed to avoid this issue. Unawareness structures first
formulate the family of state spaces, and give diﬀerent state spaces to diﬀerent
agents. Players’ unawareness is represented by diﬀerent subjective games. Other
literature on unawareness structures or similar structures includes Li (2009) andHeinsalu (2012). In a recent study, Fukuda (2021) compared the two approaches.
CURB Notions
Basu and Weibull (1991) first introduce CURB notions to standard game models. CURB notions in dynamic models are discussed by Hurkens (1995), Young(1998), and Grandjean et al. (2017). Voorneveld et al. (2005) discuss the axiom and properties of minimal CURB sets. Pruzhansky (2003) shows that inextensive games with perfect information and finite horizon, there exists only25

one minimal CURB set. Benisch et al. (2010) provide algorithms for computing CURB sets. Asheim et al. (2016) discuss epistemic robustness of CURB in
epistemic models.

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