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Correlated equilibrium, conformity andstereotyping in social groups
Edward CartwrightDepartment of Economics
Keynes College, University of Kent,Canterbury CT2 7NP, [email protected]
Myrna WoodersDepartment of Economics, Vanderbilt University
Nashville, TN 37235 [email protected]
May 2012
Keywords: correlated equilibrium, conformity, stereotyping, socialgroups, within-group fairness, homophily, social norms
Abstract
We argue that a social norm and the coordination of behavior withinsocial groups can be expressed by a correlated equilibrium. Given a socialgroup structure (a partition of individuals into social groups), we proposefour conditions that one may expect of a correlated equilibrium consistentwith social norms. These are: (a) within-group anonymity (conformitywithin groups), (b) group independence (no conformity between groups),(c) homophily (individuals in the same group have similar attributes), and(d) predictable group behavior (ex-post stability). We demonstrate thatcorrelated equilibrium satisfying (a)-(c) exist very generally and equilib-rium satisfying (a)-(d) exist in games with many players. We also considerstereotyped beliefs - beliefs that all individuals in a social group can beexpected to behave in the same way - and show that stereotyping is notcostly to the person who stereotypes but may or may not be bene�cial tosociety
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1 Introduction
A social norm can be thought of as a shared belief system of rules and standardsthat guide behavior (Gibbs, 1965; Cialdini and Trost, 1998). An individual con-forms if he chooses to follow the norms of the society to which he belongs.Norms of behavior are important in situations as diverse as public good provi-sion, resource allocation, and the assignment of property rights. In order that asocial norm exist and be stable it must be in the interests of individuals withinthe society or group to conform to the rules and standards given by the norm.This implies that there must be incentives to conform, either because of intrinsicmotivations, such as generosity, and/or extrinsic motivations, such as a fear ofpunishment. It also means, in game theoretic terminology, that conforming tothe norm is individually rational and that any social norm must be equivalentto a Nash equilibrium of some underlying game. Questioning, therefore, theexistence and multiplicity of Nash equilibria that satisfy certain conditions caninform on the properties of social norms (Akerlof, 1980; Elster 1989; Bernheim,1994; Bendor and Swistak 2001; Wooders, Cartwright and Selten, 2006).Our motivation is a belief that current game theoretic models of conformity
are not able to capture common types of social norms. In justifying this beliefwe note that the rules and standards given by a norm are often quite subtle.For example, Sugden (1989) describes an unwritten rule about the gathering ofdriftwood after a storm on the Yorkshire coast. The �rst person to �nd somedriftwood can make a pile out of the driftwood and simply place two rocks ontop the pile. These two rocks establish his property rights to the wood, evenif left on the beach unattended, for up to two high tides. The crucial featureof this norm and many others is that conformity to the norm requires di¤erentpeople to take di¤erent actions and for actions to be conditional on some externalsignal. Conditional on the random (or not necessarily random) event of who�nds the driftwood �rst, one person will claim the driftwood and others willnot. The idea that conformity to social norms may result in di¤erent individualstaking di¤erent actions appears to be well established in social psychology andsociology. Indeed, it is the basic concern of role theory (Biddle, 1986). Gametheoretic works typically require individuals to choose the same action. Themain objective of this paper is to introduce and analyze a game theoretic modelof conformity where, because actions are conditional on role, individuals canperform di¤erent actions but still conform to the same norm. We shall alsoindicate how this might be applied to the analysis of public good provision.We shall argue that correlated equilibrium provides a natural way to capture
conformity to social norms. This is because correlated equilibrium, in general-izing Nash equilibrium, allows individuals actions to be statistically dependenton some random signals external to the model (Aumann 1974, 1987). Indeed,one interpretation of correlated equilibrium is that a mediator instructs peopleto take actions according to some commonly known probability distribution.We think of the mediator as a social norm that assigns roles to individuals ina society. If, assuming others will do so as well, it is in the interests of eachindividual to assume the role assigned to him by the norm, then the probability
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distribution over roles is a correlated equilibrium (Aumann 1987; Forges 1986;Dhillon and Mertens, 1996).1 Thus, we can think of a correlated equilibrium asa Nash equilibrium in an �extended game�where the mediator or social norm isa player.Given our interpretation of correlated equilibrium, we can think of every
player as using the strategy �if my role is to play action x then play action x�.Thus, a correlated equilibrium can capture the idea that players use the samestrategy while potentially performing di¤erent actions. That actions are condi-tioned on signals or roles also recognizes how conformity can lead to coordinatedactions within social groups (Johnson and Johnson 1977; Selten 1980). Clearly,however, some care is needed because merely using the concept of correlatedequilibrium has created a setting where every player uses the same strategy. Wedo need, therefore, to be sure that �play the same strategy�equates with �be-haves in the same way�and �conforms�. To do this we shall impose conditionson how the mediator or social norm distributes roles in order to guarantee thata correlated equilibrium can be interpreted as consistent with a social norm.The conditions we impose are explained and motivated in detail in Section 3,but we can give a brief overview here. The �rst conditions we impose requirethat people conforming to the same norm have the same ex-ante probability ofbeing assigned each role and have similar attributes. We say that a correlatedequilibrium with these properties satis�es behavioral conformity. A further con-dition we impose is that the aggregate behavior of those conforming to a socialnorm be predictable. In this case we say the correlated equilibrium is consistentwith a social norm.The �rst question we address with the model is whether there exists a cor-
related equilibrium that satis�es behavioral conformity and is consistent with asocial norm. Our �rst result (Theorem 1) provides a family of games for whichnear to every Nash equilibrium there is a correlated equilibrium satisfying behav-ioral conformity. The only requirement we impose is continuity with respect toplayer attributes. This appears a very mild requirement and we therefore showthat a correlated equilibrium satisfying behavioral conformity exists very gener-ally. Our second main result (Theorem 2), demonstrates that, with a Lipschitzcontinuity condition on utility functions, and many players, near to any Nashequilibrium there is a correlated equilibrium consistent with a social norm. Themain additional requirement is that the game have many, anonymous players.We show, therefore, that a correlated equilibrium consistent with a social normexists in games with many players. It is noteworthy that we are able to obtainthe existence of a correlated equilibrium satisfying behavioral conformity undermuch milder conditions than one can obtain the existence of a Nash equilibriumsatisfying behavioral conformity. We are able to do this because the mediator
1Given that roles may be correlated across individuals, the set of correlated equilibriais generally larger than the set of Nash equilibria. This is partly responsible for the set ofcorrelated equilibria having many appealing properties; for example, the set of correlatedequilibrium is nonempty, compact, convex and easy to describe (Aumann 1974, 1987). Seealso Hart (2005) for a discussion of recent work (in collaboration with Mas-Colell) on howadaptive learning leads to correlated equilibrium play.
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correlates actions. We see this as strong evidence that correlated equilibrium isa good way to capture conformity and social norms. It also demonstrates thepotential for social norms and conformity to exist without any threat of sanc-tions for non-conformity. Using the typology of Gibbs (1965) this puts emphasison conventions and customs and reiterates that sanctions should not be seen asa de�ning characteristic of social norms.The second question we shall address is that of stereotyping. Perceptions,
and not necessarily reality, may matter in terms of whether individuals thinkof outcomes as fair and are therefore willing to conform and play their assignedroles (Fiske 1998). For example, if an outcome is perceived as equitable, it maynot matter whether it is in fact equitable. It also may be the case that individualplayers perceive all members of a social group as similar or the same and thusstereotype others according to their social group memberships. To take thesetwo considerations into account we turn to subjective correlated equilibrium. Asubjective correlated equilibrium extends the notion of correlated equilibriumby allowing players to have di¤ering beliefs about the probability with whichroles are distributed. We say that beliefs are stereotyped if each player expectsplayers in the same social group to behave in the same way. Within the model,stereotyping may be interpreted as a form of bounded rationality and allows forsimpler correlating devices. We demonstrate (Theorem 3) that stereotyping canbe consistent with correlated equilibrium. We also demonstrate that a personwho stereotypes could not do better by having non-stereotyped beliefs. Thismeans that there is no incentive for people who stereotype to revise or correcttheir beliefs. We do show, however, that stereotyping can signi�cantly lower(or increase) the payo¤ of the people who are stereotyped. The model does,therefore, show why stereotyping can persist despite potential social problems.Note that we shall not question in this paper from where di¤erent roles
emerge or evolve. We shall merely take as given observable characteristics thatare independent of the game but can serve as signals to correlate behavior viasome formal or informal institution (Goode 1960). �First to arrive at the beach�is one example of such a signal. Gender, age, occupation and social status moregenerally, are other possible signals. We observe that individuals can and docorrelate their behavior to mutual advantage using such roles and signals, evenif doing so requires di¤erent individuals to perform di¤erent actions and receiveinequitable rewards (e.g. Schelling 1960; Hayek 1982; Sugden 1989). It has alsobeen long recognized that conformity, often subconscious, to established rulesand norms of behavior facilitates such correlation of behavior (Hayek 1960, 1982;Sheri¤ 1966;Tajfel 1978; Johnson and Johnson 1987; Brown 2000). The onlything we shall assume, therefore, is that signals exist which are ex-ante irrelevantfor payo¤s but can help people correlate. For example, players in a public goodgame may use social status as a signal of who should contribute.We proceed as follows: Section 2 introduces the model, Section 3 the proper-
ties of social groups, Section 4 provides the main results and Section 5 concludes.
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2 Model and notation
A game � is given by a triple (N;A; fuigi2N ) consisting of a �nite player setN = f1; ::; ng, a �nite set of K actions A = f1; :::;Kg, and a set of payo¤functions fuigi2N . An action pro�le consists of a vector a = (a1; :::; an) whereai 2 A denotes the action of player i. The set of action pro�les is given by AN .For each i 2 N the payo¤ function ui maps AN into the real line R.A strategy in game � is given by a randomization � = (�(1); :::; �(K)) over
the set of actions where �(k) denotes the probability that the player will playaction k 2 A. Let � = �(A) denote the set of strategies. A strategy pro�leconsists of a vector � = (�1; :::; �n) where �i 2 � denotes the strategy of playeri. We assume von-Neumann Morgenstern expected utility functions and, witha slight abuse of notation, denote by ui(�) the expected payo¤ to player i givenstrategy pro�le �. Strategy pro�le � is a Nash "-equilibrium for some realnumber " � 0 if
ui(�) � ui(�; ��i)� "
for all i 2 N and � 2 �.2
2.1 Pregames
We make use of a non-cooperative pregame, which allows us to consider familiesof games derived from a common underlying structure and to formalize a notionof �similar� games.3 A non-cooperative pregame consists of a set of playerattributes, a set of actions, and a universal preference function. Given a �niteset of players, a game is induced by ascribing a point in attribute space to eachplayer. The utility function of a player in the induced game is determined byhis attribute and the universal payo¤ function. A pregame need not imply anyassumptions on the induced games. It does, however, provide a useful frameworkin which to treat a family of games, all induced from a common strategic setting.More formally, let be an attribute space, a compact, convex subset of a
metric space. Let A be a set of actions. Let W denote the set of all mappingswith �nite support from � A into the non-negative set of integers Z+.4 Amember of W is called a weight function and, for any w 2 W , w(!; k) is inter-preted as a number of players with attribute ! who choose action k: A universalpreference function h is a mapping from � A �W into the set of real num-bers R. In interpretation h(!; k; w) is the payo¤ to a player of attribute ! if
2As is standard, we use ��i to denote vector (�1; :::; �i�1; �i+1; :::; �n).3Noncooperative pregames appear in Cartwright and Wooders (2009) and Wooders et al.
(2006). Similar sorts of concepts have a long history in economics and game theory. Forexample, in the context of an exchange economies, a �pre-economy� is a space of preferencesand a set of possible endowment vectors. An economy is then determined by a set of economicagents and a function ascribing a preference relation and an endowment vector to each player.In cooperative games, a pregame is a set of player attributes and a function de�ning a payo¤possibilities set for all possible �nite sets of players described by their attributes.
4A function w from � A has �nite support if w(!; k) 6= 0 for only a �nite number ofelements (!; k) in �A:
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he plays action k and the actions of all players are summarized by w 2 W . Anon-cooperative pregame is a triple G = (; A; h).Given a pregame G = (; A; h), let N = f1; :::; ng be a �nite set and let
� be a mapping from N to , called an attribute function. The pair (N;�)is a population where N is a set of players and � provides a description of theplayers in terms of their attributes. Given action pro�le a 2 AN ; weight functionw�;a 2W is relative to a if
w�;a(!; k) = jfi 2 N : �(i) = ! and ai = kgj
for all k 2 A and all ! 2 ; that is, given a, w�;a(!; k) states the number ofplayers with attribute ! who choose action k. An induced game �(N;�) cannow be de�ned:
�(N;�) = (N;A; fu�i gi2N )
whereu�i (a)
def= h(!; ai; w�;a)
for all ! 2 �(N) and a 2 AN . We note that players who are ascribed the sameattribute have the same payo¤ function. Also, note that the utility function of aplayer depends on the numbers of players with each attribute that have choseneach strategy.
2.2 Correlated equilibrium
Given an induced game �(N;�) a mediator, interpreted as �society�, assigns(or suggests) a role from set A to each player before the game is played. Themediator is modeled by a correlating device given by a probability distributionp over action pro�les. Thus, p(a) denotes the probability that players will beassigned roles consistent with action pro�le a. We shall denote by p(a�ijai) theprobability of role assignments being consistent with a, conditional on player ihaving role ai. We shall denote by pi the marginal distribution of p, where pi(k)denotes the probability that player i is assigned role k.5 Let P denote the setof possible correlating devices.A correlating device p is a correlated equilibrium for game �(N;�) if no
player can do better by deviating from his assigned role. That is, knowing pand knowing his assigned role (and expecting all other players to conform totheir assigned roles) each player does best by conforming to his assigned role.We shall be interested in an approximate correlated equilibrium where no playercan gain more than some " by not conforming. Formally, for any " � 0 we saythat correlating device p is a correlated "-equilibrium of game �(N;�) if andonly if X
a�i
p(a�ijai)u�i (ai; a�i) �Xa�i
p(a�ijai)u�i (k; a�i)� " (1)
5Formally, pi(k) =Pa2AN :ai=k p(a) and, when pi(ai) > 0, it holds that p(a�ijai) =
p(a)=pi(ai)
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for all i 2 N and any actions k; ai 2 A.6 We refer to a correlated 0-equilibrium asa correlated equilibrium. If � is a Nash "-equilibrium of game �(N;�) then thereexists a correlated "-equilibrium in which the mediator independently assignsplayer i action (or role) k with probability �i(k) for each i 2 N .
3 Social groups
We next de�ne and motivate the properties that make correlated equilibriumconsistent with social norms. Recall that our objective is to impose conditionsthat mean �play the same strategy� equates with �behaves in the same way�and �conforms�. To begin we introduce the idea of a social group. Given agame �(N;�) a social group structure is given by a partition � = fN1; :::; NGgof the player set into G subsets for some integer G. Each Ng is a social group.The diameter of social group structure �, is de�ned by
D�;� := maxNg2�
maxi;j2Ng
fdist(�(i); �(j))g
the maximum di¤erence in attribute between players in the same social group.The �rst condition we shall consider is that of within group anonymity.
This requires that any two individuals within the same group have the sameprobability of being assigned each role. Thus, not only will individuals in thesame group use the strategy, �if my role is to play action x then play x�, theywill also have the same chance of being told their role is to play action x. Thismeans that, ex-ante, before they know their assigned roles, any two individualsin the same social group are expected to behave in the same way. For example,when two Yorkshire beachcombers independently choose to go to the beach insearch of driftwood they may be equally likely to be the �rst to arrive at theright stretch of beach and will thus be assigned the role �claim the wood�.In order to de�ne within group anonymity we introduce the notion of a
permutation of an action pro�le. Action pro�le a0 is a permutation of anotheraction pro�le a,relative to social group structure �, if the number of players ineach social group assigned each role is the same.7 Given social structure � andaction pro�le a let P�(a) denote the set of action pro�les that are permutationsof a.
De�nition 1: Given population (N;�), social group structure� and correlatingdevice p we say that correlating device p satis�es within-group anonymity if p
6Formally, we only require (1) to hold for actions ai that player i can be assigned with pos-itive probability. A correlated equilibrium thus de�ned is consistent with the usual de�nitionof correlated equilibrium (e.g. de�nition 2.4B of Fudenberg and Tirole (1998)). An approx-imate correlated equilibrium equates to a natural approximation to a correlated equilibriumas usually de�ned, although �role�is often termed �signal�. Note, however, that the use of theterm "-correlated equilibrium by Myerson (1986) has a di¤erent meaning to the one here.
7More precisely, if #�(a; k; g) = jfi 2 Ng : ai = kgj denotes the number of players in groupNg who play action k then action pro�le a0 is a permutation of a if #�(a; k; g) = #�(a0; k; g)for all k and Ng .
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treats players from the same social group identically. That is, given any twoaction pro�les a and a0 2 AN , if a0 2 P�(a) then:
p(a) = p(a0).
Within-group anonymity captures the notion that players within the same socialgroup are expected to behave in the same way. In particular, it implies anequality of opportunity (or of responsibility) within groups because any twoplayers belonging to the same social group have the same probability of beingassigned each role within the group.8 This can be seen as fair and, as we shallsee below, results in an equality of expected payo¤. The importance of fairnessand within group equity is well known (Johnson and Johnson, 1987; Tajfel,1978; Brown, 2000).Our second condition, group independence, requires that the distribution of
roles be statistically independent between di¤erent groups and thus rules outany correlation of actions across groups and draws the boundaries between socialgroups.
De�nition 2: Given population (N;�), social group structure� and correlatingdevice p, the device p satis�es group independence if there is no correlation ofroles between groups, that is, for any a 2 AN
p(a) = p1(a�1)p2(a�2):::pG(a�G)
where pg(a�g ) is the marginal probability that members of group �g have theroles given by a.
Correlation between social groups is typically unlikely (Tajfel 1978) and so itseems important to rule this out; for example, players in di¤erent social groupsmay not as easily identify or communicate with each other as those in the samesocial group. Note, however, that a lack of correlation does not imply a lack ofcoordination between groups as induced by an equilibrium.Our third condition, homophily, serves to further distinguish between groups.
Speci�cally, it captures the notion that individuals within the same group arelikely to have similar attributes. That is, they are likely to share similar prefer-ences and have similar e¤ects on others. This is a property commonly observedin social groups (Brown 2000; Currarini, Jackson and Pin 2009).
De�nition 3: Given population (N;�), a social group structure� = fN1; :::; NGgsatis�es homophily if the attribute space can be partitioned into G convexsubsets 1; :::;G such that, for any i 2 N , if i 2 Ng then �(i) 2 g.
Informally, homophily requires that any player whose attribute is intermediatebetween two members of a particular social group should also belong to that
8For instance, if i; j 2 Ng , then pi(k) = pj(k) for all k 2 A.
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same social group. Note this does not necessarily imply that two players withsimilar attributes belong to the same social group. It does, however, suggestthat any two players within a social group do have some similarity in attributes.To brie�y recap, within-group anonymity implies similarity in behavior within
groups, group independence implies boundaries between groups while homophilyimplies that group membership correlates with attribute. Together, we shallthink of these as the main criteria for behavioral conformity.
De�nition 4: Given population (N;�), social group structure � and correlatingdevice p satisfy behavioral conformity if p satis�es within-group anonymity andgroup independence and � satis�es homophily.
We now introduce a �nal condition, predictable group behavior. This re-quires that the behavior of the social group be known ex-ante, even if the be-havior of any one individual within the group is not known.
De�nition 5: Given population (N;�), social group structure� and correlatingdevice p, device p satis�es predictable group behavior if, for each group Ng 2 �,the number of players in the group who will play each action is known forsure ex-ante. Formally, for any action pro�les a and a0, if p(a); p (a0) > 0 thena0 2 P�(a).
The notion of a norm suggests that in some way the behavior of a social groupshould be predictable. Predictable group behavior imposes a strong form ofpredictability in which the behavior of the group is known for sure. Also, ifa norm is to be followed over time (which seems an essential property of thenotion of a norm) then it is preferably that players should have an incentiveto conform ex-post, as well as ex-ante, to the assignment of roles. Predictablegroup behavior, as we shall see, results in ex-post stability, meaning that playerswould indeed want to conform even when they know the roles assigned to allother players.
De�nition 6: Given population (N;�), we say that social group structure� andcorrelating device p are consistent with a social norm if they satisfy behavioralconformity and predictable group behavior.
4 The existence of a correlated equilibrium con-sistent with a social norm
In this Section we present results demonstrating the existence of a correlatedequilibrium satisfying behavioral conformity and of a correlated equilibria con-sistent with a social norm. All results are proved in an appendix.One would expect that we need to impose some assumptions in order to
obtain existence of a correlated equilibrium satisfying behavioral conformity. Acontinuity condition, ensuring that players with similar attributes are similar
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as players in induced games, is required (see Wooders et al. 2006). FollowingWooders et. al. (2006) we introduce a Lipschitz continuity assumption, withLipschitz constant of one:
De�nition 7: A pregame G = (; A; h) satis�es continuity in attributes if, givenany " > 0 and any two games �(N;�) and �(N; e�) for which dist(�(i); e�(i)) <" for all i 2 N; for any j 2 N and for any action pro�le a it holds that��u�j (a)� ue�j (a)�� < ":Continuity in attributes dictates that, given action choices, if the attribute func-tion changes only slightly, then payo¤s change only slightly. The assumptionof a Lipschitz constant of one has the bene�t of signi�cantly simplifying thenotation and analysis, but is not essential to our results.9
Our �rst result demonstrates that, given continuity in attributes, near toany Nash equilibrium there exists a correlated equilibrium satisfying behavioralconformity. We say that correlated equilibrium p is "-near to Nash equilibrium� if ������
Xa
p(a)u�i (a)�1
jNgjXj2Ng
u�j (�)
������ � " (2)
for all i 2 Ng and all Ng. Thus, the expected payo¤ of each player, given p, isclose to the average payo¤ of players in her group, given �. Note that nearnessnecessarily implies players within a social group have similar expected payo¤s.10
Theorem 1: Consider a pregame G = (; A; h) that satis�es continuity inattributes. Also consider any population (N;�), any social group structure�, and any Nash equilibrium �. There exists a correlated 2D�;�-equilibriump of game �(N;�) such that p satis�es within group anonymity and groupindependence and is D�;�-near to �.
One noteworthy point about Theorem 1 is that the social group structure canbe taken as exogenous and independent of the attribute function and game.Speci�cally, Theorem 1 states that given any social group structure, in whichplayers in the same social group have su¢ ciently similar attributes, and near toany Nash equilibrium, there is a correlated equilibrium and social group struc-ture that satisfy within group anonymity and group independence. This meansthat homophily (and, therefore, behavioral conformity) can trivially be satis�edby focussing on any social group structure that satis�es homophily. Things arenot so simple when looking for Nash equilibria rather than correlated equilib-ria; only particular Nash equilibria and social group structures are consistentwith behavioral conformity (Wooders et al. 2006). This is important, beyondmere technicalities because it appears desirable that a change in the nature of
9 Indeed, it can be done without loss of generality if one allows a rescaling of the metric onthe attribute space.10Speci�cally,
���Pa p(a)u�i (a)�
Pa p(a)u
�j (a)
��� � 2".10
interaction (i.e. the game) need not require a change in social group structure.We have shown that correlated equilibrium allows su¢ cient �exibility for this.If a social group contains players who are substantially dissimilar, then D�;�
may be very large. Using, however, the compactness of the attribute space wecan obtain existence of a correlated equilibrium satisfying behavioral conformitywith a small value for D�;�.
Corollary 1: Consider a pregame G = (; A; h) that satis�es continuity inattributes. For any real number " > 0, a bound G(") can be put on the numberof social groups such that, for any population (N;�), there exists a social groupstructure �, with no more than G(") groups, and a correlated "-equilibrium pof induced game �(N;�) that satisfy behavioral conformity.
4.1 Consistency with a social norm and ex-post stability
While Theorem 1 requires behavioral conformity it does not require predictablegroup behavior. Thus, the equilibrium considered in the Theorem need not beex-post stable (as formulated in Kalai 2004, for example), meaning that oncea player has observed the roles assigned to others he may have an incentive tochange his action.11 If so, the equilibrium is not ex-post stable and is incon-sistent with the notion of a social norm. This is exempli�ed by the followingsimple example.
Example 1: Players have to choose between two locations B and C. Theattribute space is given by fX;Rg where a player with crowding type X iswealthy and has high preference for local public goods. A player with crowdingtype R is poor. We suppose that there is only one wealthy player. The poor likeliving in the same location as the wealthy; the payo¤ of a player with attributeR is equal to 1 if he matches the choice of the wealthy player and 0 otherwise.The wealthy player, by contrast, prefers to avoid the poor; his payo¤ is equal tothe proportion of poor players whose choice of location he mismatches. Theorem1 applies and so we can construct a correlated equilibrium. But any correlatedequilibrium of this game has the wealthy player mixing between B and C andpoor players �in aggregate�mixing between B and C. Ex-post, once every playerhas chosen a location there must be at least one player who would wish to changehis location.�
There is nothing wrong with one player constituting a social group but itis di¢ cult to interpret the actions of the wealthy player in terms of a socialnorm. This is primarily because social norms suggest predictability in aggregate
11To use a Rawlsian thought experiment one can see that within-group anonymity resultsin players in the same group expecting to get the same payo¤. This would be an acceptablesocial contract under Rawls�s reasoning (Rawls 1972). The criticism often made, however,of the Rawls notion of social contract (e.g. Binmore 1989) is that ex-post outcomes need beneither fair nor individually rational, leading to questions of whether such a notion representsan appropriate form of social contract.
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behavior and consequent ex-post stability. This motivates our next result onexistence of correlated equilibria that satisfy predictable group behavior andare thus consistent with a social norm. If there exists a pure strategy Nashequilibrium, we do not require any additional assumptions to obtain the result.Otherwise, we can demonstrate existence under a continuity condition, globalinteraction, from Wooders et. al. (2006).
De�nition 8: The pregame G = (; A; h) satis�es global interaction when, forany " > 0, any game �(N;�), and any two action pro�les a and m, there exists� > 0 such that if
1
jN jXk
X!2�(N)
jw�;a(!; k)� w�;m(!; k)j < �
then��u�j (a)� u�j (m)�� < " for any j 2 N where aj = mj .
Global interaction implies that the strategy choice of an individual playercannot have a signi�cant e¤ect on the payo¤ of any other player in large games.This, and su¢ ciently many players, allows puri�cation of mixed strategy equi-libria (Cartwright and Wooders 2009).
Theorem 2: Let G = (; A; h) be a pregame satisfying continuity in attributes.Consider any population (N;�), any social group structure � and any Nashequilibrium �.(a) If � is a pure strategy Nash equilibrium then there exists a correlated 2D�;�-equilibrium p of game �(N;�) such that p satis�es within group anonymity,group independence, predictable group behavior and is D�;�-near to �.(b) If pregame G satis�es global interaction, then for any real number " > 0there is a real number �(") such that if jN j > �(") there exists a correlated2D�;� + "-equilibrium p of game �(N;�) such that p satis�es within groupanonymity, group independence, predictable group behavior and is D�;�-nearto �.
Theorem 2 treats any social group structure, meaning that homophily can triv-ially be satis�ed. Thus, it demonstrates that in a general class of game thereexists an approximate correlated equilibrium consistent with a social norm.
Corollary 2: Consider a pregame G = (; A; h) that satis�es continuity inattributes and global interaction. For any real number " > 0 there is a realnumber �(") and an integer G(") such that, for any population (N;�) wherejN j > �("), there exists a social group structure �, with no more than G(")social groups, and a correlated "-equilibrium p of induced game �(N;�) thatare consistent with a social norm.
Many papers have considered the properties of equilibrium in large games (Khanand Sun 2002). The paper most relevant for our purposes is Kalai (2004). Using
12
the law of large numbers, Kalai demonstrates that in large games all Nashequilibria are highly likely to be (approximately) ex-post stable because ex-postoutcomes are likely to be similar to what was expected ex-ante. This happensdespite all players choosing their actions independently. In Theorem 2 andcorollary 2 we obtain an equilibrium in which ex-post outcomes are exactly whatwas expected ex-ante.12 This is possible because the correlation of actions, andthe associated reduction in uncertainty, can lead to higher equilibrium payo¤s.
4.2 Subjective beliefs and stereotyping
We now adapt our framework to allow us to consider stereotyping. To do sowe relax the assumption that players know the correlating device p. Instead,players are modeled as having subjective beliefs about the device. For example,a player may have beliefs about the likely status of the other players and thesebeliefs may determine what the player thinks others will do. The beliefs maybe incorrect but nevertheless an action may be chosen that depends on thesebeliefs. Speci�cally, there is a given set of beliefs f�igi2N , where �i, givenby a probability distribution over the set of action pro�les, denotes the beliefsof player i. Thus, �i(a) denotes the probability that player i puts on playershaving been assigned roles according to action pro�le a and �i(a�ijai) denotesthe probability that player i puts on roles being assigned according to actionpro�le a given that he is assigned role ai.We say that the set of beliefs f�igi2N constitutes a subjective correlated
"-equilibrium if Xa�i
�i(a�ijai)u�i (a) � �i(a�ijai)u�i (k; a�i)� "
for each i 2 N and ai; k 2 A. This revises the de�nition of a correlated equilib-rium (as given by (1)) in the natural way by requiring no individual i to expecta payo¤ gain from changing strategy given his beliefs �i.It is well known that once subjective beliefs are allowed it becomes di¢ cult
to tie down the set of correlated equilibria (Aumann 1974, 1987; Brandenburgerand Dekel 1987). A framework of social identity, however, suggests certainproperties, including stereotyping, that one might expect beliefs to satisfy. Wepropose a de�nition of stereotyped beliefs in which a player expects playersin the same social group to behave identically. We do assume, however, thata player does not stereotype himself. This requires a slight reformulation ofwithin-group anonymity.Consider permutations of an action pro�le a for which player i�s action does
not change. More precisely, given game �(N;�), social group structure �, actionpro�le a and player i (and the set P�(a) of action pro�les that are permutations12Predictable group behavior, and Theorem 2, could be extended by relaxing predictable
group behavior to require only that aggregate behavior be approximately predictable. To doso, however, would require cumbersome notation and further approximation arguments so weprefer the current predictable group behavior condition.
13
of a) let P�i (a) denote the subset of P�(a) where a0i = ai. We can now de�nestereotyped beliefs.
De�nition 9: Given population (N;�) and social group structure � we saythat beliefs f�igi2N are stereotyped if �i(a) = �i(a
0) for any player i 2 N andaction pro�les a and a0 where a0 2 P�i (a).
It is a simple extension of Theorems 1 and 2 to show the existence of asubjective correlated "-equilibrium if beliefs are stereotyped. More interestingis whether stereotyping involves costs to players. The following result showsthat it need not. To state the result we need a revised de�nition of "-near. Wesay that set of beliefs f�igi2N are "-near to Nash equilibrium � if�����X
a
�i(a)u�i (a)� u�i (�)
����� � " (3)
for all i 2 N . Thus, a player�s expected payo¤ given his beliefs are within " ofthe Nash equilibrium payo¤.
Theorem 3: Consider a pregame G = (; A; h) that satis�es continuity inattributes. Also consider any population (N;�), any social group structure �,and any Nash equilibrium �. There exists a set of stereotyped beliefs f�igi2Nthat are a subjective 2D�;�-correlated equilibrium and D�;� near to �.
Stereotyping can therefore be consistent with equilibrium and not change theexpected payo¤ of the player who is stereotyping. One point worth highlightingis that while stereotyping may not change the expected payo¤ of the personstereotyping it can change the payo¤ of the stereotyped. This is illustrated bythe following example.
Example 2: Players have to choose between locations B and C. The attributespace is [0; 1] � fX;Rg where, as before, X denotes �wealthy�and R denotes�ordinary member of the public�. We suppose that there is only one wealthyplayer. Every member of the public gets payo¤ 1 if the wealthy player chooseslocation B (which may a¤ord the wealthy player less privacy, for example) and 0if the wealthy player chooses location C. Clearly members of the public want thewealthy player to choose B. Member of the public i has attribute �(i) = (!;R)where ! 2 [0; 1] is a measure of status. Let !B and !C denote the averagestatus of ordinary members of the public in locations B and C. The wealthyplayer prefers to have higher status neighbors but has a slight preference for Cover B; his payo¤ is !B if he chooses location B and !C + � for some small � ifhe chooses location C.The Nash equilibria of most interest are those where the higher status mem-
bers of the public choose location B in order that the wealthy player will chooselocation B. In this case all members of the public get payo¤ 1. If, however, thewealthy player stereotypes then it may be that she would choose location C and
14
all members of the public get payo¤ 0. To provide a speci�c example, supposethat player 1 is the wealthy player, player 2 has status 0:5, players 3; :::; n havestatus 0:49 and � = 0:005. There exists a Nash equilibrium where players 1 and2 choose location B and all others choose location C. Suppose, however, thatplayer 1 has stereotyped beliefs. This would mean that player 1 expects onemember of the public to choose location B but each member of the public isconsidered equally probable to be this player. The expected average status ofplayers in location B and C is clearly the same, and so player 1 should chooselocation C.�
In this example the wealthy player is not signi�cantly a¤ected by the factthat she stereotypes. This is because she stereotypes players that are simi-lar. That the wealthy player stereotypes can, however, result in a change inincentives that may lead her to change her action. In particular, the wealthyplayer is basically indi¤erent between the locations but this means her actualchoice could be sensitive to stereotyping. If she changes her action this may notsigni�cantly a¤ect her payo¤ but may dramatically a¤ect the payo¤s of others.
4.3 Public good game examples
To conclude the analysis we shall brie�y explore how our framework can beapplied to looking at public good games. Our intention is not to treat the issuein depth, but merely raise some interesting possibilities that can be explored infuture work. Public good games seem a particular apt application because onetypically obtains a large set of Nash equilibria and social norms may provide animportant means for players to coordinate. To illustrate we work through anexample �tting the framework of Bergstrom, Blume and Varian (1986).Consider a game �(N;�). Each player is endowed with amount 1 of money.
Simultaneously, players decide what fraction of their endowment to contributetowards a public good.13 Let ci 2 [0; 1] denote the contribution of player i,let C =
Pi2N ci denote total contributions and let C�i = C � ci. The utility
function of player i 2 N is given by
ui(ci; C) =1� cim
+ �(i)
rC
n
where 2 [0; 1] is the attribute space and m � 0 is a constant. Attributethus captures preference for the public good. Clearly, one can generalize fromthis game to a pregame; the pregame satis�es continuity in attributes. To givea speci�c example, suppose that n = 8;m = 4; �(i) = 0 for i = 1; 2; 3; 4 and�(i) = 1 for i = 5; 6; 7; 8. The attribute function is such that half of the playersget maximum bene�t from the public good and half get no bene�t. The partition� into social groups N1 = f1; 2; 3; 4g and N2 = f5; 6; 7; 8g seems natural andsatis�es homophily.
13We assume a �nite but �ne action set. For example, a player could be endowed with$1; 000; 000 and can contribute in cents.
15
Game �(N;�) has a large set of pure strategy Nash equilibria. All of theseequilibria require players 5; 6; 7 and 8 to contribute between them a total of 0:5towards the public good. Social norms may potentially be one way to coordinateon a particular distribution. In the terminology of our model, the mediatorcan suggest the role �contribute 0:5�to one of players 5; 6; 7 or 8 and suggest�contribute 0�to all other players. This can clearly be done in a way to satisfywithin group anonymity, group independence, and predictable group behavior.In practice, some trait of the player, such as social status, may be used toallocate roles (de Kwaadsteniet, and van Dijk 2010). For example, the playerwith highest social status in social group N2 is allocated the role of contributing0:5.Consider now a slightly di¤erent game in which �(i) = 0:99 for i = 5; 6,
with all other details remaining the same.14 Pure strategy Nash equilibria ofthis game require players 7 and 8 to contribute, between them, a total of 0:5 to-wards the public good. In other words, the fact that players 5 and 6 get a slightlylower return from the public good means that they will never contribute in equi-librium. The partition into social groups N1 = f1; 2; 3; 4g and N2 = f5; 6; 7; 8gmay still seem, however, natural in that it groups together similar players. Ifso, the mediator can behave as previously and randomly select one player fromgroup N2 who is given the role �contribute 0:5�. This would be consistent witha social norm. Note, however, that we only obtain an approximate, correlated0:02-equilibrium; players 5 or 6 may have the highest social status, but if giventhe role �contribute 0:5�would have an incentive to contribute a little less than0:5.15 Foregoing this potential little gain may be worthwhile in the interests ofsocial cohesion.It is a common feature of public good games that there are many equilibria
distinguished by the way in which total contributions are shared up amongstplayers. It is the case in the game above, and games of this type (Bergstrom,Blume and Varian 1986). It is also the case in threshold public good games(Palfrey and Rosenthal 1984, Rapoport 1987).16 The experimental evidenceshows that subjects are relatively poor at coordinating and providing publicgoods at the equilibrium level when there is decentralized interaction (Crosonand Marks 2000) but do better if there is the chance for non-binding preplaycommunication (van de Kragt et al. 1983). In applied settings, social normsmay be an important way that groups coordinate. If so, we believe that ourmodel is well placed to capture this.
14That is, n = 8;m = 4; �(i) = 0 for i = 1; 2; 3; 4 and �(i) = 1 for i = 7; 8.15Predicting this, players 7 and 8 would also have an incentive to deviate from an assigned
role of �contribute 0�.16See Cavaliere (2001) for a study of correlated equilibrium in the provision of threshold
public goods. There correlation is viewed as the consequence of communication.
16
5 Concluding remarks
This paper models conformity and social norms in settings where di¤erent indi-viduals can perform di¤erent actions but can still be seen as conforming to thesame norm. We argued that correlated equilibrium is an appealing way to modelsuch conformity. In doing so we proposed conditions one would want to imposeon the nature of correlation �within-group anonymity, group independence, ho-mophily, and predictable group behavior �and have demonstrated the existenceof a correlated equilibrium satisfying these properties. One way to interpret thisresult is that social interaction can act as a form of equilibrium-selection devicethat selects correlated equilibria satisfying certain properties.Besides satisfying behavioral conformity and predictable group behavior
there are a number of motivations to bound the number of societies. Theseare discussed at some length in Wooders et. al. (2006). We note here thatthe notion of a social group may already suggest that the group contains manymembers (in fact, bounding the number of social groups only implies that, ifthere are many players, some groups must contain many members). A relatedand interesting side issue is that of optimal social group size. Within-groupanonymity and predictable group behavior require that players in the same so-cial group behave identically and therefore suggests that small, homogenous,social groups are advantageous. In our framework, correlation of actions allows,however, increased payo¤s. In order to realize these gains and maintain groupindependence it would seem that larger social groups are advantageous. Thiscreates countervailing gains and losses to larger social groups, which suggest anoptimal group size determined by the heterogeneity of players and the potentialgains from correlating actions.In interpreting our model it is important to question how actions may become
correlated. It was not our intention in this paper to answer that question andso we have been quiet on this issue but now we will make a few remarks. First,in some social contexts there may be someone who indeed does tell people whatto do and can directly correlate actions. Second, it has been observed that, inthe absence of a formal device, correlation of actions can emerge spontaneouslywithin groups (e.g. Schelling 1960; Hayek 1982; Sugden 1989). This can beachieved through conditioning actions on random signals such as gender, age,exam results etc. On a more theoretical level, Hart and Mas-Colell have shownhow naive learning heuristics such as regret matching can lead to aggregate playcorresponding to a correlated equilibrium (see Hart 2005). This all suggeststhat correlation of actions within social groups is not unrealistic. In particular,while the correlation required by predictable group behavior or within-groupanonymity may be asking too much it may be possible for social groups toobtain correlated equilibrium that approximates predictable group behavior andwithin-group anonymity. The reward for doing so can be preferable outcomes.The possibility of subjective beliefs and stereotyping suggests an alternative
interpretation of our results. If beliefs are subjective and stereotyped then thereneed not be any correlation of actions but just a belief that there is correlation.The focus, therefore, shifts from how actions could be correlated to whether it
17
can be consistent with equilibrium for players to expect correlation even if thereis none. We demonstrated that stereotyping, even if it causes erroneous beliefs,can be consistent with equilibrium. Furthermore, a player�s payo¤ is largelyinvariant to whether he stereotypes. Stereotyping can, however, in�uence thepayo¤s of those being stereotyped. It should be emphasized that we obtain thisresult because a player only stereotypes members of groups consisting of similarplayers. This raises the question of how a player would form his beliefs aboutthe actions expected of others.We conclude by noting that our research provides a theoretical motivation
for further public good game experiments. In Section 4.3 we brie�y related ourapproach to that of �standard�public good games studied in the experimentalliterature. At this point we note that in these standard games, both actionsand attributes are one-dimensional. For example, possible actions may be howmuch money to contribute while attribute is status, both taken from some setof real numbers. Our model does not make these restrictions. Typically theprovision of a public good will require contributions in multiple dimensions. Forexample, consider the public good as the success of an Economics departmentin a University. There are various roles that must be ful�lled �Chair of theDepartment, Director of Graduate Studies, Director of Undergraduate Studies,researcher and teacher, and so on. Administrative posts within the Departmentare largely �lled by those individuals who are willing to take on the assignedroles. To our knowledge, there have been no experiments investigating whethersubjects will use some correlating device �status, years of service, nationality,contributions to charitable associations, involvement in social activities � todetermine which individuals will undertake what roles.
6 Appendix
The proofs of all the Theorems follow from the same simple arguments. In thissection we shall talk through these arguments and provide all proofs. Through-out the following we take as given a pregame G = (; A; h) that satis�es continu-ity in attributes. Until otherwise stated we shall also take as given a population(N;�) and social group structure � = fN1; ::; NGg.A function mapping from N to N is said to be a permutation of players if
is one-to-one and (i) 2 Ng whenever i 2 Ng for all i 2 N . That is we permutethe labels of players within the same society. Given a permutation of players and action pro�le a we denote by a the action pro�le where a i = a (i) for alli 2 N . With this we can make the following observation which should requireno proof.
Lemma 1: Consider any action pro�le a. If a0 2 P�(a) then there exists a (notnecessarily unique) permutation of players such that a0 = a (i.e. a0i = a
i for
all i 2 N). Furthermore, if is a permutation of players then a 2 P�(a).
Thus, if action pro�le a0 is a permutation of a then for every player i thereexists some player (i), who belongs to the same social group as i, such that
18
i, according to a0, plays the same action that (i) plays, according to actionpro�le a.The following result, which is an application of continuity in attributes,
shows an approximate equivalence between a permutation of actions and a per-mutation of utilities. A simple illustration is provided after the proof.
Lemma 2: Consider any action pro�le a and permutation of players ,���u�i (k; a �i)� u� (i)(k; a� (i))��� < D�;� (4)
for any k 2 A.
Proof : Given the population (N;�) let (N; e�) be the population in whiche�( (i)) = �(i) for all i. That is, attribute function e� assigns to player (i) thesame attribute as � assigns to i. By continuity in attributes���ue� (i)(k; a� (i))� u� (i)(k; a� (i))��� < D�;� (5)
for all i 2 N and any a 2 AN . We know that a i = a (i) for all i 2 N . Theinequality (4) follows.�
To illustrate Lemma 2 consider a population (N;�) with four players, N =f1; 2; 3; 4g and a social group structure � consisting of N1 = f1; 2; 3g andN2 = f4g. Consider the permutation of players (1) = 2; (2) = 3; (3) = 1and (4) = 4. Given action pro�le a = (a1; a2; a3; a4) we obtain that a =(a2; a3; a1; a4). Observe that a
�1 = (a3; a1; a4) and a� (1) = (a1; a3; a4): Lemma
2 implies that��u�1 (k; a �1)� u�2 (k; a�2)�� = ju�1 (k; (a3; a1; a4))� u�2 (k; (a1; a3; a4))j < D�;�:Note that there are two components to this inequality: (i) it compares thepayo¤ of player 1 to that of player 2, when they play the same action, and (ii)the identity of the players in the social group playing each action has changed.This illustrates how players with a similar attribute need have both similarutility functions and e¤ect others utility in a similar way.Lemma 2 concerns a permutation of actions. The next step is to take this to
a permutation of strategies and permuted correlating device. Given a correlatingdevice p and permutation of players we denote by p the correlating devicewhere p (a ) = p(a) for all a 2 AN . One way to interpret the device p is thatit randomly determines assigns roles a according to the correlating device p butthen assigns player i the role of player (i), that is, it gives player i role a (i)instead of player (i). This is a generalization of permuting strategies. Thefollowing result follows easily from Lemma 2 and shows that a permutation ofroles leads to an approximate permutation of expected payo¤s.
19
Lemma 3: Let p be any correlating device, any permutation of players andi any player,������Xa�i
p (a�ija i )u�i (k; a�i)�Xa� (i)
p(a� (i)ja (i))u� (i)(k; a� (i))
������ < D�;� (6)
for any k 2 A and a.
Proof : Given that p (a ) = p(a) and a i = a (i) for all a and i we have thatp (a �ija
i ) = p(a� (i)ja (i)) for all a and i. Thus,
Pa�i
p (a�ija i )u�i (k; a�i)can be rewritten
Pa� (i)
p(a� (i)ja (i))u�i (k; a �i). Equation (6) can, therefore,
be restated������Xa� (i)
p(a� (i)ja (i))hu�i (k; a
�i)� u� (i)(k; a� (i))
i������ < D�;�and so applying Lemma 2 gives the desired result.�
Now consider a correlated equilibrium p� of game �(N;�) and suppose thatwe permute the strategies of players. Speci�cally, let be a permutation ofplayers and consider correlating device p�
. Lemma 3 implies that if player (i)had no incentive to deviate from his assigned role given correlating device p�
then player i could gain at most 2D�;� from deviating from his assigned rolegiven correlating device p�
. This leads to the following result.
Lemma 4: Let p� be any correlated "-equilibrium of game �(N;�) and let be any permutation of players. Correlating device p�
is a correlated 2D�;�+"-equilibrium of game �(N;�).
Proof : If p� is a correlated "-equilibrium thenXa� (i)
p�(a� (i)ja (i))u� (i)(a) �Xa� (i)
p�(a� (i)ja (i))u� (i)(k; a� (i))� "
for all i 2 N , k 2 A and a (i). Applying Lemma 3 implies thatXa�i
p�
(a�ija i )u�i (a) �Xa�i
p�
(a�ija i )u�i (k; a�i)� 2D�;� � "
for all i 2 N , k 2 A and a i as desired.�
Thus, given a correlated equilibrium (or Nash equilibrium) we can permuteplayers and obtain an approximate correlated (or Nash) equilibrium. Withthis we are basically done. All that remains is to determine the social groupstructure and obtain existence of a correlated equilibrium. It should be clear,
20
however, that we could consider any social group structure in which playershave su¢ ciently similar attributes and apply Lemma 4. Also, the existence ofa correlated equilibrium is well known in games with a �nite strategy set.
Proof of Theorem 1: Let � denote the set of permutations of players (con-sistent with �). Let p� denote the correlating device associated with Nashequilibrium �, i.e. p�(a) = �1(a1) � ::: � �n(an). By Lemma 4 we know thateach p�
is a correlated 2D�;�-equilibrium. Consider correlating device p0 givenby
p0(a) =1
j�jX 2�
p�
(a): (7)
It is well known that the set of correlated equilibria is convex. Extending suchresults to approximate correlated equilibria is simple. It follows that p0 is acorrelated 2D�;�-equilibrium. Also, p0 satis�es within group anonymity by con-struction and roles are distributed independently across players by device p�
and in every p�
which means that p0 must satisfy group independence. Finally,Lemma 3 gives relation (2). �
Proof of Theorem 2: For any " > 0, Theorem 1 of Wooders et. al. (2006)demonstrates that in games with su¢ ciently many players and global interactionthere exists a Nash "-equilibrium in pure strategies �. Thus, we can eithertake as given by assumption an equilibrium � in pure strategies or, with globalinteraction, by the prior theorem an "-equilibrium � in pure strategies. Let p�
denote the correlating device associated with Nash equilibrium �. Note thatp(a�) = 1 for some action pro�le a�. Clearly p� satis�es predictable groupbehavior. As in the proof of Theorem 1 let � denote the set of permutationsof players (consistent with �) and let p0 be de�ned as in (7). We can use thesame arguments as in the proof of Theorem 1 to see that p0 is a correlated "-equilibrium satisfying behavioral conformity. It should be clear that the devicep0 also satis�es predictable group behavior.�
Proof of Theorem 3: Fix a player i 2 N . Let p� denote the correlating deviceassociated with Nash equilibrium �. Let �i denote the set of permutations ofplayers in which i is not permuted, that is, (i) = i. De�ne beliefs �i where
�i(a) =1
j�ijX 2�i
p�
(a): (8)
Beliefs f�igi2N are stereotyped and following Lemma 3 and the logic of proofof Lemma 4, are D�;�-near to �. Furthermore, they are a subjective 2D�;�-correlated equilibrium.
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