Discussion Paper No.366
Coordination and Imitation under UnawarenessYoshihiko Tada
Chuo University
Economic Reserch Institute,
Associate Researcher

February 2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Coordination and Imitation under Unawareness∗
Yoshihiko Tada†‡
February 25, 2022

Abstract
This study focuses on discovered versions of coordination games withunawareness, and proposes a novel solution concept under unawarenessnamed a successful-coordination equilibrium. In games with unawareness, coordination might fail because the models assume that agents areunaware of their realized actions. Then, when they observe the opponents’actions which they are unaware of, by teaching or asking their opponents
how to play the opponents’ actions, they might try to coordinate successfully. Although previous works propose models of discoveries, where
players observe opponents’ actions that they are unaware of and add theseactions to their subjective games, they do not propose models of imitation that agents could add to the choices they did not have themselves,
which was played by the opponents. We provide models of discoveries andimitations, showing that their revised games by discovering and imitatingmust propose a successful-coordination equilibrium.
JEL classification: C70; C72; D80; D83
Keywords: Unawareness; Generalized Nash Equilibrium; Cognitive Stability; Coordination; Imitation

1

Introduction

This study focuses on coordination games with unawareness (or more strictly,
symmetrical games with unawareness), discusses discoveries and imitations,and refines a cognitively stable generalized Nash equilibrium by proposing a
successful-coordination equilibrium. Games with unawareness assume unaware
players. Each player may be unaware of some actions, the opponents’ view of∗ A previous version of this paper titled “Considering repeated coordination games withunawareness” (in Japanese) was presented at the 2017 Master Course Research Seminar at
Chuo University. The author thanks Toichiro Asada and Hirokazu Takizawa for their insightfulcomments on this study.
† Graduate School of Economics, Chuo University, 742-1, Higashinakano, Hachioji-shi,
Tokyo 192-0393, Japan‡ yoshihiko.tada.4@gmail.com

1

games, or both.1 Let us focus on a coordination game. In standard coordination games (without unawareness), an equilibrium means that coordination is
successful. However, in coordination games with unawareness, an equilibriummight not mean that coordination is successful. Let us consider two players, Alice and Bob. They have two actions, X and Y . Suppose that Alice is aware of X
but unaware of Y , whereas Bob is aware of Y but unaware of X. In other words,
Alice can choose X but cannot choose Y , and Bob can choose Y but cannot
choose X. Additionally, assume that both players commonly believe that Aliceis aware of only X but unaware of Y , and Bob is aware of only Y but unawareof X. Then, Alice plays X and Bob plays Y , where Alice believes that Bob
chooses Y while Bob believes that Alice chooses X. Both players’ beliefs about
the opponents’ plays are correct even if both players miss coordination. Then,
players try to improve the coordination failure. For example, Alice might teach
Bob a way to play X or ask how to play Y , and then they try to coordinate.
To discuss cases like the above example, we need to models of discoveries
and imitations. Recently, Schipper (2021) and Tada (2022) propose models of
discoveries, where players observe opponents’ actions, that they are unaware of,and add the actions to their subjective games.2 In the above example, Aliceis unaware of Y . With Bob playing Y , she observes and becomes aware ofit. She then adds Y to her subjective game. However, since they discuss only
discoveries, there are no models focusing on imitations. We present a modelfor observing and imitating such actions called the imitative discovered game.
This model assumes that each player can observe and imitate ways of playing
the opponents’ actions, that is, each player can choose the opponents’ played
actions even if they could not choose them before observing such actions.
After analyzing imitative discovered games, we propose a successful-coordination
equilibrium. In previous works, a generalized Nash equilibrium with stable belief
hierarchies introduced by Sasaki (2017) or a self-confirming equilibrium introduced by Schipper (2021) to games with unawareness is used as a steady state
equilibrium meaning that all agents confirm their correct beliefs about the opponents’ plays. It seems that when players implement a generalized Nash equilibrium with stable belief hierarchies or self-confirming equilibrium, they play
the same equilibrium in the next stage game. However, some generalized Nashequilibrium with stable belief hierarchies or self-confirming equilibrium induces
coordination failure in a coordination game with unawareness. For example, Alice and Bob commonly believes that Alice is aware of X and unaware of Y and
Bob is aware of Y and unaware of X. Then, both players can believe only that
Alice plays X and Bob plays Y . Since Alice can implement only X and Bob can
implement only Y , their beliefs are correct. Therefore, the play and beliefs are ageneralized Nash equilibrium with stable belief hierarchies and a self-confirming
1 As pointed out by Schipper (2014), unawareness is the lack of conception rather than thelack of information. He also provided a historical survey about unawareness and games withunawareness.
2 There are diﬀerences between their models. Schipper (2021) uses Heifetz, Meier, andSchipper’s (2013) model and discusses models of discoveries in extensive-form games. Tada
(2022) uses a model similar to Perea (2018) and focus on only simultaneous-move games.

2equilibrium. However, in the equilibrium, coordination obviously fails. Therefore, players might revise their subjective games. This case means that belief
stability is diﬀerent to stability of subjective games meaning that players do
not have incentives not to revise their subjective games. This paper proposesa self-confirming equilibrium, where players only coordinate successfully. In a
successful-coordination equilibrium, not only players’ beliefs are correct but also
their subjective games need not be revised for successful coordination. That is,
the equilibrium notion is a refinement of generalized Nash equilibria with stablebelief hierarchies and self-confirming equilibrium in games with unawareness.
We show that an imitative discovered game based on any coordination gamewith unawareness must have a successful-coordination equilibrium. Furthermore, we introduce notions of block games to coordination games with unawareness and reconstruct a block game called a coordination block game inwhich the players’ action sets are the same. In an updated subjective game,some actions might be redundant because nobody plays them. Then, players
might exclude such actions. A coordination block game is a smaller game that
excludes unplayed actions. Each player would select an equilibrium among such
coordination blocks in the game.

Related Literature
Feinberg (2021), Heifetz, Meier, and Schipper (2013), Halpern and Rêgo (2014),
and Meier and Schipper (2014) are pioneering works on games with unawareness.
Specifically, Heifetz, Meier, and Schipper (2013) and Halpern and Rêgo (2014)
formulated extensive games with unawareness.
Meier and Schipper (2014) formulated Bayesian games with unawarenessthat are a generalization of Bayesian games. Another type-based model was
proposed by Perea (2018) as a special case of the model by Meier and Schipper(2014). Note that the two models have diﬀerent specifications. Meier and Schipper (2014) assumed that player’s types are directly associated with beliefs abouta structure of a game, whereas Perea (2018) did not make such an assumption.
In his model, player’s types are associated with beliefs about the structure of agame, but the types cannot be associated with the structure of the game itself.
The main solution concepts in games with unawareness have two approaches:equilibrium notions (Feinberg 2021; C̆opic̆ and Galeotti 2006; Ozbay 2007;
Halpern and Rêgo 2014; Rẽgo and Halpern 2012; Grant and Quiggin 2013;
Meier and Schipper 2014 Sasaki 2017; Schipper 2021; Kobayashi and Sasaki(2021)) and rationalizability notions (Heifetz, Meier, and Schipper 2013, 2021;
Perea 2018; Guarino 2020). Recently, Tada (2022) proposed a generalization of
the closedness-under-rational-behavior (CURB) notion proposed by Basu andWeibull (1991), which is one of the set-wise notions.
A split of our model is based on the literature about growing awareness, updating awareness, and discoveries, such as Karni and Vierø(2013, 2017), Schipper (2021), Galanis and Kotronis (2021), and Tada (2022). Studies in the literature indicate that agents additionally know information about states, events,
consequences, actions, and so on, which they were previously unaware of. Note3

that our model refers to the growing awareness of the ways in which opponents
play their game rather than the growing awareness of opponents’ plays.
The rest of this paper is organized as follows. Section 2 presents the preliminaries of the study. In Section 3, we define a successful-coordination equilibrium
and show the properties of this equilibrium. Section 4 proposes imitative discovered games and coordination block games. Finally, Section 5 presents theconclusions of the study.

2

Preliminaries

This section proposes a coordination game with unawareness, a more strictly
symmetrical game with unawareness, and a cognitively stable generalized Nashequilibrium. Let G = (I, A, u) be a standard n-person coordination game.
I = {1, . . . , n} is a finite set of players, and I−i = I \ {i}. A = ×i∈I Ai , where
Ai is a nonempty finite set of actions of i and A1 = · · · = An . Let ai ∈ Ai
be i’s action. u = (ui )i∈I , where ui : A → R is the utility function of i. For
any a = (a1 , . . . , an ) ∈ A, u1 (a) = · · · = un (a) > 0 if a1 = · · · = an , while
u1 (a) = · · · = un (a) = 0 otherwise.
We define coordination games with unawareness based on the studies byKobayashi and Sasaki (2021) and Tada (2022), which are similar to that of Perea(2018).3 4 For any standard coordination game G, let V = ×i∈I (2Ai \ {∅}) bethe set of possible views of G. Similar to most previous works, we assume thatthe set of players is commonly known and each player’s utility for each action
profile does not depend on awareness. Let v ∈ V and Avi be the set of actions of
i in v = ×j∈I Avj . Here, when player i is given v, i is aware of a ∈ v and unawareof a ∈ A \ v. Let Γ = (G, (Ti )i∈I , (vi )i∈I , (bi )i∈I ) be a coordination game withunawareness, which is described as follows: for each i ∈ I,
• Ti is a finite and nonempty set of i’s type, one of which is the type t∗i .
• vi : Ti → V is i’s view function.
• bi : Ti → T−i is the belief function of i, where T−i = ×j∈I\{i} Tj . Ifbi (ti ) = (tj )j∈I\{i} , then for each j ∈ I \ {i}, vj (tj ) ⊆ vi (ti ).
3 There is a diﬀerence between our approach and that of Perea (2018). Perea (2018) did
not fix the belief hierarchies on views, and he dealt with probabilistic beliefs on awareness.
By contrast, we assume that the “actual type” of players is fixed and given. Moreover, Perea(2018) did not deal with probabilistic beliefs.
4 A type-based approach under unawareness is similar to that of Harsanyi (1967). However,as pointed out by Meier and Schipper (2014) and Perea (2018), Harsanyi’s (1967) framework
cannot present players’ unawareness of actions. In his model, a player’s action that they areunaware of is assigned to extremely low payoﬀs. Then, the rational player does not choose theaction. However, if the player is not rational, they might play such an action. It contradicts
the condition in which the player is unaware of such actions. By contrast, in a type-based
approach under unawareness, each player cannot choose the actions that they are unaware of,even if they are irrational. Moreover, if the player is unaware of another player’s actions, thisplayer cannot reason that they can play the action.

4

Let us call G an objective game. An objective game can be interpreted asa “true game” in Γ. i’s type, ti , describes their view of the game and belief
about the opponent’s types. Given ti , vi (ti ) = v implies that i is aware of v
and unaware of A \ v, and bi (ti ) = (tj )j∈I\{i} means that at ti , i believes thatthe other types are (tj )j∈I\{i} . Simultaneously, i believes that each view of j is
vj (tj ). Let bi (ti )(j) be j’s type in bi (ti ). Each player may be unaware of some
types of players, including their own.
In contrast to the previous literature, for simplicity, we define the strategies
with a focus on pure actions only. For any i ∈ I, let si : Ti → Ai . Then, givenv (t )
ti , si (ti ) ∈ Ai i i is i’s local action at ti . Let si = (si (ti ))ti ∈Ti be i’s generalized
strategy, and let s = (si )i∈I be a generalized strategy profile. For any s, si (t∗i )v (t∗ )
is i’s actual play. The set of players’ actual play Ai i i may be a proper subsetv (t∗ )
of i’s full action set Ai . Then, the player i cannot implement ai ∈ Ai \ Ai i i .
Here, we define the generalized pure Nash equilibrium proposed by Halpern
and Rêgo (2014) as follows: s∗ is a generalized (pure) Nash equilibrium if, for
any i ∈ I and ti ∈ Ti ,s∗i (ti ) ∈ arg max ui (x, (s∗j (bi (ti )(j)))j∈I−i ).
v (ti )x∈Ai iA generalized Nash equilibrium is interpreted as an equilibrium in beliefs.
However, as shown by Schipper (2014), because games with unawareness suppose the unawareness of players’ actions, a generalized Nash equilibrium might
consist of wrong beliefs. Then, players who have such wrong beliefs might revise their subjective games, and they might choose diﬀerent actions from theimmediately preceding stage in the game. To avoid such issues, Sasaki (2017)
proposed a notion of cognitive stability or stable belief hierarchies. This notion
represents that in an equilibrium satisfying cognitive stability or stable belief
hierarchies, all participants’ beliefs about the opponents’ plays are correct. Although Sasaki (2017) distinguished a notion of stable belief hierarchies from a
notion of cognitive stability, he showed that the two notions are equivalent. Let
us define cognitive stability as follows: A generalized Nash equilibrium s∗ iscognitively stable if for any i ∈ I and ti ∈ Tis∗i (ti ) = s∗i (t∗i ).
Cognitive stability represents whether all players’ beliefs about the opponents’play are correct. In a cognitively stable generalized Nash equilibrium, all players’
local actions are the same. It means that each player’s belief is correct.

3

Successful-Coordination Equilibrium

Cognitive stability is the concept of checking the correctness of beliefs in aplayed equilibrium. In any cognitively stable generalized Nash equilibrium, allplayers confirm that their beliefs are correct; while in cognitively unstable generalized Nash equilibrium, some players confirm that their beliefs are incorrect.
5

In many classes of games with unawareness, it seems that in a cognitive stable
generalized Nash equilibrium all players need not revise their subjective views,
and in a cognitively unstable generalized Nash equilibrium some players needs
to revise their subjective view. In other words, cognitive stability seems to be aconcept of the stability of subjective views. However, in some games with unawareness, especially coordination games with unawareness, players who playwith a cognitively stable equilibrium might revise their subjective views.
Example 1. Consider two people, Alice, and Bob. They face the following
coordination game, which is an objective game.
Alice / Bob
X
v =
Y
0

X
1, 1
0, 0

Y
0, 0
1, 1

In v 0 , there exist two pure Nash equilibria, (X, X) and (Y, Y ).
Here, let us assume the following about Alice’s belief in this game.
• Alice can implement her action X, but she cannot do the other action Y
because she does not know how to play Y .
• Alice knows that Bob can choose his actions X and Y if he knows how toplay them.
• Alice knows that Bob can choose only Y , and she knows that Bob does
not know how to play X; hence, she knows that Bob cannot choose X.
• Alice supposes that Bob believes that it is common knowledge that Alicecan choose only X, Bob can choose only Y , and the others’ actions cannot
be played.
In addition, let us assume the following about Bob’s belief in this game.
• Bob can implement his action Y , but he cannot do the other action X
because he does not know how to play X.
• Bob knows that Alice can choose her actions X and Y if she knows how
to play them.
• Bob knows that Alice can choose only X, and he knows that she does notknow how to play Y ; hence, he knows that Alice cannot choose Y .
• Bob supposes that Alice believes that it is common knowledge that Alicecan choose only X, Bob can choose only Y , and the others’ actions cannot
be played.
Then, Alice’s first-order view of this game is as follows:
v1 =Alice / Bob
X
6

X
1, 1

Y
0, 0

;

Bob’s first-order view of this game is as follows:v2 =Alice / Bob
X
Y

Y
0, 0
1, 1

; and

Both players’ second- or higher-order views of this game are as follows:v3 =Alice / Bob
X

Y
0, 0

.

In this example, the mathematical formulation is as follows. Denote Aliceby A and Bob by B. Suppose that TA = {t∗A , tA } and TB = {t∗B , tB } such thatvA (t∗A ) = v 1 and bA (t∗A ) = tB ;vA (tA ) = v 3 and bA (tA ) = tB ;vB (t∗B ) = v 2 and bB (t∗B ) = tA ; andvB (tB ) = v 3 and bB (tB ) = tA .
Suppose that each player is rational. Each player doesn’t need to believethat the opponent is rational. Alice then performs the best response to (X, Y )in v 3 as X in v 1 . In addition, Bob performs the best response to (X, Y ) in v 3 as
Y in v 2 . Their beliefs and decisions consist of the following generalized strategy
profile: s∗ = ([sA (t∗A ) = X, sA (tA ) = X], [sB (t∗B ) = Y, sB (tB ) = Y ]). The
generalized strategy profile is a cognitively stable generalized Nash equilibrium.
In the play, both players confirm the correctness of their beliefs.
However, each player knows that the equilibrium play is not a Nash equilibrium in each first-order subjective view. Alice is aware of the Nash equilibrium(X, X) in v 1 , and Bob is aware of the Nash equilibrium (Y, Y ) in v 2 . In thegeneralized Nash equilibrium, coordination is not successful. □This example shows that we need to distinguish between stability of beliefs
and stability of subjective views in a coordination game. It seems that previous
works interpreted unawareness of actions as unawareness of the existence of theactions. Therefore, previous works might have considered the absence of any
issue because Alice and Bob are unaware of the objective game v 0 . In otherwords, as both players are unaware of what they face in the coordination game,they cannot know that their coordination is unsuccessful.
However, Alice and Bob may revise their subjective views if we interpret
unawareness of actions as the unawareness of how to implement the actions.
Even if players are aware of the existence of actions, they cannot perform thembecause they do not know how to do so. Let X be the action “going to Alice’s
house,” Let Y be the action “going to Bob’s house.” In other words, assume
that Example 1 is a meeting game. Then, under our interpretation, this example
plays out as follows: Both players know the objective game v 0 . That is, theyknow that they must play the coordination game. However, since Alice does notknow how to go to Bob’s house, she does not perform that action. Similarly,

7since Bob does not know how to go to Alice’s house, he does not perform that
action. Moreover, the two are aware of the opponent’s unawareness. That is,
Alice is aware that Bob is unaware of how to get to her house, and Bob is awarethat Alice is unaware of how to get to his house. Both Alice and Bob then stayhome, and they believe that the other person has also done the same. This
implementation is a cognitively stable generalized Nash equilibrium. However,under our interpretation, they are aware of not only the fact that the cognitively
stable generalized Nash equilibrium is not a coordination play but also why theyplay the non-coordination play. That is, they are aware that they are unawareof how to get to each other’s house. Then, Alice may ask Bob how to get to his
house, or she may guide him about how to get to her house. Further, Bob may
ask Alice how to get to her house, or he may guide Alice about how to get tohis house. Then, their subjective views would be revised, and they would notplay the same cognitively stable generalized Nash equilibrium.
For games with unawareness other than coordination games, when a cognitively stable generalized Nash equilibrium is played, all players would play thesame cognitively stable generalized Nash equilibrium in the next stage game.
However, for coordination games with unawareness, under the above interpretation, in some cognitively stable generalized Nash equilibrium players might
not implement the same equilibrium in the next stage game, like Example 1.
To consider only equilibria that deal with successful coordination, we propose a
successful-coordination equilibrium that is a refinement of the cognitively stable
generalized Nash equilibrium.
Definition 1. In a coordination game with unawareness Γ, s∗ is a successfulcoordination equilibrium if
1. for any i ∈ I and ti ∈ Ti ,s∗i (ti ) ∈ arg max ui (x, (s∗j (bi (ti )(j)))j∈I−i );v (ti )x∈Ai i

2. for any i ∈ I and ti ∈ Ti , s∗i (ti ) = s∗i (t∗i ); and3. s1 (t∗1 ) = · · · = sn (t∗n ).
The first and second conditions respectively require our definitions of generalized Nash equilibria and cognitive stability to be applied. The third condition
requires that the coordination be successful. In a coordination game with unawareness, the third condition is necessary to avoid instances like that illustratedin Example 1.
We can easily deduce the following remarks:
Remark 1. In every coordination game without unawareness, any successfulcoordination equilibrium is a Nash equilibrium, and vice versa.
Remark 2. A successful-coordination equilibrium may not exist. See Example
1.
8Remark 3. Every successful-coordination equilibrium is a cognitively stable
generalized Nash equilibrium. This is obvious from their definitions. However,the converse does not hold. See Example 1.
A successful-coordination equilibrium has the following properties. Although
our results are obvious, each property suggests the existence of successful coordination equilibria.
Proposition 1. In a coordination game with unawareness Γ, for any i ∈ I,v (t∗ )
if Ai i i = Ai , then every cognitively stable generalized Nash equilibrium is a
successful-coordination equilibrium.
Before proving this proposition, we refer to Sasaki’s (2017) proposition. Although his proposition includes a generalized mixed strategy profile, we restrict
his result to pure strategies.
Proposition 2 (Sasaki 2017). In a simultaneous move game with unawarenessv (t∗ )
Γ, for any i ∈ I, if Ai i i = Ai , then in any cognitively stable generalized Nashequilibrium, the actual plays of all players are Nash equilibria in G.
v (t∗ )

Proof. Suppose that for any i ∈ I, Ai i i = Ai . A cognitively stable generalized
Nash equilibrium s∗ is given. That is, for any (i, ti ) ∈ I×Ti , ui (s∗i (ti ), (s∗j (bi (ti )(j)))j∈I−i ) =ui (s∗i (ti ), (s∗j (t∗j ))j∈I−i ) = ui ((s∗j (t∗j ))j∈I ), that is, every participant’s actual play
best responds to the others’ actual play. Suppose ((s∗j (tj∗ ))j∈I ) is not a Nashequilibrium in G, that is, there exist (i, ai ) ∈ I × Ai such that ai ̸= s∗i (t∗i ) andui (ai , (s∗j (t∗j ))j∈I−i ) > ui ((s∗j (t∗j ))j∈I ). However, since s∗ is a cognitively stable
generalized Nash equilibrium, it is a contradiction. Hence, ((s∗j (t∗j ))j∈I ) is aNash equilibrium in G. Here, because t∗i denotes i’s actual type, ((s∗j (t∗j ))j∈I )refers to all players’ actual plays.
v (t∗ )

Proof of Proposition 1. Suppose that for any i ∈ I, Ai i i = Ai . A coordination game with unawareness is a special case of static game with unawareness.
Therefore, by Proposition 2, in every cognitively stable generalized Nash equilibrium s∗ , the actual play of all players (s∗i (t∗i ))i∈I is a Nash equilibrium inthe standard coordination game G. In any standard coordination game, a Nashequilibrium a∗ = (a∗1 , . . . a∗n ) satisfies a∗1 = · · · = a∗n . By definition of an actual
play, for any i ∈ I, since si (t∗i ) = a∗i , s∗1 (t∗1 ) = · · · = s∗n (t∗n ). Therefore, s∗satisfies every condition of Definition 1.
Proposition 1 suggests that if every player is aware of every player’s action,then the converse of Remark 2 holds.
∩ ∩Proposition 3. Suppose that ∩i∈I ∩ti ∈Ti vi (ti ) ̸= ∅ in a coordination gamewith unawareness Γ. If some a ∈ i∈I ti ∈Ti vi (ti ) is a Nash equilibrium in G,then there exists a successful-coordination equilibrium.
∩ ∩Proof.∩Suppose that i∈I ti ∈Ti vi (ti ) ̸= ∅ in Γ and that some a = (a1 , . . . , an ) ∈∩i∈Iti ∈Ti vi (ti ) is a Nash equilibrium in G. As Γ is a coordination game, the9Nash equilibrium in G satisfies a1 = · · · = an . For any (i, ti ) ∈ I × Ti , letsi (ti ) = ai . Then, the generalized strategy profile s satisfies the conditions ofDefinition 1. That is, the s is a successful-coordination equilibrium.
Proposition 3 is a special case of Sasaki (2017, Proposition 2).

4

Discovery and Imitation of Actions

This section presents a model of discoveries and imitations under unawarenessbased on the model of endogenously discovered games (Schipper 2021; Tada2022) as follows.5 6
Definition 2. Γ′ = (G, (Ti′ )i∈I , (vi′ )i∈I , (b′i )i∈I ) is an imitative discovered gamewith 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 thatv (t ) ∪1. vi′ (t′i ) = ×j∈I [Aj i i∗∗k∈I sk (tk )], where tk is k’s actual type; and2. for any (j, tj ) ∈ I−i × Tj with bi (ti )(j) = tj , there exists t′j such thatv (t ) ∪b′i (t′i )(j) = t′j and vj′ (t′j ) = ×k∈I [Akj j h∈I sh (t∗h )], where t∗h is k’s actual
type.
When all players observe each other’s plays, the first condition suggests thateach player not only gains knowledge of the opponents’ feasible actions, butalso discovers (or ‘learns’) a way of playing such actions. The second condition
suggests that as supposed by each player, every player commonly believes thatall players gain knowledge of the others’ feasible actions and discover a way ofplaying such actions.
Example 1 (Continued.) Suppose Alice and Bob play s∗ = ([sA (t∗A ) =X, sA (tA ) = X], [sB (t∗B ) = Y, sB (tB ) = Y ]). Then, according to the first condition, Alice adds Bob’s action Y to not only Bob’s choice but also Alice’s choicein her subjective view v 1 ; and Bob adds Alice’s action X to not only Alice’s
choice but also Bob’s choice in his subjective view v 2 . Moreover, as both players
suppose that each of them commonly believes that they gain knowledge of eachother’s feasible actions and discovers a way of playing such actions according
to the second condition, both players add actions X and Y to their respective
choice in each other’s second or any higher-order view v 3 . Then, each agent’s
first and any higher-order view is replaced with v 0 .
′′This imitative discovered game Γ′ = (G, (TA′ , TB′ ), (vA, vB), (b′A , b′B )) is formulated as follows:
5 Karni and Vierø(2013, 2017) discussed cases in which agents discover their own newfeasible actions. However, in their model, such actions are not endogenously discovered but
rather exogenously discovered. In other words, such actions are given to agents by modelers.
6 Unlike Schipper (2021) and Tada (2022)), we do not deal with discovery processes. As
indicated by one of the main results, only one update of imitation is required in our model.

10TA = {t′A } and TB = {t′B };vA (t′A ) = v 0 and bA (t′A ) = t′B ; andvB (t′B ) = v 0 and bB (t′B ) = t′B .
In Γ′ , Alice and Bob can choose two actions X and Y . □Interestingly, any imitative discovered game has the following property:
Proposition 4. Given any n-person coordination game with unawareness andany generalized strategy profile, the imitative discovered game has a successfulcoordination equilibrium.
Proof. It is obvious.
The above proposition means that it leads to an existence of a successfulcoordination equilibrium in which each player revises their subjective view withjust one. In Example 1, there exist two successful coordination equilibria in Γ′ :′s′1 = (sA(t′A ) = X, sB (t′B ) = X) and s′2 = (s′A (t′A ) = Y, sB (t′B ) = Y ).
After players discover revised subjective games in the imitative discovered
game, which set of actions do the players pay attention to? It seems to be
redundant that a player rationalizes actions based on their subjective view. Let
us consider the following example:
Example 2. Consider the following objective game played by Colin (C) andDavid (D):Colin / David
α
β
vO =
γ
δ

α
1, 1
0, 0
0, 0
0, 0

β
0, 0
1, 1
0, 0
0, 0

γ
0, 0
0, 0
1, 1
0, 0

δ
0, 0
0, 0
0, 0
1, 1

.

Here, suppose that Colin believes the following view is a common belief:vC =Colin / David
α

α
1, 1

β
0, 0

.

In contrast, David believes the following view as a common belief:vD =Colin / David
β
δ

γ
0, 0
0, 0

δ
0, 0
1, 1

.

Let us formulate this game Γ = (G, (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 (tC) = tD ;given tC , vC (tC ) = vD and bC (tC ) = t∗D ;11

given t∗D , vD (t∗D ) = vD and bD (t∗D ) = tC ; andgiven tD , vD (tD ) = vC and bD (tD ) = t∗C .
Suppose that both players implement a generalized strategy profile s∗ =([sC (t∗C ) = α, sC (tC ) = β], [sD (t∗D ) = γ, sD (tD ) = α]). In the strategy profile,′′′the actual play is (α, γ). Then, the imitative discovered game Γ′ = (G, (TC′ , TD), (vC, vD), (b′C , b′D ))is formulated as follows:′
TC′ = {t′C , t′′C } and TD= {t′D , t′′D };′′′′given tC , vC (tC ) = vC and b′C (t′C ) = t′′D ;′′given t′′C , vC
(t′′C ) = vD
and b′C (t′′C ) = t′D ;
′
′
′
′
′
) = t′′C ; andgiven tD , vD (tD ) = vD and b′D (tD
′′
′
′′
′
′
′′
given tD , vD (tD ) = vC and bD (tD ) = t′C , where′vC=

Colin / David
α
γ

α
1, 1
0, 0

Colin / David
α
′
β
vD
=
γ
δ

α
1, 1
0, 0
0, 0
0, 0

β
0, 0
0, 0
γ
0, 0
0, 0
1, 1
0, 0

γ
0, 0
1, 1
δ
0, 0
0, 0
0, 0
1, 1

, and

.

′
, (α, α) and (γ, γ),Then, Colin knows that there exist two Nash equilibria in vC′, (α, α), (γ, γ),
whereas David knows that there exist three Nash equilibria in vD′
and (δ, δ). Note that Colin is not aware that David’s view is vD and that David′
is not aware that Colin’s view is vC.
Each player must select one of those equilibria in each other’s view. Here,let us focus on David. Although he knows there are three equilibria, it seems
odd that he includes all equilibria in his choices because δ is played by neither
Colin nor David. □After the players imitate the opponents’ plays and revise their views, theymight exclude redundant actions that nobody plays. Then, the players mightreconstruct their subjective views to exclude such actions. To provide such
representation, we use block game notions proposed by Myerson and Weibull(2015). A block is a Cartesian product of nonempty subsets of players’ actions.
Let us first focus on actions that each player observes and imitates, and thenlet us define a coordination block game as follows.7
Definition 3. Given any coordination game without unawareness G = (I, A, u)
and any block T = ×i∈I Ti ∈ V , GT = (I, T, uT ) is a coordination block game if
7 Tada (2022) proposed a similar notion as a realizable CURB block game. In a realizable
CURB block game, the block is CURB.

12

1. Ti = · · · = Tn ; and2. uT = (uTi )i∈I , where uTi (a) = ui (a) for any i ∈ I and a ∈ T .
Here, T is called a coordination block.
Then, the following proposition holds.
Proposition 5. Given any n-person coordination game Γ and any∪
generalized
strategy profile s, let T ∈ V be a block such that for any i ∈ I, ATi = j∈I sj (t∗j ).
Then, the block game GT = (I, T, uT ) is a coordination block game.
Proof. It is obvious.
In the case of Example 2, as Colin and David play (α, γ), they focus on α
and γ. That is, David excludes δ from his choices. Then, the coordination blockis {α, γ} × {α, γ}, and the coordination block game isColin / David
α
vT =
γ

α
1, 1
0, 0

γ
0, 0
1, 1

.

Hence, David can restrict his choices. Then, in their equilibrium selection, bothplayers focus on α and γ.
In some coordination games (with unawareness), some players are unaware ofsome of the choices, and the set of choices might be too large. Hence, in the firstplay, players might not be able to select some specific (successful coordination)equilibrium, or they might not be able to restrict action sets to a coordination
block. However, by discovering and imitating only the actions taken in the firstplay, players can restrict their actions to some specific coordination blocks.

5

Conclusion

We focus on coordination games with unawareness and propose models, where
players imitate the opponents’ play for successful coordination. In some coordination games with unawareness, no successful-coordination equilibrium might
exist, although there exists a cognitively stable generalized Nash equilibrium.
However, in the real word, it does not seem that people leave coordination failure behind. They teach or ask the opponents for a way to play their actions,and imitate these actions.
This study proposes a model of discoveries and imitations, where each player
observes all the players’ plays and imitates the approaches of such plays, andshows that the revised game must have a successful-coordination equilibrium.
Moreover, when each player focuses on the played actions by imitating theapproaches to such played actions, they might reconstruct a coordination block
game, where each player’s action set consists of every component of the played
action profile.

13

We assume that each player can imitate the approaches of the opponents’plays. However, imitation is not always successful. Players might not understand how to play the opponents’ actions. For example, a teacher teaches how
to study, but students might not know how to study. When you are summoned
to a strange coﬀee shop, it can be diﬃcult to get there even if you are told how
to get there. Future research needs to examine when imitation is possible.

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