Discussion Paper No.371

Coordination and Imitation under UnawarenessYoshihiko Tada
Chuo University

May 2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Coordination and Imitation under Unawareness∗
Yoshihiko Tada†‡§First version: February 25, 2022
This version May 9, 2022

Abstract
This study focuses on discovered versions of coordination games withunawareness, and proposes a novel solution concept under unawarenesscalled the successful-coordination equilibrium. In games with unawareness, coordination might fail because the underlying assumption is thatagents are unaware of their realized actions. When the agents observe theopponents’ actions which they are unaware of, by teaching or asking theiropponents how to play the opponents’ actions, they might try to coordinate successfully. In this study, each player can observe the opponents’actions and imitate them. Then, all players with revised subjective games
must have a successful-coordination equilibrium so that coordination is
successful.
JEL classiﬁcation: C70; C72; D80; D83
Keywords: Unawareness; Coordination; Imitation; Generalized Nash Equilibrium; Cognitive Stability

1Introduction

This study aims to:• focus on coordination games with unawareness, speciﬁcally, symmetricalgames with unawareness,• introduce a successful-coordination equilibrium to coordination games withunawareness, and∗ A previous version of this note 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‡ Graduate School of Strategic Management, Chuo University, 1-13-27, Kasuga, Bunkyo-ku,
Tokyo 112-8551, Japan§ yoshihiko.tada.4@gmail.com

1• model a discovered game with imitation and relate it to a successfulcoordination equilibrium.
Games with unawareness model ignorance are those where players are unaware of each other’s actions. Players might not be unaware of their own feasible
actions or the opponents’ actions, but their beliefs about the opponents’ play
are diﬀerent from their actual plays. When they face each other in some coordination game with unawareness, if their beliefs are asymmetric, coordination
might fail. Let us consider the following example. In a coordination game, there
are two players, Alice and Bob, and two actions, X and Y . Here, suppose that
Alice can play only X and Bob can play only Y . Moreover, let us suppose thatboth players commonly believe this. Thus, their beliefs are (X, Y ), the play is acoordination failure and their beliefs are correct. Hence, the players understandthat their coordination is not a success.
In this case, the failure is diﬀerent from a standard coordination failure. Ina standard model, coordination failure implies an equilibrium selection mistake.
By contrast, in this example, coordination failure is derived from unawarenessof their own actions, even though they are aware that the opponents are awareof such actions. Hence, they might be aware of their coordination failure1 .
Thus, it is natural to assume that the players will readjust by resolving theirignorance. For example, Alice may teach Bob a way to play X, or ask him about
a way to play Y . This study investigates the case in which players observe theopponent’s actions that they could not play and supposes that they imitate
their actions. We reconstruct the game based on the above assumptions and let
imitative discovered games be the reconstructed game.
To consider a successful coordination under unawareness, we introduce anovel equilibrium concept, successful-coordination equilibrium. A successfulcoordination equilibrium is a speciﬁc solution concept in coordination gameswith unawareness, which deals only with successful coordination. The solution concept was characterized to show that there must exist a successfulcoordination equilibrium in any imitative discovered game.
The rest of the study is organized as follows. In section 2, we model acoordination game (speciﬁcally, a symmetrical game) with unawareness. Next,
we introduce a successful-coordination equilibrium to coordination games withunawareness and characterize it. Section 4 models imitative discovered games
and shows that any imitative discovered game has a successful-coordination
equilibrium. Additionally, we introduce a block game notion to coordination
games with unawareness. In an initial coordination game, players exclude theredundant actions that were not played in the stage game in a coordination
block game. Finally, Section 5 provides the following two discussions:
1. We discuss the relationship of successful-coordination and cognitively stable generalized Nash equilibria. A cognitively stable generalized Nashequilibrium is a generalization of a Nash equilibrium. The equilibrium
1 As shown below, such play is possible in the Nash equilibrium based on correct beliefs.

2

is interpreted as equilibrium in correct beliefs. However, in some coordination games with unawareness, a cognitively stable generalized Nashequilibrium sometimes induces coordination failure. This shows that correctness of beliefs is diﬀerent from the accuracy of subjective games.
Future research can compare the correctness of beliefs to the correctnessof subjective games under asymmetric perception. This note provides anintroduction of this comparison.
2. This note investigates the assumptions about discoveries and imitations.
However, the assumptions are speciﬁc to unawareness. In this section, we
examine the two assumptions in detail.

2

Preliminaries

This section proposes a coordination game with unawareness, a more strictly
symmetrical game with unawareness. 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 functionof 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).2 3 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.
2 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, and we do notdeal with probabilistic beliefs.
3 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. 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.

3

• 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 ).
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, player i cannot implement ai ∈ Ai \ Ai i i .

3

Successful-Coordination Equilibrium

This section proposes a novel equilibrium concept in (coordination) games withunawareness, named successful-coordination 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 condition requires that players best respond to their beliefs about
the opponent’s play, and the second condition that all players’ beliefs are correct.4 The third condition requires that the coordination should be successful.
We can easily deduce the following remark and proposition5 .
Remark 1. In every coordination game without unawareness, any successfulcoordination equilibrium is a Nash equilibrium, and vice versa.
∩ ∩Proposition 1. 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.
4 As described later, the ﬁrst condition is a deﬁnition of generalized Nash equilibria andthe second condition is a deﬁnition of cognitive stability.
5 Proposition 1 is a special case of Sasaki (2017, Proposition 2).

4∩ ∩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, theNash 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, s is a successful-coordination equilibrium.

4

Discovery and Imitation of Actions

The previous section deﬁnes a successful-coordination equilibrium. However, a
successful-coordination equilibrium may not exist in some coordination gameswith unawareness.
Remark 2. The following example shows how a successful-coordination equilibrium may not exist.
Example 1. Consider two people, Alice, and Bob. They face the following
coordination game, which is an objective game.
v0 =Alice / Bob
X
Y

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 .

5• 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

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 ]).
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. Hence, they areaware that coordination is a failure. □The above problem can be ignored if one considers the act to be symbolic.
However, given the speciﬁc situation, it is impossible to ignore. Let X be theaction “going to Alice’s house,” and let Y be the action “going to Bob’s house.”
In other words, assume that Example 1 is a meeting game. Then, Alice may
ask Bob how to get to his house, or she may guide him about how to get to herhouse. Further, Bob may ask Alice how to get to her house, or he may guide
Alice about how to get to his house.

6To resolve the above issue, let us present a model of discoveries and imitations under unawareness based on the model of endogenously discovered games(Schipper 2021; Tada 2022) as follows.6 7
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
and any sequence of players i1 , i2 , . . . , with sequence of types ti1 , ti2 , . . . , whereti1 = ti , there exists t′i ∈ Ti′ and a sequence of types in Γ′ , t′i1 , t′i2 , . . . , wheret′i1 = t′i , such that for any h ≥ 1,vi (tih ) ∪vi′h (t′ih ) = ×j∈I [Aj hsupp(sk (tk ))],

k∈I

where t∗k is k’s actual type in Γ. Note that for some Γ, Γ′ may be T ̸⊆ T ′ andT ′ ̸⊆ T , or T ∩ T ′ = ∅.
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, Aliceadds Bob’s action Y to not only Bob’s choice but also Alice’s choice in 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 supposethat each of them commonly believes that they gain knowledge of each other’s
feasible actions and discovers a way of playing such actions according to thesecond condition, both players add actions X and Y to their respective choicein each other’s second or any higher-order view v 3 . Then, each agent’s ﬁrst andany higher-order view is replaced with v 0 .
′′), (b′A , b′B )) is for, vB
This imitative discovered game Γ′ = (G, (TA′ , TB′ ), (vA
mulated as follows:
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:
6 Karni and Vierø(2013, 2017) discussed cases in which agents discover their own newfeasible actions. However, in their model, such actions are not exogenously discovered but
rather endgenously discovered. In other words, such actions are given to agents by modelers.
7 Unlike Schipper (2021) and Tada (2022)), we do not deal with discovery processes. As
indicated by one of the main results, only one imitation update is required in our model.

7

Proposition 2. Given any n-person coordination game with unawareness andany generalized strategy proﬁle, the imitative discovered game has a successfulcoordination equilibrium.
Proof. Suppose that s∗ is played by all agents. For any i ∈ I, ai = s∗i (t∗i ) is
observed and imitated by them. Hence, for some i ∈ I, a = (ai , . . . , ai ) is a
successful-coordination equilibrium.
The above proposition means that it leads to the 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 = (s′A (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

.

By 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 ;∗given tD
, 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:8′′TC′ = {tC
, 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 ;′′′′given tD , vD (tD ) = vD and b′D (t′D ) = t′′C ; and′′′′given tD
, vD
(t′′D ) = vC
and b′D (t′′D ) = 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 that there are three equilibria, it
seems odd that he includes all equilibria in his choices because δ is played byneither 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 the 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.8
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
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.
Thus, the following proposition holds.
8 Tada (2022) proposed a similar notion as a realizable CURB block game. In a realizable
CURB block game, the block is CURB.

9

Proposition 3. 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 isvT =Colin / David
α
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 ﬁrst
play, 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 ﬁrst
play, players can restrict their actions to some speciﬁc coordination blocks.

5

Discussion

5.1

Relationship with Generalized Nash Equilibrium

A self-conﬁrming equilibrium is related to a generalized Nash equilibrium. Thissection considers relationships among a self-conﬁrming equilibrium, generalized
Nash equilibrium, and cognitive stability. We ﬁrst deﬁne the generalized pure
Nash equilibrium proposed by Halpern and Rêgo (2014) as follows.
Deﬁnition 4. s∗ is a generalized (pure) Nash equilibrium if, for any i ∈ I andti ∈ T i ,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 assume 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 a10notion of cognitive stability, he showed that the two notions are equivalent. Let
us deﬁne cognitive stability as follows.
Deﬁnition 5. A generalized Nash equilibrium s∗ is cognitively 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. To facilitate comparison with a successful-coordination equilibrium, thedeﬁnition of a cognitively stable generalized Nash equilibrium is rewritten as
follows:
Remark 3. s∗ is a cognitively stable generalized Nash equilibrium if for any
i ∈ I and ti ∈ Ti ,1. s∗i (ti ) ∈ arg maxx∈Avi (ti ) ui (x, (s∗j (bi (ti )(j)))j∈I−i ); andi2.

s∗i (ti ) = s∗i (t∗i ).

It is evident that two conditions of the deﬁnition of a cognitively stable
generalized Nash equilibrium are similar in the ﬁrst and second condition ofDeﬁnition 1. Hence, the following remark is true.
Remark 4. Under unawareness, every successful-coordination equilibrium is acognitively stable generalized Nash equilibrium.
This is clear from Deﬁnition 1 and Remark 4. However, the opposite does
not hold true, that is, in some cognitively stable generalized Nash equilibrium,
coordination might be a failure. As seen in example 1, a generalized strategy
proﬁle s∗ = ([sA (t∗A ) = X, sA (tA ) = X], [sB (t∗B ) = Y, sB (tB ) = Y ]) satisﬁes a
deﬁnition of a cognitively stable generalized Nash equilibrium, but the strategy
proﬁle does not satisfy a deﬁnition of a 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.
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 needto revise their subjective view. In other words, cognitive stability seems to bea concept of the stability of subjective views.
However, in coordination games with unawareness, since coordination might
be a failure, players who play with a cognitively stable equilibrium might revise
their subjective views as shown in example 1. Therefore, we need to distinguishbetween stability of beliefs and stability of subjective views in a coordination
game.
11

Next, we examine the mathematical relationships between successful-coordination
equilibrium and cognitively stability. This relationship has the following properties.
Proposition 4. 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 and vice versa.
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 5 (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 (t∗j ))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 4. 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 5, 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.
The opposite clearly holds true. 4.

5.2

Unawareness versus Covery

Studies on unawareness distinguish between two approaches; one is lack of conception (e.g., Heifetz, Meier, and Schipper 2006); and the other is lack of knowledge (e.g., Geanakoplos 2021).
1. Lack of conception: The cholera bacterium was discovered by Koch in
1884. However, the cholera bacteria itself had existed before 1884, butpeople were unaware of its existence. Therefore, prior to 1884, people
infected with cholera bacteria battled the disease without realizing they
were infected.

122. Lack of knowledge: As was recently discovered, some COVID-19-infectedpatients do not show symptoms. Asymptomatic patients like them remain unaware that they are asymptomatically infected unless they undergo PCR testing, even when they learn that there are asymptomaticpatients with COVID-19.
Games with unawareness model lack of conception in terms of awareness ofthe games’ situation. When players are unaware that they might face a coordination game, they believe that they face a standard game with unawarenessexcluding coordination games. However, games with unawareness model lackof knowledge in terms of awareness of the action set. Previous studies haveinterpreted unawareness of action sets as a lack of knowledge, but not as a lackof conception. For example, if the meeting place is a well-known place like the
Big Ben, people can get there without needing directions. However, if it is nota widely known place, people may not know how to get there, even if they are
told how to. Many people may then ask the opponents to change the meeting
place to a more recognizable location. Knowing the choices inﬂuences decisionmaking, but whether the choices are understandable and feasible also inﬂuence
decision-making. From the above perspective, it is important how accurately
players can observe the opponents’ actions.
This study assumes that if the agents observe the opponent’s actions thatthey were unaware of, then they can understand and imitate those actions.
However, as shown above, it is not necessarily that players understand theopponent’s actions. When the opponents choose unnoticed actions that players
were unaware of, their play can be classiﬁed and discussed as follows.
Unawareness that the opponents have already made a decision
When the opponents implement actions that the agent is unaware of, they cannot recognize such actions. Then, there may be two cases, as enumerated below:• Games are not completed: Shiso Kanakuri, who was a marathon runnerin the 1912 Summer Olympics, fell sick with sunstroke during the competition. He did not wake up until the day after the competition, andso, he abstained. However, his decision to abstain was not communicatedto the Olympic Committee. Shiso’s competition record was not stopped
until March 21, 1967, when he oﬃcially reached the ﬁnish line. In otherwords, his decisions were not tied to the outcome of the game.
• Not realizing being in a game situation: Companies advertise and consumers decide whether to buy products based on the advertisements. However, companies may use subliminal eﬀects in their advertising. The consumers ignorant of advertising strategies aimed at subliminal eﬀects may
not realize that they are being put in a game situation with the company.

13

Awareness of the opponents’ decision making, but unawareness about
what the opponents have decided
Players are aware that their opponents are making a decision when the opponents take actions that they are unaware of. However, it is not always possible
to know exactly how the opponents have played.
• Misrecognition: Let us assume there are three entrances: East, West, andSouth. Out of the three, you know the East and West entrances, but do
not know the South entrance. If I go to the South entrance and you go
to the East entrance, we cannot meet. Then, you may be misled into
thinking that I went to the West entrance, because you do not know theSouth entrance.
• Not recognized as symbols: Famous sites such as the White House and Big
Ben sound familiar even to ﬁrst-time visitors to the area. However, if you
have never heard of a company or a niche restaurant, you may not know
where to ﬁnd it even by looking at a map. For example, Lake Kawaguchi,
located at the foot of Mt. Fuji, is one of the most famous lakes in Japan,
but you might not know of it, even if you knew where Mt. Fuji was. Then,
you may not understand if I were at Lake Kawaguchi.
Awareness about what the opponents have played, but unawarenessof the way of play
Even if we know exactly what the opponent’s choices are, we may not be able
to imitate them. We may never get there, even in case of famous places likethe White House, Big Ben, and Mt. Fuji. When watching a game of soccer orbaseball, only a limited number of people can imitate the players’ moves. For
us to imitate the opponents’ behavior, we also need to recognize how they didit.
However, it is not necessary to recognize it in an exact manner. If it is a tallbuilding like the Big Ben, it will stand out and we can go there. In other words,
the ability to imitate the behavior of others depends on the ease of imitation.

Of the three types described above, the ﬁrst and second are caused by lackof conception; the third is due to lack of knowledge. The study of discovery
processes is clearly the third9 . As you can see from the three types above, there
is a very strong assumption under unawareness that they know exactly what
the opponents are doing and imitate it.
As pointed out by Schipper (2014), unawareness means lack of conception
rather than lack of knowledge. By contrast, in almost all the games with unawareness, unawareness of actions means lack of knowledge rather than lack of
9 Moreover, most games with unawareness seem to assume the third type.

14

conception. Hence, most previous works seem to discuss games with “covery”
rather than games with unawareness.10

5.3

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 andSchipper (2014) assumed that player’s types are directly associated with beliefs
about the structure of a game, whereas Perea (2018) did not make such anassumption. In his model, player’s types are associated with beliefs about thestructure of a game, 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. Notethat our model refers to the growing awareness of the ways in which opponents
play their game rather than the growing awareness of opponents’ plays.

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10 The spirit of the lack of conception of actions can be seen in several prior studies, forinstance, Heifetz, et al. (2013) and Halpern and Rêgo (2014). They discuss awareness of
unawareness of actions. In their frameworks, players know that the opponents have someaction, but they do not know what the opponents’ action is.

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16

[MS14]

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2014. doi:10.1007/s00199-013-0788-5.

[MW15] Roger Myerson and Jörgen Weibull. Tenable strategy blocks and settled equilibrium. Econometrica, 83:943–976, 2015. doi:10.3982/
ECTA11048.
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Presses Universitaires de Louvain, pages 231–238, 2007.
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Leandro C. Rêgo and Joseph Y. Halpern. Generalized solution concepts in games with possibly unaware players. International Journal ofGame Theory, 41:131–155, 2012. doi:10.1007/s00182-011-0276-8.

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17

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