Generalized potentials and robust sets of equilibria

Journal of Economic Theory 124 (2005) 45–78www.elsevier.com/locate/jet

Generalized potentials and robust sets ofequilibria

Stephen Morrisa,∗, Takashi UibaDepartment of Economics, Yale University, Cowles Foundation, P.O. Box 208281, New Haven,

CT 06520-8281, USAbFaculty of Economics, Yokohama National University, Japan

Received 24 April 2003; final version received 3 June 2004Available online 11 February 2005

Abstract

This paper introduces generalized potential functions of complete information games and studies therobustness of sets of equilibria to incomplete information. A set of equilibria of a complete informationgame is robust if every incomplete information game where payoffs are almost always given by thecomplete information game has an equilibrium which generates behavior close to some equilibriumin the set. This paper provides sufficient conditions for the robustness of sets of equilibria in termsof argmax sets of generalized potential functions. These sufficient conditions unify and generalizeexisting sufficient conditions. Our generalization of potential games is useful in other game theoreticproblems where potential methods have been applied.© 2004 Elsevier Inc. All rights reserved.

JEL classification:C72; D82

Keywords:Incomplete information; Potential; Refinements; Robustness

1. Introduction

Outcomes of a game with common knowledge of payoffs may be very differentfrom outcomes of the game with a “small” departure from common knowledge, as

This paper combines an earlier paper by Ui of the same title that introduced generalized potentials; and apaper by Morris on “Potential Methods in Interaction Games” that introduced characteristic potentials and localpotentials.

∗ Corresponding author. Fax: +1 203 432 6167.E-mail addresses:[email protected](S. Morris),[email protected](T. Ui).

0022-0531/$ - see front matter © 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jet.2004.06.009

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46 S. Morris, T. Ui / Journal of Economic Theory 124 (2005) 45–78

demonstrated by Rubinstein[27] and Carlsson and van Damme[4]. This observation leadKajii and Morris [12] to study which equilibria of complete information games are notmuch affected by weakening the assumption of common knowledge; they studied the ro-bustness of equilibria to incomplete information. An equilibrium of a complete informationgame is robust if every incomplete information game with payoffs almost always givenby the complete information game has an equilibrium which generates behavior close tothat equilibrium.

Kajii and Morris [13] demonstrated that robustness can be seen as a very strongrefinement of Nash equilibria. The refinements literature examines what happens to a givenNash equilibrium in perturbed versions of the complete information game. A weak class ofrefinements requires only that the Nash equilibrium continues to be an equilibrium in somenearby perturbed game. The notion of perfect equilibria due to Selten[28] is the leading ex-ample of this class. A stronger class requires that the Nash equilibrium continues to be playedin all perturbed nearby games. The notion of stable equilibria due to Kohlberg and Mertens[14] or that of strictly perfect equilibria due to Okada[21] are leading examples of this class.Robustness belongs to the latter, stronger, class of refinements. Moreover, robustness to in-complete information allows an extremely rich set of perturbed games. In particular, whileKohlberg and Mertens[14] allowed only independent action trembles across players, thedefinition of robustness leads to highly correlated trembles and thus an even stronger refine-ment. Indeed, Kajii and Morris[12] constructed an example in the spirit of Rubinstein[27] toshow that even a game with a unique Nash equilibrium, which is strict, may fail to have anyrobust equilibrium.

Kajii and Morris [12] and Ui [34] provided sufficient conditions for the robustness ofequilibria. Kajii and Morris [12] introduced the concept ofp-dominance wherep = (p1, . . . , pn) is a vector of probabilities.1 An action profile is ap-dominant equi-librium if each player’s action is a best response whenever he assigns probability at leastpi to his opponents choosing actions according to the action profile. Kajii and Morris[12]showed that ap-dominant equilibrium with

∑i pi < 1 is robust. Ui[34] considered robust

equilibria of potential games, a class of complete information games possessing potentialfunctions. As defined by Monderer and Shapley[18], a potential function is a function onthe action space that incorporates information about players’ preferences over the actionspace that is sufficient to determine all the equilibria. Ui[34] showed that the action profilethat uniquely maximizes a potential function is robust.

These two results are developed in quite different frameworks, and on the face of it therelationship between the two is not clear. The purpose of this paper is to provide a newsufficient condition for robustness, which unifies and generalizes the sufficient conditionsprovided by Kajii and Morris[12] and Ui [34]. Furthermore, the condition can be usedto provide new sufficient conditions for robustness and applies not only to the robustnessof equilibria but also the robustness ofsetsof equilibria. This paper introduces general-ized potential functions and provides the condition in terms of argmax sets of generalizedpotential functions.

1 Morris et al.[19] earlier presented results about the case where each player had the samepi .

S. Morris, T. Ui / Journal of Economic Theory 124 (2005) 45–78 47

We start by defining the robustness concept as a set valued one,2 the robustness of setsof equilibria to incomplete information. A set of equilibria of a complete information gameis robust if every incomplete information game with payoffs almost always given by thecomplete information game has an equilibrium which generates behavior close to someequilibrium in the set. If a robust set is a singleton then the equilibrium is robust in the senseof Kajii and Morris[12,13]. Because some games have no robust equilibria, it is natural toask if a set of equilibria is robust.

We then introduce generalized potential functions. A generalized potential function is afunction on a covering of the action space, a collection of subsets of the action space suchthat the union of the subsets is the action space. It incorporates some information aboutplayers’ preferences over the collection of subsets. We call each element of the domainof a generalized potential function an action subspace. If an action subspace maximizes ageneralized potential function and the generalized potential function has a unique maximumthen we call the action subspace a generalized potential maximizer (GP-maximizer).

The main results state that there exists a correlated equilibrium assigning probability 1to a GP-maximizer and that the set of such correlated equilibria is robust. This immediatelyimplies that if a GP-maximizer consists of one action profile then the action profile is a robustequilibrium. It should be noted that a robust set induced by the GP-maximizer condition isnot always minimal. A robust set is minimal if no robust set is a proper subset of the robustset. In this paper, we do not explore the problem of how to identify minimal robust sets.

It is not so straightforward to find GP-maximizers from the definition. One reason is that,as we will see later, a complete information game may have multiple generalized potentialfunctions with different domains. We restrict attention to generalized potential functionswith two special classes of domains. One class of domains are unordered partitions of actionspaces. We introduce best-response potential functions as functions over the partitions suchthat the best response correspondence of the function defined over the partition coincideswith that of a complete information game. Potential functions of Monderer and Shapley[18] form a special class of best-response potential functions with the finest partitions.3 Weshow that a best-response potential function is a generalized potential function. The otherclass of domains are those induced by ordered partitions of action spaces. We introducemonotone potential functions as functions over the partitions such that the best responsecorrespondence of the function defined over the partition and that of a complete informationgame has some monotonic relationship with respect to the order relation of the partition.We show that a monotone potential function naturally induces a generalized potential func-tion where the domain consists of intervals of the ordered partition. We then show that ap-dominant equilibrium with

∑i pi < 1 is the induced GP-maximizer, by which the discus-

sion of Kajii and Morris[12] and that of Ui[34] are unified. We also provide new sufficientconditions for action profile sets to be GP-maximizers, review some recent applications thatuse these sufficient conditions and give some new examples showing how the generalizedpotential analysis can be used when the methods of Kajii and Morris[12] and Ui[34] fail.

2 Kohlberg and Mertens[14] were the first to propose making sets of equilibria the objects of a theory ofequilibrium refinements.

3 Morris and Ui[20] demonstrated that the class of best-response potential functions with the finest partitionsare much larger than the class of potential functions.


The unification of the potential maximizer condition and thep-dominance conditionmay be of interest in other contexts. For example, both conditions are widely used in evolu-tionary contexts. For stochastic evolutionary dynamics, the potential maximizer conditionis discussed by Blume[2,3] and Ui [32], and thep-dominance condition is discussed byEllison [7] and Maruta[16]. For perfect foresight dynamics á la Matsui and Matsuyama[17], the potential maximizer condition is discussed by Hofbauer and Sorger[10,11], and thep-dominance condition is discussed by Oyama[23]. Interestingly, a recent paper by Oyamaet al. [24] shows that singleton GP-maximizers induced by monotone potential functionssatisfy the stability conditions of perfect foresight dynamics. This implies that “general-ized potential” methods may work in other contexts, unifying the potential maximizer andp-dominance conditions. This is not surprising, we believe, because GP-maximizers aredefined so that they inherit some properties of potential maximizers. We show in this paperthat GP-maximizers inherit the robustness property of potential maximizers, while Oyamaet al.[24] show that GP-maximizers inherit the stability properties of potential maximizers.

Rosenthal[26]was the first to use potential functions in non-cooperative game theory.4 Heused potential functions as tools for finding pure-action Nash equilibria.5 Recent studiessuch as Blume[2,3], Ui [32,34], and Hofbauer and Sorger[10,11]used potential functionsas tools for finding Nash equilibria satisfying some criteria for equilibrium selection. Sincea narrow class of games admit potential functions, attempts have been made to introducetools for a broader class of games. Monderer and Shapley[18] introduced ordinal potentialfunctions6 and generalized ordinal potential functions. Voorneveld[35] introduced best-response potential functions,7 which are different from best-response potential functions inthis paper. These functions inherit ordinal aspects of potential functions and serve as toolsfor the former use (finding pure-action equilibria). They are in clear contrast to generalizedpotential functions in this paper, which serve as tools for the latter use (refining equilibria).

The organization of this paper is as follows. Section2 defines robust sets of equi-libria. Section3 introduces generalized potential functions. Section4 provides the mainresults. Section5 discusses best-response potential functions and Section6 discussesmonotone potential functions. Section7reports examples of generalized potential functions.Section8 concludes the paper.

2. Robust sets

A complete information game consists of a finite set of playersN, a finite action setAi

for i ∈ N , and a payoff functiongi : A → R for i ∈ N whereA = ∏i∈N Ai . We write

A−i = ∏j =i Aj anda−i = (aj )j =i ∈ A−i . We also write, forS ∈ 2N , AS = ∏

i∈S Ai andaS = (ai)i∈S ∈ AS . Because we will fixN andA throughout the paper, we simply denote acomplete information game byg = (gi)i∈N .

4 Hart and Mas-Colell[9] have introduced potential functions incooperativegame theory. The potential func-tions of Monderer and Shapley[18] can be regarded as an extension of the potential functions of Hart andMas-Colell[9] to non-cooperative games, as demonstrated by Ui[33].

5 In traffic network theory, non-atomic games similar to the finite games of Rosenthal[26] are studied and non-atomic potential functions are used to calculate pure-action Nash equilibria. See Oppenheim[22], for example.

6 See also Kukushkin[15].7 Ui [32] considered similar functions in the context of stochastic evolutionary games.


An action distribution ∈ (A) is a correlated equilibrium ofg if, for eachi ∈ N ,∑a−i∈A−i

(ai, a−i )gi(ai, a−i )∑

a−i∈A−i

(ai, a−i )gi(a′i , a−i )

for all ai, a′i ∈ Ai . 8 An action distribution ∈ (A) is a Nash equilibrium ofg if it is a

correlated equilibrium and, for alla ∈ A, (a) = ∏i∈N i (ai) wherei ∈ (Ai). We also

say thata ∈ A is a Nash equilibrium if ∈ (A) with (a) = 1 is a Nash equilibrium.Consider an incomplete information game with the set of playersN and the action space

A. Let Ti be a countable set of types of playeri ∈ N . The state space isT = ∏i∈N Ti .

We writeT−i = ∏j =i Tj andt−i = (tj )j =i ∈ T−i . Let P ∈ (T ) be the prior probability

distribution onTwith∑

t−i ∈T−iP (ti , t−i ) > 0 for all i ∈ N andti ∈ Ti . A payoff function

of playeri ∈ N is a bounded functionui : A × T → R. Because we will fixT, N, andAthroughout the paper, we simply denote an incomplete information game by(u, P ) whereu = (ui)i∈N .

A (mixed) strategy of playeri ∈ N is a mappingi : Ti → (Ai). We writei for the setof strategies of playeri. The strategy space is = ∏

i∈N i . We write−i = ∏j =i j and

−i = (j )j =i ∈ −i . We writei (ai |ti ) for the probability ofai ∈ Ai giveni ∈ i andti ∈ Ti . For ∈ and−i ∈ −i , we write(a|t) = ∏

i∈N i (ai |ti ) and−i (a−i |t−i ) =∏j =i j (aj |tj ), respectively. LetP ∈ (A) be such thatP (a) = ∑

t∈T P (t)(a|t) forall a ∈ A. We callP an action distribution generated by.

A strategy profile ∈ is a Bayesian Nash equilibrium of(u, P ) if, for eachi ∈ N ,∑t−i∈T−i

∑a∈A

P (t−i |ti )(a|t)ui(a, t)

∑

t−i∈T−i

∑a−i∈A−i

P (t−i |ti )−i (a−i |t−i )ui((a′i , a−i ), t)

for all ti ∈ Ti anda′i ∈ Ai whereP (t−i |ti ) = P (ti, t−i )/

∑t−i∈T−i

P (ti , t−i ). Let Ui() =∑t∈T

∑a∈A P (t)(a|t)ui(a, t) be the payoff of strategy profile ∈ to playeri ∈ N .

Then, ∈ is a Bayesian Nash equilibrium of(u, P ) if and only if, for eachi ∈ N ,Ui()Ui(′

i , −i ) for all ′i ∈ i .

For giveng, consider the following subset ofTi :

Tui

i = ti ∈ Ti | ui(a, (ti , t−i )) = gi(a)

for all a ∈ A, t−i ∈ T−i with P (ti, t−i ) > 0.Whenti ∈ T

ui

i is realized, payoffs of playeri are given bygi and he knows his payoffs. Wewrite T u = ∏

i∈N Tui

i .

Definition 1. An incomplete information game(u, P ) is anε -elaborationof g if P (T u) =1 − ε for ε ∈ [0, 1].

Payoffs of a 0-elaboration are given byg with probability 1 and every player knows hispayoffs. It is straightforward to see that if a 0-elaboration has a Bayesian Nash equilibrium

8 For any finite or countable setS, (S) denotes the set of all probability distributions onS.


∈ then an action distribution generated by, P ∈ (A), is a correlated equilibriumof g. Kajii and Morris[12, Corollary 3.5]showed the following property ofε-elaborations,which we will use later.

Lemma 1. Let (uk, P k)∞k=1 be such that(uk, P k) is anεk-elaboration ofg andεk → 0

ask → ∞. Let k be a Bayesian Nash equilibrium of(uk, P k) and letkP k be an action

distribution generated byk. ThenkP k ∞k=1 has a subsequence which converges to some

correlated equilibrium ofg.

We say that a set of correlated equilibria ofg is robust if, for smallε > 0, everyε-elaboration ofg has a Bayesian Nash equilibrium ∈ such thatP ∈ (A) is close tosome equilibrium in the set.

Definition 2. A set of correlated equilibria ofg, E ⊆ (A), is robust to all elaborationsing if, for every > 0, there existsε > 0 such that, for allε ε, everyε-elaboration(u, P )

of g has a Bayesian Nash equilibrium ∈ such that maxa∈A |(a)−P (a)| for some ∈ E .

If E is a singleton then the equilibrium inE is robust in the sense of Kajii andMorris [12].

Kajii and Morris [13] considered a weaker version of the robustness of equilibria thanthat of Kajii and Morris[12]. 9 We consider the corresponding version of the robustnessof sets of equilibria. A typeti ∈ Ti\T

ui

i is committedif player i of this type has a strictlydominant actionati

i ∈ Ai such thatui((atii , a−i ), (ti , t−i )) > ui((ai, a−i ), (ti , t−i )) for all

ai ∈ Ai\atii , a−i ∈ A−i , andt−i ∈ T−i with P (ti, t−i ) > 0. An ε-elaboration ofg is

canonical if every ti ∈ Ti\Tui

i is committed for alli ∈ N .

Definition 3. A set of correlated equilibria ofg, E ⊆ (A), is robust to canonical elab-orations in g if, for every > 0, there existsε > 0 such that, for allε ε, everycanonicalε-elaboration(u, P ) of g has a Bayesian Nash equilibrium ∈ such thatmaxa∈A |(a) − P (a)| for some ∈ E .

If E is a singleton then the equilibrium inE is robust in the sense of Kajii andMorris [13].

In Section4, we will provide two sufficient conditions for the robustness of sets ofequilibria, one for the robustness to all elaborations and the other for the robustness tocanonical elaborations, respectively.

For either of the robustness concepts, ifE is robust then a set of correlated equilibriaE ′with E ⊆ E ′ is also robust. A robust setE is minimal if no robust set is a proper subset ofE . In this paper, we do not explore the problem of how to identify minimal robust sets.

Kajii and Morris[12] provided two sufficient conditions for the robustness of singletonequilibria. One applies to games with unique correlated equilibria.

9 The difference between them is an open question.


Theorem 1. If a∗ ∈ A is a unique correlated equilibrium ofg, thena∗ is robust to allelaborations ing.

The other applies to games withp -dominant equilibria such that∑

i∈N pi < 1.

Definition 4. Let p = (pi)i∈N ∈ [0, 1]N . Action profilea∗ ∈ A is ap-dominant equilib-rium of g if, for all i ∈ N andi ∈ (A−i ) with i (a

∗−i )pi ,∑a−i∈A−i

i (a−i )gi(a∗i , a−i )

∑a−i∈A−i

i (a−i )gi(ai, a−i )

for ai ∈ Ai .

Theorem 2. If a∗ ∈ A is ap-dominant equilibrium ofgwith∑

i∈N pi < 1, thena∗ isrobust to all elaborations ing.

Ui [34] provided a sufficient condition for the robustness of singleton equilibria in po-tential games introduced by Monderer and Shapley[18].

Definition 5. A function f : A → R is aweighted potential functionof g if there existswi > 0 such that

gi(ai, a−i ) − gi(a′i , a−i ) = wi

(f (ai, a−i ) − f (a′

i , a−i ))

(1)

for all i ∈ N , ai, a′i ∈ Ai , anda−i ∈ A−i . A complete information gameg is aweighted

potential gameif it has a weighted potential function. Whenwi = 1 for i ∈ N , we callf apotential function andg a potential game.

Theorem 3. Let g be a potential game with a potential function f. Suppose thata∗ =arg maxa∈A f (a). Thena∗ is robust to canonical elaborations ing.

Sufficient conditions provided by Kajii and Morris[12] and Ui[34] do not apply to thegameg given by the following table.

g0 1 2

0 3, 2 2, 3 0, 0

1 2, 3 3, 2 0, 0

2 0, 0 0, 0 1, 1

This game has multiple equilibria and thus Theorem1 does not apply. This game doesnot have a potential function and thus Theorem3 does not apply. This game has one strictNash equilibrium2 = (2, 2). For one player to choose 2, he must believe that his opponentchooses 2 with probability at least 2/3. This implies that, if2 is ap-dominant equilibrium,then it must be true thatp1 + p24/3. Thus, Theorem2 does not apply.


3. Generalized potentials

Suppose thatg has a weighted potential functionf. Then∑a−i∈A−i

i (a−i )(gi(ai, a−i ) − gi(a

′i , a−i )

)= wi

∑a−i∈A−i

i (a−i )(f (ai, a−i ) − f (a′

i , a−i ))

for all i ∈ N , ai, a′i ∈ Ai , andi ∈ (A−i ). Thus, we have

arg maxai∈Ai

∑a−i∈A−i

i (a−i )gi(ai, a−i ) = arg maxai∈Ai

∑a−i∈A−i

i (a−i )f (ai, a−i ) (2)

for all i ∈ N and i ∈ (A−i ). We generalize (2) to define generalized potentialfunctions.10

Before providing a formal definition, we present an example. Considerg discussed inthe previous section as an example. Remember thatAi = 0, 1, 2 for i ∈ N = 1, 2.We define a collection of subsets ofAi , Ai = 0, 1, 0, 1, 2 for i ∈ N , and defineA = X1 × X2 | X1 ∈ A1, X2 ∈ A2. ConsiderF : A → R given by the following table.

F

0, 1 0, 1, 20, 1 2 0

0, 1, 2 0 1

The functionF has the following property: fori ∈ (Aj ) andi ∈ (Aj ) with i (0) +i (1)i (0, 1),

Xi ∩ arg maxai∈Ai

∑aj ∈Aj

i (aj )gi(ai, aj ) = ∅

for all Xi ∈ arg maxX′

i∈Ai

∑Xj ∈Aj

i (Xj )F (X′i × Xj )

wherei = j . As we will see later,F is a generalized potential function ofg.To provide the formal definition, we first introduce the domain of a generalized potential

function denoted byA. For eachi ∈ N , let Ai ⊆ 2Ai \∅ be a covering ofAi . That is,Ai is a collection of non-empty subsets ofAi such that

⋃Xi∈Ai

Xi = Ai . The domainof a generalized potential function isA = ∏i∈N Xi | Xi ∈ Ai for i ∈ N. We writeA−i = ∏j =i Xj | Xj ∈ Aj for j = i andX−i = ∏

j =i Xj ∈ A−i . Note thatA andA−i

are coverings ofA andA−i , respectively. We callX ∈ A an action subspace.We then introduce, fori ∈ (A−i ), a corresponding subset of(A−i ) denoted by

i(A−i ). Imagine that playeri believes thata−i ∈ A−i is chosen in two steps: first,X−i ∈

A−i is chosen according toi ∈ (A−i ), and then,a−i ∈ X−i is chosen according to some

10 The existence of a functionf such that property (2) is satisfied is in fact a necessary but not a sufficient conditionfor g to be a weighted potential game. See the discussion in Section 5 and Morris and Ui[20].


X−i

i ∈ (A−i ) such thatX−i

i assigns probability 1 toX−i , i.e.,∑

a−i∈X−iX−i

i (a−i ) = 1.Then, the induced belief of playeri overA−i is i ∈ (A−i ) such that

i (a−i ) =∑

X−i∈A−i

i (X−i )X−i

i (a−i )

for all a−i ∈ A−i . We writei(A−i ) for the set of the beliefs of playeri overA−i induced

by the above rule:

i(A−i ) = i ∈ (A−i )|i (a−i ) =

∑X−i∈A−i

i (X−i )X−i

i (a−i ) for a−i ∈ A−i ,

X−i

i ∈ (A−i ) with∑

a−i∈X−i

X−i

i (a−i ) = 1 for X−i ∈ A−i.

Definition 6. A function F : A → R is ageneralized potential functionof g if, for alli ∈ N , i ∈ (A−i ), andi ∈ i

(A−i ),

Xi ∩ arg maxa′

i∈Ai

∑a−i∈A−i

i (a−i )gi(a′i , a−i ) = ∅

for every

Xi ∈ arg maxX′

i∈Ai

∑X−i∈A−i

i (X−i )F (X′i × X−i )

such thatXi is maximal in the argmax set ordered by the set inclusion relation. An actionsubspaceX∗ ∈ A is ageneralized potential maximizer(GP-maximizer) ifF (X∗) > F (X)

for all X ∈ A\X∗.It is clear thatF : A → R in the above example is a generalized potential function

becausei(Aj ) ⊆ i ∈ (Aj ) | i (0) + i (1)i (0, 1) wherei = j .

At the extreme, considerF : A → R such thatAi = Ai for all i ∈ N . Note thatA = A. Clearly, every complete information game has a generalized potential functionof this type. At the other extreme, considerF : A → R such thatAi = ai | ai ∈ Aifor all i ∈ N . Note thatA = a | a ∈ A. A weighted potential game has a generalizedpotential function of this type, which we prove in Section5.

Lemma 2. If g is a weighted potential game with a weighted potential function f thenghas a generalized potential functionF : A → R such thatAi = ai | ai ∈ Ai for alli ∈ N andF (a) = f (a) for all a ∈ A.

Before closing this section, we give a characterization ofi(A−i ).

Lemma 3. For all i ∈ (A−i ), i ∈ i(A−i ) if and only if∑

a−i∈B−i

i (a−i )∑

X−i∈A−iX−i⊆B−i

i (X−i )

for all B−i ∈ 2A−i .


This lemma is an immediate consequence of the result of Strassen[30], which is wellknown in the study of Dempster–Shafer theory.11 Dempster–Shafer theory considers non-additive probability functions called belief functions. Everyi ∈ (A−i ), called a basicprobability assignment, defines a corresponding belief functionv

i

i : 2A−i → [0, 1] suchthat

vi

i (B−i ) =∑

X−i∈A−iX−i⊆B−i

i (X−i )

for all B−i ∈ 2A−i . It is known that the correspondence betweeni andvi

i is one-to-one.An additive probability functioni ∈ (A−i ) is said to be compatible with a belief functionvi

i if

i (B−i )vi

i (B−i )

for all B−i ∈ 2A−i . 12 Strassen[30] proved that, for alli ∈ (A−i ), i is compatible withvi

i if and only if i ∈ i(A−i ), which is exactly Lemma3.

4. Main results

Suppose thatg has a generalized potential functionF : A → R with a GP-maximizerX∗. Let EX∗ be the set of correlated equilibria ofg that assign probability 1 toX∗:

EX∗ = ∈ (A) | is a correlated equilibrium ofg such that∑

a∈X∗(a) = 1.

The setEX∗ contains at least one Nash equilibrium. To see this, observe that

X∗i ∈ arg max

Xi∈Ai

F (Xi × X∗−i ).

By the definition of generalized potential functions,

X∗i ∩ arg max

ai∈Ai

∑a−i∈A−i

i (a−i )gi(ai, a−i ) = ∅

for everyi ∈ (A−i ) with∑

a−i∈X∗−ii (a−i ) = 1. This implies that the best response

correspondence ofg restricted toX∗ has non-empty values. Thus, we can show the existenceof Nash equilibria inEX∗ in the standard way using Kakutani fixed point theorem.

Our main results state thatEX∗ is robust. We present two theorems below. In Theorem4,we consider all generalized potential functions and provide a sufficient condition for the

11 Dempster[5,6] and Shafer[29].12 In literature of non-additive probabilities written by economists,i is called a core ofv

ii

because it is a core

when we regardB−i ∈ 2A−i as a coalition.


robustness to canonical elaborations. In Theorem5, we consider a special class of general-ized potential functions such thatAi ∈ Ai for all i ∈ N and provide a sufficient conditionfor the robustness to all elaborations.

Theorem 4. If g has a generalized potential functionF : A → R with a GP-maximizerX∗, thenEX∗ is non-empty and robust to canonical elaborations ing.

Theorem 5. If g has a generalized potential functionF : A → R with a GP-maximizerX∗ such thatAi ∈ Ai for all i ∈ N , thenEX∗ is non-empty and robust to all elaborationsin g.

If EX∗ is a singleton, then it is a minimal robust set and the equilibrium inEX∗ is robustin the sense of Kajii and Morris[12,13]. Clearly, if a GP-maximizer consists of one actionprofile, thenEX∗ is a singleton. It is straightforward to see thatEX∗ of the example in theprevious section is also a singleton where the GP-maximizer consists of four action profiles.

It should be noted thatEX∗ is not always a minimal robust set. For example, if a generalizedpotential function is such thatAi = Ai for all i ∈ N , thenEX∗ is the set of all correlatedequilibria.13 The above theorems are useful only when we have non-trivial generalizedpotential functions.

In the remainder of this section, we prove Theorems4 and5 simultaneously. The proofis presented in four steps.

For the first step, let(u, P ) be anε-elaboration ofgand consider collections of mappings

i = i : Ti → Ai | for all ti ∈ Ti\Tui

i , i (ti ) ∈ Ai contains

every undominated action of typeti, = : T → A | (t) =

∏i∈N

i (ti ) for all t ∈ T wherei ∈ i for all i ∈ N,

where we say thatai ∈ Ai is an undominated action of typeti if it is not a strictly dominatedaction of typeti . We say thatai ∈ Ai is a strictly dominated action of typeti if there existsa′

i ∈ Ai such thatui((a′i , a−i ), (ti , t−i )) > ui((ai, a−i ), (ti , t−i )) for all a−i ∈ A−i and

t−i ∈ T−i with P (ti, t−i ) > 0. Note that is non-empty if and only if, for alli ∈ N andti ∈ Ti\T

ui

i , there existsXi ∈ Ai such thatXi contains every undominated action of typeti . As considered in Theorem4, if (u, P ) is canonical and playeri of type ti ∈ Ti\T

ui

i hasa strictly dominant actionati

i ∈ Ai then is non-empty becauseAi is a covering ofAi andthere existsXi ∈ Ai such thatati

i ∈ Xi . As considered in Theorem5, if Ai ∈ Ai for alli ∈ N then is non-empty becauseAi contains every action. To summarize, we have thefollowing lemma.

Lemma 4. If (u, P ) is canonical then is non-empty. IfAi ∈ Ai for all i ∈ N then isnon-empty.

13 Kajii and Morris[12] noted the robustness of the set of all correlated equilibria.


For the second step, letV : → R be such that

V () =∑t∈T

P (t)F ((t))

for all ∈ and consider the set of its maximizers∗ = arg max∈ V ().

Lemma 5. If is non-empty then∗ is non-empty. If∗ ∈ ∗ then∑t∈T , ∗

(t)=X∗P (t)1 − ε,

where is a positive constant.

Proof. Let k ∈ ∞k=1 be such that

limk→∞ V (k) = sup

∈V ().

LetQk ∈ (T ×A) be such thatQk(t, X) = P (t)(k(t), X) for all (t, X) ∈ T ×A where : A × A → 0, 1 is such that(X′, X) = 1 if X′ = X and(X′, X) = 0 otherwise.Then ∑

(t,X)∈T ×AQk(t, X)F (X) =

∑(t,X)∈T ×A

P (t)(k(t), X)F (X)

=∑t∈T

P (t)F (k(t)) = V (k).

We regardQk∞k=1 as a sequence of probability measures on a discrete metric spaceT × A. Note that, for everyε > 0, there exists a finite subsetSε ⊂ T such that∑

(t,X)∈Sε×A Qk(t, X) = P (Sε) > 1 − ε for all k1. This implies thatQk∞k=1 is tight

becauseSε ×A is finite and thus compact. Accordingly, by Prohorov’s theorem,14 Qk∞k=1has a weakly convergent subsequenceQkl ∞l=1 such thatQkl → Q∗ as l → ∞. It isstraightforward to see that there exists∗ ∈ such that

Q∗(t, X) = liml→∞ Qkl (t, X) = P (t) lim

l→∞ (kl (t), X) = P (t)(∗(t), X)

for all (t, X) ∈ T × A. Then

sup∈

V () = liml→∞ V (kl )

= liml→∞

∑(t,X)∈T ×A

Qkl (t, X)F (X)

=∑

(t,X)∈T ×AQ∗(t, X)F (X) = V (∗).

Therefore,∗ ∈ ∗ and thus∗ is non-empty.

14 See Billingsley[1], for example.


Let F ∗ = F (X∗), F ′ = maxX∈A\X∗ F (X), and F ′′ = minX∈A F (X). Note thatF ∗ > F ′ F ′′. Let ∈ be such thati (ti ) = X∗

i for all ti ∈ Tui

i andi ∈ N . We have

V (∗) V ()

=∑t∈T u

P (t)F ((t)) +∑

t∈T \T u

P (t)F ((t))

P (T u)F ∗ + (1 − P (T u)

)F ′′ = (1 − ε)F ∗ + εF ′′.

We also have

V (∗) =∑

t∈T , ∗(t)=X∗

P (t)F (∗(t)) +∑

t∈T , ∗(t)=X∗

P (t)F (∗(t))

∑

t∈T , ∗(t)=X∗

P (t)F ∗ +1 −

∑t∈T , ∗

(t)=X∗P (t)

F ′.

Combining the above inequalities, we have

(1 − ε)F ∗ + εF ′′ ∑

t∈T , ∗(t)=X∗

P (t)F ∗ +1 −

∑t∈T , ∗

(t)=X∗P (t)

F ′

and thus ∑t∈T , ∗

(t)=X∗P (t)1 − ε,

where = (F ∗ − F ′′)/(F ∗ − F ′) > 0.

In order to get intuition about the set∗, compareV () with

Ui() =∑t∈T

P (t)

(∑a∈A

(a|t)ui(a, t)

).

If ε > 0 is close to 0, thenui = gi with probability close to 1. In this case, the relationshipbetweenV and(u, P ) is similar to that betweenF andg. We already show that there existsan equilibrium ofg assigning probability 1 to the maximizer ofF, i.e.,X∗. In the third stepbelow, we will show that there exists an equilibrium of(u, P ) assigning probability 1 tosome maximizer ofV, i.e.,∗ ∈ ∗.

Let be partially ordered by the relation⊆ such that ⊆ ′ for , ′ ∈ if and only ifi (ti ) ⊆ ′

i (ti ) for all ti ∈ Ti andi ∈ N .

Lemma 6. If ∗ ⊆ is non-empty, then it contains at least one maximal element. If∗ isa maximal element of∗, then(u, P ) has a Bayesian Nash equilibrium∗ ∈ such that∗(t) ∈ (A) assigns probability1 to the action subspace∗(t) ∈ A for all t ∈ T , i.e.,∑

a∈∗(t) ∗(a|t) = 1 for all t ∈ T .


Proof. If every linearly ordered subset of∗ has an upper bound in∗, then∗ containsat least one maximal element by Zorn’s Lemma. Let′ ⊆ ∗ be linearly ordered. Fixt = (ti)i∈N ∈ T . For eachi ∈ N , observe that

Xi | Xi = ′i (ti ), ′ ∈ ′ ⊆ Ai

is linearly ordered by the set inclusion relation. Since this set is finite, it has a maximumelement, which is equal to

⋃′

i∈′i′

i (ti ) ∈ Ai . Clearly, there exists(i,t) ∈ ′ such that

(i,t)i (ti ) = ⋃

′i∈′

i′

i (ti ). Consider(i,t) | i ∈ N ⊆ ′. Since this set is linearly ordered

and finite, it has a maximum element(j,t). Simply denote it by〈t〉, which satisfies〈t〉i (ti ) =⋃

′i∈′

i′

i (ti ) for all i ∈ N . For ε > 0, consider〈t〉 | t ∈ T , P (t) > ε ⊆ ′. Since this

set is linearly ordered and finite, it has a maximum element〈s〉. Simply denote it byε,which satisfiesε

i (ti) = ⋃′

i∈′i′

i (ti ) for all ti ∈ Ti andi ∈ N such thatP (t) > ε. Let

∈ be such thati (ti ) = ⋃′

i∈′i′

i (ti ) for all ti ∈ Ti andi ∈ N . Note that is an upper

bound of′. Sinceε(t) = (t) for t ∈ T with P (t) > ε, it must be true that

|V () − V (ε)| maxX,X′∈A

|F (X) − F (X′)| ×∑

t∈T , P (t)ε

P (t).

This implies that limε→0 |V () − V (ε)| = 0. Note thatV (ε) = max∈ V () because

ε ∈ ∗. Therefore,V () = max∈ V () and thus ∈ ∗, which completes the proof ofthe first half of the lemma.

We prove the second half. Let∗ ∈ ∗ be a maximal element. Let∗i ∈ i be such that

∗(t) = ∏i∈N ∗

i (ti ) for all t ∈ T and write∗−i (t−i ) = ∏

j =i ∗j (tj ).

We write∗i = i ∈ i | ∑ai∈∗

i (ti )i (ai |ti ) = 1 for all ti ∈ Ti, ∗ = ∏

i∈N ∗i , and

∗−i = ∏j =i ∗

j . We show that there exists a Bayesian Nash equilibrium∗ ∈ ∗.

Let i : ∗−i → 2∗i be such thati (−i ) = arg maxi∈i

Ui(i , −i ) ∩ ∗i for all

−i ∈ ∗−i ; and let : ∗ → 2∗be such that() = ∏

i∈N i (−i ) for all ∈ ∗. Notethat is the best response correspondence of(u, P ) restricted to∗.

We show that has non-empty values. This is true if and only if, for alli ∈ N ,−i ∈ ∗−i ,andti ∈ Ti ,

∗i (ti ) ∩ arg max

ai∈Ai

∑t−i∈T−i

∑a−i∈A−i

P (t−i |ti )−i (a−i |t−i )ui((ai, a−i ), t) = ∅. (3)

Suppose thatti ∈ Ti\Tui

i . Then (3) is true because∗i (ti ) contains every undominated

action of typeti .Suppose thatti ∈ T

ui

i . Rewrite the left-hand side of (3) as


ai∈Ai

∑t−i∈T−i

∑a−i∈A−i

P (t−i |ti )−i (a−i |t−i )ui((ai, a−i ), t)


= ∗i (ti ) ∩ arg max

ai∈Ai

∑a−i∈A−i

∑

t−i∈T−i

P (t−i |ti )−i (a−i |t−i )

gi(ai, a−i )

= ∗i (ti ) ∩ arg max

ai∈Ai

∑a−i∈A−i

tii (a−i )gi(ai, a−i ), (4)

wheretii ∈ (A−i ) is such that

tii (a−i ) =

∑t−i∈T−i

P (t−i |ti )−i (a−i |t−i )

for all a−i ∈ A−i . Because∗ is a maximal element of∗,

∗i (ti ) ∈ arg max

Xi∈Ai

∑t−i∈T−i

P (t−i |ti )F (Xi × ∗−i (t−i ))

= arg maxXi∈Ai

∑X−i∈A−i

∑t−i∈T−i

∗−i(t−i )=X−i

P (t−i |ti )

F (Xi × ∗

−i (t−i ))

= arg maxXi∈Ai

∑X−i∈A−i

tii (X−i )F (Xi × X−i ),

wheretii ∈ (A−i ) is such that

tii (X−i ) =

∑t−i∈T−i

∗−i(t−i )=X−i

P (t−i |ti )

for all X−i ∈ A−i . This implies that iftii ∈

tii

(A−i ) then


ai∈Ai

∑a−i∈A−i

tii (a−i )gi(ai, a−i ) = ∅ (5)

by the definition of generalized potential functions. To see thattii ∈

tii

(A−i ), rewrite

tii (a−i ) as

tii (a−i ) =

∑X−i∈A−i

∑t−i∈T−i

∗−i(t−i )=X−i

P (t−i |ti )−i (a−i |t−i )

=∑

X−i∈A−i

tii (X−i )

ti ,X−i

i (a−i ),


where

ti ,X−i

i (a−i ) =

∑t−i∈T−i

∗−i(t−i )=X−i

P (t−i |ti )−i (a−i |t−i )

tii (X−i )

if tii (X−i ) = 0,

1

|X−i | if tii (X−i ) = 0 anda−i ∈ X−i ,

0 if tii (X−i ) = 0 anda−i ∈ X−i .

Because−i ∈ ∗−i and thus∑

a−i∈∗−i (t−i )−i (a−i |t−i ) = 1 for all t−i ∈ T−i , we have

ti ,X−i

i ∈ (A−i ) with∑

a−i∈X−iti ,X−i

i (a−i ) = 1. This implies thattii ∈

tii

(A−i ) and

thus (5). Therefore, (3) is true by (4) and (5).We have shown that has non-empty values. We can show that∗ is compact15 and

convex and that has a closed graph and convex values. By Kakutani–Fan–Glicksbergfixed point theorem, has a fixed point∗ ∈ ∗, which is a Bayesian Nash equilibrium of(u, P ).

We now report the fourth and final step. An immediate implication of the above lemmasis the following. If(u, P ) is canonical (the case considered in Theorem4), or if Ai ∈ Ai forall i ∈ N (the case considered in Theorem5), then(u, P ) has a Bayesian Nash equilibrium∗ ∈ such that

∑a∈∗

(t) ∗(a|t) = 1 for all t ∈ T and∑a∈X∗

∗P (a) =

∑a∈X∗

∑t∈T

P (t)∗(a|t)

∑

t∈T , ∗(t)=X∗

P (t)∑

a∈X∗∗(a|t)

=∑

t∈T , ∗(t)=X∗

P (t)1 − ε, (6)

where∗ is a maximal element of∗. Thus, to complete the proof, it is enough to show that,for every > 0, there existsε > 0 such that, for allε ε and everyε-elaboration with aBayesian Nash equilibrium∗ satisfying (6), there exists ∈ EX∗ such that maxa∈A |(a)−∗

P (a)|.Seeking a contradiction, suppose otherwise. Then, for some > 0, there exists a sequence

(uk, P k)∞k=1 such that:

• (uk, P k) is anεk-elaboration ofg andεk → 0 ask → ∞.• (uk, P k) has a Bayesian Nash equilibrium∗k with

∑a∈X∗ ∗k

P (a)1 − εk.• max

a∈A|(a) − ∗k

P (a)| > for all ∈ EX∗ or EX∗ = ∅.

By Lemma1, ∗kP ∞k=1 has a subsequence∗kl

P ∞l=1 such that

liml→∞ max

a∈A|(a) − ∗kl

P (a)| = 0

15 A strategy subspace∗ is compact with the topology of weak convergence defined in ∈ (T ×A) | ∈ ∗,(t, a) = P (t)(a|t) for all (t, a) ∈ T × A.


where ∈ (A) is a correlated equilibrium ofg. Because∑a∈X∗

(a) = liml→∞

∑a∈X∗

∗kl

P (a) liml→∞ (1 − εkl ) = 1,

we have ∈ EX∗ . This is a contradiction, which completes the proof of the theorems.

5. Unordered domains

We restrict attention to the class of generalized potential functions such that domainsare partitions of action spaces. LetPi ⊆ 2Ai \∅ be a partition ofAi . We write P =∏i∈N Xi | Xi ∈ Pi for i ∈ N and P−i = ∏j =i Xj | Xj ∈ Pj for j = i, whichare partitions ofA andA−i , respectively. The partition element ofPi containingai ∈ Ai

is denoted byPi(ai). Similarly, the partition element ofP containinga and that ofP−i

containinga−i are denoted byP (a) andP−i (a−i ), respectively. We say that a functionv : A → R is P-measurable ifv(a) = v(a′) for a, a′ ∈ A with a′ ∈ P (a).

Definition 7. A P-measurable functionv : A → R is abest-response potential functionof g if, for eachi ∈ N ,

Xi ∩ arg maxa′

i∈Ai

∑a−i∈A−i

i (a−i )gi(a′i , a−i ) = ∅

for all Xi ∈ Pi andi ∈ (A−i ) such that

Xi ⊆ arg maxa′

i∈Ai

∑a−i∈A−i

i (a−i )v(a′i , a−i ).

A partition elementX∗ ∈ P is abest-response potential maximizer(BRP-maximizer) ifv(a∗) > v(a) for all a∗ ∈ X∗ anda /∈ X∗.

For example, consider the special case wherePi is the finest partition, i.e.,Pi =aiai∈Ai

for all i ∈ N . Then, it is straightforward to see that a functionv : A → R

is a best-response potential function ofg if and only if

arg maxa′

i∈Ai

∑a−i∈A−i

i (a−i )v(a′i , a−i ) ⊆ arg max

a′i∈Ai

∑a−i∈A−i

i (a−i )gi(a′i , a−i )

for all i ∈ N andi ∈ (A−i ). 16 For example, a weighted potential function is a best-response potential function by (2). However, a best-response potential function is not alwaysa weighted potential function, even if there are no dominated actions, as demonstrated byMorris and Ui[20]. Thus the class of best-response potential functions is much larger thanthe class of weighted potential functions.

16 A best-response potential function considered by Voorneveld[35] is a function satisfying this condition forthe class of beliefs such thati (a−i ) = 0 or 1. Thus, best-response potential functions in this paper form a specialclass of those in Voorneveld[35].


A best-response potential functionv induces a generalized potential function. LetF :A → R be such thatA = P andF (P (a)) = v(a) for all a ∈ A. Note thatP-measurabilityof v implies thatF is well defined. SinceA−i is a partition ofA−i ,i ∈ i

(A−i ) if and onlyif∑

a−i∈X−ii (a−i ) = i (X−i ) for all X−i ∈ A−i by Lemma3. Thus, fori ∈ (A−i )

andi ∈ i(A−i ),∑

X−i∈A−i

i (X−i )F (X′i × X−i ) =

∑a−i∈A−i

i (a−i )v(a′i , a−i )

if X′i = Pi(a

′i ). This implies that, if

Xi ∈ arg maxX′

i∈Ai

∑X−i∈A−i

i (X−i )F (X′i × X−i ),

then

Xi ⊆ arg maxa′

i∈Ai

∑a−i∈A−i

i (a−i )v(a′i , a−i )

and thus

Xi ∩ arg maxa′

i∈Ai

∑a−i∈A−i

i (a−i )gi(a′i , a−i ) = ∅

for all i ∈ i(A−i ) by the definition of best-response potential functions. Therefore,

F : A → R is a generalized potential function. This proves Lemma2 and immediatelyimplies the following result by Theorem4, which generalizes Theorem3.

Proposition 1. If g has a best-response potential functionv : A → R with a BRP-maximizerX∗, thenEX∗ is non-empty and robust to canonical elaborations ing.

6. Ordered domains

Let Pi be a partition ofAi such thatPi is linearly ordered by the order relation i

for i ∈ N . Let Zi andZi be the smallest and the largest elements ofPi , respectively. Thecorresponding product order relation overP is denoted byN , and that overP−i is denotedby −i , respectively. IfPi(ai) iZi for ai ∈ Ai andZi ∈ Pi , we simply writeai iZi .For Xi ⊆ Ai , we say thatai ∈ Xi is minimal inXi if ai iPi(xi) for all xi ∈ Xi and thatai ∈ Xi is maximal inXi if ai iPi(xi) for all xi ∈ Xi .

Definition 8. Let X∗ ∈ P be given. AP-measurable functionv : A → R with v(a∗) >

v(a) for all a∗ ∈ X∗ anda /∈ X∗ is amonotone potential functionof g if, for all i ∈ N andi ∈ (A−i ), there exists

ai ∈ arg maxa′

i iX∗i

∑a−i∈A−i

i (a−i )gi(a′i , a−i ),

ai ∈ arg maxa′

i iX∗i

∑a−i∈A−i

i (a−i )v(a′i , a−i )


such thatPi(ai) iPi(ai), and symmetrically, there exists

ai ∈ arg maxa′

i iX∗i

∑a−i∈A−i

i (a−i )gi(a′i , a−i ),

ai ∈ arg maxa′

i iX∗i

∑a−i∈A−i

i (a−i )v(a′i , a−i )

such thatPi(ai) iPi(ai). A partition elementX∗ ∈ P is called amonotone potentialmaximizer(MP-maximizer).

We restrict attention to a complete information gameg satisfying strategic complemen-tarities or a monotone potential functionv satisfying strategic complementarities in thefollowing sense.

Definition 9. A complete information gameg satisfiesstrategic complementaritiesif, foreachi ∈ N ,

gi(ai, a−i ) − gi(a′i , a−i )gi(ai, a′−i ) − gi(a

′i , a′−i )

for all ai, a′i ∈ Ai anda−i , a′−i ∈ A−i such thatPi(ai) >i Pi(a

′i ) andP−i (a−i ) >−i

P−i (a′−i ). A function v : A → R satisfies strategic complementarities if an identical

interest gameg with gi = v for all i ∈ N satisfies strategic complementarities.

Note that if the partitionPi is the finest one, then the order relation i naturally induces anorder relation over the action setAi and the above definition of strategic complementaritiesreduces to the standard one.

A monotone potential functionv with an MP-maximizerX∗ induces a generalizedpotential function with a GP-maximizerX∗ if g or v satisfies strategic complementarities.Let A be such that

Ai = [Z′i , Z′′

i ] | Z′i , Z′′

i ∈ Pi , Z′i iX

∗i iZ

′′i

for i ∈ N where[Z′i , Z′′

i ] ⊆ Ai is such that

[Z′i , Z′′

i ] =⋃

Z′i iZi iZ

′′i

Zi .

Note that[Zi, Zi] = Ai ∈ Ai . For Z′−i , Z′′−i ∈ P−i with Z′−i −iZ′′−i andZ′, Z′′ ∈ P

with Z′ N Z′′, we write

[Z′−i , Z′′−i] =∏j =i

[Z′j , Z′′

j ] =⋃

Z′−i −iZ−i −iZ′′−i

Z−i ,

[Z′, Z′′] =∏i∈N

[Z′i , Z′′

i ] =⋃

Z′ N Z N Z′′Z.

Then, we have

A−i = [Z′−i , Z′′−i] | Z′−i , Z′′−i ∈ P−i , Z′−i −iX∗−i −iZ

′′−i,A = [Z′, Z′′] | Z′, Z′′ ∈ P, Z′ N X∗ N Z′′.


Note that, for[Z′i , Z′′

i ] ∈ Ai and[Z′−i , Z′′−i] ∈ A−i , [Z′, Z′′] = [Z′i , Z′′

i ]×[Z′−i , Z′′−i] ∈ A.Let F : A → R be such that

F ([Z′, Z′′]) = V (Z′) + V (Z′′),

whereV : P → R is such thatV (P (a)) = v(a) for all a ∈ A, which is well defined byP-measurability ofv. Note thatF (X∗) > F (X) for all X ∈ A\X∗. By showing thatFis a generalized potential function, we claim the following result.

Proposition 2. Suppose thatg has a monotone potential functionv : A → R with anMP-maximizerX∗. If g or v satisfies strategic complementarities, thenEX∗ is non-emptyand robust to all elaborations ing.

Proof. By Theorem5, it is enough to show thatF : A → R given above is a generalizedpotential function ofg with a GP-maximizerX∗.

Fori ∈ (A−i ), let Z∗i , Z∗∗

i ∈ Pi be such that

[Z∗i , Z∗∗

i ] ∈ arg max[Z′

i ,Z′′i ]∈Ai

∑[Z′−i ,Z

′′−i ]∈A−i

i ([Z′−i , Z′′−i])F ([Z′i , Z′′

i ] × [Z′−i , Z′′−i])

and[Z∗i , Z∗∗

i ] is maximal in the argmax set ordered by the set inclusion relation. We provethat

[Z∗i , Z∗∗

i ] ∩ arg maxxi∈Ai

∑a−i∈A−i

i (a−i )gi(xi, a−i ) = ∅ (7)

for all i ∈ i(A−i ).

First, we calculate∑[Z′−i ,Z

′′−i ]∈A−i

i ([Z′−i , Z′′−i])F ([Z′i , Z′′

i ] × [Z′−i , Z′′−i])

=∑

[Z′−i ,Z′′−i ]∈A−i

i ([Z′−i , Z′′−i])V (Z′i × Z′−i )

+∑

[Z′−i ,Z′′−i ]∈A−i

i ([Z′−i , Z′′−i])V (Z′′i × Z′′−i )

=∑

Z′−i −iX∗−i

∑

Z′′−i −iX∗−i

i ([Z′−i , Z′′−i])V (Z′

i × Z′−i )

+∑

Z′′−i −iX∗−i

∑


i ([Z′−i , Z′′−i])V (Z′′

i × Z′′−i ).


Thus, we have

Z∗i = min

arg max

Z′i iX

∗i

∑Z−i∈P−i

′i (Z−i )V (Z′

i × Z−i )

,

Z∗∗i = max

arg max

Z′′i iX

∗i

∑Z−i∈P−i

′′i (Z−i )V (Z′′

i × Z−i )

,

where ′i ,

′′i ∈ (P−i ) are such that

′i (Z−i ) =

∑Z′′−i −iX

∗−i

i ([Z−i , Z′′−i]) if Z−i −iX∗−i ,

0 otherwise,

′′i (Z−i ) =

∑Z′−i −iX

∗−i

i ([Z′−i , Z−i]) if Z−i −iX∗−i ,

0 otherwise.

Next, consideri ∈ i(A−i ). Let i ∈ (P−i ) be such that

i (Z−i ) =∑

a−i∈Z−i

i (a−i )

for all Z−i ∈ P−i . We show that ′′i first order stochastically dominates i and i first

order stochastically dominates ′i . We say thatQ−i ⊆ P−i is a decreasing subset ofP−i if

Z−i ∈ Q−i andZ′−i −iZ−i together implyZ′−i ∈ Q−i . The definition of the stochasticdominance relation says that ′′

i first order stochastically dominates i if, for any decreasingsubsetQ−i ⊆ P−i ,∑

Z−i∈Q−i

i (Z−i )∑

Z−i∈Q−i

′′i (Z−i ). (8)

It is known that ′′i first order stochastically dominates i if and only if, for any increasing

function17 Gi : P−i → R,∑Z−i∈P−i

i (Z−i )Gi(Z−i )∑

Z−i∈P−i

′′i (Z−i )Gi(Z−i ).

We show (8) for two cases separately,X∗−i /∈ Q−i andX∗−i ∈ Q−i . If X∗−i /∈ Q−i , thenZ−i −iX

∗−i is false for allZ−i ∈ Q−i and thus

∑Z−i∈Q−i

i (Z−i )∑

Z−i∈Q−i

′′(Z−i ) = 0

17 We say thatGi : P−i → R is increasing ifGi(Z−i )Gi(Z′−i

) for Z−i −iZ′−i

.


because ′′i (Z−i ) = 0 unlessZ−i −iX

∗−i . If X∗−i ∈ Q−i , Lemma3 implies that

∑Z−i∈Q−i

i (Z−i ) =∑

Z−i∈Q−i

∑

a−i∈Z−i

i (a−i )

=∑

a−i∈⋃Z−i∈Q−iZ−i

i (a−i )

∑

[Z′−i,Z′′−i

]∈A−i

[Z′−i,Z′′−i

]⊆⋃Z−i∈Q−iZ−i

i ([Z′−i , Z′′−i])

=∑

Z′′−i−i X∗−i

Z′′−i∈Q−i

∑


i ([Z′−i , Z′′−i])

=∑

Z−i −i X∗−iZ−i∈Q−i

′′i (Z−i )

=∑

Z−i∈Q−i

′′i (Z−i ).

Therefore, ′′i first order stochastically dominates i . Symmetrically, we can show that i

first order stochastically dominates ′i .

Using the stochastic dominance relation, we show that

[Z∗i , X∗

i ] ∩ arg maxxi iX

∗i

∑a−i∈A−i

i (a−i )gi(xi, a−i ) = ∅, (9)

[X∗i , Z∗∗

i ] ∩ arg maxxi iX

∗i

∑a−i∈A−i

i (a−i )gi(xi, a−i ) = ∅, (10)

which imply (7). ForZ−i ∈ P−i , let Z−i

i ∈ (A−i ) be such that

Z−i

i (a−i ) =

i (a−i )

i (Z−i )if i (Z−i ) > 0 anda−i ∈ Z−i ,

1

|Z−i | if i (Z−i ) = 0 anda−i ∈ Z−i ,

0 if a−i /∈ Z−i .

Note that∑

a−i∈Z−iZ−i

i (a−i ) = 1 andi (a−i ) = ∑Z−i∈P−i

i (Z−i )Z−i

i (a−i ) for alla−i ∈ A−i . Thus,∑

a−i∈A−i

i (a−i )gi(xi, a−i ) =∑

Z−i∈P−i

i (Z−i )∑

a−i∈A−i

Z−i

i (a−i )gi(xi, a−i ).

Let ′i ∈ (A−i ) be such that

′i (a−i ) =

∑Z−i∈P−i

′i (Z−i )

Z−i

i (a−i ).


Then, we have∑a−i∈A−i

′i (a−i )v(xi, a−i ) =

∑Z−i∈P−i

′i (Z−i )

∑a−i∈A−i

Z−i

i (a−i )v(xi, a−i )

=∑

Z−i∈P−i

′i (Z−i )V (Pi(xi) × Z−i ).

This implies thatZ∗i = Pi(a

′i ) where

a′i ∈ arg max

xi iX∗i

∑a−i∈A−i

′i (a−i )v(xi, a−i )

is minimal in the argmax set. Let

ai ∈ arg maxxi iX

∗i

∑a−i∈A−i


be minimal in the argmax set and let

bi ∈ arg maxxi iX

∗i

∑a−i∈A−i

i (a−i )gi(xi, a−i ),

b′i ∈ arg max

xi iX∗i

∑a−i∈A−i

′i (a−i )gi(xi, a−i )

be maximal in the argmax sets, respectively. Sincev is a monotone potential function, itmust be true thatPi(ai) iPi(bi) andPi(a

′i ) iPi(b

′i ). Suppose thatg satisfies strategic

complementarities. For anyxi ∈ Ai with Pi(xi) <i Pi(b′i ),

gi(b′i , a−i ) − gi(xi, a−i )gi(b

′i , a′−i ) − gi(xi, a′−i )

wheneverP−i (a−i ) >−i P−i (a′−i ). This implies that∑

a−i∈A−i

Z−i

i (a−i )(gi(b

′i , a−i ) − gi(xi, a−i )

)

∑

a−i∈A−i

Z′−i

i (a−i )(gi(b

′i , a−i ) − gi(xi, a−i )

)

wheneverZ−i >−i Z′−i . In other words,∑

a−i∈A−iZ−i

i (a−i )(gi(b

′i , a−i ) − gi(xi, a−i )

)is increasing inZ−i . Since i first order stochastically dominates ′

i , it must be true that∑a−i∈A−i

i (a−i )(gi(b

′i , a−i ) − gi(xi, a−i )

)=

∑Z−i∈P−i

i (Z−i )∑

a−i∈A−i

Z−i

i (a−i )(gi(b

′i , a−i ) − gi(xi, a−i )

)

∑Z−i∈P−i

′i (Z−i )

∑a−i∈A−i

Z−i

i (a−i )(gi(b

′i , a−i ) − gi(xi, a−i )

)=

∑a−i∈A−i

′i (a−i )

(gi(b

′i , a−i ) − gi(xi, a−i )

)0.


This implies thatPi(b′i ) iPi(bi). Therefore,Z∗

i = Pi(a′i ) iPi(b

′i ) iPi(bi) and thus (9)

is true. Suppose thatv satisfies strategic complementarities. By the similar discussion, foranyxi ∈ Ai with Pi(xi) <i Pi(a

′i ),∑

a−i∈A−i

i (a−i )(v(a′

i , a−i ) − v(xi, a−i ))

=∑

Z−i∈P−i

i (Z−i )∑

a−i∈A−i

Z−i

i (a−i )(v(a′

i , a−i ) − v(xi, a−i ))

∑

Z−i∈P−i

′i (Z−i )

∑a−i∈A−i

Z−i

i (a−i )(v(a′

i , a−i ) − v(xi, a−i ))

=∑

a−i∈A−i

′i (a−i )

(v(a′

i , a−i ) − v(xi, a−i ))

> 0.

This implies thatPi(a′i ) iPi(ai). Therefore,Z∗

i = Pi(a′i ) iPi(ai) iPi(bi) and thus (9)

is true.To summarize, if eitherg or v satisfies strategic complementarities, (9) is true. Similarly,

we can show that (10) is true. Therefore, we obtain (7).

We can obtain the simpler form of the MP-maximizer condition if a complete informationgame satisfies diminishing marginal returns. We say that a complete information gamesatisfies diminishing marginal returns if every player’s payoff function is concave withrespect to his own action. LetZ+

i ∈ Pi be the smallest element larger thanZi = Zi , andZ−

i ∈ Pi be the largest element smaller thanZi = Zi .

Definition 10. A complete information gameg satisfiesdiminishing marginal returnsif,for eachi ∈ N anda−i ∈ A−i ,

gi(a+i , a−i ) − gi(ai, a−i )gi(ai, a−i ) − gi(a

−i , a−i )

for ai /∈ Zi ∪ Zi , a+i ∈ Pi(ai)

+, anda−i ∈ Pi(ai)

−.

In the case of diminishing marginal returns, we will see that the MP-maximizer conditionreduces to the following simpler condition.

Definition 11. Let X∗ ∈ P be given. AP-measurable functionv : A → R with v(a∗) >

v(a) for all a∗ ∈ X∗ anda /∈ X∗ is a local potential functionof g if, for eachi ∈ N andZi >i X∗

i ,

maxa′

i∈Z−i

∑a−i∈A−i

i (a−i )gi(a′i , a−i ) max

a′i∈Zi

∑a−i∈A−i


for all i ∈ (A−i ) such that∑a−i∈A−i

i (a−i )v(a−i , a−i )

∑a−i∈A−i

i (a−i )v(ai, a−i ),


wherea−i ∈ Z−

i andai ∈ Zi ; and symmetrically, for eachi ∈ N andZi <i X∗i ,

maxa′

i∈Z+i

∑a−i∈A−i

i (a−i )gi(a′i , a−i ) max

a′i∈Zi

∑a−i∈A−i


for all i ∈ (A−i ) such that∑a−i∈A−i

i (a−i )v(a+i , a−i )

∑a−i∈A−i

i (a−i )v(ai, a−i ),

wherea+i ∈ Z+

i and ai ∈ Zi . A partition elementX∗ ∈ P is called alocal potentialmaximizer(LP-maximizer).

We show that if a complete information game satisfies diminishing marginal returns, thena local potential function is a monotone potential function, by which we claim the followingresult.

Proposition 3. Suppose thatg has a local potential functionv : A → R with an LP-maximizerX∗. If g satisfies diminishing marginal returns, and if g or v satisfies strategiccomplementarities, thenEX∗ is non-empty and robust to all elaborations ing.

Proof. By Proposition2, it is enough to show that ifgsatisfies diminishing marginal returns,then a local potential functionv is a monotone potential function. Let

ai ∈ arg maxxi iX

∗i

∑a−i∈A−i

i (a−i )gi(xi, a−i )

be maximal in the argmax set and let

ai ∈ arg maxxi iX

∗i

∑a−i∈A−i


be minimal in the argmax set. We prove thatPi(ai) iPi(ai). If Pi(ai) = Zi , thenPi(ai) iPi(ai). If Pi(ai) = Zi , then Pi(ai)

− exists, and it must be true that, for alla−

i ∈ Pi(ai)−,∑

a−i∈A−i

i (a−i )(v(ai, a−i ) − v(a−

i , a−i ))

> 0.

Sincev is a local potential function, for somea∗i ∈ Pi(ai),∑

a−i∈A−i

i (a−i )(gi(a

∗i , a−i ) − gi(a

−i , a−i )

)0

for all a−i ∈ Pi(ai)

−. Sinceg satisfies diminishing marginal returns, we must have∑a−i∈A−i

i (a−i )(gi(xi, a−i ) − gi(x

−i , a−i )

)0


for all xi iPi(ai)− andx−

i ∈ Pi(xi)−. This implies that∑

a−i∈A−i


∑a−i∈A−i


for all xi iPi(ai)−. Therefore, it must be true thatPi(ai) iPi(a

∗i ) = Pi(ai).

Symmetrically, let

ai ∈ arg maxxi iX

∗i

∑a−i∈A−i


be minimal in the argmax set and let

ai ∈ arg maxxi iX

∗i

∑a−i∈A−i


be maximal in the argmax set. By the symmetric argument, we can prove thatPi(ai) i

Pi(ai).Combining the above arguments, we conclude that a local potential functionv is a mono-

tone potential function.

Proposition3 has an important implication in the special case where

Pi = a∗i , Ai\a∗

i with a∗

i iAi\a∗i for all i ∈ N and an LP-maximizer isa∗. 18 Note that a complete

information game satisfies diminishing marginal returns in the trivial sense. It is straightfor-ward to see that a functionv : A → R is a local potential function with an LP-maximizera∗ if and only if

• v(a∗) > v(a) for a = a∗,• for all i ∈ N , v(ai, a−i ) = v(a′

i , a−i ) for ai, a′i ∈ Ai\a∗

i anda−i ∈ A−i ,• for all i ∈ N , if∑

a−i∈A−i

i (a−i )v(a∗i , a−i )

∑a−i∈A−i

i (a−i )v(ai, a−i ), (11)

then ∑a−i∈A−i


∑a−i∈A−i

i (a−i )gi(ai, a−i ) (12)

for ai = a∗i .

One can show that ifg has ap-dominant equilibriuma∗ with∑

i∈N pi < 1 (seeDefinition 4), theng has a local potential functionv of this type. Thus, Theorem2 is animmediate consequence of Proposition3, the above discussion and the following lemma.

18 We have elsewhere labelled this class of local potential functions as “characteristic potential functions” becausethere exists one-to-one correspondence betweenv : A → R and : 2N → R by the rule(S) = v(a) if andonly if ai = a∗

ifor i ∈ S andai = a∗

ifor i ∈ S.


Lemma 7. For p = (pi)i∈N ∈ [0, 1]N with∑

i∈N pi < 1,ghas ap-dominant equilibriuma∗ if and only ifg has a local potential functionv : A → R with an LP-maximizera∗such that

v(a) =

1 −∑i∈N pi if a = a∗,

−∑i∈S pi if ai = a∗i for i ∈ S andai = a∗

i for i /∈ S.

In addition, v satisfies strategic complementarities.

Proof. Since

v(a∗i , a−i ) − v(ai, a−i ) =

1 − pi if a−i = a∗−i ,−pi otherwise

for ai = a∗i , v satisfies strategic complementarities. Note thatv is P-measurable and

v(a∗) > v(a) for a = a∗. Note also that (11) is equivalent to∑a−i∈A−i

i (a−i )(v(a∗

i , a−i ) − v(ai, a−i ))= i (a

∗−i )(1 − pi) +∑

a−i =a∗−i

i (a−i )(−pi)

= i (a∗−i ) − pi 0

for ai = a∗i . Thus, v is a local potential function ofg if and only if i (a

∗−i )pi

implies (12), which is true if and only ifa∗ is ap-dominant equilibrium. This completes theproof.

A local potential function in Lemma7 can be extended to the more general case where

Pi = X∗i , Ai\X∗

i with X∗

i iAi\X∗i for all i ∈ N . Considerv : A → R such that

v(a) =

1 −∑i∈N pi if a ∈ X∗,

−∑i∈S pi if ai ∈ X∗i for i ∈ S andai /∈ X∗

i for i /∈ S

with∑

i∈N pi < 1. If g has a local potential functionv given above with an LP-maximizerX∗, thenX∗ can be regarded as a set-valued extension ofp-dominance. In fact, Tercieux[31]extended the notion ofp-dominance to a set-valued one,p-best response set, and demon-strated that ap-best response setX∗ is characterized by a local potential functionv givenabove with an LP-maximizerX∗. 19

Local potential functions have the following characterization, which is easier to apply infinding local potential functions. Remember that, in weighted potential functions, the payoffdifference condition (1) leads to the belief condition (2). 20 The condition in the followinglemma provides the payoff difference condition which leads to the belief condition inDefinition11.

19 We are grateful to Olivier Tercieux for discussions clarifying the relation between LP-maximizers andp-bestresponse sets, which led to small change in the formulation of the LP-maximizer condition.

20 See Morris and Ui[20] for the duality argument between beliefs and payoff differences.


Lemma 8. LetX∗ ∈ P be given. AP-measurable functionv : A → R with v(a∗) > v(a)

for all a∗ ∈ X∗ anda /∈ X∗ is a local potential function ofg if, for eachi ∈ N , there existsi (a

−i , ai)0 for ai ∈ Zi withZi >i X∗

i anda−i ∈ Z−

i such that

gi(a−i , a−i ) − gi(ai, a−i )i (a

−i , ai)

(v(a−

i , a−i ) − v(ai, a−i ))

for all a−i ∈ A−i , and symmetrically, there existsi (a+i , ai)0 for ai ∈ Zi withZi <i X∗

i

anda+i ∈ Z+

i such that

gi(a+i , a−i ) − gi(ai, a−i )i (a

+i , ai)

(v(a+

i , a−i ) − v(ai, a−i ))

for all a−i ∈ A−i .

Proof. Suppose thatv satisfies the condition in the lemma. Then,∑a−i∈A−i

i (a−i )(gi(a

−i , a−i ) − gi(ai, a−i )

) i (a

−i , ai)

∑a−i∈A−i

i (a−i )(v(a−

i , a−i ) − v(ai, a−i ))

.

Clearly, if

∑a−i∈A−i

i (a−i )(v(a−

i , a−i ) − v(ai, a−i ))0,

then

∑a−i∈A−i

i (a−i )(gi(a

−i , a−i ) − gi(ai, a−i )

)0.

Thus,v satisfies the first half of the condition in Definition11. By the symmetric argu-ment, we can show thatv also satisfies the second half. Therefore,v is a local potentialfunction.

If Pi is the finest partition for alli ∈ N , then the converse of the above lemma is alsotrue. Forai ∈ Ai , let a+

i ∈ Ai be the smallest element larger thanai anda−i ∈ Ai be the

largest element smaller thanai .

Lemma 9. Suppose thatPi = aiai∈Aifor all i ∈ N . Let a∗ ∈ A be given. A function

v : A → R with v(a∗) > v(a) for all a = a∗ is a local potential function ofg if and onlyif, for eachi ∈ N , there existsi (a

−i , ai)0 for ai >i a∗

i anda−i such that

gi(a−i , a−i ) − gi(ai, a−i )i (a

−i , ai)

(v(a−

i , a−i ) − v(ai, a−i ))


for all a−i ∈ A−i , and symmetrically, there existsi (a+i , ai)0 for ai <i a∗

i anda+i such

that

gi(a+i , a−i ) − gi(ai, a−i )i (a

+i , ai)

(v(a+

i , a−i ) − v(ai, a−i ))

for all a−i ∈ A−i .

Proof. It is enough to show the “only if” part. Suppose thatv is a local potential function.To show thati (a

−i , ai) andi (a

+i , ai) exist, we use Farkas’ lemma.21 Farkas’ lemma says

that, for finite dimensional vectorsa0,a1, . . . ,am ∈ Rn, the following two conditions areequivalent.

• If (a1.y), . . . , (am.y)0 for y ∈ Rn, then(a0.y)0.• There existsx1, . . . , xm 0 such thatx1a1 + · · · + xmam = a0.

Sincev is a local potential function andPi = aiai∈Aifor all i ∈ N , if∑

a−i∈A−i

i (a−i )(v(a−

i , a−i ) − v(ai, a−i ))0,

then ∑a−i∈A−i

i (a−i )(gi(a

−i , a−i ) − gi(ai, a−i )

)0.

This implies that, if(ya−i)a−i∈A−i

∈ RA−i is such that

−∑

a−i∈A−i

ya−i

(v(a−

i , a−i ) − v(ai, a−i )) 0,

− ya−i0 for a−i ∈ A−i ,

then

−∑

a−i∈A−i

ya−i

(gi(a

−i , a−i ) − gi(ai, a−i )

)0.

By Farkas’ lemma, there existx 0 andxa−i0 for a−i ∈ A−i such that

−x(v(a−

i , a−i ) − v(ai, a−i ))−

∑a′−i∈A−i

xa′−ia′−i (a−i )

= − (gi(a

−i , a−i ) − gi(ai, a−i )

)for all a−i ∈ A−i wherea′−i : A−i → R is such thata′−i (a−i ) = 1 if a−i = a′−i and

a′−i (a−i ) = 0 otherwise. Thus,

gi(a−i , a−i ) − gi(ai, a−i )x

(v(a−

i , a−i ) − v(ai, a−i ))

and we can choosei (a−i , ai) = x. Symmetrically, we can show the existence ofi (a

+i , ai),

which completes the proof.

21 See textbooks of convex analysis such as Rockafellar[25].


7. Examples

As well as unifying the sufficient conditions for the robustness of equilibria providedby Kajii and Morris[12] and Ui [34], our generalized potential approach generates othersufficient conditions for the robustness of equilibria where the earlier results do not apply.In this section, we discuss examples applying these new sufficient conditions.

7.1. 3 × 3Games

We first discuss how to use Lemma9. Consider the followingg andv.

g0 1 2

0 5, 5 3, 3 −4, 01 3, 3 7, 7 4, 62 0, −4 6, 4 6, 6

v

0 1 20 5 4 11 4 6 52 1 5 7

Assuming the finest partitions of players’ action sets,gsatisfies diminishing marginal returnsandv satisfies strategic complementarities. In addition,

gi(1, aj ) − gi(0, aj ) = 2(v(1, aj ) − v(0, aj )

),

gi(2, aj ) − gi(1, aj ) = v(2, aj ) − v(1, aj ).

Thus, by Lemma9, v is a local potential function with an LP-maximizer(2, 2), and byProposition3, (2, 2) is a robust equilibrium.

Frankel et al.[8] study LP-maximizers of two player three action games with symmetricpayoffs. They report a slightly involved but complete characterization of the unique singletonLP-maximizer for this class.22 Oyama et al.[24] report a complete characterization of theunique singleton MP-maximizer in generic two player three action games with symmetricpayoffs satisfying strategic complementarities. These characterizations, however, cannot beextended beyond three action games: Frankel et al.[8] establish the non-existence of a sin-gleton LP-maximizer in an open set of two playerfouraction games with symmetric payoffssatisfying strategic complementarities and diminishing marginal returns. An interesting (butopen) question is whether the methods in this paper could be used to characterize a minimalnon-singleton local potential maximizer in two player many action games with symmetricpayoffs satisfying strategic complementarities and diminishing marginal returns.

As noted above, recent results in Tercieux[31] showing the robustness of “p-best responsesets” can be shown to be a special case of our generalized potential results. In the example wecited at the end of Section 2, the GP-maximizer0, 1×0, 1 is a “(1

3, 13)-best response set”

in Tercieux’s sense. Tercieux[31] provides further examples of action profile sets satisfyingthe(p1, p2)-best response property, withp1 + p2 < 1, for two player three action games.

22 Frankel et al.[8] describe an extension of the LP-maximizer condition that allows for continuous action gamesand use it to provide sufficient conditions for an action profile to be selected as the “noise independent selection”of a global game. Equilibria that are robust to incomplete information will always be the “noise independentselection” of a global game.


7.2. Binary action games

For i ∈ N = 1, . . . , n, let Ai = 1, 2 and Pi = 1, 2 wherePi is linearlyordered by the rule1 i2. Note thatg satisfies diminishing marginal returns in thetrivial sense. By Lemma9, v : A → R is a local potential function with an LP-maximizer1 = (1, . . . , 1) if and only if v(1) > v(a) for all a = 1 and there existsi 0 such thatgi(1, a−i ) − gi(2, a−i )i (v(1, a−i ) − v(2, a−i )) for all a−i ∈ A−i andi ∈ N .

To illustrate the condition, consider a unanimity gameg such that

gi (a) =

yi if a = 1,

zi if a = 2,

0 otherwise,

whereyi, zi > 0 for all i ∈ N . Note thatg satisfies strategic complementarities. A func-tion v : A → R is a local potential function with an LP-maximizer1 if and only ifv(1) > v(a) for all a = 1 and there existsi 0 such thatyi i (v(1) − v(2,1−i )),−zi i (v(1,2−i ) − v(2)), and 0i (v(1, a−i ) − v(2, a−i )) for a−i = 1−i ,2−i , for alli ∈ N . Becausezi > 0, we must havei > 0 andv(1,2−i )−v(2) < 0. Then, we can showthat the above condition implies thatyi/i > zj /j for all i = j . In other words,1 is anLP-maximizer only if there existsi > 0 for i ∈ N such thatyi/i > zj /j for all i = j .

We show this wheni = 1 andj = n. Let ak ∈ Ank=0 be such that, for eachk, ak

i = 1 ifi > k andak

i = 2 if i k. Note thata0 = 1 andan = 2. We have

y1/1 v(a0) − v(a1),

0 v(ak−1) − v(ak) for k ∈ 2, . . . , n − 1,−zn/n v(an−1) − v(an).

Thus,y1/1 − zn/n ∑nk=1

(v(ak−1) − v(ak)

) = v(a0) − v(an) = v(1) − v(2) > 0.It should be noted that there exist an open set of games that do not have any local potential

function. For example, all games in the neighborhood of the following unanimity game donot have a local potential function with an LP-maximizer1 or 2. Let N = 1, 2, 3,y1 = 6, y2 = y3 = 1, z1 = z2 = z3 = 2. If 1 is an LP-maximizer, then it must betrue that 1/2 > 2/3 and 1/3 > 2/2, which implies that 1> 4. Thus,1 is notan LP-maximizer. If2 is an LP-maximizer, then it must be true that 2/2 > 6/1 and2/1 > 1/2, which implies that 4> 6. Thus,2 is not an LP-maximizer.

7.3. Non-singleton LP-maximizers in three action games

For i ∈ N = 1, . . . , n, let Ai = 0, 1, 2 andPi = 0, 1, 2 wherePi is linearlyordered by the rule0, 1 i2. Note thatg satisfies diminishing marginal returns in thetrivial sense. By Lemma8, aP-measurable functionv : A → R is a local potential functionwith an LP-maximizerX∗ = 0, 1N if v(a∗) > v(a) for all a∗ ∈ X∗ anda /∈ X∗, andthere exists0

i , 1i 0 such that

gi(0, a−i ) − gi(2, a−i ) 0i (v(0, a−i ) − v(2, a−i )) ,

gi(1, a−i ) − gi(2, a−i ) 1i (v(1, a−i ) − v(2, a−i ))

for all a−i ∈ A−i andi ∈ N .


For example, consider the following game:

gi (a) =

yi(a) if a ∈ X∗,

zi if a = 2,

0 otherwise,

whereyi : X∗ → R is such thatyi(a) > 0 for all a ∈ X∗ and zi > 0. Note thatg satisfies strategic complementarities. AP-measurable functionv : A → R is a localpotential function with an LP-maximizerX∗ if v(a∗) > v(a) for all a∗ ∈ X∗ anda /∈ X∗,and there existsai

i 0 for ai ∈ 0, 1 such thatyi(a)ai

i (v(a) − v(2, a−i )) for a−i ∈X∗−i , −zi ai

i (v(ai,2−i ) − v(2)), and 0ai

i (v(a) − v(2, a−i )) for a−i /∈ X∗−i ∪ 2−i,for all i ∈ N . Note thatai

i > 0 andv(ai,2−i ) − v(2) < 0 becausezi > 0.In general, a robust set induced by the LP-maximizer,EX∗ , is not a singleton. Consider

Example 3.1 of Kajii and Morris[12]. Let N = 1, 2, 3 andzi = 1 for all i ∈ N . Let therestricted game(yi)i∈N be the cyclic matching pennies game; each player’s payoffs dependonly on his own action and the action of his “adversary.” Player 3’s adversary is player 2,player 2’s adversary is player 1, and player 1’s adversary is player 3. Thus, for example,player 1’s payoffs are completely independent of player 2’s action. Every player tries tochoose action different from his adversary’s. Player 1’s restricted payoff function is suchthaty1(1, 0, a3) = y1(0, 1, a3) = 3 andy1(1, 1, a3) = y1(0, 0, a3) = 2 for all a3 ∈ 0, 1.The other players’ restricted payoff functions are given similarly.

Kajii and Morris [12] showed that no singleton action profile is robust. However,v :A → R such that

v(a) =

2 if a ∈ X∗,

1 if a = 2,

0 otherwise

is a local potential function andX∗ is an LP-maximizer. Thus,EX∗ is a robust set.

8. Concluding remarks

This paper introduces generalized potential functions and provides sufficient conditionsfor the robustness of sets of equilibria. Special cases of the conditions unify the sufficientconditions for the robustness of equilibria provided by Kajii and Morris[12] and Ui [34],and provide new sufficient conditions.

The generalized potential technique introduced in this paper may be useful in analyzingquestions other than the robustness of incomplete information, as already suggested by thework of Oyama et al.[24]. In addition, there are a number of open questions about therobustness of equilibria. First, are robust equilibria unique if they exist? Kajii and Morris[12] showed that a strictlyp-dominant equilibrium with

∑i∈N pi < 1 is the unique robust

equilibrium; and we do not have examples of generic games with multiple robust equilibria.However, we do not know an argument showing that robust equilibria of generic gamesmust be unique if they exist. Second, how can we tell if a robust set is aminimalrobust set?Finally, is there a gap between robustness to all elaborations and robustness to allcanonical


elaborations? The generalized potential technique might be employed to answer each ofthese basic questions about the robustness of equilibria.

Acknowledgments

We are grateful to Atsushi Kajii, Daisuke Oyama, Bill Sandholm, and Olivier Tercieuxfor helpful comments and discussions. Ui acknowledges Financial Support from the ZenginFoundation for Studies on Economics and Finance and the Grant-in-Aid for Encouragementof Young Scientists.

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Generalized potentials and robust sets of equilibria