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Journal of Economic Theory � ET2196

journal of economic theory 71, 443�484 (1996)

Equilibrium in Beliefs under Uncertainty*

Kin Chung Lo

Department of Economics, University of Alberta,Edmonton, Alberta, Canada T6G 2H4

Received March 10, 1995; revised December 13, 1995

Existing equilibrium concepts for games make use of the subjective expectedutility model axiomatized by Savage [28] to represent players' preferences.Accordingly, each player's beliefs about the strategies played by opponents arerepresented by a probability measure. Motivated by experimental findings suchas the Ellsberg Paradox demonstrating that the beliefs of a decision maker maynot be representable by a probability measure, this paper generalizes equilibriumconcepts for normal form games to allow for the beliefs of each player to be repre-sentable by a closed and convex set of probability measures. The implications ofthis generalization for strategy choices and welfare of players are studied. Journalof Economic Literature Classification Numbers: C72, D81. � 1996 Academic Press, Inc.

1. INTRODUCTION

Due to its simplicity and tractability, the subjective expected utilitymodel axiomatized by Savage [28] has been the most important theory inanalysing human decision making under uncertainty. In particular, it isalmost universally used in game theory. Using the subjective expectedutility model to represent players' preferences, a large number of equi-librium concepts have been developed. The central one being NashEquilibrium.

On the other hand, the descriptive validity of the subjective expectedutility model has been questioned, for example, because of Ellsberg's [10]famous mind experiment, a version of which follows. Suppose there are twourns. Urn 1 contains 50 red balls and 50 black balls. Urn 2 contains 100balls. Each ball in urn 2 can be either red or black but the relative propor-tions are not specified. Consider the four acts listed in Table I. Ellsbergargues that the typical preferences for the acts are f1 tf2 o f3 tf4 , wherethe strict preference f2 o f3 reflects an aversion to the ``ambiguity'' or

article no. 0129

4430022-0531�96 �18.00

Copyright � 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

* This is a revised version of Chapter 1 of my Ph.D. thesis at the University of Toronto.I especially thank Professor Larry G. Epstein for pointing out this topic, and for providingsupervision and encouragement. I am also grateful to Professors Eddie Dekel, R. M. Neal,Mike Peters, and Shinji Yamashige for valuable discussions and to an associate editor andtwo referees for helpful comments. Remaining errors are my responsibility.


``Knightian uncertainty'' associated with urn 2. Subsequent experimentalstudies generally support that people are averse to ambiguity. (A summarycan be found in Camerer and Weber [6].) Such aversion contradicts thesubjective expected utility model, as is readily demonstrated for theEllsberg experiment. In fact, it contradicts any model of preferences inwhich underlying beliefs are represented by a probability measure.(Machina and Schmeidler [23] call such preferences ``probabilisticallysophisticated.'' In this paper, I reserve the term ``Bayesian'' for subjectiveexpected utility maximizer.)

The Ellsberg Paradox has motivated generalizations of the subjectiveexpected utility model. In the multiple priors model axiomatized by Gilboaand Schmeidler [15], the single prior of Savage is replaced by a closed andconvex set of probability measures. The decision maker is said to be uncer-tainty averse if the set is not a singleton. He evaluates an act by computingthe minimum expected utility over the probability measures in his set of priors.

Although the Ellsberg Paradox only involves a single decision maker fac-ing an exogenously specified environment, it is natural to think thatambiguity aversion is also common in decision making problems wheremore than one person is involved. Since existing equilibrium notions ofgames are defined under the assumption that players are subjective expec-ted utility maximizers, deviations from the Savage model to accommodateaversion to uncertainty make it necessary to redefine equilibrium concepts.

This paper generalizes Nash Equilibrium and one of its variations innormal form games to allow the beliefs of each player to be representableby a closed and convex set of probability measures as in the Gilboa�Schmeidler model. The paper then employs the generalized equilibriumconcepts to study the effects of uncertainty aversion on strategic interactionin normal form games.

Note that in order to carry out a ceteris paribus study of the effects ofuncertainty aversion on how a game is played, the solution concept we usefor uncertainty averse players should be different from that for Bayesianplayers only in terms of attitude towards uncertainty. In particular, thesolution concepts should share, as far as possible, comparable epistemicconditions. That is, the requirements on what the players should knowabout each other's beliefs and rationality underlying the new equilibriumconcepts should be ``similar'' to those underlying familiar equilibrium

TABLE I

f1 Win 8100 if the ball drawn from urn 1 is blackf2 Win 8100 if the ball drawn from urn 1 is redf3 Win 8100 if the ball drawn from urn 2 is blackf4 Win 8100 if the ball drawn from urn 2 is red

444 KIN CHUNG LO


concepts. This point is emphasized throughout the paper and is used todifferentiate the equilibrium concepts proposed here from those proposedby Dow and Werlang [9] and Klibanoff [18], also in an attempt togeneralize Nash Equilibrium in normal form games to accommodate uncer-tainty aversion.

The paper is organized as follows. Section 2 contains a brief review of themultiple priors model and a discussion of how it is adapted to the contextof normal form games. Section 3 defines Nash Equilibrium and one of itsvariants. Section 4 defines and discusses the generalized equilibrium con-cepts used in this paper. Section 5 makes use of the equilibrium conceptsdefined in Section 4 to investigate how uncertainty aversion affects players'strategy choices and welfare. Section 6 identifies how uncertainty aversionis related to the structure of a game. Section 7 discusses the epistemic con-ditions of the equilibrium concepts for uncertainty averse players used inthis paper and compares them with those underlying the correspondingequilibrium notions for subjective expected utility maximizing players. Sec-tion 8 provides a comparison with Dow and Werlang [9] and Klibanoff[18]. The comparison also serves to clarify the implications for adoptingdifferent approaches for developing equilibrium notions for games withuncertainty averse players. Section 9 argues that the results in previoussections hold even if we drop the particular functional form of the utilityfunction proposed by Gilboa and Schmeidler [15] but retain some of itsbasic properties. Some concluding remarks are offered in Section 10.

2. PRELIMINARIES

2.1. Multiple Priors Model

In this section, I provide a brief review of the multiple priors model anda discussion of some of its properties that will be relevant in later sections.For any topological space Y, adopt the Borel _-algebra 7Y and denote byM(Y) the set of all probability measures over Y.1 Adopt the weak* topol-ogy on the set of all finitely additive probability measures over (Y, 7Y) andthe induced topology on subsets. Let (X, 7X) be the space of outcomes and(0, 70) the space of states of nature. Let F be the set of all boundedmeasurable functions from 0 to M(X).2 That is, F is the set of two-stage,horse-race�roulette-wheel acts, as in Anscombe and Aumann [1]. Forf, g # F and : # [0, 1], :f+(1&:) g#h, where h(|)=:f (|)+(1&:)g(|) \| # 0. f # F is called a constant act if f (|)=p \| # 0; such an act

445EQUILIBRIUM IN BELIEFS

1 The only exception is that when Y is the space of outcomes X, M(X) denotes the set ofall probability measures over X with finite supports.

2 See Gilboa and Schmeidler [15, pp. 149�150].


involves (probabilistic) risk but no uncertainty. For notational simplicity,I also use p # M(X) to denote the constant act that yields p in every stateof the world, x # X, the degenerate probability measure on x, and | # 0,the event [|] # 70 . The primitive p is a preference ordering over acts.The relations of strict preference and indifference are denoted by o andt , respectively.

Gilboa and Schmeidler [15] impose a set of axioms on p that arenecessary and sufficient for p to be represented by a numerical functionhaving the following structure: there exists an affine function u: M(X) � Rand a unique, nonempty, closed and convex set 2 of finitely additive prob-ability measures on 0 such that for all f, g # F,

fpg � minp # 2 | u b f dp�min

p # 2 | u b g dp. (2.1.1)

It is convenient, but in no way essential, to interpret 2 as ``representing thebeliefs underlying p ''; I provide no formal justification for such an inter-pretation.

The difference between the subjective expected utility model and themultiple priors model can be illustrated by a simple example. Suppose0=[|1 , |2] and X=R. Consider an act f#( f (|1), f (|2)). If the decisionmaker is a Bayesian and his beliefs over 0 are represented by a probabilitymeasure p, the utility of f is

p(|1) u( f (|1))+p(|2) u( f (|2)).

On the other hand, if the decision maker is uncertainty averse with the setof priors

2=[ p # M([|1 , |2]) | pl�p(|1)�ph with 0�pl<ph�1],

then the utility of f is

{pl u( f (|1))+(1&pl) u( f (|2))ph u( f (|1))+(1&ph) u( f (|2))

if u( f (|1))�u( f (|2))if u( f (|1))�u( f (|2)).

Note that given any act f with u( f (|1))>u( f (|2)), (|1 , pl ; |2 , 1&pl)3

can be interpreted as local probabilistic beliefs at f in the following sense.There exists an open neighborhood of f such that for any two acts g andh in the neighborhood,

gph � pl u(g(|1))+(1&pl) u(g(|2))�pl u(h(|1))+(1&pl) u(h(|2)).

446 KIN CHUNG LO

3 Throughout this paper, I use ( y1 , p1 ; ...; ym , pm) to denote the probability measure whichattaches probability pi to yi .


That is, the individual behaves like an expected utility maximizer in thatneighborhood with beliefs represented by (|1 , pl ; |2 , 1&pl). Similarly,(|1 , ph ; |2 , 1&ph) represents the local probabilistic beliefs at f ifu( f (|1))<u( f (|2)). Therefore, the decision maker who ``consumes''different acts may have different local probability measures at those acts.

There are three issues regarding the multiple priors model that will berelevant when the model is applied to normal form games. The first con-cerns the decision maker's preference for randomization. According to themultiple priors model, preferences over constant acts, that can be identifiedwith objective lotteries over X, are represented by u( } ) and thus conformwith the von Neumann Morgenstern model. The preference ordering overthe set of all acts is quasiconcave. That is, for any two acts f, g # F withftg, we have :f+(1&:) gp f for any : # (0, 1). This implies that thedecision maker may have a strict incentive to randomize among acts.

The second concerns the notion of null event. Given any preferenceordering p over acts, define an event T/0 to be p -null as in Savage[28]: T is p -null if for all f, f $, g # F,

_ f (|)g(|)

if | # Tif | � T &t_ f $(|)

g(|)if | # Tif | � T & .

In words, an event T is p -null if the decision maker does not care aboutpayoffs in states belonging to T. This can be interpreted as the decisionmaker knows (or believes) that T can never happen. If p is expectedutility preferences, then T is p -null if and only if the decision makerattaches zero probability to T. If p is represented by the multiple priorsmodel, then T is p -null if and only if every probability measure in 2attaches zero probability to T.

Finally, the notion of stochastic independence will also be relevantwhen the multiple priors model is applied to games having more thantwo players. Suppose the set of states 0 is a product space 01_ } } } _0N.In the case of a subjective expected utility maximizer, where beliefs arerepresented by a probability measure p # M(0), beliefs are said to bestochastically independent if p is a product measure: p=_N

i=1 mi , wheremi # M(0i) \i. In the case of uncertainty aversion, the decision maker'sbeliefs over 0 are represented by a closed and convex set of probabilitymeasures 2. Let marg0i 2 be the set of marginal probability measures on0i as one varies over all the probability measures in 2. That is,

marg0i 2#[mi # M(0i) | _p # 2 such that mi=marg0i p].

Following Gilboa and Schmeidler [15, pp. 150�151], say that the decisionmaker's beliefs are stochastically independent if

2=closed convex hull of [_Ni=1 mi | mi # marg0i 2 \i ].



That is, 2 is the smallest closed convex set containing all the productmeasures in _N

i=1 marg0i 2.

2.2. Normal Form Games

This section defines n-person normal form games where players'preferences are represented by the multiple priors model. Throughout, theindices i, j, and k vary over distinct players in [1, ..., n]. Unless specifiedotherwise, the quantifier ``for all such i, j, and k'' is to be understood. Asusual, &i denotes the set of all players other than i. Player i 's finite purestrategy space is Si with typical element si . The set of pure strategy profilesis S#_n

i=1 Si . The game specifies an outcome function gi : S � X forplayer i. Since mixed strategies induce lotteries over X, we specify an affinefunction ui : M(X) � R to represent payer i 's preference ordering overM(X). A set of strategy profiles, outcome functions, and utility functionsdetermine a normal form game (Si , gi , ui)

ni=1.

Let M(Si) be the set of mixed strategies for player i with typical element_i . The set of mixed strategy profiles is therefore given by _n

i=1 M(Si)._i (si) denotes the probability of playing si according to the mixed strategy_i , _&i (s&i) denotes >j{i _j (sj) and _&i is the corresponding probabilitymeasure on S&i #_ j{i Sj . Note that when players are Bayesians, _i issometimes interpreted as the probabilistic conjecture held by i 's opponentsabout i 's pure strategy choice. This paper adopts the view that uncertaintyaverse players have a strict incentive to randomize.4 Therefore, _i

represents player i 's conscious randomization. For example, suppose afactory employer has two pure strategies s=monitor worker 1 ands$=monitor worker 2. His decision problem is to choose a (possiblydegenerate) random device to determine which worker he is going tomonitor. (See Section 4.2 below for arguments for and against thisapproach.)

Assume that player i is uncertain about the strategy choices of all theother players. Since the objects of choice for player j is the set of mixedstrategies M(Sj), the relevant state space for player i is _j{i M(Sj),endowed with the product topology. Each mixed strategy of player i can beregarded as an act over this state space. If player i plays _i and the otherplayers play _&i , i receives the lottery that yields outcome gi (si , s&i) withprobability _i (si) _&i (s&i). Note that this lottery has finite support becauseS and therefore [gi (s)]s # S are finite sets. It is also easy to see that the actcorresponding to any mixed strategy is bounded and measurable in the senseof the preceding subsection. Consistent with the multiple priors model,

448 KIN CHUNG LO

4 Also note that uncertainty aversion is not the only reason for players to have a strictincentive to randomize. In Crawford [7] and Dekel et al. [8], players may also strictly preferto randomize even though they are probabilistically sophisticated.


player i 's beliefs over _j{i M(Sj) are represented by a closed and convexset of probability measures B� i . Therefore, the objective of player i is tochoose _i # M(Si) to maximize

minpi # B� i

|_j{i M(Sj)

:si # Si

:s&i # S&i

ui (gi (si , s&i)) _i (si) _&i (s&i) dpi (_&i).

Define the payoff function ui : S � R as follows: ui (s)#ui (gi (s)) \s # S.A normal form game can then be denoted alternatively as (Si , ui)

ni=1 and

the objective function of player i can be restated in the form

minpi # B� i

|_j{i M(Sj)

:si # Si

:s&i # S&i

ui (si , s&i) _i (si) _&i (s&i) dpi (_&i). (2.2.1)

In order to produce a simpler formulation of player i 's objective func-tion, note that each element in B� i is a probability measure over a set ofprobability measures. Therefore, the standard rule for reducing two-stagelotteries leads to the following construction of Bi �M(S&i):

Bi#{pi # M(S&i) | _pi # B� i such that

pi (s&i)=|_j{ i M(Sj)

_&i (s&i) dpi (_&i) \s&i # S&i= .

The objective function of player i can now be rewritten as

minpi # Bi

ui (_i , pi), (2.2.2)

where ui (_i , pi)#�si # Si �s&i # S&i ui (si , s&i) _i (si) pi (s&i).Convexity of B� i implies that Bi is also convex. Further, from the perspec-

tive of the multiple priors model (2.1.1), (2.2.2) admits a natural interpreta-tion whereby S&i is the set of states of nature relevant to i and Bi is his setof priors over S&i . Because of the greater simplicity of (2.2.2), the equi-librium concepts used in this paper will be expressed in terms of (2.2.2) andBi instead of (2.2.1) and B� i . The above construction shows that doing thisis without loss of generality. However, the reader should always bear inmind that the former is derived from the latter and I will occasionally goback to the primitive level to interpret the equilibrium concepts.

3. EQUILIBRIUM CONCEPTS FOR BAYESIAN PLAYERS

This section defines equilibrium concepts for Bayesian players. Thedefinition of equilibrium proposed by Nash [24] can be stated as follows:



Definition 1. A Nash Equilibrium is a mixed strategy profile [_i*]ni=1

such that

_*i # BRi (_*&i)#argmax_i # M(Si)

ui (_i , _*&i).

Under the assumption that players are expected utility maximizers,Nash proves that any finite matrix game of complete information hasa Nash Equilibrium. It is well known that there are two interpretationsof Nash Equilibrium. The traditional interpretation is that _i* is the actualstrategy used by player i. In a Nash Equilibrium, it is best for player i touse _i* given that other players choose _*&i . The second interpretation isthat _i* is not necessarily the actual strategy used by player i. Instead itrepresents the marginal beliefs of player j about what pure strategy playeri is going to pick. Under this interpretation, Nash Equilibrium is usuallystated as an n-tuple of probability measures [_i*]n

i=1 such that

si # BRi (_*&i) \si # support of _*i .

Its justification is that given that player i 's beliefs are represented by _*&i ,BRi (_*&i) is the set of strategies that maximize the utility of player i. Soplayer j should ``think'', if j knows i 's beliefs, that only strategies inBRi (_*&i) will be chosen by i. Therefore, the event that player i will choosea strategy which is not in BRi (_*&i) should be ``null'' (in the sense of Sec-tion 2.1) from the point of view of player j. This is the reason for imposingthe requirement that every strategy si in the support of _*i , which representsthe marginal beliefs of player j, must be an element of BRi (_*&i).

The ``beliefs'' interpretation of Nash Equilibrium allows us to see clearlythe source of restrictiveness of this solution concept. First, the marginalbeliefs of player j and player k about what player i is going to do arerepresented by the same probability measure _*i . Second, player i 's beliefsabout what his opponents are going to do are required to be stochasticallyindependent in the sense that the probability distribution _*&i on thestrategy choices of the other players is a product measure. We are thereforeled to consider the following variation.

Definition 2. A Bayesian Beliefs Equilibrium is an n-tuple of probabil-ity measures [bi]n

i=1 where bi # M(S&i) such that5

margSi bj # BRi (bi)#argmax_i # M(Si)

ui (_i , bi).

450 KIN CHUNG LO

5 To avoid confusion, note that this is not Harsanyi's Bayesian Equilibrium for games ofincomplete information with Bayesian players.


It is easy to see that if [_i*]ni=1 is a Nash Equilibrium, then [_*&i]n

i=1 isa Bayesian Beliefs Equilibrium. Conversely, a Bayesian Beliefs Equilibrium[bi]n

i=1 constitutes a Nash Equilibrium [_i*]ni=1 if bi=_*&i . Note that

in games involving only two players, the two equilibrium concepts areequivalent in that a Bayesian Beliefs Equilibrium must constitute a NashEquilibrium.

However, when a game involves more than two players, the definition ofBayesian Beliefs Equilibrium is more general. For instance, in a BayesianBeliefs Equilibrium, players i and k can disagree about what player j isgoing to do. That is, it is allowed that margSj bi {margSj bk .

Example 1. Marginal Beliefs Disagree. Suppose the game involvesthree players. Player 1 only has one strategy [X]. Player 2 only has onestrategy [Y]. Player 3 has two pure strategies [L, R]. The payoff toplayer 3 is a constant. [b1=YL, b2=XR, b3=XY] is a Bayesian BeliefsEquilibrium. However it does not constitute a Nash Equilibrium becauseplayers 1 and 2 disagree about what player 3 is going to do. K

Second, in a Bayesian Beliefs Equilibrium, player i is allowed to believethat the other players are playing in a correlated manner. As argued byAumann [3], this does not mean that the other players are actually coor-dinating with each other. It may simply reflect that i believes that thereexist some common factors among the players that affect their behaviour;for example, player i knows that all other players are professors ofeconomics.

Example 2. Stochastically Dependent Beliefs. Suppose the gameinvolves three players. Player 1 has two pure strategies [U, D]. Player 2has two pure strategies [L, R]. Player 3 has two pure strategies [T, B].The payoffs of players 1 and 2 are constant. The payoff matrix for player 3is as shown in Table II. (For all n-person games presented in this paper,the payoff is in terms of utility.) It is easy to see that

b1=(LT, 0.5; RT, 0.5),

b2=(UT, 0.5; DT, 0.5),

TABLE II

UL UR DL DR

T &10 3 4 &10B 0 0 0 0



and

b3=(UR, 0.5; DL, 0.5)

constitute a Bayesian beliefs Equilibrium. Moreover the marginal beliefs ofthe players agree. However it does not constitute a Nash Equilibrium. Thereason is that player 3's beliefs about the strategies of players 1 and 2 arestochastically dependent. If player 3 believes that the strategies of player 1and player 2 are stochastically independent, player 3's beliefs are possibly(UL, 0.25; UR, 0.25; DL, 0.25; DR, 0.25) and T would no longer be his bestresponse. K

4. EQUILIBRIUM CONCEPTS FOR UNCERTAINTYAVERSE PLAYERS

4.1. Equilibrium Concepts

This section defines generalizations of Nash Equilibrium and BayesianBeliefs Equilibrium to allow players' preferences to be represented by themultiple priors model. The proposed equilibrium concepts preserve allessential features of their Bayesian counterparts, except that players' beliefsare not necessarily represented by a probability measure. Further discus-sion is provided in Section 4.2.

The generalization of Bayesian Beliefs Equilibrium is presented first.

Definition 3. A Beliefs Equilibrium is an n-tuple of sets of probabilitymeasures [Bi]n

i=1 where Bi �M(S&i) is a nonempty, closed, and convexset such that

margSi Bj �BRi (Bi)#argmax_i # M(Si)

minpi # Bi

ui (_i , pi).

When expressed in terms of B� i , a Beliefs Equilibrium is an n-tuple ofclosed and convex sets of probability measures [B� i]n

i=1 such that

_i # BRi (B� i) \_i # .pj # B� j

support of margM(Si) pj ,

where BRi (B� i) is the set of strategies which maximize (2.2.1).6

The interpretation of Beliefs Equilibrium parallels that of its Bayesiancounterpart. Given that player i 's beliefs are represented by B� i , BRi (B� i) is

452 KIN CHUNG LO

6 To be even more precise, a Beliefs Equilibrium is an n-tuple of closed and convex sets ofprobability measures [B� i]n

i=1 such that the complement of BRi (B� i) is a set of margM(Si) pj -measure zero for every pj # B� j .


the set of strategies that maximize the utility of player i. So player j should``think,'' if j knows i 's beliefs, that only strategies in BRi (B� i) will be chosenby i. Therefore, the event that player i will choose a strategy that is not inBRi (B� i) should be ``null'' (in the sense of Section 2.1) from the point ofview of player j. This is the reason for imposing the requirement that everystrategy _i in the union of the support of every probability measure inmargM(si) B� j , which represents the marginal beliefs of player j about whatplayer i is going to do, must be an element of BRi (B� i).

It is obvious that every Bayesian Beliefs Equilibrium is a Beliefs Equi-librium. Say that a Beliefs Equilibrium [Bi]n

i=1 is proper if not every Bi isa singleton.

Recall that Nash Equilibrium is different from Bayesian Beliefs Equi-librium in two respects: (i) The marginal beliefs of the players agree and(ii) the overall beliefs of each player are stochastically independent. Anappropriate generalization of Nash Equilibrium to allow for uncertaintyaversion should also possess these two properties. Consider therefore thefollowing definition.

Definition 4. A Beliefs Equilibrium [Bi]ni=1 is called a Beliefs Equi-

librium with Agreement if there exists _ni=1 7i �_n

i=1 M(Si) such thatBi=closed convex hull of [_&i # M(S&i) | margSj _&i # 7j].

We can see as follows that this definition delivers the two properties``agreement'' and ``stochastic independence of beliefs.'' As explained inSection 2.2, player i 's beliefs are represented by a closed and convex setof probability measures B� i on _j{i M(Sj). I require the marginal beliefsof the players to agree in the sense that margM(Sj) B� i=margM(Sj) B� k . Tocapture the idea that the beliefs of each player are stochastically independ-ent, I impose the requirement that B� i contains all the product measures.That is,

B� i=closed convex hull of [_j{i mj | mj # margM(Sj) B� i].

Bi is derived from B� i as in Section 2.2. By construction, we havemargSj Bi=margSj Bk=convex hull of 7j and Bi takes the form required inthe definition of Beliefs Equilibrium with Agreement.

Note that Beliefs Equilibrium and Beliefs Equilibrium with Agreementcoincide in two-person games. Further, for n-person games, if [bi]n

i=1 is aBayesian Beliefs Equilibrium with Agreement, then [bi]n

i=1 constitutes aNash Equilibrium.

To provide further perspective and motivation, I state two variations ofBeliefs Equilibrium and explain why they are not the focus of this paper.Given that any strategy in BRi (Bi) is equally good for player i, it isreasonable for player j to feel completely ignorant about which strategy i



TABLE III

L R

U 3, 2 &1, 2D 0, 4 0, &100

will pick from BRi (Bi). This leads us to consider the following strengtheningof Beliefs Equilibrium:

Definition 5. A Strict Beliefs Equilibrium is a Beliefs Equilibrium withmargSi Bj=BRi (Bi).

A Beliefs Equilibrium may not be a Strict Beliefs Equilibrium, asdemonstrated in the following example.7 The example also shows that aStrict Beliefs Equilibrium does not always exist, which is obviously aserious deficiency of this solution concept.

Example 3. Nonexistence of Strict Beliefs Equilibrium. The gamein Table III only has one Nash Equilibrium, [U, L]. It is easy to checkthat it is not a Strict Beliefs Equilibrium. In fact, there is no Strict BeliefsEquilibrium for this game. K

An opposite direction is to consider weakening the definition of BeliefsEquilibrium.

Definition 6. A Weak Beliefs Equilibrium is an n-tuple of beliefs[Bi]n

i=1 such that margSi Bj & BRi (Bi){<.

It is clear that any Beliefs Equilibrium is a Weak Beliefs Equilibrium. Theconverse is not true. If margSi Bj �3 BRi (Bi), there are some strategies (inj 's beliefs about i) that player i will definitely not choose. However, playerj considers those strategies ``possible.'' On the other hand, margSi Bj &

BRi (Bi){< captures the idea that player j cannot be ``too wrong.'' WeakBeliefs Equilibrium is also not the focus of this paper because we do notexpect much strategic interaction if the players know so little about theiropponents. However, it will be discussed further in Section 7 (Proposi-tion 8) and Section 8, where its relation to the equilibrium conceptsproposed by Dow and Werlang [9] and Klibanoff [18] is discussed.

4.2. Discussion

a. Mixed strategies as objective randomization vs subjective beliefs. Aspointed out earlier, a mixed strategy of a player is traditionally regarded as

454 KIN CHUNG LO

7 A parallel statement for Bayesian players is that a Nash Equilibrium may not be a StrictNash Equilibrium.


his conscious randomization. In recent years, a modern view of mixedstrategies has emerged according to which players do not randomize. Eachplayer chooses a definite action but his opponents may not know whichone, and the mixture represents their conjecture about his choice.

Note that in games with Bayesian players, the two views are ``observa-tionally indistinguishable'' in the sense that they lead to the same set ofNash Equilibria. However, this is not necessarily the case for games withuncertainty averse players. For example, consider the game in Table IV.(For all two person games presented in this paper, player 1 is the rowplayer and player 2 is the column player.) If player 1 is Bayesian, D isnever optimal no matter whether he randomizes or not. Now suppose thatplayer 1 is uncertainty averse. If he has a preference ordering representedby (2.2.2), then whatever his beliefs, the utility of the mixed strategy(U, 0.5; C, 0.5) is equal to 5 and the utility of D is only equal to 1. There-fore, D is also never optimal. On the other hand, if we assume that player 1does not randomize and therefore has the choice set [U, C, D], then hewill strictly prefer to play D rather than U and C if his beliefs are, say,B1=M([U, C, D]).

The above example demonstrates the necessity to re-examine the twoviews of mixed strategies when we consider games with uncertainty averseplayers. Such a re-examination is provided below. It also serves to justifythe adoption of the traditional view in this paper.

One justification of the modern view is that the normal form game understudy is repeated over time, where each player's pure strategy choices areindependent and identically distributed random variables. A mixed strategyequilibrium can therefore be interpreted as a stochastic steady state.However, since uncertainty is presumably eliminated asymptotically, thisrepeated game scenario is of limited relevance for the present study ofgames with uncertainty averse players.

The standard objection to the traditional view also does not necessarilyextend to games with uncertainty averse players. The argument against thetraditional view is that since expected utility is linear in probabilities, Bayesianplayers do not have a strict incentive to randomize (see, for instance,Brandenburger [5, p. 91]). However, when preferences deviate from theexpected utility model, there may exist a strict incentive to randomize.

TABLE IV

L R

U 10, 1 0, 1C 0, 1 10, 1D 1, 1 1, 1



To see this, let us first go back to the context of single-person decisiontheory. Recall that p is a preference ordering over the set of acts F, whereeach act f maps 0 into M(X). The interpretation of f is as follows. First ahorse race determines the true state of nature | # 0. The decision maker isthen given the objective lottery ticket f (|). He spins the roulette wheel asspecified by f (|) to determine the actual prize he is going to receive. Alsorecall that for any two acts f, f $ # F and : # [0, 1], :f+(1&:) f $ refers tothe act which yields the lottery ticket :f (|)+(1&:) f $(|) in state |.

Suppose p is strictly quasiconcave as in the Gilboa�Schmeidler modeland the decision maker has to choose between f and f $. Suppose furtherthat he perceives that nature moves first; that is, a particular state |* # 0has been realized but the decision maker does not know what |* is. If thedecision maker randomizes between choosing f and f $ with probabilities :and 1&:, respectively, he will receive the lottery :f (|)+(1&:) f $(|)when |*=|. This is precisely the payoff of the act :f+(1&:) f $ in state|. That is, randomization enables him to enlarge the choice set from[ f, f $] to [:f+(1&:) f $ | : # [0, 1]]. Correspondingly, there will ``typi-cally'' be an : # (0, 1) such that :f+(1&:) f $ is optimal according to p .8

On the other hand, suppose the decision maker moves first and naturemoves second. If the decision maker randomizes between choosing f andf $ with probabilities : and 1&:, respectively, he faces the lottery( f, :; f $, 1&:) that delivers act f with probability : and f $ with probability1&:. Therefore, randomization delivers the set [( f, :; f $, 1&:) | : #[0, 1]] of objective lotteries over F. Note that [( f, :; f $, 1&:) | : #[0, 1]] is not in the domain of F and so the Gilboa�Schmeidler model issilent on the decision maker's preference ordering over this set.

The above discussion translates to the context of normal form games withuncertainty averse players as follows. The key is whether player i perceiveshimself as moving first or last. The assumption of strict incentive to ran-domize can be justified by the assumption that each player perceives himselfas moving last. On the other hand, if we assume that each player perceiveshimself as moving first, and has an expected utility representation forpreferences over objective lotteries on F, then there will be no strict incentiveto randomize. Since the perception of each player about the order of strategychoices is not observable and there does not seem to be a compelling theoreti-cal case for assuming either order, it would seem that either specification ofstrategy space merits study. (See also Dekel et al. [8] for another instancewhere the perception of the players about the order of moves is important.)

456 KIN CHUNG LO

8 Note that this only explains why the decision maker may strictly prefer to randomize. Wealso need to rely on the dynamic consistency argument proposed by Machina [21] to ensurethat the decision maker is willing to obey the randomization result after the randomizingdevice is used. See also Dekel et al. [8, p. 241] for discussion of this issue in the context ofnormal form games.


Another objection to the assumption of strict incentive to randomize thatmight be raised in the context of uncertainty is that it contradicts ``Ellsbergtype'' behaviour. The argument goes as follows. Suppose a decision makercan choose between f3 and f4 listed in Table I. If the decision maker ran-domizes between f3 and f4 with equal probability, it will generate the act12 f3+ 1

2 f4 which yields the lottery [8100, 12; 80, 1

2] in each state. Therefore,12 f3+ 1

2 f4 is as desirable as f1 or f2 . This implies that the decision makerwill be indifferent between having the opportunity to choose an act from[ f1 , f2] or from [ f3 , f4] and the Ellsberg Paradox disappears! The discus-sion in previous paragraphs gives us the correct framework to handle thisobjection. Randomization between f3 and f4 with equal probability willgenerate the act 1

2 f3+ 12 f4 only if either the decision maker is explicitly told

or he himself perceives that he can first draw a ball from the urn but notlook at its colour, then toss a fair coin and choose f3( f4) when head (tail)comes up. (Raiffa [27, p. 693]). Also, the preference pattern f1 tf2 o

f3 tf4 is already sufficient to constitute one version of the EllsbergParadox. In this version, consideration of randomization is irrelevant.Therefore, assuming a strict incentive to randomize does not make everyversion of the Ellsberg Paradox disappear.

Finally, one standard defence of the interpretation of mixed strategies asobjective randomization also makes sense in games with uncertainty averseplayers. That is, one may imagine a hypothetical ``guide to playing games.''Such a guide can certainly recommend a mixed strategy or a set of mixedstrategies to each player.

b. Knowledge of beliefs. In common with the equilibrium concepts forBayesian players presented in Section 3, Beliefs Equilibrium (with Agreement)assumes that each player knows his opponents' beliefs. Three possiblejustifications for this assumption are as follows. First, players' beliefs arederived from statistical information (which is not necessarily precise enoughto be characterized by an objective probability measure). For instance, asalesman possesses statistical information about the bargaining behaviourof customers. If a customer also knows the information, then he knows thebeliefs of the salesman (Aumann and Brandenburger [4, p. 1176]). Second,players may learn about their opponents' beliefs through pre-game com-munication. For instance, suppose a player has two pure strategies X andY. He may announce that he will choose X with probability between 0 and1. In fact, Example 5 in Section 5 below illustrates that it may be strictlybetter for a player to make such a ``vague'' announcement. This point isalso discussed by Greenberg [16]. Finally, players' beliefs may be derivedfrom public recommendation. A ``guide'' suggests a set of strategies to eachplayer publicly. After receiving the suggestion, each player chooses astrategy which is unknown to his opponents.



Admittedly, the above justifications may not be entirely convincing. Forexample, when two players receive the same statistical information whichdoes not take the form of an objective probability measure, it is demandingto assume that their beliefs agree. However, without the assumption ofagreement, it would be harder to justify that players know each other'sbeliefs, even though beliefs are derived from the same source of informa-tion. Nevertheless, note that the agreement assumption is equally strong ifwe restrict players to be Bayesians. In fact, if we allow players' beliefs todisagree, it seems even harder for Bayesian beliefs to be mutual knowledge.How can player i know the unique subjective probability measure represen-ting the beliefs of player j? To conclude, I acknowledge the limitation ofthe above story and, following Aumann and Brandenburger [4, p. 1176],only intend to show that a player may well know another's conjecture insituations of economic interest.

c. Knowledge of rationality. As do their Bayesian counterparts, BeliefsEquilibrium (with Agreement) assumes mutual knowledge of rationality.That is, player j's beliefs about player i 's behaviour are focused on i 's bestresponse given i 's true beliefs. This can be justified by the assumption thatplayers learn their opponents' rational behaviour from past observations.We can assume that past observations are obtained from previous plays ofthe same normal form game (which has not been repeated sufficiently oftento eliminate all uncertainty about strategy choices).9 Alternatively, we canassume that players' knowledge of opponents' rationality is derived fromother sources. For example, before players i and j play a normal formgame, i has observed that j was rational when j played a different gamewith player k.

4.3. Relationship with Maximin Strategy and RationalizabilityFinally it is useful to clarify the relationship between Beliefs Equilibrium

defined in Section 4.1 and some familiar concepts in the received theory ofnormal form games.

Definition 7. The strategy _*i is a maximin strategy for player i if

_*i # argmax_i # M(Si)

minpi # M(S&i)

ui (_i , pi).

The following result is immediate:

Proposition 1. If [M(S&i)]ni=1 is a Beliefs Equilibrium, then every

_i # M(Si) is a maximin strategy.

458 KIN CHUNG LO

9 Mukerji [21] argues that uncertainty about opponents' strategy choices can even persistas a steady state in the repeated game scenario.


Definition 8. Set 1 0i #M(Si) and recursively define10

1 ni =[_i # 1 n&1

i | _p # M(_j{i supp 1 n&1j )

such that ui (_i , p)�ui (_$i , p) \_$i # 1 n&1i ].

For player i, the set of Correlated Rationalizable Strategies is Ri #��n=0 1 n

i .We call RBi #��

n=1 M(_j{i supp 1 n&1j ) the set of Rationalizable Beliefs.

These notions are related to Beliefs Equilibrium by the next proposition.

Proposition 2. Suppose [Bi]ni=1 is a Beliefs Equilibrium. Then

BRi (Bi)�Ri and Bi �RBi .

Proof. Set 1� 0i #M(Si) and recursively define

1� ni =[_i # 1� n&1

i | _P�M(_j{i supp 1� n&1j )

such that minp # P

ui (_i , p)�minp # P

ui (_$i , p) \_$i # 1� n&1i ].

By definition, 1 0i =1� 0

i . It is obvious that 11i �1� 1

i . Any element _i not in11

i does not survive the first round of the iteration in the definition ofcorrelated rationalizability. Since correlated rationalizability and iteratedstrict dominance coincide (see Fudenberg and Tirole [14, p. 52]), theremust exist _*i # 1 0

i such that ui (_*i , p)>ui (_i , p) \p # M(_j{i supp 1 0j ). This

implies minp # P ui (_*i , p)>minp # P ui (_i , p) \P�M(_j{i supp 1� 0j ). There-

fore, _i � 1� 1i and we have 11

i =1� 1i . The argument can be repeated to estab-

lish 1 ni =1� n

i \n.BRi (Bi) is rationalized by Bi , that is, BRi (Bi)�1� 1

i . According to thedefinition of Beliefs Equilibrium, margSi Bj �BRi (Bi)�1� 1

i . This implies_j{i margSj Bi �_j{i 1� 1

j and therefore Bi �M(_j{i supp 1� 1j ). The argu-

ment can be repeated to establish BRi (Bi)�1� ni and Bi �M(_j{i supp 1� n

j )\n. K

5. DOES UNCERTAINTY AVERSION MATTER?

5.1. Questions

In Section 4, I have set up a framework that enables us to investigatehow uncertainty aversion affects strategic interaction in the context of nor-mal form games. My objective here is to address the following two specificquestions:


10 The notation supp 1 n&1j stands for the union of the supports of the probability measures

in 1 n&1j .


1. As an outside observer, one only observes the actual strategychoice but not the beliefs of each player. Is it possible for an outsideobserver to distinguish uncertainty averse players from Bayesian players?

2. Does uncertainty aversion make the players worse off (better off)?

To deepen our understanding, let me first provide the answers to the abovetwo questions in the context of single-person decision making and conjecturethe possibility of extending them to the context of normal form games.

5.2. Single-Person Decision Making

The first question is: as an outside observer, can we distinguish an uncer-tainty averse decision maker from a Bayesian decision maker? The answeris obviously yes if we have ``enough'' observations. (Otherwise the EllsbergParadox would not exist!) However, it is easy to see that if we onlyobserve an uncertainty averse decision maker who chooses one act from aconvex constraint set G�F, then his choice can always be rationalized (aslong as monotonicity is not violated) by a subjective expected utility func-tion. For example, take the simple case where 0=[|1 , |2]. The feasibleset of utility payoffs C#[(u( f (|1)), u( f (|2))) | f # G] generated by G willbe a convex set in R2. Suppose the decision maker chooses a point c # C.To rationalize his choice by an expected utility function, we can simplydraw a linear indifference curve which is tangent to C at c, with slopedescribing the probabilistic beliefs of the expected utility maximizer.

The above answer is at least partly relevant to the first question posedin Section 5.1. That is because in a normal form game, an outside observeronly observes that each player i chooses a strategy from the set M(Si). Animportant difference, though, is that the strategy chosen by i is a bestresponse given his beliefs and these are part of an equilibrium. Therefore,it is possible that the consistency condition imposed by the equilibriumconcept can enable us to break the observational equivalence.

The second question addresses the welfare consequences of uncertaintyaversion: does uncertainty aversion make a decision maker worse off(better off)? There is a sense in which uncertainty aversion makes a decisionmaker worse off. For simplicity, suppose again that X=R. Suppose thatinitially, beliefs over the state space 0 are represented by a probabilitymeasure p and next that beliefs change from p to the set of priors 2 withp # 2. Given f # F, let CE2( f ) be the certainty equivalent of f, that is,u(CE2( f ))=minp # 2 � u b f dp. Similar meaning is given to CEp( f ). Thenuncertainty aversion makes the decision maker worse off in the sense that

CEp( f )�CE2( f ).

That is, the certainty equivalent of any f when beliefs are represented by pis higher than that when beliefs are represented by 2.

460 KIN CHUNG LO


Note that in the above welfare comparison, I am fixing the utility functionof lotteries u. This assumption can be clarified by the following restatement:Assume that the decision maker has a fixed preference ordering p* overM(X) which satisfies the independence axiom and is represented numericallyby u. Denote by p and p $ the orderings over acts corresponding to thepriors p and 2, respectively. Then the above welfare comparison presumes thatboth p and p $ agree with p* on the set of constant acts, that is, for anyf, g # F with f (|)=p and g(|)=q for all | # 0, f pg � f p $ g � pp*q.

At this point, it is not clear that the above discussion extends to thecontext of normal form games. When strategic considerations are present,one might wonder whether it is possible that if player 1 is uncertaintyaverse and if player 2 knows that player 1 is uncertainty averse, then thebehaviour of player 2 is affected in a fashion that benefits player 1 relativeto a situation where 2 knows that 1 is a Bayesian.11 When both players areuncertainty averse and they know that their opponents are uncertaintyaverse, can they choose a strategy profile that Pareto dominates equilibriagenerated when players are Bayesians?

5.3. Every Beliefs Equilibrium Contains a Bayesian Beliefs EquilibriumIn this section, the two questions posed in Section 5.1 are addressed

using the equilibrium concepts Bayesian Beliefs Equilibrium and BeliefsEquilibrium. The answers are implied by the following proposition.

Proposition 3. If [Bi]ni=1 is a Beliefs Equilibrium, then there exist

bi # Bi , i=1, ..., n, such that [bi]ni=1 is a Bayesian Beliefs Equilibrium and

BRi (Bi)�BRi (bi).

Proof.12 It is sufficient to show that there exists bi # Bi such thatBRi (Bi)�BRi (bi). This and the fact that [Bi]n

i=1 is a Beliefs Equilibriumimply

margSi bj # margSi Bj �BRi (Bi)�BRi (bi).

Therefore, [bi]ni=1 is a Bayesian Beliefs Equilibrium.


11 A well-known example where this kind of reasoning applies is the following. An expectedutility maximizer who is facing an exogenously specified set of states of nature always prefersto have more information before making a decision. However, this is not necessarily the caseif the decision maker is playing a game against another player. The reason is that if player 1chooses to have less information and if player 2 ``knows'' it, the strategic behaviour of player 2may be affected. The end result is that player 1 may obtain a higher utility by throwing awayinformation. (See the discussion of correlated equilibrium in Chapter 2 of Fudenberg andTirole [14].)

12 Though I prove a result (Proposition 12) below for more general preferences, I providea separate proof here because the special structure of the Gilboa�Schmeidler model permitssome simplification.


We have that ui ( } , pi) is linear on M(Si) for each pi and ui (_i , } ) islinear on Bi for each _i . Therefore, by Fan's Theorem (Fan [13]),

u�# max

_i # M(Si)minpi # Bi

ui (_i , pi)=minpi # Bi

max_i # M(Si)

ui (_i , pi).

By definition, _i # BRi (Bi) if and only if minpi # Bi ui (_i , pi)=u�. Therefore,

ui (_i , pi)�u�

\pi # Bi \_i # BRi (Bi). (5.3.1)

Take bi # argminpi # Bi max_i # M(Si) ui (_i , pi). Then conclude that

ui (_i , bi)�u�= max

_i # M(Si)ui (_i , bi) \_i # M(Si). (5.3.2)

Combining (5.3.1) and (5.3.2), we have

ui (_i , bi)=u�

\_i # BRi (Bi), that is, BRi (Bi)�BRi (bi). K

Example 4. Illustrating Proposition 3. Consider the game inTable V. The sets of probability measures B1=M([C1 , C2 , C3 , C4]) andB2=[R1] constitute a Beliefs Equilibrium. It contains the Bayesian BeliefsEquilibrium [b1=(C1 , 0.5; C2 , 0.5), b2=R1]. Also note that BR1(B1)=[ p # M([R1 , R2 , R3 , R4]) | p(R3)=p(R4)] and BR1(b1)=M([R1 , R2 ,R3 , R4]). This shows that the inclusion property BRi (Bi)�BRi (bi) inProposition 3 can be strict. The example also demonstrates that a ProperBeliefs Equilibrium may contain more than one Bayesian Beliefs Equi-librium. For instance, [b$1=C3 , b$2=R1] is another Bayesian Beliefs Equi-librium. However BR1(b$1)=[R1]. Therefore, not every Bayesian BeliefsEquilibrium [b$i]n

i=1 contained in a Beliefs Equilibrium [Bi]ni=1 has the

property BRi (Bi)�BRi (b$i). K

For games involving more than two players, a Beliefs Equilibrium ingeneral does not contain a Nash Equilibrium. This is already implied bythe fact that a Bayesian Beliefs Equilibrium is itself a Beliefs Equilibrium butnot a Nash Equilibrium. However, since Bayesian Beliefs Equilibrium andNash Equilibrium are equivalent in two person games, Proposition 3 hasthe following corollary.

TABLE V

C1 C2 C3 C4

R1 0, 1 0, 1 1, 1 0, 1R2 0, 1 0, 1 0, 1 1, 1R3 1, 1 &1, 1 0, 1 0, 1R4 &1, 1 1, 1 0, 1 0, 1

462 KIN CHUNG LO


Corollary of Proposition 3. In a two-person game, if [B1 , B2] is aBeliefs Equilibrium, then there exists _*j # Bi such that [_*1 , _*2 ] is a NashEquilibrium and BRi (Bi)�BRi (_*j ).

Proposition 3 delivers two messages. The first regards the prediction ofhow the game will be played. Suppose [Bi]n

i=1 is a Beliefs Equilibrium. Theassociated prediction regarding strategies played is that i chooses some_i # BRi (Bi). According to Proposition 3, it is always possible to find atleast one Bayesian Beliefs Equilibrium [bi]n

i=1 contained in [Bi]ni=1 such

that the observed behaviour of the uncertainty averse players (the actualstrategies they choose) is consistent with utility maximization given beliefsrepresented by [bi]n

i=1. This implies that an outsider who can only observethe actual strategy choices in the single game under study will not be ableto distinguish uncertainty averse players from Bayesian players. (I willprovide reasons to qualify such observational equivalence in the nextsection.)

We can use Proposition 3 also to address the welfare consequences ofuncertainty aversion, where the nature of our welfare comparisons isspelled out in Section 5.2. If [Bi]n

i=1 is a Beliefs Equilibrium, it contains aBayesian beliefs Equilibrium [bi]n

i=1 , and therefore,

max_i # M(Si)

minpi # Bi

ui (_i , pi)� max_i # M(Si)

ui (_i , bi).

The left-hand side of the above inequality is the ex ante utility of player iwhen his beliefs are represented by Bi and the right-hand side is ex anteutility when beliefs are represented by bi . The inequality implies that exante, i would prefer to play the Bayesian Beliefs Equilibrium [bi]n

i=1 to theBeliefs Equilibrium [Bi]n

i=1 . In this ex ante sense, uncertainty aversionmakes the players worse off.13

5.4. Uncertainty Aversion Can Be Beneficial When Players Agree

The comparisons above addressed the effects of uncertainty aversionwhen the equilibrium concepts used, namely Beliefs Equilibrium andBayesian Beliefs Equilibrium, do not require agreement between agents.Here I re-examine the effects of uncertainty aversion when agreement isimposed, as incorporated in the Beliefs Equilibrium with Agreement andNash Equilibrium solution concepts.


13 For the Bayesian Beliefs Equilibrium [bi]ni=1 constructed in the proof of Proposition 3,

we actually have

max_i # M(Si)

minpi # Bi

ui (_i , pi)= max_i # M(Si)

ui (_i , bi).

This of course, does not exclude the possibility that there may exist other Bayesian BeliefsEquilibria contained in [Bi]n

i=1 such that the equality is replaced by a strict inequality.


TABLE VI

XC2 YC2 XD2 YD2

C1 c c e eD1 a b d d

For two-person games, the Corollary of Proposition 3 still applies sinceagreement is not an issue given only two players. However, for gamesinvolving more than two players, the following example demonstrates thatit is possible to have a Beliefs Equilibrium with Agreement not containingany Nash Equilibrium.

Example 5. Uncertainty Aversion Leads to Pareto Improvement. Thegame presented in this example is a modified version of the prisoners'dilemma. The game involves three players, 1, 2, and N. Player N can beinterpreted as ``nature.'' The payoff of player N is a constant and his setof pure strategies is [X, Y]. Players 1 and 2 can be interpreted as twoprisoners. The set of pure strategies available for players 1 and 2 are[C1 , D1] and [C2 , D2], respectively, where C stands for ``co-operation''and D stands for ``defection.'' The payoff matrix for player 1 is shown inTable VI and that for player 2 is shown in Table VII. Assume that thepayoffs satisfy

a>c>b and c>d>e and 2c<a+b. (5.4.1)

Note that the game is the prisoner's dilemma game if the inequalitiesa>c>b in (5.4.1) are replaced by a=b>c. (When a=b>c, the payoffsof players 1 and 2 for all strategy profiles do not depend on nature's move.)This game is different from the standard prisoners' dilemma in one respect.In the standard prisoners' dilemma game, the expression a=b>c says thatit is always better for one player to play D given that his opponent playsC. In this game, the expression a>c>b says that if one player plays D andone plays C, the player who plays D may either gain or lose. The inter-pretation of the inequalities c>d>e in (5.4.1) is the same as that in thestandard prisoners' dilemma. That is, it is better for both players to playC rather than D. However, a player should play D given that his opponent

TABLE VII

XC1 YC1 XD1 YD1

C2 c c e eD2 b a d d

464 KIN CHUNG LO


plays D. Note that the last inequality 2c<a+b in (5.4.1) is implied bya=b>c in the prisoners' dilemma game. The inequality 2c<a+b can berewritten as (a&c)&(c&b)>0. For player 1, for example, (a&c) is theutility gain from playing D1 instead of C1 if the true state is XC2 . (c&b)is the corresponding utility loss if the true state is YC2 . Therefore, theinterpretation of 2c<a+b is that if you know your opponent plays C, thepossible gain (loss) for you to play D instead of C is high (low).

Assume that players 1 and 2 know each other's action but they areuncertain about nature's move. To be precise, suppose that the beliefs ofthe players are

BN =[C1C2]

B1=[ p # M([XC2 , YC2]) | pl�p(XC2)�ph with 0�pl<ph�1]

B2=[ p # M([XC1 , YC1]) | pl�p(XC1)�ph with 0�pl<ph�1].

The construction of [BN , B1 , B2] reflects the fact that the players agree.For example, the marginal beliefs of players 1 and 2 regarding [X, Y]agree with

2=[ p # M([X, Y]) | pl�p(X)�ph with 0�pl<ph�1]. (5.4.2)

Given [B N , B1 , B2], the payoffs of each pure strategy profile for players 1and 2 are shown in Table VIII. Recall that the payoff of player N is a con-stant. Therefore, both X and Y are optimal and we only need to considerplayers 1 and 2. [B N , B1 , B2] is a Beliefs Equilibrium with Agreement and

BR1(B1)=[C1] and BR2(B2)=[C2]

if and only if pl<c&ba&b

and ph>a&ca&b

.

Note that our assumptions guarantee that

c&ba&b

>0 anda&ca&b

<1,

so there exist values for pl and ph consistent with the above inequalities.However, [B N , B1 , B2] does not contain a Nash Equilibrium. To see

this, suppose that players 1 and 2 are Bayesians who agree that

TABLE VIII

C2 D2

C1 c, c e, ph b+(1&ph)aD1 pl a+(1&pl ) b, e d, d



TABLE IX

C2 D2

C1 c, c e, *b+(1&*)aD1 *a+(1&*) b, e d, d

p(X)=*=1&p(Y) for any * # [0, 1]. Then they are playing the game inTable IX.

The strategy profile [C1 , C2] is a Nash Equilibrium if and only if

c�*a+(1&*) b and c�*b+(1&*) a.

There exists * # [0, 1] such that [C1, C2] is a Nash Equilibrium if and only if

c� 12 (a+b),

which contradicts the last inequality in (5.4.1). Therefore, it is neveroptimal for both Bayesian players to play C and any Nash Equilibriumrequires both players to play D and therefore that both players receive dwith certainty. In the Beliefs Equilibrium with Agreement constructed above,on the other hand, both players play C and receive c>d with certainty.

To better understand why uncertainty aversion leads to a better equi-librium in this game, let us go back to the beliefs [B1 , B2] of players 1 and2. As explained in Section 2.1, although the global beliefs of players 1 and2 on [X, Y] are represented by the same 2 in (5.4.2), the local probabilitymeasures for different acts may be different. For example, the localprobability measure on [X, Y] at the act corresponding to D1 is(X, pl ; Y, 1&pl) and for D2 it is (X, ph ; Y, 1&ph), respectively. In the senseof local probability measures, therefore, players 1 and 2 disagree on therelative likelihood of X and Y when they are consuming the acts D1 andD2 , respectively. This allows playing D to be undesirable for both players.

The example delivers three messages. First, it shows that in a gameinvolving more than two players, uncertainty aversion can lead to an equi-librium that Pareto dominates all Nash Equilibria. Second, interpretingplayer N in the above game as ``nature,'' the game becomes a two-persongame where the players are uncertain about their own payoff functions.Therefore, uncertainty aversion can be ``beneficial'' even in two-persongames. Third, the beliefs profile [BN , B1 , B2] continues to be a BeliefsEquilibrium with Agreement if, for instance, the payoff of player N is inde-pendent of his own strategy, but is the highest when player 1 plays C1 andplayer 2 plays C2 . In this case, even if players can communicate, player Nhas a strict incentive not to announce his own strategy.14 K

466 KIN CHUNG LO

14 Greenberg [16] develops an example independently. The intuition of his example is verysimilar to that of Example 5 in this paper.


TABLE X

L C R

U 1, 1 1, 1 10, 1M 1, 1 1, 1 10, 1D 1, 10 1, 10 10, 10

6. WHY DO WE HAVE UNCERTAINTY AVERSION?

The next question I want to address is: When should we expect (or not)the existence of an equilibrium reflecting uncertainty aversion? In the con-text of single-person decision theory, we do not have much to say aboutthe origin or precise nature of beliefs on the set of states of nature.However, we should be able to say more in the context of game theory.The beliefs of the players should be ``endogenous'' in the sense of dependingon the structure of the game. For example, it is reasonable to predict thatthe players will not be uncertainty averse if there is an ``obvious way'' toplay the game.

The following two examples identify possible reasons for players to beuncertainty averse.

Example 6. Nonunique Equilibria. In the game in Table X, any strategyprofile is a Nash Equilibrium. Any [B1 , B2] is a Beliefs Equilibrium.Uncertainty aversion in this game is due to the fact that the players do nothave any idea about how their opponents will play. K

Example 7. Nonunique Best Responses. In the game in Table XI,[U, L] is the only Nash Equilibrium. However, it is equally good forplayer 1 to play D if he believes that player 2 plays L. Under this cir-cumstance, it may be too demanding to require player 2 to attach probabil-ity one to player 1 playing U. At the other extreme, where 2 is totallyignorant of 1's strategy choice, we obtain the Proper Beliefs Equilibrium[B1=[L], B2=M([U, D])]. K

This example shows that the existence of a unique Nash Equilibrium isnot sufficient to rule out an equilibrium with uncertainty aversion.However, I can prove the following:

TABLE XI

L R

U 0, 1 1, 0.5D 0, 1 0, 2



TABLE XII

L R

U 2, 1 1, 2D 2, 2 0, 2

Proposition 4. If the game has a unique Bayesian Beliefs Equilibriumand it is also a strict Nash Equilibrium, then there does not exist a ProperBeliefs Equilibrium.

Proof. Let [bi]ni=1 be the unique Bayesian Beliefs Equilibrium. Since it

is also a strict Nash Equilibrium of the game, there exists s*i # Si such thatbi=s*&i . Let [Bi]n

i=1 be a Beliefs Equilibrium. According to Proposition 3,s*&i # Bi . Using Proposition 3 and the definition of Beliefs Equilibrium,we have margSi Bj �BRi (Bi)�BRi (s*&i)=[s*i ]. This implies [Bi]n

i=1=[s*&i]n

i=1. K

Corollary of Proposition 4. In a two-person game, if the game has aunique Nash Equilibrium and it is also a Strict Nash Equilibrium, then theredoes not exist a Proper Beliefs Equilibrium.

A Proper Beliefs Equilibrium can be ruled out also if the game isdominance solvable.

Proposition 5. If the game is dominance solvable, then there does notexist a Proper Beliefs Equilibrium.15 (See Table XII.)

Proof. Let [Bi]ni=1 be a Beliefs Equilibrium. According to Proposi-

tion 2, BRi (Bi)�Ri and Bi �RBi . Since iterated strict dominance andcorrelated rationalizability are equivalent, a dominance solvable game hasa unique pure strategy profile [s*i ]n

i=1 such that Ri=s*i and RB i=s*&i .Therefore, BRi (Bi)=s*i and Bi=s*&i . K

7. DECISION THEORETIC FOUNDATION

Recently, decision theoretic foundations for Bayesian solution conceptshave been developed. In particular, Aumann and Brandenburger [4]develop epistemic conditions for Nash Equilibrium. The purpose of this

468 KIN CHUNG LO

15 We may also want to ask the reverse question: Are the conditions stated in Propositions4 and 5 necessary for the absence of Proper Beliefs Equilibrium? The game in Table XII hastwo Nash Equilibria (and therefore it is not dominance solvable). They are [U, R] and[D, L]. However, there does not exist a Proper Beliefs Equilibrium.


line of research is to understand the knowledge requirements needed tojustify equilibrium concepts. Although research on the generalization ofNash Equilibrium to allow for uncertainty averse preferences has alreadystarted (see Section 8 below), serious study of the epistemic conditions forthose generalized equilibrium concepts has not yet been carried out.

In this section, I provide epistemic conditions for the equilibrium con-cepts proposed in this paper. The main finding is that Beliefs Equilibrium(Beliefs Equilibrium with Agreement) and Bayesian Beliefs Equilibrium(Nash Equilibrium) presume similar knowledge requirements. This supportsthe interpretation of results in previous sections as reflecting solely the effectsof uncertainty aversion.

Before I proceed, I acknowledge that although the partitional informa-tion structure used below is standard in game theory (see, for instance,Aumann [3] and Osborne and Rubinstein [25, p. 76]), it is more restric-tive than the interactive belief system used by Aumann and Brandenburger[4]. In their framework, ``know'' means ``ascribe probability 1 to'' which ismore general than the ``absolute certainty without possibility of error'' thatis being used here (Aumann and Brandenburger [4, p. 1175]). Apart fromthis difference, the two approaches share essentially the same spirit. Thereis a common set of states of the world. A state contains a description ofeach player's knowledge, beliefs, strategy, and payoff function.16

Formally, the following notation is needed. Let 0 be a common finite setof states of nature for the players with typical element |. Each state | # 0consists of a specification for each player i of

v Hi (|)�0, which describes player i 's knowledge in state | (whereHi is a partitional information function)

v 2i (|), a closed and convex set of probability measures on Hi (|),the beliefs of player i in state |

v fi (|) # M(Si), the mixed strategy used by player i in state |

v ui (|, } ): S � R, the payoff function of player i in state |.

To respect the partitional information structure, the payoff functionui : 0_S � R and the strategy fi : 0 � M(Si) are required to be adapted toHi . Given fi , fi (|)(si) denotes the probability that i plays si according tothe strategy fi in state |. For each | # 0, player i's beliefs over S&i are


16 Another feature of the interactive belief system (Aumann and Brandenburger [4,p. 1164]) that is shared by the model here is that players' prior beliefs are not part of thespecification. Note that the common prior assumption in their paper is imposed only for theirTheorem B (p. 1168).


represented by a closed and convex set of probability measures Bi (|) thatis induced from 2i (|) in the following way:

Bi (|)#{pi # M(S&i) | _qi # 2i (|) such that pi (s&i)

= :| # Hi (|)

qi (|) `j{i

fj (|)(sj) \s&i # S&i= .

The above specification is common knowledge among the players. Player iis said to know an event E at | if Hi (|)�E. Say that an event is mutualknowledge if everyone knows it. Let H be the meet of the partitions of allthe players and H(|) the element of H which contains the element |. Anevent E is common knowledge at | if and only if H(|)�E. Say that playeri is rational at | if his strategy fi (|) maximizes utility as stated in (2.2.2)when beliefs are represented by Bi (|).

The following proposition describes formally the knowledgerequirements for [Bi]n

i=1 to be a Beliefs Equilibrium. If each Bi is asingleton, then a parallel result for Bayesian Beliefs Equilibrium is obtained.(The version of Proposition 6 for Bayesian Beliefs Equilibrium for two-person games can be found in Theorem A in Aumann and Brandenburger[4, p. 1167].) To focus on the intuition, all the propositions in this sectionare only informally discussed. Their proofs can be found in the appendix.

Proposition 6. Suppose that at some state |, the rationality of theplayers, [ui]n

i=1 , and [Bi]ni=1 are mutual knowledge. Then [Bi]n

i=1 is aBeliefs Equilibrium.

The idea of Proposition 6 is not difficult. At |, player i knows j 's beliefsBj (|), payoff function BRj (|), and that j is rational. Therefore, anystrategy fj (|$) for player j with |$ # Hi (|), that player i thinks is possible,must be player j 's best response given his beliefs. That is,fj (|$) # BRj (|)(Bj (|)) \|$ # Hi (|). Since the preference ordering of playerj is quasiconcave, any convex combination of strategies in the set[ fj (|$) | |$ # Hi (|)] must also be a best response for player j. Byconstruction, margSj Bi (|) is a subset of the convex hull of[ fj (|$) | |$ # Hi (|)]. This implies margSj Bi (|)�BRj (|)(Bj (|)) andtherefore [Bi]n

i=1 is a Beliefs Equilibrium.In a Beliefs Equilibrium with Agreement, the beliefs [Bi]n

i=1 of theplayers over the strategy choices of opponents are required to have theproperties of agreement and stochastic independence. Since Bi is derivedfrom 2i , it is to be expected that some restrictions on 2i are needed for[Bi]n

i=1 to possess the desired properties. In the case where players areexpected utility maximizers so that 2i (|) is a singleton for all |,

470 KIN CHUNG LO


Theorem B in Aumann and Brandenburger [4, p. 1168] shows that byrestricting [2i]n

i=1 to come from a common prior, mutual knowledge ofrationality and payoff functions and common knowledge of beliefs are suf-ficient to imply Nash Equilibrium. In the case where players are uncer-tainty averse, the following proposition says that by restricting each playeri to being completely ignorant at each | about the relative likelihood ofstates in Hi (|), exactly the same knowledge requirements that imply NashEquilibrium also imply Beliefs Equilibrium with Agreement.

Proposition 7. Suppose that 2i (|)=M(Hi (|)) \|. Suppose that atsome state |, the rationality of the players and [ui]n

i=1 are mutualknowledge and that [Bi]n

i=1 is common knowledge. Then [Bi]ni=1 is a Beliefs

Equilibrium with Agreement.

The specification of 2i (|) as the set of all probability measures on Hi (|)reflects the fact that player i is completely ignorant about the relativelikelihood of states in Hi (|). It is useful to explain the role played by thisparametric specialization of beliefs. Given any state | and any event E, thebeliefs [ p(E) | p # M(Hi (|))] of player i about E at | must satisfy one andonly one of the following three conditions.

1. p(E)=1 \p#M(Hi (|)) if and only if Hi (|)�E (player i knows E).

2. [ p(E) | p # M(Hi (|))]=[0, 1] if and only if Hi (|)�3 E andHi (|) & E{< (player i does not know E or 0"E).

3. p(E)=0 \p # M(Hi (|)) if and only if Hi (|) & E=< (player iknows 0"E.)

Therefore, player j knows i's beliefs about E if and only if one and onlyone of the following is true.

1. j knows that i knows E.

2. j knows that i does not know E or 0"E.

3. j knows that i knows 0"E.

As a result, mutual knowledge of beliefs about E implies agreement ofbeliefs about E. The common knowledge assumption in Proposition 7 isused only to derive the property of stochastically independent beliefs.17

Note that Theorem B in Aumann and Brandenburger [4] is not a spe-cial case of Proposition 7. The common prior assumption imposed by theirtheorem coincides with the restriction on 2i imposed by Proposition 7 onlyin the case where Hi (|)=[|] \|. Therefore, the examples they provide toshow the sharpness of their theorem do not apply to Proposition 7. To


17 In particular, unlike Theorem B in Aumann and Brandenburger [4], the proof ofProposition 7 does not rely on the ``agreeing to disagree'' result of Aumann [2].


serve this purpose, I provide Example 8 to show that Proposition 7 is tightin the sense that mutual knowledge rather than common knowledge of[Bi]n

i=1 is not sufficient to guarantee a Beliefs Equilibrium with Agreement.

Example 8. Mutual Knowledge of Beliefs Is Not Sufficient for Agree-ment. The game consists of three players. The set of states of nature is0=[|1 , |2 , |3 , |4]. The players' information structures are

H1=[[|1 , |2], [|3 , |4]],

H2=[[|1 , |3], [|2 , |4]],

and

H3=[[|1 , |2 , |3], [|4]].

Their strategies are listed in Table XIII.Suppose 2i (|)=M(Hi (|)) \|. At |1 , the beliefs [Bi (|1)]3

i=1 of theplayers are mutual knowledge. For example, since B1(|)=M([UT, DT])\| # 0, B1(|1) is common knowledge and therefore mutual knowledge at|1 . According to the proof of Proposition 7, marginal beliefs of the playersagree. For example, margS2

B1 (|1) = margS2B3 (|1) = M([U, D]) .

However, B3(|1)=M([LU, LD, RU]) is not common knowledge at |1 .Player 3 does not know that player 1 knows player 3's beliefs. At |1 ,player 3 cannot exclude the possibility that the true state is |3 . At |3 ,player 1 only knows that player 3's beliefs are represented by eitherB3(|3)=M([LU, LD, RU]) or B3(|4)=[RD]. Note that B3(|1) does nottake the form required in the definition of Beliefs Equilibrium withAgreement. K

Finally, although the notion of Weak Beliefs Equilibrium (Definition 6)is not the main focus of this paper, it is closely related to the equilibriumconcepts proposed by the papers discussed in Section 8 below. Accordingto the following proposition, complete ignorance and rationality at a state| are sufficient to imply Weak Beliefs Equilibrium.

Proposition 8. Suppose that at some state |, 2i (|)=M(Hi (|)) andthat players are rational. Then [Bi]n

i=1 is a Weak Beliefs Equilibrium.

TABLE XIII

|1 |2 |3 |4

f1 L L R Rf2 U D U Df3 T T T T

472 KIN CHUNG LO


By looking at the definition of Weak Beliefs Equilibrium more carefully,Proposition 8 is hardly surprising. For instance, given any n-person normalform game, it is immediate that [M(S&i)]n

i=1 is a Weak Beliefs Equilibrium.Note that Proposition 8 requires only that the players be rational; they donot need to know that their opponents are rational. They also do not needto know anything about the beliefs of their opponents.

8. RELATED LITERATURE

In this section, I compare my equilibrium concepts with those proposed byDow and Werlang [9] and Klibanoff [18].18 Since the latter employs thesame strategy space as in this paper, let me first conduct a direct comparison.

8.1. Klibanoff [18]

Klibanoff [18] also adopts the multiple priors model to representplayers' preferences in normal form games with any finite number ofplayers and proposes the following solution concept:19

Definition 9. ([_i]ni=1 , [Bi]n

i=1) is an Equilibrium with UncertaintyAversion if the following conditions are satisfied:

1. _&i # Bi .

2. minpi # Bi ui (_i , pi)�minpi # Bi ui (_$i , pi) \_$i # M(Si).

_i is the actual strategy used by player i and Bi is his beliefs aboutopponents' strategy choices. Condition 1 says that player i 's beliefs cannotbe ``too wrong.'' That is, the strategy profile _&i chosen by other playersshould be considered ``possible'' by player i. Condition 2 says that _i is abest response for i given his beliefs Bi .


18 A brief review of other related papers is provided below. There are two other papers ongeneralizations of Nash Equilibrium. Lo [19] proposes Cautious Nash Equilibrium which,when specialized to the multiple priors model, refines the equilibrium concept in Dow andWerlang [9]. However, its main focus is on relaxing mutual knowledge of rationality, ratherthan uncertainty aversion. Mukerji [21] proposes the equilibrium concept Equilibrium in=-ambiguous Beliefs. The equilibrium concept only admits players' utility functions having aspecific form but otherwise is identical to that in Dow and Werlang [9]. Epstein [11] andMukerji [21] generalize rationalizability. The former requires common knowledge ofrationality but the latter does not. For normal form games of incomplete information, Epsteinand Wang [12] establish the general theoretical justification for the Harsanyi style formula-tion for non-Bayesian players. Lo [20] provides a generalization of Nash Equilibrium inextensive form games. All the above papers either adopt the multiple priors model or considera class of preferences that includes the multiple priors model as a special case.

19 The equilibrium concept presented here is a simplified version. Klibanoff [18] assumesthat players' beliefs are represented by lexicographic sets of probability measures.


In addition, the following refinement of Equilibrium with UncertaintyAversion is offered in his paper. ([_i]n

i=1 , [Bi]ni=1) is an Equilibrium with

Uncertainty Aversion and Rationalizable Beliefs if it is an Equilibrium withUncertainty Aversion and, in addition, Bi �RBi (defined in Definition 8).That is, player i believes that his opponents' strategy choices are correlatedrationalizable.

Although Klibanoff 's equilibrium concepts involve both the specificationof beliefs and the actual strategies used by the players, while the equi-librium concepts in my paper involve only the former, the main differencesbetween them can be summarized in terms of beliefs in four aspects asshown in Table XIV. It enables us to conclude that Beliefs Equilibrium withAgreement is a refinement of Klibanoff 's equilibrium concepts.

Proposition 9. If [Bi]ni=1 is a Beliefs Equilibrium with Agreement, then

for any _i # margSi Bj , it is the case that ([_i]ni=1 , [Bi]n

i=1) is an Equi-librium with Uncertainty Aversion and Rationalizable Beliefs.

Since knowledge of rationality and beliefs is the most essential propertyunderlying the equilibrium concepts, let us focus on two-person gameswhere agreement and stochastic independence are not the issues. Thefollowing example illustrates that an Equilibrium with Uncertainty Aversionand Rationalizable Beliefs may not be a Beliefs Equilibrium.

Example 9. Refinement of Klibanoff [18]. The game in Table XV isdeliberately constructed so that every strategy of every player survivesiterated elimination of strictly dominated strategies. Therefore, Klibanoff 'sstandard equilibrium concept coincides with his own refinement. It is easyto check that [_1 , _2 , B1 , B2]=[D, R, M(S2), M(S1)] is a Equilibriumwith Uncertainty Aversion (and Rationalizable Beliefs). This equilibriumpredicts that D and R to be the unique best response for players 1 and 2,respectively. As a result, player 1 receives 2 and player 2 receives 5.

TABLE XIV

Equilibrium with Uncertainty Aversion Beliefs Equilibriumand Rationalizable Beliefs with Agreement

Knowledge ofrationality and beliefs margSi

Bj & BRi (Bi){< margSiBj�BRi (Bi)

Agreement ofmarginal beliefs margSj

Bi & margSjBk{< margSj

Bi=margSjBk

Stochastically Bi contains at least one product Bi contains all productindependent beliefs measure in _j{i margSj

Bi measures in _j{i margSjBi

Rationalizable beliefs Bi�RB i Bi�RBi (Proposition 2)

474 KIN CHUNG LO


TABLE XV

L C R

U 10, 10 1.99, 10 10, 10D 2, 4 2, 4 2, 5

It is reasonable that player 2 will play R. The reason is that R is as goodas L and C if 2 plays U and it is strictly better than L and C if 2 plays D.However, if player 1 realizes this, 1 should play U, and as a result, bothplayers will receive 10. Note that [B1 , B2]=[M(S2), M(S1)] is not aBeliefs Equilibrium. Moreover, no Beliefs Equilibrium in this game willpredict D to be player 1's unique best response. K

Finally, for any two-person game, Klibanoff's standard equilibrium conceptis equivalent to the notion of Weak Beliefs Equilibrium (Definition 6).

Proposition 10. [B1 , B2] is a Weak Beliefs Equilibrium if and only if thereexists _i # Bj such that [_1 , _2 , B1 , B2] is an Equilibrium with UncertaintyAversion.

8.2. Dow and Werlang [9]

Dow and Werlang [9] consider two-person games and assume thatplayers' preference orderings over acts are represented by the convex capacitymodel proposed by Schmeidler [29]. Any such preference ordering is amember of the multiple priors model (Gilboa and Schmeidler [15]). Theirequilibrium concept can be restated using the multiple priors model asfollows.

Definition 10. [B1 , B2] is a Nash Equilibrium Under Uncertainty if thefollowing conditions are satisfied:

1. There exists Ei �Si such that pj (Ei)=1 for at least one pj # Bj .

2. minpi # B i ui (si , pi)�minp i # B i ui (s$i , pi) \si # Ei \s$i # Si .

Dow and Werlang [9] interpret condition 1 as saying that player j``knows'' that player i will choose a strategy in Ei . Condition 2 says thatevery si # Ei is a best response for i, given that Bi represents i 's beliefsabout the strategy choice of player j.

Unlike Klibanoff and this paper, Dow and Werlang restrict players tochoosing pure rather than mixed strategies. It is therefore important toreiterate the justification for using one strategy space instead of the other.According to the discussion in Section 4.2, the use of pure vs mixedstrategy spaces depends on the perception of the players about the order ofstrategy choices. The adoption of a mixed strategy space in Klibanoff [18]



and in this paper can be justified by the assumption that each player perceiveshimself as moving last. On the other hand, we can understand the adoptionof a pure strategy space in Dow and Werlang [9] as assuming that eachplayer perceives himself as moving first and has an expected utilityrepresentation for preferences over objective lotteries on acts. Furthercomparison of Dow and Werlang [9] and this paper is provided in thenext subsection.

8.3. Epistemic Conditions

I suggested in the introduction that in order to carry out a ceteris paribusstudy of the effects of uncertainty aversion on how a game is played, weshould ensure that the generalized equilibrium concept is different fromNash Equilibrium only in one dimension, players' attitude towardsuncertainty. In particular, the generalized solution concept should sharecomparable knowledge requirements with Nash Equilibrium. According tothis criterion, I argue in Section 7 that the solution concepts I propose areappropriate generalizations of their Bayesian counterparts.

Dow and Werlang [9] and Klibanoff [18] do not provide epistemicfoundations for their solution concepts and a detailed study is beyond thescope of this paper. However, I show below that in the context of two-personnormal form games, exactly the same epistemic conditions that supportWeak Beliefs Equilibrium as stated in Proposition 8, namely, completeignorance and rationality, also support Nash Equilibrium Under Uncertaintyand Equilibrium with Uncertainty Aversion. Therefore, the sufficient conditionsfor players' beliefs to constitute an equilibrium in these two senses do notrequire the players to know anything about the beliefs and rationality oftheir opponents.

The weak epistemic foundation for their equilibrium concepts is readilyreflected by the fact that given any two-person normal form game, [M(S2),M(S1)] is always a Nash Equilibrium Under Uncertainty and there alwaysexist _1 and _2 such that (_1 , _2 , M(S2), M(S1)) is an Equilibrium withUncertainty Aversion. The equilibrium notions in these two papers thereforedo not fully exploit the difference between a game, where its payoff structure(e.g., dominance solvability) may limit the set of ``reasonable'' beliefs, anda single-person decision making problem, where any set of priors (or singleprior in the Bayesian case) is ``rational.'' In fact, Dow and Werlang[9, p. 313] explicitly adopt the view that the degree of uncertainty aversionis subjective, as in the single-agent setting, rather than reasonably tied tothe structure of the game. As a result, their equilibrium concept delivers acontinuum of equilibria for every normal form game (see their theorem onp. 313).

Let me now proceed to the formal statements. Recall that Klibanoff 'sstandard equilibrium concept can be readily rewritten as a Weak Beliefs

476 KIN CHUNG LO


Equilibrium (Proposition 10). It follows that Proposition 8 provides theepistemic conditions underlying Klibanoff's equilibrium concept as simplifiedhere.

Since Dow and Werlang [9] adopt a pure strategy space, I keep allnotation from Section 7 but redefine

v fi : 0 � Si

v Bi (|)#[ pi # M(Sj) | _qi # 2i (|) such thatpi (sj)=�| # H i(|) & [|$ | f j (|$)=s j] qi (|) \sj # Sj]

v BRi (Bi)#argmaxsi # Si minpi # Bi ui (si , pi).

That is, fi (|) is the pure action used by player i at | and Bi (|) is the setof probability measures on Sj induced from 2i (|) and fj . It represents thebeliefs of player i at | about j 's strategy choice.

Proposition 11. Suppose that at some state |, 2i (|)=M(Hi (|)) andthat the players are rational. Then [B1 , B2] is a Nash Equilibrium UnderUncertainty.

When a decision maker's beliefs are represented by a probability measure p,an event E is p -null if p(E)=0. It is well recognized that when preferencesare not probabilistically sophisticated, there are alternative ways of definingnullity. The equilibrium concepts in Dow and Werlang [9], Klibanoff[18], and this paper can all be regarded as generalizations of NashEquilibrium if the ``right'' notion of nullity is adopted. To see this, firstassume that each player does not have a strict incentive to randomize.Take S&i to be the state space of player i and Si to be a subset of actswhich map S&i to R. Suppose that pi represents the preference orderingof player i over Si . Player i is rational if he chooses si such that si pi si

\si # Si . The following is an appropriate restatement of Nash Equilibriumin terms of preferences:

Definition 11. [ pi , pj] is a Nash Equilibrium if the following condi-tions are satisfied:

1. There exists 1i �Si such that the complement of 1i is pj -null.

2. si pi si \si # 1i \si # Si .

In words, [ pi , pj] is a Nash Equilibrium if the event that player i isirrational is pj -null. Suppose pi and pj are represented by the multiplepriors model. Let Bi and Bj be the sets of probability measures underlyingpi and pj , respectively. Then [Bi , Bj] is a Beliefs Equilibrium (withAgreement) if and only if [ pi , pj] satisfies Definition 11, with Si replacedby M(Si) and using the definition of nullity as stated in Section 2.1. The



equilibrium concepts of Dow and Werlang [9] and Klibanoff [18] areequivalent to Definition 11 if the notion of nullity in Dow and Werlang[9] is adopted: an event is pj -null if it is attached zero probability by atleast one probability measure in Bj .

20

The above discussion may lead the reader to think that the epistemicconditions provided for the equilibrium concepts in Dow and Werlang [9]and Klibanoff [18] are biased by using the notion of knowledge that isonly appropriate for the equilibrium concepts proposed in this paper.Therefore, it is worth reiterating that the conditions stated in Propositions8 and 11 do not require the players to know anything about theiropponents' beliefs and rationality. Therefore, the notion of knowledge to beadopted is irrelevant. Moreover, Propositions 8 and 11 do not even exploitthe fact that the information structure is represented by partitions. The twopropositions and their proof continue to hold as long as the beliefs ofplayer i at | are represented by the set of all probability measures over anevent Hi (|)�0 with the property that | # Hi (|). Therefore, the conclusionthat complete ignorance and rationality imply the two equilibrium conceptsremains valid even in the absence of partitional information structures.

9. MORE GENERAL PREFERENCES

The purpose of this section is to show that even if we drop the particularfunctional form proposed by Gilboa and Schmeidler [15] but retain someof its basic properties, counterparts of this paper's equilibrium conceptsand results can be formulated and proven.

Let us first go back to the context of single-person decision theory anddefine a class of utility functions that generalizes the multiple priors model.Recall the notation introduced in Section 2.1, whereby p is a preferenceordering over the set of acts F, where each act maps 0 into M(X). Imposethe following restrictions on p : Suppose that p restricted to constantacts conforms to expected utility theory and so is represented by an affineu: M(X) � R. Suppose that there exists a nonempty, closed, and convex setof probability measures 2 on 0 such that p is representable by a utilityfunction of the form

f [ U( f )#V \{| u b f dp | p # 2=+ (9.1)

478 KIN CHUNG LO

20 To see that Klibanoff 's equilibrium concept satisfies Definition 11 when Dow andWerlang's definition of nullity is adopted, restate Weak Beliefs Equilibrium in terms ofB� i : [B� i , B� j ] is a Weak Beliefs Equilibrium if there exists b� j # B� j such that _i # BRi (B� i )\_i # support of b� j . Set 1i in Definition 11 to be the support of b� j .


for some function V: R2 � R. Assume that p is monotonic in the sensethat for any f, g # F, if � u b f dp>� u b g dp \p # 2, then fog. Say that p

is multiple priors representable if p satisfies all the above properties.Quasiconcavity of p will also be imposed occasionally.

Two examples are provided here to clarify the structure of the utilityfunction U in (9.1). Suppose there exists a probability measure + overM(0) and a concave and increasing function h such that

U( f )=|M(0)

h \|0u b f dp+ d+.

In this example, the set of probability measures 2 corresponds to the supportof +. The interpretation of the above utility function is that the decisionmaker views an act f as a two-stage lottery. However, the reduction ofcompound lotteries axiom may not hold (Segal [30]). Note that this utilityfunction satisfies quasiconcavity. Another example for U which is notnecessarily quasiconcave is the Hurwicz [17] criterion;

U( f )=: minp # 2 |

0u b f dp+(1&:) max

p # 2 |0

u b f dp,

where 0�:�1.Adapting the model to the context of normal form games as in

Section 2.2, the objective function of player i is V([ui (_i , bi) | bi # Bi]). Allequilibrium notions can be defined precisely as before. I now prove thefollowing extension of Proposition 3.

Proposition 12. Consider an n-person game. Suppose that the preferenceordering of each player is multiple priors representable and quasiconcave. If[Bi]n

i=1 is a Beliefs Equilibrium, then there exists bi # Bi such that [bi]ni=1

is a Bayesian Beliefs Equilibrium and BRi (Bi)�BRi (bi).

Proof. As in the proof of Proposition 3, it suffices to show that, givenBi , there exists bi # Bi such that BRi (Bi)�BRi (bi).

I first show that for each _i # BRi (Bi), there exists bi # Bi such that_i # BRi (bi). Suppose that this were not true. Then there exists _i # BRi (Bi)such that for each bi # Bi , we can find _$i # M(Si) with ui (_i , bi)<ui (_$i , bi).This implies that there exists _i* # M(Si) such that ui (_i* , bi)>ui (_i , bi)\bi # Bi . (See Lemma 3 in Pearce [26, p. 1048].) Since the preference ofplayer i is monotonic, player i should strictly prefer _i* to _i when his beliefsare represented by Bi . This contradicts the fact that _i # BRi (Bi).

Quasiconcavity of preference implies that BRi (Bi) is a convex set.Therefore, there exists an element _i # BRi (Bi) such that the support of _i

is equal to the union of the support of every probability measure in



BRi (Bi). Since _i # BRi (bi) for some bi # Bi and ui ( } , bi) is linear on M(Si),this implies that si # BRi (bi) \si # support of _i . This in turn impliesBRi (Bi)�BRi (bi). K

Besides Proposition 3, it is not difficult to see that Proposition 2 alsoholds if the preference ordering of each player is multiple priors representable.(The monotonicity of the preference ordering for player i ensures that 1 n

i =1� n

i \n in the proof of Proposition 2.) Propositions 4 and 5 are also validbecause their proofs depend only on Propositions 2 and 3. Finally, all theresults in Section 6, except Proposition 6 which also requires preferences tobe quasiconcave, are true under the assumption of multiple priors representablepreferences.

10. CONCLUDING REMARKS

Let me first summarize the questions addressed in this paper:

1. What is a generalization of Nash Equilibrium (and its variants) innormal form games that allows for uncertainty averse preferences?

2. What are the epistemic conditions for those equilibrium concepts?

3. Can an outside observer distinguish uncertainty averse playersfrom Bayesian players?

4. Does uncertainty aversion make the players worse off (better off)?

5. How is uncertainty aversion related to the structure of the game?

Generalizations of Nash Equilibrium have already been proposed byDow and Werlang [9] and Klibanoff [18] to partly answer questions 3,4, and 5. One important feature of the equilibrium concepts presented inthis paper that is different from Dow and Werlang [9] but in commonwith Klibanoff [18] is the adoption of mixed instead of pure strategyspace. They can both be justified by different perceptions of the playersabout the order of strategy choices. On the other hand, I can highlight thefollowing relative merits of the approach pursued here. A distinctive featureof the solution concepts proposed in my paper is their epistemic foundation,which resemble as closely as possible those underlying the correspondingBayesian equilibrium concepts. As pointed out by Dow and Werlang [9,p. 313], their equilibrium concepts are only ``presented intuitively ratherthan derived axiomatically.'' In my paper, some epistemic conditions arealso provided for the equilibrium concepts proposed by Dow and Werlang[9] and Klibanoff [18]. The weakness of their equilibrium concepts isrevealed by the fact that the epistemic conditions do not involve anystrategic considerations. This point was demonstrated in Section 8.3 where

480 KIN CHUNG LO


I noted that in any normal form game, regardless of its payoff structure,the beliefs profile [M(S2), M(S1)] constitutes an equilibrium in their sense.

APPENDIX

Proof of Proposition 6. Fix | at the state where the rationality of theplayers, [ui]n

i=1 and [Bi]ni=1, are mutual knowledge. Player i knows player

j 's beliefs means

Hi (|)�[|$ # 0 | Bj (|$)=Bj (|)].

Player i knows player j is rational means

Hi (|)�[|$ # 0 | fj (|$) # BRj (|$)(Bj (|$))].

Note that BRj varies with the state because uj does. Player i knows playerj 's payoff function means

Hi (|)�[|$ # 0 | BRj (|$)=BRj (|)].

Therefore,

Hi (|)�[|$ # 0 | Bj (|$)=Bj (|)]

& [|$ # 0 | fj (|$) # BRj (|$)(Bj (|$))]

& [|$ # 0 | BRj (|$)=BRj (|)]

�[|$ # 0 | fj (|$) # BRj (|)(Bj (|))].

This implies

[ fj (|$) | |$ # Hi (|)]�BRj (|)(Bj (|)).

The fact that the preference of player j is quasiconcave implies thatBRj (|)(Bj (|)) is a convex set. Therefore, we have

convex hull of [ fj (|$) | |$ # Hi (|)]�BRj (|)(Bj (|)).

By construction of Bi (|),

margS j Bi (|)�convex hull of [ fj (|$) | |$ # Hi (|)]�BRj (|)(Bj (|)).

This shows that [Bi]ni=1 is a Beliefs Equilibrium. K

Proof of Proposition 7. The conditions stated in Proposition 7 implythose in Proposition 6. Therefore, it is immediate that [Bi]n

i=1 is a BeliefsEquilibrium.



By construction of Bi (|) and the assumption 2i (|)=M(Hi (|)) \|, wehave

margS j Bi (|)=convex hull of [ fj (|$) | |$ # Hi (|)] \|.

Now fix | at the state where the rationality of the players and [ui]ni=1

are mutual knowledge and [Bi]ni=1 is common knowledge. Player k knows

player i 's beliefs implies that Bi (|$)=Bi (|) \|$ # Hk(|) and therefore,

convex hull of [ fj (|") | |" # Hi (|$)]

=margS j Bi (|$)

=margS j Bi (|)

=convex hull of [ fj (|") | |" # Hi (|)] \|$ # Hk(|).

Let 7kj �[ fj (|$) | |$ # Hk(|)] be the set of extreme points of

margSj Bk(|). I claim that 7kj �margSj Bi (|) (and therefore margSj Bk(|)

�margS j Bi (|)). Suppose that it were not true. Then there exists _j # 7kj

such that _j � margSj Bi (|$) \|$ # Hk(|). Therefore,

_j � .|$ # Hk(|)

[ fj (|") | |" # Hi (|$)]$[ fj (|$) | |$ # Hk(|)]$7kj % _j ,

which is a contradiction. Since i, j, and k are arbitrary, we havemargSj Bi (|)=margS j Bk(|) and, in particular, 7i

j=7kj #7j .

It only remains to show that

Bi (|)=convex hull of [_&i # M(S&i) | margS j _&i # 7j]

Bi (|) takes the form as required if and only if for each _&i # _j{i 7 j ,there exists |$ # Hi (|) such that fj (|$)=_j \j{i.

Suppose that the condition stated above were not satisfied. Without lossof generality, assume that for player 1, there exists _&1 # _j{1 7 j such thatfor each |" # H1(|), there exists j{1 where fj (|"){_j . This implies that_&1 � B1(|). B1(|) is common knowledge at | implies that B1(|")=B1(|)\|" # H(|). Therefore, _&1 � B1(|") \|" # H(|). Therefore, for each|" # H(|), there exists j{1 where fj (|"){_j .

Now consider player 2. The last sentence in the previous paragraphimplies that for |$ # H(|) such that f2(|$)=_2 , for each |" # H2(|$),there exists j # [3, ..., n] such that fj (|"){_j . Therefore, >n

j=3 _j �marg_n

j=3 S j B2(|$). Again B2(|) is common knowledge at | # 0 impliesB2(|")=B2(|) \|" # H(|). Therefore, >n

j=3 _j � marg_nj=3 S j B2(|")

\|" # H(|) and we can conclude that for each |" # H(|), there existsj # [3, ..., n] such that fj (|"){_j .

482 KIN CHUNG LO


Repeat the same argument for players 3, ..., n to conclude that for each|" # H(|), fn(|"){_n . This contradicts the fact that _n # 7n . K

Proof of Proposition 8. By construction of Bi (|) and the assumption2i (|)=M(Hi (|)), it follows that margSj Bi (|)=convex hull of [ fj (|$) | |$# Hi (|)]. In particular, fj (|) # margSj Bi (|). At |, the fact that player jis rational implies fj (|) # BRj (|)(Bj (|)). Therefore, margSj Bi (|) &

BRj (|)(Bj (|)){<. K

Proof of Proposition 11. Set Ej=fj (|). By construction of Bi (|) andassumption 2i (|)=M(Hi (|)), it follows that Bi (|)=M([ fj (|$) | |$ #Hi (|)]). In particular, there exists a probability measure in Bi (|)which attaches probability one to Ej . Therefore, Ej satisfies condition 1 inDefinition 10. At |, the fact that player j is rational implies fj (|) #BRj (|)(Bj (|)). Therefore, condition 2 in Definition 10 is also satisfied.This completes the proof that [B1 , B2] is a Nash Equilibrium UnderUncertainty. K

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