The robust selection of rationalizability

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Journal of Economic Theory 151 (2014) 448–475

www.elsevier.com/locate/jet

The robust selection of rationalizability ✩

Yi-Chun Chen a, Satoru Takahashi a, Siyang Xiong b,∗

a Department of Economics, National University of Singapore, 117570, Singaporeb Department of Economics, University of Bristol, BS8 1TN, United Kingdom

Received 9 May 2013; final version received 7 November 2013; accepted 2 February 2014

Available online 11 February 2014

Abstract

We propose a notion of selecting rationalizable actions by perturbing players’ higher-order beliefs, whichwe call robust selection. Similarly to WY selection [28], robust selection generalizes the idea behind theequilibrium selection in the email game [27] and the global game [3]. In contrast to WY selection, however,we require selection to be robust to misspecifications of payoffs. Robust selection is a strong notion in thesense that, among types with multiple rationalizable actions, “almost all” selections are fragile; but it isalso a weak notion in the sense that any strictly rationalizable action can be robustly selected. We show thatrobust selection is fully characterized by the curb collection, a notion that generalizes the curb set in [2].We also use the curb collection to characterize critical types [12] in any fixed finite game.© 2014 Elsevier Inc. All rights reserved.

JEL classification: C70; C72; D80

Keywords: Equilibrium refinement; WY selection; Robust selection; E-mail game; Global game

✩ We thank an Editor, an Associate Editor and two anonymous referees for their insightful comments which havegreatly improved the paper. We also thank Eddie Dekel, Songying Fang, Amanda Friedenberg, Matthew Jackson, TakashiKunimoto, Qingmin Liu, George Mailath, Marcin Peski, Tomas Rodriguez-Barraquer, Xu Tan, Jonathan Weinstein,Muhamet Yildiz, Charles Zheng, and conference/seminar participants at CityU HK, Games 2012, Penn, Rice, SeoulNational, Texas A&M, Western Ontario, and the 2012 GCOE conference at Hitotsubashi University for very helpfulcomments. We gratefully acknowledge financial support from the National Science Foundation (SES-1227620) and theSingapore Ministry of Education Academic Research Fund Tier 1. All remaining errors are our own.

* Corresponding author.E-mail addresses: [email protected] (Y.-C. Chen), [email protected] (S. Takahashi), [email protected]

(S. Xiong).

http://dx.doi.org/10.1016/j.jet.2014.02.0010022-0531/© 2014 Elsevier Inc. All rights reserved.

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Y.-C. Chen et al. / Journal of Economic Theory 151 (2014) 448–475 449

1. Introduction

One challenge that game theory faces is the prevalence of multiple equilibria, which substan-tially limits the theory’s predictive power. Two seminal papers introduced the idea of sharpeningpredictions: [27] on the email game and [3] on the global game. Both papers demonstrate thatin a complete-information coordination game with two strict equilibria, we can select one of theequilibria as the unique rationalizable action by perturbing players’ higher-order beliefs.1

Weinstein and Yildiz [28] (hereafter, WY) substantially generalize this observation and provethe following surprising result: in a fixed game that satisfies a richness assumption on payoffs,every rationalizable action can be selected as the unique rationalizable action for any (Harsanyi)type by perturbing the type’s higher-order beliefs. Based on this result, WY conclude that norefinement of rationalizability retains its validity when we have only partial knowledge of theplayers’ incomplete information.2 More precisely, any rationalizable action (which may be ruledout by some refinement) could be the unique prediction, provided that the implicit belief assump-tions of a model are relaxed in the “right” way. Consequently, the multiplicity of equilibria is anartifact of the modeling assumptions.

In this paper, we depart from WY by proposing a new selection notion, which we call robustselection. Specifically, we observe that the selection in [3] (hereafter, global-game selection)exhibits a robustness feature that is not shared by the selection adopted in WY (hereafter, WYselection), namely, the robustness to small misspecifications in payoffs. Abstracting from thisfeature, we say that an action is robustly selected in a game if we can perturb players’ higher-order beliefs in such a way that the action is selected as the unique rationalizable action in everynearby game.

The robustness to payoff misspecifications is important because every game in economicmodels is at best an approximation of reality. Similarly to the precise specification of play-ers’ higher-order beliefs, the precise specification of payoffs is postulated only for analyticaltractability; it would be unrealistic to assume that a modeler can keep track of all the payoffsprecisely, especially when the game is complicated.3 We further show that robustly selecting anaction amounts to selecting the action as the unique ε-rationalizable action by perturbing play-ers’ higher-order beliefs (Proposition 1). Robust selection thus incorporates the robustness toε-rationality.

Based on the notion of robust selection, our results differ substantially from those in WY.First, contrary to their result that any rationalizable action can be WY-selected, we prove that in“almost all” cases, types have multiple rationalizable actions only due to a payoff tie, and that itis impossible to robustly select any rationalizable action for such types (Propositions 3 and 4).Furthermore, an action can be robustly selected if it is strictly rationalizable, and only if it is astrict best reply (Proposition 5). Finally, we prove that robust selection is fully characterized by a

1 The idea has inspired a variety of applied works using robustness to incomplete information as a criterion for equi-librium selection. See, for instance, [24,4,14], and [18]. See also [23] for an extensive literature review.

2 See pp. 390–391 in WY for a discussion relating their approach to the two pioneering works on the robustness ofrefinements: [16] and [8].

3 Measurement errors are bound to occur because of (i) the limited cognitive ability of human beings and (ii) ourcurrent technology. For instance, a precise measurement of a real number could conceivably be used to store an infiniteamount of digital data; however, because such a storage technology is not yet available, we have to rely on traditionalstorage devices that have finite limits.

450 Y.-C. Chen et al. / Journal of Economic Theory 151 (2014) 448–475

novel notion that we call the curb collection (Theorem 1), which generalizes the curb set definedin [2].

Our results convey two economic messages: one is aligned with the message in WY andthe other goes in the opposite direction. First, similarly to WY, we argue that it is problematic toselect predictions by perturbing players’ higher-order beliefs, but our reason is different from thatof WY. WY prove that any rationalizable action can be WY-selected; hence, there are too manyselections and “selecting” any one of them seems ad hoc. In contrast, we show here that theregenerically exists no robust selection; hence, selecting predictions by perturbing higher-orderbeliefs has a limited scope.

Second, WY show that any rationalizable action can be WY-selected, and hence, the global-game selection is not different from selecting any other rationalizable action. In contrast, ourresults demonstrate that the selection in global games exhibits desirable properties that are notshared by some other WY selections. In particular, our Proposition 5 implies that a weakly dom-inated action can never be robustly selected, despite being rationalizable. This further impliesthat in a second-price auction with discrete bids, although any bid can be WY-selected, biddingthe true value is the unique prediction that can be robustly selected (Example 3).

The remainder of the paper is organized as follows. Section 2 illustrates the notion of ro-bust selection and its robustness feature. Section 3 defines preliminaries. Section 4 proves thegeneric impossibility of robust selection. Section 5 fully characterizes robust selection. Section 6discusses the implications of our results and related issues. Appendix A contains all the proofsomitted from the main text.

2. Robustness of robust selection

In this section, we illustrate the notion of robust selection. Similarly to WY, we use (Harsanyi)types to formulate incomplete information. Each type of a player specifies a belief (possibly,but not necessarily, derived from a common prior) about the payoff-relevant parameters andthe opponents’ types. The belief of a player’s type thus encodes the player’s belief about thepayoff-relevant parameters (i.e., the first-order belief), the player’s belief about the other players’beliefs about the payoff-relevant parameters (i.e., the second-order belief), and so on.

For tractability, a modeler often uses a simple type t to represent a player’s higher-orderbeliefs, i.e., a type living in a “simple” type space such as a type space with complete information,a finite type space, or a type space with a common prior. Because of the implicit assumptionsincorporated into these type spaces (e.g., the common knowledge of payoffs or a common prior),the type t may have multiple rationalizable actions. The starting point of WY as well as thispaper is to admit that t is merely an idealization of the reality denoted by t ′, which is a perturbedversion of t . We study whether a rationalizable action of t can be made a uniquely rationalizableaction for t ′.

More precisely, fix a set of players and a set of action profiles. A game G consists of a set ofpayoff-relevant parameters, where each parameter identifies a profile of payoff functions definedon the action profiles. We say that an action a is selected by a sequence of types {tm} in G if a isthe only rationalizable action for every tm in G. We say that an action a can be WY-selected for atype t in G if there exists a sequence of types {tm} that approximates t up to any finite order andselects a in G. While WY perturbs types, they fix the game G. This can be interpreted as that themodeler has precise information about the game being played, but is unsure of her model aboutthe players’ higher-order beliefs about the payoff-relevant parameters.


We depart from WY with the observation that WY selection generally hinges on the specifi-cation of G being precise. We use an example to illustrate this point and contrast the examplewith the global-game selection.

Example 1 (Tie-breaking). There is only one player, who chooses between two actions a and b.The player’s payoff depends on a parameter θ ∈ {θ0, θa} and the action chosen, as specifiedbelow.4

G1:action payoff

a 0b 0θ = θ0

action payoffa 0b −1θ = θa

In this single-player decision problem, a type is simply the player’s belief about θ . For m =1,2, . . . ,∞, define type tm as

tm[θ0] = 1 − 1

mand tm[θa] = 1

m,

where tm[θ ] denotes the probability that type tm assigns to θ . Clearly, actions a and b are bothrationalizable for t∞, and the sequence of types {tm} converges to t∞. Moreover, a is uniquelyrationalizable for every tm in G1. That is, a is WY-selected for t∞ in G1 via the sequence oftypes {tm}. However, the sequence of types {tm} no longer selects a if we change the payoffs asfollows (where γ0 > 0):

G1:action payoff

a 0b γ0θ = θ0

action payoffa 0b −1θ = θa

In fact, there is no sequence of types that converges to t∞ and selects a in G1.

Example 2 (The global game). There are two players whose payoffs depend on an unknownparameter θ as follows:

G2:Attack No Attack

Attack θ, θ θ − 1,0No Attack 0, θ − 1 0,0

Suppose that θ is uniformly distributed on the interval [−1,2], and player i receives a noisysignal xi ≡ θ + ξi , where (ξ1, ξ2) is distributed independently of θ and uniformly on [− 1

m, 1

m]2.

Let ti,m denote the type of player i who observes xi = 34 , given m. The type ti,∞ has common

knowledge of θ = 34 and it has two rationalizable actions, “Attack” and “No Attack.” As m → ∞,

type ti,m can be viewed as a small perturbation of the common knowledge scenario, and “Attack”is the unique rationalizable action for ti,m with any sufficiently large but finite m. In this sense, wesay that “Attack” is WY-selected for type ti,∞ in G2 via the sequence of types {ti,m}. Furthermore,

4 For simplicity, we consider here a single-agent game, but the idea can be easily extended to multi-player games (seeSection 4).


this remains the case even if we modify the payoffs as follows (where γl ∈ (− 110 , 1

10 ) for everyl = 1,2, . . . ,8):

G2:Attack No Attack

Attack θ + γ1, θ + γ2 θ − 1 + γ3, γ4No Attack γ5, θ − 1 + γ6 γ7, γ8

Namely, in G2, still both “Attack” and “No Attack” are rationalizable for ti,∞, and “Attack” isthe unique rationalizable action for every ti,m with a sufficiently large but finite m. Thus, thesame sequence of types {ti,m} still selects “Attack” in any such G2.5

Compared to the tie-breaking selection in Example 1, the global-game selection in Example 2is robust to payoff misspecifications in the sense that the sequence of types {ti,m} selects theaction “Attack” in all nearby games. We generalize this observation and say that an action a canbe robustly selected for a type t in G if there exists a sequence of types {tm} that approximatest and selects a in every game that is sufficiently close to G (Definition 3). It is important toinvestigate the robustness to such payoff misspecifications since measurement errors in payoffsare bound to occur in practice (see footnote 3).

3. Preliminaries

Throughout the paper, for any metrizable space Y , we use �(Y) to denote the space ofprobability measures on the Borel σ -algebra of Y . We endow �(Y) with the weak∗-topology.Moreover, we endow a product space with the product topology, a subspace with the relativetopology, and a finite set with the discrete topology. Let |E| denote the cardinality of a set E.

3.1. The model of incomplete information

Fix a finite set of players N . A model is a tuple (Θ,T , κ), where Θ is a finite set of payoff-relevant parameters, T = ∏

i∈N Ti is a compact metric space, and each ti ∈ Ti (called a type ofplayer i) is associated with a belief κti ∈ �(Θ × T−i ). Assume that ti �→ κti is a continuousmapping.

Given a type ti in a model (Θ,T , κ), we can compute the first-order belief of ti (i.e., his beliefabout Θ) by setting t1

i equal to the marginal distribution of κti on Θ . We can also compute thesecond-order belief of ti (i.e., his belief about (θ, t1−i )) by setting

t2i [E] = κti

[{(θ, t−i ):

(θ, t1−i

) ∈ E}]

, ∀E ⊂ Θ × (�(Θ)

)|N |−1.

We can compute the entire (Θ-based) hierarchy of beliefs (t1i , t2

i , . . . , tki , . . .) by proceeding inthis way.

We collect all such hierarchies and construct the universal type space T ∗i based on Θ . This has

the property that ti = (t1i , t2

i , . . .) ∈ T ∗i if and only if there exists some type t ′i in some model such

that tki = t ′ ki for every k. Endowed with the product topology, T ∗i is a compact metrizable space

and admits a homeomorphism κ∗i :T ∗

i → �(Θ ×T ∗−i ) (see [22]). Thus we can regard (Θ,T ∗, κ∗)

5 In the rest of the paper, we will assume that the parameter θ takes only finitely many values. We can easily modifyExample 2 in order to satisfy this assumption.


as a model, where κ∗ti

≡ κ∗i (ti) for every ti ∈ T ∗

i . Moreover, the hierarchy of beliefs of ti ∈ T ∗i

in the model (Θ,T ∗, κ∗) is given by ti itself, and this is why we use tki to denote both the k-thcomponent of ti and the k-th order belief of ti in (Θ,T ∗, κ∗). With a further abuse of notations,we say that a sequence of types {ti,m} on T ∗

i converges to a (not necessarily universal) type ti insome model, denoted as ti,m → ti , if for every k, tki,m → tki in the weak∗-topology.

A type ti ∈ T ∗i is said to be a complete-information type if there exist some θ0 ∈ Θ and some

t−i ∈ T ∗−i such that κ∗tj[θ0, t−j ] = 1 for every j ∈ N , i.e., this type has common knowledge of

θ = θ0. We denote this type by tθ0i .

3.2. The game and solution concept

Each player i has a finite set of actions Ai with |Ai | � 2, which we fix throughout the paper.Let A = ∏

i∈N Ai denote the set of action profiles. A game is a tuple G = (Θ, (ui)i∈N), whereui : Θ × A → R is player i’s payoff function.

Following WY, we adopt the solution concept of interim correlated rationalizability (ICR)proposed in [9,10]. For any π ∈ �(Θ × A−i ) and any ε ∈ R, we use BRi (π,G, ε) to denote theset of ε-best replies to π in game G = (Θ, (ui)i∈N). That is,

BRi (π,G, ε) ={ai ∈ Ai :

∑θ,a−i

[ui(θ, ai, a−i ) − ui

(θ, a′

i , a−i

)]π[θ, a−i] � −ε,∀a′

i �= ai

}.

Given a game G = (Θ, (ui)i∈N) with a model (Θ,T , κ), a conjecture of player i is a measurablefunction σ−i : Θ × T−i → �(A−i ). Based on σ−i , a type ti forms a distribution on Θ × A−i ,denoted by πti ,σ−i

, where

πti ,σ−i[θ, a−i] =

∫T−i

σ−i (θ, t−i )[a−i]κti [θ, dt−i], ∀(θ, a−i ) ∈ Θ × A−i .

Moreover, given any ε ∈ R, the ε-ICR actions of type ti in game G, denoted by S∞i [ti ,G, ε], are

defined as

S∞i [ti ,G, ε] =

∞⋂k=0

Ski [ti ,G, ε],

where S0i [ti ,G, ε] = Ai , and inductively, for each integer k � 1, ai ∈ Sk

i [ti ,G, ε] if and only ifthere is a conjecture σ−i of player i such that

(a) σ−i (θ, t−i )[a−i] > 0 ⇒ aj ∈ Sk−1j [tj ,G, ε], ∀j �= i;

(b) ai ∈ BRi (πti ,σ−i,G, ε).

For t−i = (tj )j �=i , we write S∞−i[t−i ,G, ε] for∏

j �=i S∞j [tj ,G, ε]. We say that σ−i is an

ε-valid conjecture if and only if σ−i (θ, t−i )[S∞−i[t−i ,G, ε]] = 1. Dekel, Fudenberg and Mor-ris [9,10] prove the following properties for any ε-ICR with ε � 0, which we will use later inthe paper: (i) S∞

i [ti ,G, ε] is nonempty for any ti ; (ii) S∞i [ti ,G, ε] ⊂ S∞

i [ti ,G, ε′] if ε � ε′;(iii) ai ∈ S∞[ti ,G, ε] if and only if ai ∈ BRi (πt ,σ ,G, ε) for some ε-valid conjecture σ−i ;
i i −i


(iv) the set of ε-ICR actions of a type is fully determined by its belief hierarchy. By (iv), we willidentify a type with its belief hierarchy.6

3.3. WY selection and robust selection

We say that an action ai is selected by a sequence of types {ti,m} in G if S∞i [ti,m,G,0] = {ai}

for every m. We first define WY selection as follows.

Definition 1. Given a game G = (Θ, (ui)i∈N) with a model (Θ,T , κ), an action ai can be WY-selected for ti ∈ Ti in G if there is a sequence of types {ti,m} on the Θ-based universal type spaceT ∗

i such that ti,m → ti and ai is selected by {ti,m} in G.

Under the following richness assumption, WY show that every rationalizable action can beWY-selected.

Definition 2. A game G = (Θ, (ui)i∈N) satisfies the richness assumption if for every i ∈ N

and every ai ∈ Ai , there exists θai ∈ Θ such that ui(θai , ai, a−i ) > ui(θ

ai , a′i , a−i ) for every

a′i ∈ Ai \ {ai} and every a−i ∈ A−i .

Conversely, by the upper hemicontinuity of the rationalizability correspondence (see [9, The-orem 2]), every action that can be WY-selected must also be rationalizable. Therefore, rational-izability and WY selection coincide under the richness assumption.

To formalize the notion of robust selection, consider two parameter spaces Θ and Θ . Wesee Θ as an extension of Θ , which captures some additional uncertainties about payoffs andtypes. Given a game G = (Θ, (ui)i∈N), we say that a game G = (Θ, (ui)i∈N) is ε-close7 toG = (Θ, (ui)i∈N) via an onto mapping f : Θ → Θ if we have ‖G − G‖f � ε, where

‖G − G‖f := maxi∈N, θ∈Θ, a∈A

∣∣ui (θ , a) − ui

(f (θ), a

)∣∣.We regard f (θ) ∈ Θ as a good approximation of θ ∈ Θ because these two parameters definenearby payoffs. Furthermore, let T ∗

i and T ∗i denote the Θ-based and Θ-based universal type

spaces, respectively. Then, through the approximation via f , each higher-order belief over Θ

(i.e., ti ∈ T ∗i ) is mapped naturally to some higher-order belief over Θ (i.e., ti ∈ T ∗

i ). Denote thismapping by fi : T ∗

i → T ∗i .8 That is, respecting the approximation of Θ on Θ , we regard the type

fi(ti ) ∈ T ∗i as an approximation of a type ti ∈ T ∗

i .

6 Here we follow WY and perturb types in the universal type space. Our analysis nonetheless extends to perturbationsin any hierarchy-complete model (i.e., a model which contains all possible hierarchies of beliefs; see [15]). We thank theAssociate Editor for pointing this out to us.

7 Our notion of ε-closeness is different from the notions of ε-elaboration in [16,8], and [21] in that ε here refers topayoff perturbations rather than perturbations in prior probabilities.

8 Precisely, we define fi as follows. For every ti = (t1i, t2

i, . . .) ∈ T ∗

i, we define fi(ti ) ≡ (f 1

i(t1

i), f 2

i(t2

i), . . .) ∈ T ∗

i,

where

f 1i

(t1i

)[E] = t1i

[{θ : f (θ) ∈ E

}], ∀E ⊂ Θ,

f 2i

(t2i

)[E] = t2i

[{(θ , t1−i

):

(f (θ), f 1−i

(t1−i

)) ∈ E}]

, ∀E ⊂ Θ × �(Θ)|N |−1,

and so on. Consequently, for any ti ∈ T ∗ , we have
i


We now define the notion of robust selection.

Definition 3. Given a game G = (Θ, (ui)i∈N) with a model (Θ,T , κ), an action ai can be ro-bustly selected for ti ∈ Ti in G if there are ε > 0 and a sequence of types {ti,m} on the Θ-baseduniversal type space T ∗

i such that ti,m → ti , and for every game G = (Θ, (ui)i∈N) that is ε-closeto G via some onto mapping f : Θ → Θ and every sequence of types {ti,m} on the Θ-baseduniversal type space T ∗

i with fi(ti,m) = ti,m for all m, ai is selected by {ti,m} in G.

We interpret the notion of robust selection as follows. Suppose that a modeler faces a com-plicated game in reality, which involves a large parameter space Θ . For tractability, the modelermay use a simpler game to approximate the reality. In particular, the modeler may use a smallerparameter space Θ to approximate Θ via an onto mapping f : Θ → Θ , and accordingly, rep-resent each type ti ∈ T ∗

i by a simpler type ti = fi(ti ) ∈ T ∗i . In this way, the game G and the

type space T ∗i are regarded as a “reduced-form” representation of G and T ∗

i in reality. Due toapproximation errors, however, an action that is selected by “reduced-form” types {ti,m} in the“reduced-form” game G may not be selected by the corresponding “true” types {ti,m} in the“true” game G in reality. The notion of robust selection is defined to address this problem. Thatis, an action that can be robustly selected for the “reduced-form” type ti in the “reduced-form”game G is an action that is guaranteed to be selected by the corresponding “true” types {ti,m} inthe “true” game G in reality as long as the approximation is sufficiently good.

3.4. An alternative definition of robust selection

One drawback of the definition of robust selection is its complication, as it is based explicitlyon rationalizable actions in all nearby games. To mitigate the complication, we provide an alter-native and simpler definition of robust selection, and show that the two definitions are equivalent.

Definition 4. Given a game G = (Θ, (ui)i∈N) with a model (Θ,T , κ), an action ai can berobustly∗ selected for ti ∈ Ti in G if there are ε > 0 and a sequence of types {ti,m} on the Θ-baseduniversal type space T ∗

i such that ti,m → ti and S∞i [ti,m,G, ε] = {ai} for every m.

Proposition 1. Given a game G = (Θ, (ui)i∈N) with a model (Θ,T , κ), an action can be robustlyselected for ti ∈ Ti in G if and only if it can be robustly∗ selected for ti in G.

The following proposition is a critical step in the proof of Proposition 1.

Proposition 2. Given a game G = (Θ, (ui)i∈N) and ε > 0,

S∞i [ti ,G, ε] =

⋃{(G,f ): ‖G−G‖f � ε

2 }

⋃{ti : fi(ti )=ti }

S∞i [ti , G,0], ∀ti ∈ T ∗

i .

κ∗fi (ti )

[E] = κ∗ti

[{(θ , t−i ):

(f (θ), f−i (t−i )

) ∈ E}]

, ∀E ⊂ Θ × T ∗−i ,

where κ∗· = κ∗i(·) and κ∗· = κ∗

i(·) are the homeomorphisms between T ∗

iand �(Θ × T ∗−i

) and between T ∗i

and �(Θ ×T ∗ ), respectively.
−i


That is, an action ai is ε-rationalizable for ti ∈ T ∗i in G if and only if there exist a game G that

is ε2 -close to G via f : Θ → Θ and a type ti ∈ T ∗

i with fi(ti ) = ti such that ai is 0-rationalizablefor ti in G.

With Proposition 2, the proof of Proposition 1 is straightforward: ai is robustly∗ selectedfor ti in G (i.e., ti,m → ti and S∞

i [ti,m,G, ε] = {ai} for every m) if and only if ai is robustlyselected for ti in G (i.e., S∞

i [ti,m, G,0] = {ai} for every m, every game G that is ε2 -close to G

via f : Θ → Θ , and every ti,m with fi(ti,m) = ti,m).In light of Proposition 1, we will hereafter adopt Definition 4 as the definition of robust se-

lection. Finally, we note that whether an action can be WY-/robustly selected or not for a typedepends only on its Θ-based belief hierarchy.

4. Ties and generic impossibility of robust selection

Hereafter, we fix a payoff parameter space Θ , and except in Section 6.1, we also fix a gameG = (Θ, (ui)i∈N). We will thus simplify the notation by writing S∞

i [ti , ε] for S∞i [ti ,G, ε]. Fur-

thermore, if ε = 0, we write S∞i [ti] for S∞

i [ti ,0]. Similarly, all notations without a reference toε should be understood as an implicit reference to ε = 0; moreover, models without a referenceto Θ should be understood as based on Θ .

We start by identifying non-robust selections. To do this, we follow WY in partitioning theuniversal type space into two parts: the set of types with multiple rationalizable actions, denotedby Mi , and the set of types with a unique rationalizable action, denoted by Ui :

Mi = {ti ∈ T ∗

i :∣∣S∞

i [ti]∣∣ > 1

}and Ui = {

ti ∈ T ∗i :

∣∣S∞i [ti]

∣∣ = 1}.

Given |Ai | � 2, Mi �= ∅ if G satisfies the richness assumption. We will focus on Mi becausethere is no need to select predictions for types with a unique rationalizable action.9

Each type ti in Mi has multiple rationalizable actions either (i) because ti has many possiblevalid conjectures that are consistent with opponents’ rationalizable actions (such as the complete-information type ti,∞ in Example 2), or (ii) because ti has a payoff tie to an essentially uniquevalid conjecture. Formally, (ii) happens for a type ti if and only if ti belongs to

Bi ≡ {ti ∈ Mi : κ∗

ti[Θ × U−i] = 1

}. (1)

Note that each type in Bi has only one valid conjecture up to null events, i.e., the conjectureunder which with probability 1, every type of every opponent plays his unique rationalizableaction. In this case, multiplicity of rationalizable actions must be due to a payoff tie, whichgeneralizes the single-person decision problem in Example 1. Recall that we cannot robustlyselect any rationalizable action in Example 1. Similarly, we have the following proposition.10

Proposition 3. No type in Bi admits a robust selection.

9 In fact, by Dekel, Fudenberg and Morris [9, Lemma 1 and Theorem 2], if S∞i

[ti ] = {ai }, then there is some ε > 0such that for any sequence of types {ti,m} converging to ti , S∞

i[ti,m, ε] = {ai } for all sufficiently large m, i.e., ai is

robustly selected for ti .10 We note that having a payoff tie is not the only way to have a non-robust selection. For example, Proposition 12 inSection 6.3 shows that types of player i who assign probability one to B−i (and who may not have a payoff tie) alsoadmit no robust selection.


We will prove later a stronger result, Proposition 12 in Section 6.3, which implies Proposi-tion 3.

Note that Mi is a closed subset of the compact metric space T ∗i (see Proposition 2 in WY), and

hence Mi is a Baire space (see [1, 3.47 Theorem]).11 In a Baire space, the notion of genericitycan be formalized by residual sets.12 The following proposition says that Bi is generic in Mi .

Proposition 4. Suppose that G satisfies the richness assumption. Then, Bi is a residual set in Mi .

Propositions 3 and 4 imply that for “almost all” types with multiple rationalizable actions,selections are fragile to payoff misspecifications. This is in sharp contrast to WY. According toWY, any rationalizable action can be WY-selected, and thus all selections, including the global-game selection, are ad hoc. However, if we view G as an imperfect approximation of reality, then“almost all” selections are fragile in the sense that no action is guaranteed to be selected in the“true” game in reality.

One caveat is that the mathematical definition of residual sets should not be interpreted aseconomic relevance, and that Proposition 4 does not imply that types in Mi \Bi are economicallyirrelevant. Indeed, there are many economically interesting types that belong to Mi \Bi (e.g., thecomplete-information type ti,∞ in Example 2). We thus ask the following questions: which typedoes or does not admit a robust selection, and if it does, which action can be robustly selectedfor that type? We answer these questions in the next section.

5. Characterization

In this section, we first draw connections between robust selection and the notions of strictbest replies and strict rationalizability. We then introduce a new notion called curb collection andprove that this notion fully characterizes robust selection.

5.1. Strict rationalizability

We define a strict best reply and a strictly rationalizable action as follows. For complete-information games, our definition of strictly rationalizability is equivalent to the standard one(i.e., iterated deletion of never-strict best replies).

Definition 5. Given a model (T , κ) and a type ti ∈ Ti , an action ai is a strict best reply forti if ai ∈ S1

i [ti ,−ε] for some ε > 0; an action ai is strictly rationalizable for a type ti if ai ∈S∞

i [ti ,−ε] for some ε > 0.

We will prove the following result in Appendix A as an immediate consequence of our maincharacterization of robust selection.

Proposition 5. Given a model (T , κ) and ti ∈ Ti , (i) an action ai can be robustly selected forti only if ai is a strict best reply for ti ; (ii) given the richness assumption, an action ai can berobustly selected for ti if ai is strictly rationalizable for ti .

11 A topological space is called a Baire space if every intersection of countably many open and dense sets is dense.12 A meager set is a countable union of nowhere dense sets and the complement of a meager set is called a residual set.That is, a residual set contains a countable intersection of open and dense sets.


In light of Proposition 5, the following example demonstrates that a non-trivial robust selec-tion can be employed to obtain a meaningful prediction.

Example 3. Consider a discrete second-price auction, where bidders simultaneously bid integersbetween 0 and 10. Assume that the richness assumption holds in this game, and focus on indepen-dent and private-value types whose values are uniformly distributed on the set {0,1,2, . . . ,10}.First, any bid between 0 and 9 is a best reply if an opponent bids 10. Second, bidding 10 is a bestreply if all opponents bid 0. As a result, any bid is rationalizable, and hence can be WY-selected.In contrast, truthful bidding forms a strict equilibrium, and any other bid is weakly dominatedby truthful bidding. Hence, by Proposition 5, bidding the true value is the unique prediction thatcan be robustly selected.

However, there are actions that can be robustly selected without being strictly rationalizable,as shown in the following example.13

Example 4. Consider a two-player game:

G3:c d d ′

a 1,1 0,0 0,0b 0,0 1,1 1,1

θ = θ0

c d d ′a 0,0 0,1 0,0b 0,0 0,1 0,0

θ = θd

When it is common knowledge between the two players that θ = θ0, neither d nor d ′ is strictlyrationalizable for player 2, and hence b is not strictly rationalizable for player 1. Nevertheless,b can be robustly selected for t

θ01 . To see this, define a sequence of types {tm} as follows, with tm

being player 1’s (resp. player 2’s) type if m is odd (resp. even):

t0 ≡ tθd

2 and κ∗tm

[θ0, tm−1] ≡ 1, ∀m � 1.

Clearly, t2m−1 → tθ01 and t2m → t

θ02 . Moreover, S∞

1 [t2m−1,12 ] = {b} and S∞

2 [t2m, 12 ] = {d, d ′} for

any positive integer m. Therefore, b is robustly selected for tθ01 .

5.2. Characterization: complete-information models

Our characterization of robust selection is based on a notion called curb collection. Since thisnotion requires mathematical technicalities in general models, we first present the definition forcomplete-information models. Fix a state θ0 ∈ Θ and focus on the complete-information type t

θ0i .

Let Ai be the collection of all nonempty subsets of Ai , i.e., Ai = 2Ai \ {∅}. For each profile(Rj )j �=i with Rj ⊂ Aj , we denote by R−i ≡ {R−i = ∏

j �=i Rj : Rj ∈ Rj } the collection of allproduct subsets of A−i generated by (Rj )j �=i .

To motivate the notion of curb collections, consider the collection of sets of ε-ICR actions fortypes sufficiently close to the complete-information type t

θ0i :

Rθ0i ≡ {

Ri ∈Ai : ∃ε > 0, ∃{ti,m}∞m=0 ⊂ T ∗i s.t. ti,m → t

θ0i and Ri = S∞

i [ti,m, ε], ∀m}.

13 Nevertheless, in Section 6.1, we show that in a complete-information game with generic payoffs, an action can berobustly selected if and only if it is strictly rationalizable.


Clearly, Rθ0i fully characterizes the robust selections for t

θ0i , because ai can be robustly selected

for tθ0i if and only if {ai} ∈ Rθ0

i . However, the characterization is not satisfactory, because Rθ0i

is defined by (ε-)rationalizable actions of all types sufficiently close to tθ0i , which are difficult

to compute. Nevertheless, the beliefs and the rationalizable sets of all such types share a com-mon property. Specifically, for each Ri ∈ Rθ0

i , let tRibe a type sufficiently close to t

θ0i such that

Ri = S∞i [tRi

, ε]. Since tRibelieves with high probability that θ = θ0 and ICR sets for his op-

ponents belong to Rθ0−i , we can define a belief μRi∈ �(Rθ0−i ) as his (approximate) belief over

his opponents’ ICR sets. By the definition of ε-ICR, ai ∈ Ri if ai is a best reply to a conjecture(approximately) consistent with μRi

.Our notion of curb collection is based on the observation above. Formally, a function ϕ−i :

A−i → �(A−i ) is a conditional conjecture if

ϕ−i (R−i )[a−i] > 0 ⇒ a−i ∈ R−i .

Combining μ ∈ �(A−i ) with ϕ−i , we define a distribution πμ,ϕ−i∈ �(Θ × A−i ) as

πμ,ϕ−i[θ0, a−i] =

∑R−i∈A−i

μ[R−i] × ϕ−i (R−i )[a−i], ∀a−i ∈ A−i .

Definition 6. A profile of nonempty collections (Ri )i∈N with Ri ⊂Ai for each i ∈ N is called acurb collection at state θ0 if for every i ∈ N and every Ri ∈ Ri , there exists μRi

∈ �(A−i ) suchthat

(b) μRi[R−i] > 0 ⇒ R−i ∈R−i , and

(c) Ri ⊃ BRi (πμRi,ϕ−i

) for any conditional conjecture ϕ−i .

A profile of nonempty sets of actions (Ri)i∈N is a curb (i.e., closed under rational behavior)set at θ0 if we have Ri ⊃ BRi (π) for any π ∈ �({θ0} × R−i ) [2].14 That is, each player i isassociated with a nonempty set of actions, Ri , and all best responses of player i are within Ri

if he believes that the state is θ0 and his opponents play actions within R−i . The notion of curbcollection (Definition 6) generalizes the curb set by allowing for multiple “copies” of player i,each of which is identified with a nonempty set of actions Ri ∈ Ri , and all best responses of the“copy” Ri are within Ri if he has a belief μRi

over copies of his opponents, and believes that thestate is θ0 and copies of his opponents play actions within their associated sets.15

Compared to Rθ0i , the definition of curb collection is very simple, and based purely on best

replies derived from finite collections (Ai )i∈N . Furthermore, the union of two curb collectionsis also a curb collection.16 As a result, the largest curb collection exists, and is easily com-puted. That is, let R0

i = Ai ; for k = 0,1, . . . , let Rk+1i be the collection of all Ri ∈ Rk

i suchthat Ri ⊃ ⋃

ϕ−iBRi (πμ,ϕ−i

) for some μ ∈ �(Rk−i ). Then we have a weakly decreasing sequence

R0i ⊃R1

i ⊃ · · · . By the finiteness of actions, there exists k∗ such that Rk∗+1i =Rk∗

i ≡R∗i for all

14 This is slightly different from the original definition by Basu and Weibull [2], which requires π to be independentacross player i’s opponents.15 Indeed, (Ri)i∈N ∈ ∏

i∈N Ai is a curb set at θ0 if and only if (Ri = {Ri })i∈N is a curb collection at θ0.16 Formally, if (Ri )i∈N and (R′ )i∈N are curb collections, then (Ri ∪R′ )i∈N is a curb collection.

i i


i ∈ N . The collection (R∗i )i∈N thus obtained is the largest curb collection.17 For instance, in Ex-

ample 4, consider the complete-information model defined by θ = θ0. Let R01 = {{a}, {b}, {a, b}}

and R02 = {{c}, {d}, {d ′}, . . . , {c, d, d ′}}. Then the process stops in one step with R∗

1 ≡ R11 =

{{a}, {b}, {a, b}} and R∗2 ≡R1

2 = {{c}, {d, d ′}, {c, d, d ′}}.The following result, which is a special case of Theorem 1, characterizes robust selection

based on curb collections. It follows from Proposition 6 that in Example 4, {b} is robustly selectedfor t

θ0i , whereas neither d nor d ′ can be robustly selected for t

θ0i .

Proposition 6. Suppose that G satisfies the richness assumption. An action ai can be robustlyselected for complete-information type t

θ0i if and only if {ai} ∈ R∗

i for the largest curb collection(R∗

j )j∈N at state θ0.

The “if” direction follows by showing that there exist ε > 0 and sequences of types {τRi,m}for all Ri ∈R∗

i such that τRi,m → tθ0i as m → ∞ and S∞

i [τRi,m, ε] ⊂ Ri for all m. The “only if”

direction follows by showing that (Rθ0i )i∈N is indeed a curb collection. In the next subsection,

we formalize these directions for general models in Propositions 7 and 8, respectively.

5.3. Characterization: general models

We now extend the notion of curb collection to a general model (T , κ). We say that a measur-able function ϕ−i : Θ × T−i ×A−i → �(A−i ) is a conditional conjecture if

ϕ−i (θ, t−i ,R−i )[a−i] > 0 ⇒ a−i ∈ R−i .

Given μ ∈ �(Θ × T−i ×A−i ) and ϕ−i , we define πμ,ϕ−i∈ �(Θ × A−i ) as

πμ,ϕ−i[θ, a−i] =

∫T−i×A−i

ϕ−i (θ, t−i ,R−i )[a−i]μ[θ, dt−i , dR−i], ∀(θ, a−i ) ∈ Θ × A−i .

Definition 7. Given a model (T , κ), a profile of measurable functions (Ri : Ti → 2Ai \ {∅})i∈N

is called an ε-curb collection in (T , κ) if for every i ∈ N , there exists a measurable function

Ti ×Ai � (ti ,Ri) �→ μti ,Ri∈ �(Θ × T−i ×A−i )

such that for each ti ∈ Ti and Ri ∈Ri (ti),

(a) the marginal distribution of μti,Rion Θ × T−i is equal to κti ,

(b) μti,Ri[{(θ, t−i ,R−i ): R−i ∈R−i (t−i )}] = 1, and

(c) Ri ⊃ BRi (πμti ,Ri,ϕ−i

, ε) for any conditional conjecture ϕ−i .

The following result is our full characterization of robust selection.

17 If we begin with the collection of all singleton sets R0i

= {{ai }: ai ∈ Ai }, then this process is equivalent to the iterated

deletion of never strict best replies, and the resulting collection R∗ is equal to {{ai }: ai is strictly rationalizable}.
i


Theorem 1. Suppose that G satisfies the richness assumption. Given a model (T , κ), an actionai can be robustly selected for ti ∈ Ti if and only if there exists ε > 0 such that {ai} ∈ Ri (ti ) forsome ε-curb collection (Rj )j∈N in (T , κ).

Note that ε-curb collections are defined within the model (T , κ), and yet the notion of ro-bust selection requires examining all (infinitely many) types in the neighborhoods surrounding(T , κ). Nonetheless, Theorem 1 shows that the singleton sets in ε-curb collections in (T , κ) pindown exactly the actions that can be robustly selected for any type in (T , κ). In particular, whenthe model is finite (i.e., |T | < ∞) or even with complete information, Theorem 1 translates aninfinite-dimensional problem into a finite-dimensional problem.18

The “if” and “only if” directions of Theorem 1 are immediate consequences of Propositions 7and 8, respectively. We prove these two propositions in the next two subsections.

Proposition 7. Suppose that G satisfies the richness assumption. Let (Rj )j∈N be an ε-curbcollection in a model (T , κ) with ε > 0. If Ri ∈ Ri (ti) for some ti ∈ Ti , then there exist γ > 0and a sequence of types {ti,m} on T ∗

i such that ti,m → ti and S∞i [ti,m, γ ] ⊂ Ri for every m.

Proposition 8. Let (T , κ) be a model and ε > 0. Then, (RTi : Ti → 2Ai \ {∅})i∈N defined below

is an ε2 -curb collection:

RTi (ti) ≡ {

Ri ∈Ai : ∃{ti,m}∞m=0 ⊂ T ∗i s.t. ti,m → ti and Ri = S∞

i [ti,m, ε], ∀m}.

These results generate interesting corollaries. First, in Appendix A, we show that Proposition 5immediately follows from Theorem 1. Second, in Section 6.3, we employ Propositions 7 and 8to fully characterize the notion of critical types studied in [12] and [7] in any finite-action gamethat satisfies the richness assumption.

5.4. Proof of Proposition 7

Suppose that (Rj : Tj → 2Aj \ {∅})j∈N is an ε-curb collection in a model (T , κ). By therichness assumption, for every action ai , there exists some θai such that ai is strictly dominantfor the type tθ

ai

i which has common knowledge of θai . Since the action set is finite, there existsε′ > 0 such that S∞

i [tθai

i , ε′] = {ai} for any ai . Let γ = min(ε, ε′). Then, Proposition 7 is impliedby Claim 1 below.

Claim 1. For any j ∈ N and any integer m� 0, there exists a measurable function

Tj ×Aj � (tj ,Rj ) �→ τtj ,Rj ,m ∈ T ∗j

such that for any tj ∈ Tj and any Rj ∈Rj (tj ), we have τmtj ,Rj ,m = tmj and S∞

j [τtj ,Rj ,m, γ ] ⊂ Rj .

Proof. By the definition of ε-curb collection, for each j ∈ N , there exists a measurable function(tj ,Rj ) �→ μtj ,Rj

∈ �(Θ × T−j ×A−j ) that satisfies all conditions in Definition 7.

18 By the finiteness of action sets and the upper hemicontinuity of best replies, a collection (Ri )i∈N is a 0-curb col-lection in finite model (T , κ) if and only if it is an ε-curb collection in (T , κ) for some ε > 0. This is why we focus onthe simpler notion of (0-)curb collection in Section 5.2. However, it can be shown that the equivalence between 0- andε-curb collections no longer holds for infinite models.


We prove the claim by induction on m. For m = 0, fix an arbitrary element a(Rj ) of Rj foreach Rj ∈ Aj , i.e., a(Rj ) ∈ Rj . Since G satisfies the richness assumption, for each tj ∈ Tj and

each Rj ∈ Rj (tj ), define τtj ,Rj ,0 = tθa(Rj )

j . Obviously, τtj ,Rj ,0 is measurable in (tj ,Rj ). Also,S∞

j [τtj ,Rj ,0, γ ] = {a(Rj )} ⊂ Rj .We now assume that the claim is true for m and prove the case for m+1. We define τtj ,Rj ,m+1

as follows:

κ∗τtj ,Rj ,m+1

[F ] ≡ μtj ,Rj

[{(θ, t−j ,R−j ): (θ, τt−j ,R−j ,m) ∈ F

}], ∀F ⊂ Θ × T ∗−j ,

where τt−j ,R−j ,m = (τtk,Rk,m)k �=j . By the measurability of μtj ,Rjand τt−j ,R−j ,m, τtj ,Rj ,m+1 is

well-defined and measurable in (tj ,Rj ). Also, since the marginal distribution of μtj ,Rjon Θ ×

T−j is equal to κtj , the induction hypothesis implies that τm+1tj ,Rj ,m+1 = tm+1

j .

Now fix a γ -valid conjecture σ−j : Θ × T ∗−j → �(A−j ) for τtj ,Rj ,m+1 arbitrarily. Let ϕ−j :Θ × T−j ×A−j → �(A−j ) be given by

ϕ−j (θ, t−j ,R−j )[a−j ] = σ−j (θ, τt−j ,R−j ,m)[a−j ].Since σ−j is γ -valid and S∞−j [τt−j ,R−j ,m, γ ] ⊂ R−j by the induction hypothesis, ϕ−j is a condi-tional conjecture. Also, we have

πμtj ,Rj,ϕ−j

[θ, a−j ] =∫

T−j ×A−i

ϕ−j (θ, t−j ,R−j )[a−j ]μtj ,Rj[θ, dt−j , dR−j ]

=∫

T−j ×A−i

σ−j (θ, τt−j ,R−j ,m)[a−j ]μtj ,Rj[θ, dt−j , dR−j ]

=∫

T ∗−j

σ−j

(θ, t∗−j

)[a−j ]κ∗τtj ,Rj ,m+1

[θ, dt∗−j

] = πτtj ,Rj ,m+1,σ−j[θ, a−j ].

Therefore, S∞j [τtj ,Rj ,m+1, γ ] ⊂ ⋃

ϕ−jBRj (πμtj ,Rj

,ϕ−j, γ ) ⊂ Rj . �

5.5. Proof of Proposition 8

We assume without loss of generality that (T , κ) is embedded in the universal type space(T ∗, κ∗).

For each ti ∈ Ti and each Ri ∈ RTi (ti), let �ti,Ri

be the set of weak∗ limits of all μti,m ∈�(Θ × T ∗−i ×A−i ) such that {ti,m} → ti and S∞

i [ti,m, ε] = Ri for all m, where

μti,m[F ] = κ∗ti,m

({(θ, t−i ):

(θ, t−i , S

∞−i[t−i , ε]) ∈ F

}), ∀F ⊂ Θ × T ∗−i ×A−i . (2)

By the compactness of �(Θ × T ∗−i ×A−i ), we have �ti,Ri�=∅. Also �ti,Ri

depends on (ti ,Ri)

upper hemicontinuously. Thus, it follows from the Kuratowski–Ryll-Nardzewski selection the-orem that we have a measurable function (ti ,Ri) �→ μti ,Ri

∈ �(Θ × T ∗−i × A−i ) such thatμti ,Ri

∈ �ti,Riwhenever Ri ∈RT

i (ti).Hereafter, we fix any ti ∈ Ti and any Ri ∈ RT

i (ti), and prove (a), (b), and (c) in Defini-tion 7. Let {ti,m} be a sequence of types such that ti,m → ti and μti,m → μti,Ri

as m → ∞,and S∞[ti,m, ε] = Ri for all m.
i


Since the marginal distribution of μti,m on Θ × T ∗−i is equal to κ∗ti,m

for all m, ti,m → ti asm → ∞, and κ∗

i is continuous, the marginal distribution of μti ,Rion Θ × T ∗−i is equal to κti .

Thus (a) holds.For each ∈ N, let

F ≡ cl

{(θ, t−i ,R−i ): ∃t ′−i ∈ T ∗−i s.t. d−i

(t−i , t

′−i

)� 1

and S∞−i

[t ′−i , ε

] = R−i

},

F∞ ≡ (Θ × T−i ×A−i ) ∩⋂ ∈N

F

where d−i is the metric on T ∗−i . Note that

F∞ ⊂ {(θ, t−i ,R−i ): t−i ∈ T−i and R−i ∈ RT−i (t−i )

}, (3)

because for each (θ, t−i ,R−i ) ∈ F∞, we have t−i ∈ T−i and there exists {t−i, } ∈N → t−i andS∞−i[t−i, , ε] = R−i for all ∈ N.

Furthermore,{(θ, t−i ,R−i ): S∞−i[t−i , ε] = R−i

} ⊂ F , ∀l ∈N. (4)

Hence, (2) and (4) imply that

1 = μti,m

[{(θ, t−i ,R−i ): S∞−i[t−i , ε] = R−i

}]� μti,m[F ],

i.e., μti,m[F ] = 1. Since μti,m → μti,Rias m → ∞, we have μti ,Ri

[F ] = 1 for all l. As a re-sult, μti ,Ri

[⋂ ∈N F ] = 1. Combining this with μti ,Ri[Θ × T−i × A−i] = 1 by (a), we have

μti ,Ri[F∞] = 1, which, together with (3) implies (b).

We now prove (c). Fix ai ∈ Ai such that ai ∈ BRi (πμti ,Ri,ϕ−i

, ε2 ) for some conditional conjec-

ture ϕ−i . It suffices to show ai ∈ Ri .Pick m sufficiently large so that∑

θ,R−i

∣∣μti,m

[{θ} × T ∗−i × {R−i}] − μti,Ri

[{θ} × T ∗−i × {R−i}]∣∣

� ε

4 maxθ,ai ,a−i|ui(θ, ai, a−i )| . (5)

Fix αi ∈ �(Ai \ {ai}), let ψαi−i (θ,R−i ) ∈ R−i be one of the action profiles of player i’s oppo-

nents that favor action ai most relative to αi , i.e.,

ψαi−i (θ,R−i ) ∈ arg max

a−i∈R−i

[ui(θ, ai, a−i ) − ui(θ,αi, a−i )

].

Since we have ai ∈ BRi (πμti ,Ri,ϕ−i

, ε2 ) for some conditional conjecture ϕ−i , we have

∫Θ×T ∗−i×A−i

[ui

(θ, ai,ψ

αi−i (θ,R−i )) − ui

(θ,αi,ψ

αi−i (θ,R−i ))]

dμti ,Ri� −ε

2. (6)

Now we define σαi−i : Θ × T ∗−i → A−i as

σαi−i (θ, t−i ) = ψ

αi−i

(θ, S∞−i[t−i , ε]

) ∈ S∞−i[t−i , ε],which implies that σ

αi is an ε-valid (pure-action) conjecture. We then have
−i


∑θ,a−i

[ui(θ, ai, a−i ) − ui(θ,αi, a−i )

]π

ti,m,σαi−i

[θ, a−i]

=∫

Θ×T ∗−i×A−i

[ui

(θ, ai,ψ


(θ,αi,ψ


dμti,m

�∫

Θ×T ∗−i×A−i

[ui

(θ, ai,ψ


(θ,αi,ψ


dμti ,Ri− ε

2�−ε,

where the inequalities follow from (5) and (6), respectively.Therefore, for each αi ∈ �(Ai \ {ai}), there exists an ε-valid conjecture σ

αi−i against which ai

is an ε-better reply for ti,m than αi . Then it follows from the usual duality argument that we canfind an ε-valid conjecture, independent of αi , against which ai is an ε-best reply for ti,m. Thus,ai ∈ S∞

i [ti,m, ε] = Ri .

6. Discussion

6.1. Generic complete-information games

In our setup, we fixed the payoffs of a game. A complete-information game can be identifiedwith a profile of utility functions (ui : A → R)i∈N . Let Gc be the set of all such complete-information games endowed with the Euclidean topology. For simplicity, we write S∞

i [G] andS∞

i [G,ε] for the set of rationalizable actions and the set of ε-rationalizable actions, respectively.We define the set of games G∗ as follows:

G∗ = {G ∈ Gc: ∃ε > 0 s.t. S∞

i [G] = S∞i [G,−ε], ∀i ∈ N

}.

That is, G∗ is the set of games in which the set of strictly rationalizable actions coincides withthe set of rationalizable actions in G. We prove the following result in Appendix A.19

Proposition 9. G∗ is open and dense in Gc .

We use two concrete examples to illustrate the openness and the denseness of G∗ in Gc . First,we use the games, G4 and G′

4, defined below to illustrate the openness.

G4:c d

a 1,1 0,0b 0,0 1,1

G′4:

c d

a 1 + γ1,1 + γ2 γ3, γ4b γ5, γ6 1 + γ7,1 + γ8

Clearly, rationalizability and strict rationalizability coincide in G4, and hence we have G4 ∈ G∗.Moreover, we have G′

4 ∈ G∗ for any γ1, . . . , γ8 ∈ (− 12 , 1

2 ). Second, we use the games, G5, G′5

and G′′5, defined below to illustrate the denseness.

G5:c d

a 1,1 1,1b 0,0 1,1

G′5:

c d

a 1,1 1,1b −ζ,0 1 − ζ,1

G′′5:

c d

a 1,1 1,1 − ζ

b −ζ,0 1 − ζ,1 − ζ

19 The result has been mentioned in previous papers, e.g., [29, p. 18] and [25, footnote 20].


Clearly, G5 /∈ G∗. However, for any (small) ζ > 0, action a is the unique (strict) rationalizableaction for player 1 in G′

5; actions a and c are the unique (strict) rationalizable actions for players1 and 2 in G′′

5, respectively. In Appendix A.3, we first show that G′ ∈ G∗ if G′ is sufficientlyclose to G ∈ G∗ (i.e., openness); second, we provide a systematic way to construct a sufficientlynearby game G′ ∈ G∗ of any given game G ∈ Gc (i.e., denseness).

In light of Proposition 5, Proposition 9 implies that, in generic complete-information games,rationalizable actions, strictly rationalizable actions, actions that can be WY-selected, and actionsthat can be robustly selected all coincide. While this seems to contradict the generic impossibilityresult in Propositions 3 and 4, the setups in which the two results are derived are different. ForPropositions 3 and 4, we fix the payoffs and vary the hierarchies of beliefs, while for Proposi-tion 9, we vary the payoffs and fix the (complete-information) hierarchy of belief.

This contrast raises the question whether the generic impossibility of robust selection inPropositions 3 and 4 still holds if we vary both payoffs and beliefs. To study this issue, fix asubset of parameters {θai }i∈N, ai∈Ai

and fix the payoff function for each θai such that

ui

(θai , ai, a−i

)> ui

(θai , a′

i , a−i

), ∀a′

i ∈ Ai \ {ai}, ∀a−i ∈ A−i .

Assume that Θ ≡ Θ \ {θai }i∈N, ai∈Ai�=∅. Define Gr =R|N×Θ×A|. Then, each profile of payoff

functions (ui)i∈N ∈ R|N×Θ×A| can be identified with a game (Θ, (ui)i∈N) which satisfies therichness assumption. Let Γi = Gr × T ∗

i and define

Mi = {(G, ti) ∈ Γi :

∣∣S∞i [ti ,G,0]∣∣ > 1

}.

Also for each G ∈ R|N×Θ×A| define Ui,G = {ti ∈ T ∗i : |S∞

i [ti ,G,0]| = 1}, U−i,G =∏j �=i Uj,G, and

Bi = {(G, ti) ∈ Mi : κ∗

ti[Θ × U−i,G] = 1

}.

The following result (proved in Appendix A) extends the generic impossibility of robust se-lection in Propositions 3 and 4 to the space Mi .20

Proposition 10. Given the set Gr = R|N×Θ×A| of games based on Θ that satisfy the richnessassumption, Bi is a residual set in Mi ; moreover, for every (G, ti) ∈ Bi , ti does not admit anyrobust selection in G.

6.2. Robust selection and robust prediction

WY show that every rationalizable action can be WY-selected. Conversely, by the upper hemi-continuity of the rationalizability correspondence (see [9, Theorem 2]), every action that can beWY-selected must also be rationalizable. That is, an action can be WY-selected if and only if itis rationalizable. Our results show that robust selection is strictly stronger than rationalizabilityand strictly weaker than strict rationalizability.

Though robust selection refines rationalizability, some types may still have multiple actionsthat can be robustly selected. For example, the complete-information type in the global gamehas two strict equilibrium actions, both of which can be robustly selected. In the absence ofprecise measurement of payoffs, actions that are not robustly selected are fragile predictions.

20 It can be shown that Mi is a Polish space and thus a Baire space.


They are predictions inferior to those actions that can be robustly selected and equipped with therobustness feature.

Among those actions that can be robustly selected, which one should we pick? Equivalently,what perturbation of higher-order beliefs is the most sensible approximation of the strategicsituation that we are analyzing? The answer to this question is subject to the judgment of themodeler. For example, if we think that the global game (i.e., Example 2 in Section 2) is a goodapproximation of the strategic situation that we are facing, we would predict that player i whoobserves xi = 3

4 will choose “Attack,” whereas if we think that the alternative perturbation inExample 2 in WY is more sensible, we would predict “No Attack.” If no perturbation is moreappealing than the other, we may follow WY in saying that a prediction is appealing only whenit holds under both actions.

6.3. Robust selection and critical types

In [12] and [7], a type ti is said to be G-critical (or display strategic discontinuity in G)if there exist ε > 0, an action ai , and a sequence of types {ti,m} with ti,m → ti such that ai ∈S∞

i [ti ,G,0] and ai /∈ S∞i [ti,m,G, ε] for every m. A type is critical if it is a G-critical type for

some finite-action game G.For instance, by the upper hemicontinuity of ICR, a type who has a unique rationalizable

action in G is not G-critical despite having a robust selection. In contrast, a type who has multiplerationalizable actions and admits a robust selection in game G must be G-critical. Indeed, thecomplete-information type in [27] is a prominent example of G-critical types.

Ely and Peski [12] characterize critical types. Here we fix a finite-action game G that satisfiesthe richness assumption, and we characterize G-critical types by means of the ε-curb collectionin G, as follows (the proof is in Appendix A):

Proposition 11. Suppose that G satisfies the richness assumption. Given a model (T , κ), a typeti ∈ Ti is G-critical if and only if there exists some ε-curb collection (Rj )j∈N in (T , κ) withε > 0 such that S∞

i [ti] \ Ri �=∅ for some Ri ∈Ri (ti ).

Moreover, let Bi be defined as in (1) and NCi be the set of types in T ∗i which are not G-critical.

Note that NCi is measurable.21 We prove the following proposition in Appendix A.

Proposition 12. If κ∗ti[Θ × NC−i] = 1, then ti ∈ NCi . In particular, Bi ⊂ NCi .

In other words, types in Bi defined in Section 4 are not G-critical, although they admit WYselections. Furthermore, since every type that has multiple rationalizable actions and admits arobust selection must be critical, Proposition 12 implies Proposition 3. Also, combined with

21 To see this, for each Ri ∈ Ai and ε � 0, let Ti(Ri , ε) = {ti ∈ T ∗i

: S∞i

[ti ,G, ε] = Ri }. Since S∞i

[·,G, ε] is upperhemicontinuous, Ti(Ri , ε) is a finite intersection of open or closed subsets of T ∗

i, and hence measurable. By the definition

of NCi , we have

NCi =⋃

Ri∈Ai

(Ti(Ri ,0) \

⋃ ∈N

⋃ai∈Ri

⋃Si∈Ai , ai /∈Si

clTi

(Si ,

1

)),

which is measurable.


Proposition 4, Proposition 12 implies that G-critical types are non-generic in the set of typeswith multiple rationalizable actions.22

6.4. Set-valued robust selection

We can also define robust selection for a set of actions. We say that a set of actions Ri can berobustly selected for type ti if there is an ε > 0 and a sequence of types {ti,m} such that ti,m → tiand S∞

i [ti,m,G, ε] = Ri for every m. Our results can accommodate the set version of robustselection with minor modifications. In particular, Proposition 12 shows that no proper subsetof rationalizable actions of any type in Bi can be robustly selected. Moreover, Proposition 8shows that every set of actions that can be robustly selected must belong to a curb collection.Conversely, Proposition 7 shows that every set of actions in a curb collection contains a subsetthat can be robustly selected.

6.5. Ex ante selection versus interim selection

Both WY selection and robust selection are interim notions. That is, an action is selectedaccording to the interim (hierarchy of) beliefs of a type. An alternative approach is to studyselections from an ex ante viewpoint. In this case, a type is replaced by a type space whichis usually associated with a common prior. For example, Engl [13] considered perturbations ofpriors and proved that for any sequence of ex ante strategy profiles {σm} that converges to anequilibrium in the original game, the strategy profile σm must be an ex ante εm-equilibrium inthe perturbed games with εm → 0.23

A recent paper that is closer to our work is [20]. The paper obtains two main results: (1) forany ex ante equilibrium and any sequence of ex ante perturbations of the game (including pertur-bations of priors, type spaces, and payoff functions), there is a corresponding sequence of ex anteεm-equilibria with εm → 0 converging to that equilibrium; and (2) under a stronger notion of per-turbation, the approximating ex ante εm-equilibria can be strengthened to interim εm-equilibriain which almost all types choose εm-best replies. Jackson, Rodriguez-Barraquer and Tan [20]conclude from their results that no refinement of equilibrium is robust to slight perturbations inbest-reply behavior or to underlying preferences.

Both [20] and our paper employ solution concepts based upon ε-best replies to approximatesolution concepts based upon 0-best replies. However, [20] can approximate any equilibriumin any incomplete-information game with a sequence of εm-equilibria along any sequence oftheir interim perturbations, while such an approximation is only generically true in our setup.This discrepancy occurs for two reasons. First, we require that all types play interim ε-bestreplies, while their first result allows a set of types with positive measure to play actions differentfrom their interim ε-best replies (as long as they aggregate to an ex ante ε-best reply). Second,the stronger notion of perturbations in their second result excludes “crazy types” that behavevery differently from types in the original game. In contrast, WY and we adopt the finite-order

22 Without imposing the richness assumption, Ely and Peski [12] show that critical types, i.e., types that are G-criticalfor some G, are non-generic in the universal type space. Under the richness assumption, Ely and Peski’s non-genericityresult also follows from the non-genericity of types with multiple rationalizable actions proved by WY.23 The result of Engl [13] generalizes the earlier result of Fudenberg and Levine [17] to games of incomplete informa-tion.


perturbations which allow any belief in the tail of a perturbed hierarchy; hence, the type space ofthe perturbed type may include “crazy types.”24

The ex ante perspective is also adopted by Kajii and Morris [21]. They define a robust equi-librium as a Nash equilibrium in a complete-information game which is played with sufficientlylarge probability in some Bayesian Nash equilibrium on any common priors which are suffi-ciently close to the complete-information scenario. This approach has been extended by Oyamaand Tercieux [26] to allow for perturbations with non-common priors. The solution concept in[21] and [26] is based upon 0-best replies. Haimanko and Kajii [19] (Chassang and Takahashi [5],resp.) extend [21] to study the notion of an approximately (strictly, resp.) robust equilibriumbased upon ε-best replies (strict best replies, resp.).

Appendix A

In the appendix, we first prove Propositions 2, 4, and 9, which hold independently of otherresults in the paper. We then use Theorem 1 to derive Proposition 5 and use Propositions 7 and8 to derive Propositions 11 and 12, which imply Proposition 3 as we discussed in the main text.Finally, we use Proposition 3 and Lemmas 1 and 2 to derive Proposition 10.

A.1. Proof of Proposition 2

Proposition 2. Given a game G = (Θ, (ui)i∈N) and ε > 0,

S∞i [ti ,G, ε] =

⋃{(G,f ): ‖G−G‖f � ε

2 }

⋃{ti : fi (ti )=ti }

S∞i [ti , G,0], ∀ti ∈ T ∗

i .

Proof. First, we prove the “⊃” direction. Fix any G that is ε2 -close to G with an onto mapping

f : Θ → Θ . We inductively prove

Ski [ti , G,0] ⊂ Sk

i

[fi(ti ),G, ε

], ∀ti ∈ T ∗

i , ∀k.

Clearly, S0i [ti , G,0] = Ai = S0

i [fi(ti ),G, ε]. Suppose that the claim holds for k and we prove theclaim for k + 1. Let ai ∈ Sk+1

i [ti , G,0]. Then, there is a conjecture σ−i : Θ × T ∗−i → �(A−i ) ofplayer i such that

(a) σ−i (θ , t−i )[a−i] > 0 ⇒ aj ∈ Skj [tj , G,0],∀j �= i;

(b) ai ∈ BRi (πti ,σ−i, G,0).

Define νi ∈ �(Θ × T ∗−i × A−i ) as

νi

[E × {a−i}

] =∫

E

σ−i (θ , t−i )[a−i]κ∗ti[dθ, dt−i], ∀E ⊂ Θ × T ∗−i , ∀a−i ∈ A−i . (7)

We then induce νi ∈ �(Θ × T ∗−i × A−i ) via f as

24 This is also why the lower hemicontinuity result in Engl [13] fails to hold when he considers the notion of weakconvergence of priors which allows “crazy types” to occur with small probability.


νi

[E × {a−i}

] = νi

[{(θ , t−i , a−i ):

(f (θ), f−i (t−i )

) ∈ E}]

,

∀E ⊂ Θ × T ∗−i , ∀a−i ∈ A−i . (8)

Let σ ′−i : Θ ×T ∗−i → �(A−i ) be a conditional distribution of νi on Θ ×T ∗−i (see [11, 10.2.2 The-

orem]). Since σ−i (θ , t−i )[a−i] > 0 only if aj ∈ Skj [tj , G,0] for every j �= i, by the induc-

tion hypothesis, for κ∗fi (ti )

-almost all (θ, t−i ), σ−i (θ, t−i )[a−i] > 0 only if aj ∈ Skj [tj ,G, ε]

for every j �= i. On the κ∗fi (ti )

-measure zero set of (θ, t−i ), we modify σ ′−i to be some mea-

surable selection from Sk−i[t−i ,G, ε] to get a conjecture σ−i : Θ × T ∗−i → �(A−i ) such thatσ−i (θ, t−i )[a−i] > 0 only if aj ∈ Sk

j [tj ,G, ε] for every j �= i. Moreover, since κ∗fi (ti )

[E] =κ∗ti[{(θ , t−i ): (f (θ), f−i (t−i )) ∈ E}] for any E ⊂ Θ × T ∗−i (see footnote 8), by (7) and (8),

we have πf (ti ),σ−i[θ, a−i] = ∑

θ∈f −1(θ) πti ,σ−i[θ , a−i]. Since G is ε

2 -close to G via f and ai ∈BRi (πti ,σ−i

, G,0), it follows that ai ∈ BRi (πf (ti ),σ−i,G, ε). Therefore, ai ∈ Sk+1

i [fi(ti ),G, ε].The “⊂” direction is proved by construction. In particular, we let Θ = Θ × A and Ti =

{(ti , ai): ti ∈ T ∗i and ai ∈ S∞

i [ti ,G, ε]}. For each ti = (ti , ai) ∈ Ti , we let κtibe any belief over

Θ × T−i ⊂ Θ × A × T ∗−i × A−i that puts probability 1 on ai and whose marginal over Θ ×T ∗−i × A−i is equal to the belief that ε-rationalizes ai for ti in G. Also for each θ = (θ, ai, a−i ),we have

ui

(θ , a′

i , a′−i

) ={

ui(θ, ai, a′−i ) + ε

2 , if a′i = ai;

ui(θ, a′i , a

′−i ) − ε2 , if a′

i �= ai.

Note that ti and ti = (ti , ai) have the same hierarchy of beliefs over Θ . Also it follows that theprofile of correspondences (Ri)i∈N , where Ri(ti ) = {ai} for any ti = (ti , ai), satisfies the bestreply property in G = (Θ, (ui)i∈N) with model (Θ, T , κ), and hence ai is rationalizable forti = (ti , ai) in G. �A.2. Proof of Proposition 4

Proposition 4. Suppose that G satisfies the richness assumption. Then, Bi is a residual set.

Proof. Since G satisfies the richness assumption, both Mi and Bi are nonempty. Define

Bi,n ≡{ti ∈ Mi : κ∗

ti[Θ × U−i] > 1 − 1

n

}.

Since Bi = ⋂∞n=1 Bi,n, Proposition 4 is implied by the following two lemmas.

Lemma 1. Bi,n is open in Mi .

Proof. By Proposition 2 in WY, U−i is open. Then, by the definition of the weak∗ topology,{μ ∈ �(Θ ×T ∗−i ): μ[Θ ×U−i] > 1− 1

n} is open. Since κ∗· = κ∗

i (·) is a homeomorphism betweenT ∗

i and �(Θ × T ∗−i ), it follows that Bi,n is open. �Lemma 2. Suppose that G satisfies the richness assumption. Then, Bi is dense in Mi .

Proof. Recall that T ∗i is a compact metric space and di is the metric on T ∗

i . We now divide theproof into three steps.


We say that ti ∈ T ∗i is a finite type if there exists t ′i in a model (T , κ∗) with Tj ⊂ T ∗

j and|Tj | < ∞ for every j ∈ N .

Step 1. For any finite type ti ∈ Mi in (T , κ∗), there is some conjecture σ−i : Θ × T−i → �(A−i )

which is valid for ti and BRi (πti ,σ−i) contains more than one action.

Define

Σ−i ≡ {σ−i : Θ × T−i → �(A−i ): σ−i (θ, t−i ) ∈ �

(S∞−i[t−i]

), ∀(θ, t−i )

};Σ

ai−i ≡ {σ−i∈ Σ−i : {ai} = BRi (πti ,σ−i

)}, ∀ai ∈ Ai.

Σ−i is the set of all valid conjectures, and Σai−i is the set of valid conjectures to which ai is the

unique best reply for ti . Observe that Σ−i is a connected subset of RΘ×T−i×A−i (recall that T−i

is a finite set). Moreover, for every ai ∈ Ai , Σai−i is an open subset of Σ−i and Σ

ai−i ∩ Σa′i−i =∅ if

ai �= a′i .

Now suppose that |BRi (πti ,σ−i)| = 1 for any σ−i ∈ Σ−i . Then we have Σ−i = ⋃

ai∈AiΣ

ai−i .Since Σ−i is connected, it cannot be equal to the disjoint union of multiple nonempty opensubsets. Thus, Σ

ai−i =∅ for all but one action. This is a contradiction to ti ∈ Mi .

Step 2. For any finite type ti ∈ Mi in (T , κ∗), there is a sequence of finite types {ti,m} such thatti,m → ti and ti,m ∈ Bi for all m.

By Step 1, there is some valid conjecture σ−i : Θ × T−i → �(A−i ) for ti such thatBRi (πti ,σ−i

) has at least two actions.Since G satisfies the richness assumption, by Proposition 1 in WY, for each t−i ∈ T−i and each

a−i ∈ S∞−i[t−i], there exists a sequence of types {τt−i ,a−i ,m} on T ∗−i such that τt−i ,a−i ,m → t−i andS∞−i[τt−i ,a−i ,m] = {a−i}. We define ti,m ∈ T ∗

i as follows:

κ∗ti,m

[θ, τt−i ,a−i ,m] = κ∗ti[θ, t−i] × σ−i (θ, t−i )[a−i], ∀[θ, t−i] ∈ Θ × T−i , ∀a−i ∈ A−i .

(9)

Then, ti,m is a finite type and κ∗ti,m

(Θ ×U−i ) = 1. Also, since τt−i ,a−i ,m → t−i for each t−i ∈ T−i

and each a−i ∈ S∞−i[t−i], we have κ∗ti,m

→ κ∗ti

. Since (κ∗)−1 is continuous, we have ti,m → ti .We now show that ti,m has multiple rationalizable actions. By our construction, the valid

conjecture σ−i,m : Θ × T ∗−i → �(A−i ) for ti,m is uniquely determined by

σ−i,m(θ, τt−i ,a−i ,m)[a−i] = 1 (10)

on the support of κ∗ti,m

. Thus, for any θ ∈ Θ and any a−i ∈ A−i , we have

πti,m,σ−i,m[θ, a−i] =

∑t−i∈T−i

κ∗ti,m

[θ, τt−i ,a−i ,m] × σ−i,m(θ, τt−i ,a−i ,m)[a−i]

=∑

t−i∈T−i

κ∗ti,m

[θ, τt−i ,a−i ,m]

=∑

t−i∈T−i

κ∗ti[θ, t−i] × σ−i (θ, t−i )[a−i]

= πt ,σ [θ, a−i],
i −i


where the first equality follows from the definition of πti,m,σ−i,m; the second equality follows from

(10); the third equality follows from (9); the last equality follows from the definition of πti ,σ−i.

Thus we have πti,m,σ−i,m= πti ,σ−i

. Therefore, |BRi (πti,m,σ−i,m)| = |BRi (πti ,σ−i

)| > 1.To sum up, we find a sequence of finite types {ti,m} such that ti,m → ti and ti,m ∈ Mi , ti,m ∈ Bi ,

and κ∗ti,m

(Θ × U−i ) = 1, i.e., ti,m ∈ Bi,m for all m.

Step 3. Bi is dense in Mi .

Let di be the metric on T ∗i . Take any ti ∈ Mi and any ε > 0. First, by [6, Lemma 3], there

is some finite type t ′i ∈ T ∗i such that S∞

i [ti] = S∞i [t ′i ] and di(ti , t

′i ) < ε

2 . Then, for any ε > 0,by Step 2, there is some t ′′i ∈ Bi such that di(t

′i , t

′′i ) < ε

2 . Hence, di(ti , t′′i ) < ε. Therefore, Bi is

dense in Mi . �A.3. Proof of Proposition 9

For any G = (uj )j∈N and G′ = (u′j )j∈N in Gc, define ‖G − G′‖ = maxj∈N,a∈A |uj (a) −

u′j (a)|.

Proposition 9. G∗ is open and dense in Gc.

Proof. In this proof, we write S∞[G] = ∏j∈NS∞

j [G]. We first prove that G∗ is open. Fix anyG ∈ G∗. Thus, for some ε > 0, S∞

j [G] = S∞j [G,−ε] for every j . Furthermore, by finiteness of

actions, there is some ε′ > 0 such that S∞j [G] = S∞

j [G,ε′] for every j . Hence, there is someε∗ > 0, such that

S∞[G,−ε∗] = S∞[G] = S∞[

G,ε∗]. (11)

Then for any G′ with ‖G′ − G‖ < ε∗4 , we have

S∞[G,−ε∗] ⊂ S∞

[G′,−ε∗

2

]⊂ S∞[

G′] ⊂ S∞[G,ε∗]. (12)

Then, (11) and (12) imply that S∞[G′] = S∞[G′,− ε∗2 ]. Thus,

∥∥G′ − G∥∥ <

ε∗

4⇒ G′ ∈ G∗.

We conclude that G∗ is open.We next prove that G∗ is dense by the following claim.

Claim 2. For any ε > 0 and any G = (uj )j∈N /∈ G∗, there exits G′ ∈ Gc with ‖G′ − G‖ < ε suchthat S∞[G] � S∞[G′].

To see that Claim 2 implies that G∗ is dense, pick any G ∈ Gc . Construct a sequence of gamesin Gc as follows: (1) set G1 = G; (2) for any k � 1, if Gk ∈ G∗, we set Gk+1 = Gk ; if Gk /∈ G∗,by Claim 2, we can find some Gk+1 with ‖Gk+1 − Gk‖ < ε such that S∞[Gk]� S∞[Gk+1].

Thus, as long as, Gk′/∈ G∗ for all k′ < k, we construct a sequence of games {Gk} such that

S∞[G1]� S∞[

G2]� · · ·� S∞[Gk

].


The sets of rationalizable actions cannot shrink for more than∑

j∈N |Aj | times. Hence, Gk ∈ G∗

for some k �∑

j∈N |Aj |. Moreover, ‖G − Gk‖ = ‖G1 − Gk‖ � ε × ∑j∈N |Aj |. Since ε is

arbitrary, we conclude that G∗ is dense.We now turn to prove Claim 2. Fix any ε > 0 and G = (uj )j∈N /∈ G∗. First, observe that

G′ ∈ G∗ if and only if the set S∞[G′] satisfies the strict best reply property, i.e., for any j ∈ N

and aj ∈ S∞j [G′], there exists some π ∈ �(S∞−j [G′]) such that BRj (π,G′) = {aj }. Since G =

(uj )j∈N /∈ G∗, it follows that there exist i ∈ N and a∗i ∈ S∞

i [G] such that a∗i is not a strict best

reply to any π ∈ �(S∞−j [G]). By the usual duality argument, there exists αi ∈ �(Ai \ {a∗i }) such

that

ui(αi, a−i ) − ui

(a∗i , a−i

)� 0, ∀a−i ∈ S∞−i[G], (13)

where ui(αi, a−i ) = ∑ai∈Ai

αi[ai]ui(ai, a−i ). Second, by finiteness of actions, there is someε′ > 0 such that

S∞−i[G] = S∞−i

[G,ε′]. (14)

We define G′ = (u′j )j∈N as follows:

u′j (aj , a−j ) =

{ui(a

∗i , a−i ) − γ, if j = i and aj = a∗

i ;uj (aj , a−j ), otherwise,

(15)

where γ = 12 min{ε, ε′}. Clearly, ‖G − G′‖ = γ < ε. By (14), we have

S∞j

[G′] ⊂ S∞

j [G,2γ ] ⊂ S∞j

[G,ε′] = S∞

j [G], ∀j ∈ N. (16)

Then, (13), (15), and (16) imply that

u′i (αi, a−i ) − u′

i

(a∗i , a−i

)> 0, ∀a−i ∈ S∞−i

[G′].

As a result, a∗i is not rationalizable in G′. This together with (16) for j = i implies that S∞

i [G] �S∞

i [G′]. Thus, Claim 2 follows. �A.4. Proof of Proposition 5

Proposition 5. Given a model (T , κ) and ti ∈ Ti , (i) an action ai can be robustly selected for tionly if ai is a strict best reply for ti ; (ii) if G satisfies the richness assumption, then an action ai

can be robustly selected for ti if ai is strictly rationalizable for ti .

Proof. For (i), suppose that ai can be robustly selected for ti . Then, by the “only if” part ofTheorem 1, {ai} ∈ Ri (ti) for some ε-curb collection (Rj )j∈N .25 Thus, there exists μti ,Ri

∈�(Θ × T−i × A−i ) such that the marginal distribution of μti ,Ri

on Θ × T−i is equal to κti andBRi (πμti ,Ri

,ϕ−i, ε) = {ai} for any conditional conjecture ϕ−i . Since at least one such conditional

conjecture exists, ai is a strict best reply for ti .For (ii), suppose that ai ∈ S∞

i [ti ,−ε] for some ε > 0. Then, it follows from the Kuratowski–Ryll-Nardzewski selection theorem that

Rj (tj ) = {{aj }: aj ∈ S∞j [tj ,−ε]}, ∀tj ∈ Tj

defines an ε-curb collection. Since G satisfies the richness assumption, it follows from the “if”part of Theorem 1 that ai can be robustly selected for ti . �25 Note that the “only if” direction of Theorem 1 does not rely on the richness assumption.


A.5. Proof of Proposition 11

Proposition 11. Suppose that G satisfies the richness assumption. Given a model (T , κ), a typeti ∈ Ti is G-critical if and only if there exists some ε-curb collection (Rj )j∈N in (T , κ) withε > 0 such that S∞

i [ti] \ Ri �=∅ for some Ri ∈Ri (ti ).

Proof. Suppose that there is some ε-curb collection (Rj )j∈N and some Ri ∈ Ri (ti) such thatsome action ai ∈ S∞

i [ti] and ai /∈ Ri . Since Gsatisfies the richness assumption, by Proposition 7,there exist γ > 0 and a sequence of types {ti,m} such that ti,m → ti and S∞

i [ti,m, γ ] ⊂ Ri forevery m. Since S∞

i [ti,m, γ ] ⊂ Ri for every m, it follows that ai ∈ S∞i [ti] and ai /∈ S∞

i [ti,m, γ ] forevery m. That is, ti is G-critical.

Conversely, suppose that ti is G-critical. Then, there exist ε > 0, an action ai , and a sequenceof types {ti,m} with ti,m → ti such that ai ∈ S∞

i [ti] and ai /∈ S∞i [ti,m, ε] for every m. By finite-

ness of actions, there exist some Ri ⊂ Ai and some subsequence of {ti,mk} such that ti,mk

→ tiand S∞

i [ti,mk, ε] = Ri for all k. Hence, by Proposition 8, Ri ∈ RT

i (ti) for the ε2 -curb collection

(RTj )j∈N , and moreover, ai ∈ S∞

i [ti] and ai /∈ Ri . �A.6. Proof of Proposition 12

Proposition 12. If κ∗ti[Θ × NC−i] = 1, then ti ∈ NCi . In particular, Bi ⊂ NCi .

Proof. By Proposition 11, we prove the proposition by showing that for any ε > 0 and anyε-curb collection (Rj )j∈N in (T ∗, κ∗),

Ri ∈ Ri (ti ) ⇒ S∞i [ti] ⊂ Ri. (17)

Fix an ε > 0 and an ε-curb collection (Rj )j∈N in (T ∗, κ∗). Then, there exists a measurablefunction (tj ,Rj ) �→ μtj ,Rj

∈ �(Θ × T ∗−j ×A−j ) that satisfies all conditions in Definition 7.Fix a valid conjecture σ−i :Θ × T ∗−i → �(A−i ) for ti arbitrarily. By Proposition 11, for each

t−i ∈ NC−i and R−i ∈R−i (t−i ), we have S∞−i[t−i] ⊂ R−i .26 Let ϕ−i : Θ×T ∗−i ×A−i → �(A−i )

be a conditional conjecture that satisfies ϕ−i (θ, t−i ,R−i )[a−i] = σ−i (θ, t−i )[a−i] for any t−i ∈NC−i and any R−i ∈ R−i (t−i ). Then it follows that πμti ,Ri

,ϕ−i= πti ,σ−i

, and thus S∞i [ti] ⊂⋃

ϕ−iBRi (πμti ,Ri

,ϕ−i) ⊂ ⋃

ϕ−iBRi (πμti ,Ri

,ϕ−i, ε) ⊂ Ri . Thus (17) holds.

Finally, by the upper hemicontinuity of S∞j [·] for j �= i, we have U−i ⊂ NC−i . It follows that

Bi ⊂ NCi . �A.7. Proof of Proposition 10

Proposition 10. Given the set Gr = R|N×Θ×A| of games based on Θ that satisfy the richnessassumption, Bi is a residual set in Mi ; moreover, for every (G, ti) ∈ Bi , ti does not admit anyrobust selection in G.

Proof. We prove the result in two steps:

Step 1. Bi,n ≡ {(G, ti) ∈ Mi : κ∗ti[Θ × U−i,G] > 1 − 1

n} is open in Γi .

26 Note that the “only if” direction of Proposition 11 does not rely on the richness assumption.


Suppose that κ∗ti[Θ × U−i,G] > 1 − 1

n. For every t−i ∈ U−i,G, we know that

|S∞−i[t−i ,G,0]| = 1; then, there is some ε(t−i ) > 0 such that |S∞−i[t−i ,G, ε(t−i )]| = 1. Thus,U−i,G = ⋃∞

m=1 U−i,G,m where

U−i,G,m ≡{t−i ∈ U−i,G:

∣∣∣∣S∞−i

[t−i ,G,

1

m

]∣∣∣∣ = 1

}.

Since κ∗ti[Θ × U−i,G] > 1 − 1

n, it follows that for some m,

κ∗ti[Θ × U−i,G,m] > 1 − 1

n. (18)

Moreover, by upper hemicontinuity of ICR, for each t−i ∈ U−i,G,m, there is an open set Ot−isuch

that t−i ∈ Ot−i⊂ U−i,G,m. Thus, Θ ×U−i,G,m is an open set. By (18) and the proof of Lemma 1,

there is some open set Oti such that ti ∈ Oti and

κ∗t ′i[Θ × U−i,G,m] >

1

n, ∀t ′i ∈ Oti . (19)

By [9, Lemma 10], there is an open set OG such that G ∈ OG and |S∞−i[t−i ,G′,0]| = 1 for every

t−i ∈ U−i,G,m and G′ ∈ OG. That is,

U−i,G,m ⊂ U−i,G′ , ∀G′ ∈ OG. (20)

By (19) and (20), we conclude that (G, ti) ∈ OG × Oti and

κ∗t ′i[Θ × U−i,G′ ] >

1

n, ∀(

G′, t ′i) ∈ OG × Oti .

Hence, Bi,n is open in Γi .

Step 2. There is a residual set Bi ⊂ Mi such that for every (G, ti) ∈ Bi , ti does not admit anyrobust selection in G.

By Lemma 2, for each (G, ti) ∈ Mi with (G, ti) /∈ Bi , there is a sequence of types {ti,m} suchthat ti,m → ti and κ∗

ti,m[Θ ×U−i,G] = 1 for each m, i.e., (G, ti,m) ∈ Bi . Hence, Bi is dense in Mi .

Moreover, by Step 1, Bi,n is open in Γi . Since Bi ⊂ Bi,n, it follows that Bi,n is open and densein Mi . Since

⋂∞n=1 Bi,n = Bi , it follows that Bi is a residual set in Mi . Finally, by Proposition 3,

for every (G, ti) ∈ Bi , ti do not admit any robust selection in G. Thus, Step 2 follows. �References

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The robust selection of rationalizability

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