Discussion Paper No.370

Unawareness of Actions and Myopic Discovery
Process in Simultaneous-Move Games with

UnawarenessYoshihiko Tada
Chuo University

May 2022

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Unawareness of Actions and Myopic Discovery
Process in Simultaneous-Move Games with
Unawareness ∗Yoshihiko Tada†‡First version: February 18, 2022
This Version: May 12, 2022

Abstract
The present paper introduces a generalization of a closedness underrelation behavior (CURB) notion to simultaneous-move games with unawareness. The study, using a type-based approach, models a myopic
discovery process in which players take the best responses to opponents’preceding plays and revise their subjective games. Mainly, the study
shows that the plays of all players converge to some realizable CURB
set through any myopic discovery process. Given any simultaneous-movegame with unawareness, any myopic discovery process converges to somerevised game in which players do not need further revision of their subjective games. Moreover, we explore the relationships of the CURB notion
with other solution concepts in our model.
JEL classiﬁcation: C70; C72; D80; D83
Keywords: Game Theory; Unawareness; Discovery of Actions; Growing
Awareness; Closedness under Rational Behavior; Block Game

1Introduction

In the present paper, we mainly discuss the following model and result:• Generalizing a notion of closedness under rational behavior (CURB) tosimultaneous-move games with unawareness,∗ 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 the 2018
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, 2021, virtual. The author thanks Masakazu Fukuzumi, Toichiro Asada, Hirokazu Takizawa, Norimasa Kobayashi, and Yasuo Sasaki for their insightfulcomments on this paper.
† Graduate School of Economics, Chuo University, 742-1, Higashinakano, Hachioji-shi,
Tokyo 192-0393, Japan‡ yoshihiko.tada.4@gmail.com

1• Modeling the myopic discovery process, and• Proving the main theorem that plays of all agents converge to some CURB
set in a certain discovered game in which players do not need further
revision of their subjective games.
Although a lack or abundance of knowledge inﬂuences people’s decisionmaking, the human community is stable. Speciﬁcally, stability in people createsa speciﬁc solution (or equilibrium). However, people do not always choose aparticular solution from the beginning. People need to start by recognizing thedecision-making or interactive situation they are in and ﬁnd a speciﬁc solution
through trial and error. The main purpose of this study is to demonstrate
how agents discover and play a speciﬁc solution under a lack of knowledge orunderstanding by using discovery processes in simultaneous-move games withunawareness.
Studies of unawareness analyze decision-making and interactive situations
assuming a lack of knowledge or understanding. Models of unawareness assume
that agents are unaware of events, choices, opponents’ plays, and so on.1 In
games with unawareness, the outcomes of players’ implementations might bediﬀerent from their play expectations because the players “did” not know someof the opponents’ actions. Therefore, since the players observe the opponents’play, they are surprised by the play and may realize the error of their subjective views about the game’s situation. Thus, prior beliefs about the opponents’play may be diﬀerent from those reformulated after the opponents’ play. Consequently, the players’ next-stage game might change.
Schipper (2021) proposed models of discovery processes to analyze belief revision and replay through the revision process.2 In a discovery process, players
add opponents’ actions that they are unaware of in each stage game. Schipper(2013) focused on extensive-form games with unawareness and showed that arationalizable discovery process where every player implements rationalizable actions converges to some revised game possessing a rationalizable self-conﬁrming
equilibrium.3
This study adopts the idea of discovery processes and discusses the convergence of plays. To discuss such convergence, we introduce and generalizea CURB notion. A CURB notion is a set-wise solution concept, that is, re1 Schipper (2014) provided a historical survey. In this paper, we present related literature
in Section 5.
2 A discovery process is diﬀerent from a learning process. Learning is an update process ofprobability distributions. However, as shown by Schipper (2013), unawareness of an event is
diﬀerent from assigning probability 0 to the event. Any event that agents are unaware of is
not included in the subjective state space. Hence, the agent cannot assign any probability to
the event. If such an event occurs, the subjective state space must be expanded. In gameswith unawareness, the set of actions must also be increased when actions that players were
unaware of are played. Therefore, learning cannot be applied to games with unawareness.
Discovery processes are alternative models of learning processes that were proposed to avoid
this issue.
3 Note that Schipper (2021) did not show that a rationalizable discovery process converges
to some solution.

2

ﬁnements of the strict Nash equilibrium and rationalizability. It combines thecharacteristics of an equilibrium notion and rationalizability notion as follows.
• When players implement actions in some CURB set, they do not haveinventive deviation from the CURB set. Hence, we can apply stable convention to a CURB notion.
• If players’ belief formation consists of mixed actions, a CURB notion can
avoid disconﬁrmation diﬃculties for probability distributions.
A CURB notion possessing the above features might be a convergence of implementations by players who best respond to opponents’ immediately preceding
plays. In fact, Hurkens (1995) and Young (1998) showed that in a standard
game, Markov plays converge to a minimal CURB set, considering the possibility of error. Although we do not assume the possibility of error, we show myopic
play, namely, that all players best respond to opponents’ preceding plays andconverge to some CURB set in simultaneous-move games with unawareness.
However, in some simultaneous-move games with unawareness, some players’
subjective games might not have a CURB set that is on the objective gamebecause the player is unaware of a subset of the CURB set. Hence, we try to
generalize a CURB notion to simultaneous-move games with unawareness. The
key here is the realizable action set, that is, the Cartesian product of the set of aplayer’s realizable actions. We focus only on CURB sets on the realizable action
set, namely, realizable CURB sets. Our main theorem is that players’ myopic
play converges to some realizable CURB set.4 When myopic players implement
actions on some realizable CURB set and do not deviate from the CURB set,
then all players’ actions in the CURB set are stable.5
We ﬁrst formulate simultaneous-move games with unawareness in Section
2 based on a type-based approach. Additionally, in this section, we generalizea CURB notion. Section 3 models the myopic discovery processes and convergence of those processes and shows our main result. Certainly, we need to
discuss the relationship of a CURB notion and other solution concepts and features of discovery processes. This discussion is provided in Section 4. Other
discussions are presented in Section 5. In this section, we consider the block
game notion in (simultaneous-move) games with unawareness, adaptive plays,
and the limitations of this work, and review related literature. Our framework
is based on Perea (2018), albeit with some issues. We point out the issues andoﬀer a solution in the Appendix.

4 Note that some realizable CURB set is not CURB in some player’s subjective game.
Hence, we have to distinguish whether the set is CURB in all subjective games or not.
5 However, we do not show that plays converge to speciﬁc actions in the CURB set. Players
might choose to alternate actions over the CURB set.

3

2Preliminaries

2.1

Simultaneous-Move Games with UnawarenessThis section provides a deﬁnition of simultaneous-move games with unawareness, which are type-based models, and generalizes a CURB notion to simultaneousmove games with unawareness. Let G = (I, A, u) be a standard ﬁnite simultaneousmove 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.6 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 work, 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 block,7and Avi be the set of i’s actions in v = ×j∈I Avj . Let Av−i = ×j∈I−i Avj . Here,when 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 S⊆ Avi S. 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

6 Our deﬁnitions are similar to those of Perea (2018). Note that there are two majordiﬀerences. First, Perea’s (2018) model did not ﬁx belief hierarchies on views. We assume
that the “actual type” of players is given. Second, Perea (2018) considered probabilistic beliefs
on awareness, whereas our players always have point beliefs on their opponents’ awareness,
as is often assumed in studies of games with unawareness. Furthermore, in the Appendix, we
model a probabilistic belief diﬀerent from Perea’s (2018) framework. He assumed that eachagent assigns probabilities to all opponent types, whereas we assume that opponents’ typespaces that the agent believes might be diﬀerent when the agent is given diﬀerent types.
7 A block is a Cartesian product of non-empty subsets of actions.

4

as the “true game” in Γ.8 i’s type ti describes their view about the game andbelief 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 ifthere exists
ti1 , . . . , tih such that ti1 = ti and tih = tj . Here, we
S a subsequenceSsuppose 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

Closedness under Rational Behavior (CURB)

We generalize a CURB notion, which is one of the set-wise notions, to simultaneousmove games with unawareness. A CURB notion is a reﬁnement of a strict Nashequilibrium and a rationalizable set proposed by Basu and Weibull (1991). It
has the features of both an equilibrium and a rationalizability notion. A rationalizable action is a support of all mixed equilibria, whereas a CURB set is asubset of the supports of such equilibria. In a mixed equilibrium, players might
not be able to disconﬁrm a distribution of mixed actions, whereas a CURB
notion avoids the impossibility of disconﬁrmation. Moreover, since games withunawareness assume asymmetric subjective views, players attempt to disconﬁrm
their beliefs on the opponents’ subjective views. A CURB notion can be used
for such tests.
First, we deﬁne the generalized strategies. For any i ∈ I, let si : Ti →Svi (ti )v (t )) with si (ti ) ∈ M (Ai i i ) for all ti ∈ Ti . Then, given ti , si (ti ) ∈ti ∈Ti M (Aiv (t )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 from s.
Second, we propose a generalization of a CURB notion to simultaneousmove games with unawareness. Although Basu and Weibull (1991) ﬁrst deﬁned
a CURB notion on a standard game G, this paper deﬁnes it on each view. Given
any standard simultaneous-move game 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)

8 The term “objective game” was used by Halpern and Rêgo (2014). Feinberg (2021)
referred to such a game as the “modeler’s normal-form game” and Perea (2018) called it the“base game.” In this context, we follow the study by Halpern and Rêgo (2014).

5

be the set of i’s pure-action best responses to their beliefs on 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. Given 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.
Basu and Weibull (1991) showed that every standard game has a minimal
CURB set.
Remark 1. Given any standard game, every possible view has a minimal CURB
set.
In standard games, we only need to consider a CURB set on 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, realizable CURB sets might
be diﬀerent for standard games and games with unawareness. Hence, we must
distinguish CURB notions between the two models. In the CURB notion under unawareness, we deﬁne a CURB set on the realizable action set, called a
realizable CURB set, as follows:
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.
6

Given Γ, some realizable CURB set C ∈ V may be C ⊆ vi (ti ) for any(i, ti ) ∈ I × Ti . However, the set might not be CURB in vi (ti ) at some ti . Givena realizable CURB set, we distinguish between a case in which the realizable
CURB set is CURB in every vi (ti ) for any (i, ti ) ∈ I × Ti , and one in which it
is not as follows:
Deﬁnition 3. In a simultaneous-move game with unawareness Γ, C ∈ V is
a common (minimal) 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 for any (i, ti ) ∈ I × Ti \ {t∗i }, C is CURB in vi (ti ), and for any i,v (t∗ )C 9
βi i i (AC−i ) ⊆ Ai .
A common realizable CURB set is a subset in each subjective view, but theset might not be CURB from some view. By contrast, a common CURB set is
CURB in each subjective view other than the player’s actual view,10 , and eachplayer’s actions in the CURB set best respond to the opponents’ actions in the
CURB set in the player’s actual view.
Example 1. Let us consider that two players, Alice (A) and Bob (B), face thefollowing objective game:11
A/B
a1
vO =
a2
a3

b1
3, 3
5, 0
0, 0

b2
0, 5
1, 1
0, 0

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
a3

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:
A/B
a1
vB =
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 .
9 Tada (2020), noted that a common CURB set means only a CURB set on the full action

set. By contrast, this paper generalizes this notion by focusing on a realizable action set.
10 Here, an actual view means the view when the agent’s actual type is given.
11 This example is similar to Example 3 in Schipper (2018). An idea of our similar example
is borrowed from his example.

7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 };∗given tA
, vA (t∗A ) = vA , and bA (t∗A ) = tB ;given 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 , while 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

.

Then, there exists 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 . □3

Myopic Discovery ProcessStandard game models study the convergence to a minimal CURB set by using alearning model or an adaptation model, for example, Hurkens (1995) and Young(1998). Previous literature on standard models shows that when all the players
best respond to the opponents’ preceding previous play, their plays converge tosome minimal CURB set. We can also predict that in dynamics of simultaneousmove games with unawareness, all the agents’ implementations converge to some
generalized CURB set when they best respond to the opponents’ preceding play.
To prove this prediction, we model a myopic discovery process in this section.
A discovery process represents an update process by which each player revises
their own belief about the game’s structure and the opponents’ play. This model
was ﬁrst introduced by Schipper (2021) to games with unawareness. He analyzed a rationalizable discovery process in extensive-form models, which is basedon Heifetz et al. (2013). This study models a discovery process in simultaneousmove games with unawareness based on Perea (2018). Although our deﬁnition,
at ﬁrst glance, may seem diﬀerent from that of Schipper (2021), both are essentially the same.
8

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 , and any sequence of players i1 , i2 , . . . , ih , . . . , with a sequence of types ti1 , ti2 , . . . , tih , . . . ,where ti1 = ti , there exists t′i ∈ Ti′ and a sequence of types in Γ′ , t′i1 , t′i2 , . . . , t′ih , . . . ,where t′i1 = ti , such that for any h ≥ 1,vivi′h (t′ih ) = ×j∈I [Aj h (tih ) ∪ supp(sj (t∗j ))],
where t∗j is j’s actual type in Γ. Note that some Γ, Γ′ may be T ̸⊆ T ′ andT ′ ̸⊆ T , or T ∩ T ′ = ∅.
[]Example 2. Consider the following objective game played by Colin (C) andDavid (D):
C/D
c1
v0 =
c2
c3

d1
3, 3
5, 0
-1, 0

d2
0, 5
1, 1
0, 1

d3
0, -1
1, 0
2, 2

and v 2 =

C/D
c1
c2

;

and two possible views as follows:

v1 =

C/D
c1
c3

d2
0, 5
0, 1

d3
0, -1
2, 2

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 ;9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

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 ).
From 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. By contrast, in games with unawareness, it does not seemappropriate that every player implements and observes a mixed action because
under unawareness, the players cannot observe the frequency of their opponents’actions at each stage of the game during any discovery process.12
In a discovery process, cautious players might carefully revise their beliefs
about the game, opponents’ plays and rationalities, and pay-oﬀ uncertainty(e.g., they might play rationalizable strategies). However, in the real world,
the agents must pay a higher cost for revising such beliefs and implementing
rationalizable strategies. If the players are myopic, they do not pay a high costfor revising their beliefs. This section explains the myopic discovery process inwhich every player best responds to the opponents’ previous plays.
First, we deﬁne a strategy of myopic play in a discovered game as follows:
12 I thank an anonymous referee for pointing this out.

10Deﬁnition 6. 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.
Second, we provide a myopic discovery process.
Deﬁnition 7. 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. A discovery process P is converging to Γ if there exists h such that for any
λ ≥ h, Γλ = Γ.
From the above formulations, we can show the convergence of a CURB set
as follows:
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 that 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. Simply, 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.
Therefore, β λ (m) ⊆ C for any λ ≥ n. Since, β λ (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.
From 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 previous literature, highlighting the importance of the CURB
13 Speciﬁcally, R is a maximum ﬁxed under rational behavior (FURB) set. An action proﬁleset, C ∈ V , is a FURB set if β(C) = C.

11

set. However, the process therein converges to a general CURB set and, notnecessarily, a “minimal” one, such as in Hurkens (1995) and Young (1998).14
Example 2 (Continued). Let Γ be an initial game, that is, a game at λ = 1.
Then, the realizable action set is as follows:
C/D
c1
v =
c3

d1
3, 3
-1, 0

R

d3
0, -1
2, 2

.

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.
Let us focus on two generalized strategy proﬁles:
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 ]).
First, we focus on the former strategy proﬁle, s1 . The objective outcome is(c1 , d1 ). Since Colin is unaware of d1 , he is surprised and revises his view asfollows:
′

v1 =

C/D
c1
c3

d1
3, 3
-1, 0

d2
0, 5
0, 1

d3
0, -1
2, 2

.

′
′
′
), (b′C , b′D )),, vD), (vC
Then, at λ = 2, the discovered game Γ′ = (G, (TC′ , TD
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
′
vD
(t2∗
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 responses, 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 ]).

Neither of the players 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
14 Hurkens (1995) and Young (1998) showed the convergence of a minimal CURB set byusing the adaptive plays proposed by Young (1993). This paper discusses a discovery processwith adaptive plays in Section 5.

12

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 .
Second, 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:

v

2′

C/D
c1
=
c2
c3

d1
3, 3
5, 0
-1, 0

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
(tD ) = v 1 , and b′′D (t2D ) = t2C ∗ .
vDAt λ = 2′ , when they play myopic best responses, the generalized strategy
proﬁle is
′

′

′

′

′

′

′

′

′

s22 = ([s2C (t2C ∗ ) = c3 , s2C (t2C ) = c3 ], [s2D (t2D∗ ) = d3 , s2D (t2D ) = d3 ]).
Neither of the players 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′′ .
□4

Discovery Process and Other Solution Concepts

4.1

Relationships among Other Solution Concepts

This section discusses the relationship between a CURB notion and other solution concepts, for example, the generalized Nash equilibrium, the self-conﬁrming
equilibrium, and rationalizability.

13

4.1.1Equilibrium Notions

Previous literature discusses and proposes equilibrium notions, whereas in thisstudy we focus on generalized Nash (Halpern and Rêgo 2021 and Sasaki 2017)
and self-conﬁrming equilibria (Schipper2021 and Kobayashi and Sasaki 2021).
Given any generalized strategy proﬁle s, let µ = (µi )i∈I be the belief systemvi (ti )in s, where for any (i, ti ) ∈ I × Ti , µi (ti ) ∈ M (A−i) and µi = (µi (ti ))ti ∈Ti .
This study allows correlated beliefs. Therefore, the generalized Nash equilibriumis deﬁned as follows: in a simultaneous-move game with unawareness Γ, s∗ isa generalized Nash equilibrium 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 pointed out, a generalized Nash equilibrium is best interpreted as“an equilibrium in beliefs” (Halpern and Rêgo, 2014: 50). However, as pointed
out by Schipper (2014), a generalized Nash equilibrium can consist of wrong
beliefs. In that case, each player would revise their own beliefs about a game’s
structure and the opponents’ play, and may not play the same generalized Nashequilibrium.
To avoid this scenario, previous literature has discussed reﬁnements of general equilibria as steady-state notions, namely, the stable-belief-hierarchies notion, cognitive stability notion, and self-conﬁrming notion.
First, we consider the stable-belief-hierarchies notion and cognitive stability
notion, as proposed by 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 (t∗i ).
The former notion means that every player’s belief about the opponents’ plays
is correct, while the latter means that each player’s decision-making on theirarbitrary type is equivalent to their actual play.
Sasaki (2017) expressed the following:
Remark 2. 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 3. 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.
14

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.
The relationship between a CURB notion and a cognitive stability notion
has 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 onvi (ti ). Then, following Basu and Weibull (1991), a Nash equilibrium on vi (ti ),m∗ ∈ M (vi (ti )) exists, satisfying m∗ ≡ m′ ∈ M (C). Suppose that m′ isv (t )
not a Nash equilibrium on v. Simply, (i, ai ) ∈ I × Ai i i exists, such that
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 cognitively
stable generalized Nash equilibrium in any game with unawareness. The contraposition is that if any cognitively stable generalized Nash equilibrium does notexist, then any common CURB set does not exist. This means that if some ofthe players cannot perceive any CURB set in the realizable action set, then theplayers 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 speciﬁc play. 15
The 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.
15 I thank Masakazu Fukuzumi for this suggestion.

15

Second, let us consider the self-conﬁrming equilibrium proposed by Fudenberg and Levine (1993). Schipper (2021) generalized a rationalizable selfconﬁrming equilibrium to include extensive-form games with unawareness. Kobayashi
and Sasaki (2021) focused on only simultaneous-move games with unawarenessand discussed a rationalizable self-conﬁrming equilibrium by using epistemic
models. We discuss the k-self-conﬁrming equilibrium, which means that all theplayers in the k-th order mutually believe that all the players’ beliefs are correct.
The following deﬁnition of self-conﬁrming equilibria is based on Kobayashi and
Sasaki (2021). s∗ is a k-self-conﬁrming equilibrium if there exists belief systemµ 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 .
4.1.2

Rationalizable Notions

Rationalizability was proposed by Bernheim (1984) and Pearce (1984). Heifetz
et al. (2013b) generalized Pearce’s extensive-form rationalizability to gameswith unawareness. This notion is one of the approaches that avoids the issue ofequilibrium notions in general.
Let us give a type proﬁle (ti )i∈I , let A(h) = ×ih ∈I Aih (ih ), and let A−ih (ih ) =vj (bih (tih )(j))×j∈I−ih Aj. Here, given A(h), β k (A(h)) = β ◦ · · · ◦ β (A(h)). Then,| {z }kthis study deﬁnes rationalizability in simultaneous-move games with unawareness as follows: T
In a simultaneous-move game with unawareness Γ, for any i ∈ I,
R((tih )ih ∈I ) = k=1 β k (A(h)) is a rationalizable action set at tih , where ti1 = t∗i ,and the rationalizable strategy R = (R((tih )ih ∈I ))h∈N :N is the set of natural numbers. .16
As pointed out by Basu and Weibull (1991), the rationalizable action set is the
maximum FURB set.17
Kobayashi and Sasaki (2021) proposed a k-rationalizable self-conﬁrming
equilibrium as follows: In a simultaneous-move game with unawareness Γ, s∗ isa k-rationalizable self-conﬁrming equilibrium if it is a k-self-conﬁrming equilibrium and supp[(s∗i (t∗i ))i∈I ] ⊆ R((tih )ih ∈I ), where for any i ∈ I, h = 1, . . . , k + 1
such that ti1 = t∗i . Kobayashi and Sasaki (2021) remarked as follows:
Remark 4. s∗ is a ∞-rationalizable self-conﬁrming equilibrium if and only ifs∗ is a cognitively stable generalized Nash equilibrium.
4.1.3

ExampleThis section compares a CURB notion with other solution concepts.
16 As pointed out by Basu and Weibull (1991), the rationalizable action set is a maximum
FURB set, we can use β to deﬁne rationalizability.
17 Some rationalizable action set might not be the maximum CURB set. Given the rationalizable action set X ⊆ A in standard game, Y ⊆ A might exist such that X ⫋ Y andβ(Y ) ⫋ Y .

16

Example 1 (Continued). Let us reconsider Example 1. First, we consider
generalized (pure) Nash equilibria in the game. Four generalized pure Nashequilibria exist:
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, we consider self-conﬁrming (pure) equilibria in the game. A 0self-conﬁrming equilibrium and ∞-self-conﬁrming equilibrium exist. The 0-selfconﬁrming equilibrium is onlys5 = ([sA (t∗A ) = a1 , sA (tA ) = a1 ], [sB (t∗B ) = b1 , sB (tB ) = b1 ]).
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 .
By contrast, the ∞-self-conﬁrming equilibrium is only s4 . When comparing a cognitively stable generalized Nash equilibrium with an ∞-self-conﬁrming
equilibrium, both the equilibria are the same.
Third, we consider rationalizability. Given three tuples t1 = (t∗A , t∗B ), t2 =∗(tA , tB ), and t3 = (tA , t∗B ), the rationalizable strategy is R = (R(t1 ), R(t2 ), R(t3 )),and the pure rationalizable actions at t2 and t3 are as follows:
R(t2 ) = {a1 , a3 } × {b2 , b3 }; andR(t3 ) = {a2 , a3 } × {b1 , b3 }.
Then, at t1 ,v (t∗ )v (t∗ )

R(t1 ) = βAi A ({b2 , b3 }) × βBi B ({a2 , a3 }) = {a1 , a3 } × {b1 , b3 }.
Here, it is obvious that 0-self-conﬁrming equilibrium s5 is a 0-rationalizableself-conﬁrming equilibrium and that ∞-self-conﬁrming equilibrium s4 is an ∞rationalizable self-conﬁrming equilibrium.
Let us compare a CURB notion with the other notions. First, we comparea CURB notion with equilibrium notions. From Proposition 1, any common
CURB set includes a support for the objective outcome, induced from the cognitively stable generalized Nash equilibrium. Since the ∞-rationalizable selfconﬁrming equilibrium and cognitively stable generalized Nash equilibrium are
the same, C 2 includes a support for the objective outcome (a3 , b3 ), induced
from a cognitively stable generalized Nash equilibrium and ∞-rationalizableself-conﬁrming equilibrium, s4 . By contrast, a common realizable CURB set
C 1 , which is not a common CURB set, is a support for the objective outcome(a1 , b1 ), induced from 0-rationalizable self-conﬁrming equilibrium s5 . In C 1 or17

s5 , each player is rational but their ﬁrst-order belief is irrational play. 18 Moreover, they might be certain about the opponent’s irrationality.
Next, we 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. By 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, a CURB notion has similar characterizations to theother notions.

4.2

Discovery and Equilibrium Notions

This section discusses the relationships between discovery processes and equilibrium notions.
4.2.1

Rationalizable Discovery Process and Self-Conﬁrming Equilibrium

Schipper (2021) modeled rationalizable discovery processes in which all players
implement rationalizable actions in each stage game. We formulate rationalizable discovery processes based on Perea (2018) as follows:
Deﬁnition 8. A discovery process P = (⟨Γ1 , s0 ⟩, ⟨Γ2 , s1 ⟩, . . . , ⟨Γλ , sλ−1 ⟩, . . . )is a rationalizable discovery process if for any λ, Rλ is the set of rationalizablestrategy.
Schipper (2021) modeled a rationalizable discovery process based on Heifetz
et al. (2013b) and showed that every rationalizable discovery process (in any
extensive-form game with unawareness) converges to a self-conﬁrming game, inwhich no rational player needs further revision and possesses some (0-)rationalizableself-conﬁrming equilibrium.
However, in our framework, we do not show Schipper’s (2021) result.
Example 3. Let us consider two agents Elena and Filip and assume they facea zero-sum game:
E/F
f1
f2
e1
1, -1 -1, 1 .
v=
e2-1, 1 1, -1Here, suppose that in a zero-sum game with unawareness, there exists a view as
v′ =

E/F
e1

f1
1, -1

f2
-1, 1

,

TE = {t∗e , te , t′e } and Tf = {t∗f , tf },18 Rational play means that the players maximize their utilities.

18

given t∗e , ve (t∗e ) = v and be (t∗e ) = tf ,given te , ve (te ) = v ′ and be (te ) = t∗f ,given t′e , ve (t′e ) = v ′ and be (t′e ) = t∗f ,given t∗f , vf (t∗f ) = v ′ and bf (t∗f ) = t′e , andgiven tf , vf (tf ) = v and bf (tf ) = te .
Then, a strategy proﬁle that is uniquely rationalizable iss = ([se (t∗e ) = e1 , se (te ) = e2 , se (t′e ) = e1 ], [sf (t∗f ) = f2 , sf (tf ) = f1 ]),that is, the actual play is (e1 , f2 ). In the play, Filip’s belief is correct, whereas
Elena’s belief is wrong because she predicts that Filip plays f1 . However,both players are aware of the actual play, that is, discoveries do not occur.
Hence, in the next-stage game, they play the same s. Then, there is no nrationalizable self-conﬁrming equilibrium for any natural number n, that is, the0-rationalizable self-conﬁrming equilibrium does not exist because in Elena’s actual subjective view v, a unique self-conﬁrming equilibrium is that both players
assign probability 21 to each of their actions, while in Filip’s actual subjective
view v ′ , the unique self-conﬁrming equilibrium is (e1 , f2 ). □4.2.2

Myopic Discovery Process and Cognitive Stability

In this section, we consider cognitively stable generalized Nash equilibria in
myopic discovery processes. First, we provide a mutual CURB notion that eachplayer’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 thatCβi∗ (AC−i ) ̸⊆ Ai in the realizable action set. Since the realizable action set isv (t∗ )v (t∗ )

Cdeﬁned 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 a common CURB set exists.
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 ), wheret∗j is j’s actual type and t∗j leads to ti , C is not CURB in vi (t∗i ). This is acontradiction. Therefore, the mutual CURB set is CURB in every view in thegame. Then, from Lemma 2, since the mutual CURB set is a realizable CURB
set, the set is a common CURB set.
19

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. From Lemma 3, the mutual CURB
set is a common CURB set. Then, from Proposition 1, a cognitively stable
generalized Nash equilibrium exists.
Theorem 2. Suppose a simultaneous-move game with unawareness, Γ, has amutual CURB set. Then, a myopic discovery process converging to a discovered
game possessing a cognitively stable generalized Nash equilibrium exists.
Proof. Suppose that a mutual CURB set, C, in Γ exists. From Lemma 2,
C is a realizable CURB set. From Theorem 1, a myopic discovery process,P , converging to C exists. Since C is a common realizable CURB set from
Lemma 3, from Proposition 2, a cognitively stable generalized Nash equilibrium
exists.
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.
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.19
Next, let us consider a relationship with a Nash equilibrium in an objective
game. Sasaki (2017) discussed the relationships between a cognitively stable
generalized Nash equilibrium and a Nash equilibrium in an objective game inany simultaneous-move game with unawareness. The researcher showed thefollowing 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 cognitively
stable generalized Nash equilibrium induces an objective outcome to be a Nashequilibrium in objective game G.
19 Tada (2018) discussed a revision process in which players play a generalized Nash equilibrium in each round and conjectured 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.

20

Proof. Given any simultaneous-move game with unawareness Γ, suppose that forv (t∗ )
any i ∈ I, Ai i i = Ai , where t∗i is i’s actual type. Suppose that the generalized
strategy proﬁle, s∗ , is a cognitively stable generalized Nash equilibrium. From
Remark 3, 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.
From 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 Nashequilibrium induces an objective outcome to be a Nash equilibrium in objective
game G.

5

Discussion

5.1

A CURB Block Game and Economy of Cognitive Costs

In our model, some myopic discovery processes do not converge to a discovered
game possessing a common CURB set. Some of the players may be certain of
the opponents irrationality. However, by using the block game notion (e.g.,
Myerson and Weibull, 2015) of a smaller game than each player s subjective
game, the players can reconstruct a block game possessing a common CURB
set from a discovered game that a myopic discovery process converges to andthey can then ascertain each other’s rationality.
Let us consider a case in which a discovered game possesses a realizable
CURB set. When all the players implement a generalized strategy proﬁle sothat the objective outcome is in the realizable CURB set, if they are rational,
they do not perform actions outside the realizable CURB set. Thus, all theactions in the complementary set of the realizable CURB set are redundant
for them. Therefore, each player excludes the actions in the complementary
set to economize the cognitive costs of the true structure of the game. If theyeconomize these cognitive costs, their subjective games are the smallest games inwhich the action set is a common realizable CURB set. The following deﬁnition
represents the “economy of 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; and21• 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 Example 2, when Colin and David play s1 in the initial game and s21 in
the next stage of the game, since the objective outcome induced by s21 is (c1 , d1 ),1
1
C1
the realizable CURB block game with C 1 is GC = (I, C 1 , (uCC , uD )). Thus,
1
1in the economized game, ΓC , all the 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 Example 2, there exists a unique generalized Nash equilibriumsuch that Colin and David play c1 and d1 in each subjective game, respectively.
Thus, from 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 for 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.

5.2

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) provided adaptive play models that do
not allow participants to provide their best responses to previous plays. This
subsection discusses a generalization of the adaptive plays to simultaneous-move
games with 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 an error
rate such that ε is suﬃciently small. The generalized strategy proﬁle, s′ , is anadaptive 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
Deﬁnition 11.

22

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) used anadaptive play notion and showed the convergence to a minimal CURB set. Our
proof of 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, The supports of the objective outcome induced by adaptive plays
converge to a common minimal realizable CURB set.
Informal proof. Given any simultaneous-move game with unawareness and any
adaptive discovery process, it is necessary to focus on only the realizable action
set, as per Theorem 1. Based on Hurkens (1995) and Young (1998), adaptive
plays converge to a minimal CURB set of the realizable action set. Then, theset is a common minimal realizable CURB set.20

5.3

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 Example 2, two cognitively unstable generalized
Nash equilibria, s1 and s2 , exist in the initial game. This study does notyield any appropriate answer to the question as to which equilibrium Colinand David play when they both implement a generalized Nash equilibrium
play.
Our discovery process, and that of previous works, explains how to build
each player’s subjective game under unawareness; however, it does not
explain how to do so in an initial game with unawareness. This issue is asubject 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.

20 The exact proofs are beyond the capabilities of the author and are omitted.

23

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.
4. In Section 4, we showed that Schipper’s (2021) result might not hold in
our framework. Speciﬁcally, in our model, some rationalizable discovery
process might not converge to any (simultaneous-move) game with unawareness possessing a self-conﬁrming equilibrium. This paper proposes
an open question as follows: what are the conditions for satisfying theresult of Schipper (2021) in our framework?

5.4

Related Literature

Games with UnawarenessPioneering 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). The majority of
the literature has modeled extensive-form games with unawareness. Studies ofnormal-form models (or, more strictly, simultaneous-move models) are C̆opic̆
Galeotti (2006), Sasaki (2017), Perea (2018), and Kobayashi and Sasaki (2021).
Meier and Schipper (2014) modeled Bayesian games with unawareness and discussed relationships with standard Bayesian games.
As mentioned in the text, there are two main solution concepts; one is equilibrium notions (Feinberg 2021, C̆opic̆ Galeotti 2006, Ozbay 2007, Halpern andRêgo 2014, Rêgo and Halpern 2012, Grant and Quiggin 2013, and Meier andSchipper 2014). Halpern and Rêgo (2014) generalized a Nash equilibrium. Rêgoand Halpern (2012) and Grant and Quiggin (2013) discussed a sequential equilibrium in extensive-form games. Meier and Schipper (2014) proposed an unawareness perfect equilibrium. Sasaki 2017, Schipper 2021, and Kobayashi and
Sasaki (2021) discussed reﬁnements of equilibria notions and corrected beliefs
on an equilibrium.
The other 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 not yet been
formed. In addition, since the rationalizable action set is supported by all mixedstrategy equilibria, the diﬃculty of disproving the probability distribution in
mixed action equilibria can be eliminated.
Growing AwarenessA study of discovery processes entails an analysis of growing awareness or updating awareness. Karni and Vierø(2013, 2017) discussed decision-making under24

unawareness and proposed 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 0 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) modeled that all agents announce prices toeach other and provided generalizations of the results of Genakoplos and Polemarchakis (1982) and Ostrovsky (2012). They supposed that updating awareness is minimal and a true state is never excluded. Traders eventually agree on
the price of the security. Moreover, if the security is separable, traders agree on
the correct price and aggregate information.
Unawareness in GeneralThe ﬁrst motivation of studies of unawareness is overcoming the No-Trade Theorem presented by Milgrom and Stokey (1982). Previous works of unawarenessthat address this issue have adopted 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 of 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 they do not have that knowledge. The otherstrand of the literature on this approach includes Samet (1990) and Shin (1993).
However, in (non-)partitional models, several assumptions lead to trivial unawareness, that is, an agent cannot be unaware of any event; see Modica andRustichini (1994, 1999), Dekel et al. (1998), and Chen et al. (2012). The lattermodel is proposed to avoid this issue. Unawareness structures ﬁrst formulate thefamily 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) and Heinsalu(2012). In a recent study, Fukuda (2021) compared the two approaches.
CURB Notions
Basu and Weibull (1991) ﬁrst introduced 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) discussed theaxiom and properties of minimal CURB sets. Pruzhansky (2003) showed thatin extensive games with perfect information and ﬁnite horizons, there exists
only one minimal CURB set. Benisch et al. (2010) provided algorithms forcomputing CURB sets. Asheim et al. (2016) discussed the epistemic robustnessof CURB in epistemic models.

25

Appendix A
To discuss the ignorance, as a lack of conception, of a structure of a game,
this paper uses a model of games with unawareness, but not Bayesian games.
Bayesian games represent ignorance of actions as extremely low payoﬀs assigned
to the actions that a player does not know (Harsanyi). In this formulation,
rational players do not choose such actions. However, irrational players mightchoose those actions. Moreover, since each player’s belief about the opponents’actions is a probability distribution on all the actions in the game, each player
assigns some probability to all the actions even if modelers suppose that theplayer does not know some of the actions. As pointed out by Schipper (2013),
since the unawareness of an event is diﬀerent from a probability of 0 being
assigned to the event, we cannot consider ignorance as a lack of conception.
Hence, games with unawareness are appropriate for discussing the ignorance ofactions as a lack of conception.
This paper uses a non-probabilistic type-based approach and models simultaneousmove games with unawareness. In simultaneous-move games with unawareness,
type-based approaches are adapted by Meier and Schipper (2014) and Perea(2018). Meier and Schipper (2014) modeled Bayesian games with unawarenessbased on Heifetz et al.’s (2013a) probabilistic unawareness structures. By contrast, Perea’s (2018) type spaces are similar to Harsanyi’s (1967) type spaces.
Note that crucial diﬀerences between Meier and Schipper (2014) and Perea(2018) exist. The former assumed that players’ types are directly associatedwith views of the games, whereas the latter believed that although players’
types associated with beliefs about views of the games, the types cannot be
associated with the views themselves.
However, Perea (2018) had a problem similar to that encountered by Harsanyi(1967). He assumed probabilistic beliefs in his type-based approach. Thus, his
probability distribution about the opponents’ types, that is, the opponents’
type spaces, is mutually known. Speciﬁcally, his framework cannot distinguishbetween unawareness and assigning some probability. This study uses Perea’s(2018) style. Although we ignore this issue, we do not solve it. Can we modelgames with unawareness by using similar models to the Harsanyi–Perea style?
This study’s answer is that we can represent Harsanyi–Perea-style’s games withunawareness by making an additional assumption.
Given standard game G, let V be the set of possible views. Then, a probabilistic version of simultaneous-move games with unawareness is deﬁned asfollows:
Deﬁnition 13. Given any standard game G, let Γ = (G, (Ti )i∈I , (vi )i∈I , (bi )i∈I )be a probabilistic version of a simultaneous-move game with unawareness asfollows: for each i ∈ I,
• Ti is a ﬁnite and non-empty set of i’s type. This study assumes that oneof the types is i’s actual type, denoted by t∗i .
• vi : Ti → V is i’s view function.
26

S

′
Tj
′ ∈×
T−i
\{∅}) ∆(T−i ) is i’s belief function, where for allj∈I−i (2′′ti ∈ Ti , T−i∈ ×j∈I−i (2Tj \ {∅}) exists such that bi (ti ) ∈ ∆T−i
. Here,′′∆(T−i ) is the set of probability measures over T−i , that is, bi (ti ) is a

• bi : T i →probability measure over some subset of T−i . For each ti ∈ Ti and t−i =(tj )j∈I−i ∈ T−i , bi (t−i |ti ) ≥ 0 implies that vj (tj ) ⊆ vi (ti ) for all j ∈ I−i ,where bi (t−i |ti ) is the probability that bi (ti ) assigns to t−i . Given any
ti ∈ Ti and tj ∈ Tj , denote by bi (tj |ti ) to be the probability that bi (ti )assigns to tj .
A player is aware of a subset of the opponents’ types if and only if for any
opponents’ type in the subset, the player assigns some probability to the type.
This deﬁnition is diﬀerent from Perea’s (2018). His model assigned someprobability to every type on the type set. By contrast, any probability might not
be assigned to some type. Simply, in our model, we can represent the unawareness of types. However, under this deﬁnition, we cannot distinguish between
the unawareness of the opponents’ types and probability 0 assigned to the types
because we do not deﬁne relationships among types. To distinguish unawarenessfrom probability 0, we need an additional assumption. Given (i, ti ) ∈ I × Ti ,let ti1 , ti2 , . . . , tih , . . . be a sequence of types, where ti1 = ti and for any h ≥ 2,
tih = tj , where (j, tj ) ∈ I × Tj and bi (tj |tih−1 ) ≥ 0.
Assumption 1. Given Γ, for any (i, ti ) ∈ I × Ti , any tih and tik (h ≤ k),where i1 = ih = ik = i, and the two opponents’ type subsets T−ih , T−ik ⊆ T−i ,if bih (tih ) ∈ ∆(T−ih ) and bik (tik ) ∈ ∆(T−ik ), then T−ik ⊆ T−ih and for any
t−i ∈ T−ih \ T−ik , any probability must not be assigned to t−i .
This assumption speciﬁes the relationships among the types. When player i
is given two types, if one ti leads to the other t′i , then the opponents’ type set′′
T−i at ti is a superset of the set T−i
at t′i . Then, when T−i
is a proper subset of′
T−i , any probability is assigned to every type t′′i ∈ T−i \ T−i
at t′i . Hence, under
this assumption, we can distinguish between unawareness and probability 0.

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