Equilibria of nonatomic anonymous games
aa r X i v : . [ ec on . T H ] M a y Equilibria of nonatomic anonymous games ∗ Simone Cerreia-Vioglio a , Fabio Maccheroni a , David Schmeidler ba Universit`a Bocconi and Igier, b Tel Aviv University
April 2020
Abstract
We add here another layer to the literature on nonatomic anonymous gamesstarted with the 1973 paper by Schmeidler. More specifically, we define a newnotion of equilibrium which we call ε -estimated equilibrium and prove its ex-istence for any positive ε . This notion encompasses and brings to nonatomicgames recent concepts of equilibrium such as self-confirming, peer-confirming,and Berk–Nash. This augmented scope is our main motivation. At the sametime, our approach also resolves some conceptual problems present in Schmei-dler (1973), pointed out by Shapley. In that paper the existence of pure-strategyNash equilibria has been proved for any nonatomic game with a continuum ofplayers, endowed with an atomless countably additive probability. But, requiringBorel measurability of strategy profiles may impose some limitation on players’choices and introduce an exogenous dependence among players’ actions, whichclashes with the nature of noncooperative game theory. Our suggested solutionis to consider every subset of players as measurable. This leads to a nontrivialpurely finitely additive component which might prevent the existence of equi-libria and requires a novel mathematical approach to prove the existence of ε -equilibria. The original framework of Schmeidler.
Games with a continuum of anony-mous players were introduced by Schmeidler in [32] where he also proved the existence ∗ Acknowledgments to be added.
1f pure-strategy Nash equilibria for these games. At the time, there were models ofmarkets and cooperative games with infinitely many players, but not of noncooper-ative games. In [32], the players’ space is modelled to be the unit interval endowedwith the Borel σ -algebra and the Lebesgue measure, where there is a finite set ofactions and each player chooses an action from this set. The utility of each playerdepends on the distribution of actions across all players and the action he chooses.The interpretation is that the same game is repeated in each period. The payoff, inutils, is received at the end of the period. At the same time, because of the anonymityassumption, the strategic complications of repeated games are meaningless here. Aparadigmatic example is that of daily commuters driving downtown (or back home)and having to choose a bridge (or tunnel) to enter the city. Thus, in each period theyplay a one-shot game, analyzed in [32]. Here, the metaphysical assumption of correctlyguessing what other players will do, required for playing a Nash equilibrium strategyin one-shot games, is mitigated by two factors. The first is minor: each player has toguess correctly the distribution of the strategy (the same guess for all). The second ismajor: there is regularity in the daily traffic of commuters. Schmeidler [32] formalizesthese intuitions. The limitations of this model are discussed below. Our motivations.
The goal of our paper is to generalize the above finding inseveral directions. We are motivated by three main observations:(i) In recent years, alternative, and perhaps more realistic, notions of equilibriumhave been developed for noncooperative games with finitely many players. Atthe same time, these notions have not been considered for nonatomic anony-mous games. In particular, we have in mind equilibrium concepts which allowfor beliefs to be not necessarily correct, but nonetheless consistent with the in-formation possessed by each player whether it is endogenously or exogenouslygenerated. Thus, our goal is to bring these more realistic notions of equilibriumto nonatomic anonymous games which model exactly situations where individu-als are negligible and are not fully aware of the strategic environment surroundingthem. This renders sophisticated strategic reasoning, such as Nash equilibrium(and any of its refinements) or rationalizability, less plausible. Theorem 1 in [32] is a special case of the last theorem of Schmeidler’s Ph.D. dissertation inmathematics titled “Games with a continuum of players” ; submitted and approved in 1969, at theHebrew University in Jerusalem. The problem was inspired by the moonlighting job of the author asa member of a team advising on Tel Aviv transportation. We are not after proving any sort of “translation principle”, that is, a principle for which any This in turn clashes withthe noncooperative idea of strictly independent decision-making, since “closeplayers” tend to play “close strategies”.(iii) In modelling a large population of players in which each agent “has the samenegligible weight”, Schmeidler opted for the infinite set of points in the unitinterval endowed with the Lebesgue measure. At the same time, as noted byAumann [3], in analyzing economies with a continuum of traders, “the choice ofthe unit interval as a model for the set of [players] is of no particular significance.A planar or spatial region would have done just as well. In technical terms, [theplayers’ space] can be any measure space without atoms.” Thus, for example,one could alternatively model the players’ space as the set of natural numbersendowed with a natural density. Our goal is to take Aumann’s remark verbatimand not commit to any particular specification of the players’ space in order to seehow much of our analysis can be carried out in a general space without atoms.More formally, we suggest using Savage’s structure of nonatomic probabilitiesdefined on the power set of the space of players (Section 2).
Our contributions.
Our second and third motivation bring us to model theplayers’ space as a set T endowed with a nonatomic probability λ defined on all sub-sets T . Using a measure over the power set takes care of both Shapley’s and Aumann’scomments. In particular, by considering the power set, we allow for the most permis-sive measurable structure possible, since any profile of strategies or utilities becomesautomatically measurable. Measuring the subsets of players/coalitions according to equilibrium notion developed for a finite-players framework easily translates, in terms of existence,to a nonatomic setting. This intuition is based on Lusin’s Theorem which states that for each ε > − ε (see, e.g., Aliprantis and Border [1, Theorem 12.8]). ε -equilibria and to our first motivation. We in-troduce a concept of approximate equilibrium for nonatomic anonymous games, whichwe call ε -estimated equilibrium. This notion of ε -equilibrium encompasses severalapproximate equilibrium concepts: ε -self-confirming ( ε -SCE), ε -peer-confirming ( ε -PCE), and ε -Berk–Nash ( ε -BNE). These equilibria and their ε -versions are formallydefined and discussed in the relevant sections, Sections 3.1, 3.2, and 3.3 (see also therelated literature below). They were mostly developed for finite games and, inter alia,in this paper we extend them to nonatomic games. Nevertheless, the principles behindtheir definitions in a finite-players framework naturally translate to a nonatomic setup.The common thread behind ε -SCE, ε -PCE, and ε -BNE in an anonymous nonatomicgame is the following scheme, which is also the basis for our ε -estimated equilibria:1. Every player best-responds to his beliefs (optimality);2. The belief of every player is consistent with what he can observe ( ε -discrepancy). Where these types of equilibrium differ is how point 2 is formalized, since point 1 istranslated in the same way for all of them. In particular, in SCE, each player receives amessage which is a function of the action he takes and the distribution of actions of theother players. In equilibrium, almost every player best-responds to a distribution thatgenerates a message which is ε -close to the message generated by the true distributionof the actions. In PCE, the message each player receives is the distribution of theactions conditional on a subset of players: his peers. Thus, almost all the playersbest-respond to a distribution which is ε -close to the true distribution of actions oftheir peers, not of all the players. In both ε -SCE and ε -PCE the distributions to whichplayers best-respond are ε -close in terms of observables to the true one; thus they are endogenously generated. By contrast, in BNE, each player t entertains an exogenous set of possible distributions of actions, denoted by Q t , that he believes are accurate indescribing other players’ behavior. Moreover, he is not willing to depart from Q t . Soin equilibrium, almost every player best-responds to a distribution which is ε -close to More precisely, we require points 1 and 2 to hold for every player except a null set of them (seealso point 1 of Remark 1). Q t of the true distribution of actions, according to a statisticalmeasure.Our notion of ε -estimated equilibrium provides a framework where we can accountfor all the three different features described above: that is, the distribution of actionsused by each player in equilibrium is ε -close, whether in statistical terms or properdistance, to the set of all distributions which are compatible with the true one. Thislatter set can be exogenously determined as in BNE or endogenously generated as inSCE or PCE.In Theorem 1, under mild assumptions, we prove that ε -estimated equilibria al-ways exist. As particular cases, we obtain the existence of self-confirming ε -equilibria(Corollary 1), peer-confirming ε -equilibria (Corollary 3), and Berk–Nash ε -equilibria(Corollary 4). Despite the fact that standard Nash equilibria might fail to exist, weprove that ε -Nash equilibria do exist (Corollary 2). Finally, mimicking the notion ofrationalizable self-confirming equilibrium (see Rubinstein and Wolinsky [29]), we alsopropose a definition of rationalizable estimated equilibrium and discuss its existence(Remark 1). Related literature.
The seminal contribution of Aumann [3] (in a general equi-librium framework), followed by Schmeidler [32] (in a game-theoretic framework), ini-tiated a large literature where the negligibility of agents is modelled via a nonatomicprobability players’ space (see, e.g., Khan and Sun [23] for a survey). We will next dis-cuss the relevant literature by connecting it to our three main motivations/contributions.(i) Our definition of ε -estimated equilibrium seems to be new. At the same time, itencompasses three types of equilibrium: self-confirming (SCE), peer-confirming(PCE), and Berk–Nash (BNE) which were developed almost exclusively forgames with finitely many players, respectively, by Battigalli [6] as well as Fuden-berg and Levine [14] (SCE), Lipnowski and Sadler [24] (PCE), and Esponda andPouzo [12] (BNE). The only exceptions seem to be SCE and BNE, which werealso studied for population games, where the latter can be seen as a very specialform of nonatomic games. Moreover, we also consider ε -versions of the above Many subsequent papers extended Schmeidler’s results to more general players’ spaces, but where λ is always assumed to be countably additive and A is allowed to be infinite: see, e.g., Balder [5],Khan and Sun [21], Khan, Rath, and Sun [20], Rath [27], and the references therein. The scope of thistype of results is analyzed in Carmona and Podczeck [9]. Finally, in the same setting of Schmeidler[32], Jara-Moroni [16] extends the notion of rationalizability to nonatomic anonymous games whileRath [28] investigates the issue of existence of perfect, proper, and persistent equilibria, being all ofthem refinements of Nash equilibrium. ε -SCE of course, two approachesare available. The first assumes that: a) players best-respond to their beliefs,but b) beliefs are only ε -consistent with evidence. The second requires that: a’)players ε -best-respond to their beliefs, but b’) beliefs are perfectly consistent.For games with finitely many players, the first approach was introduced by Bat-tigalli [6] and Kalai and Lehrer [17] and [18], while the second was proposedfor pure equilibria by Azrieli [4]. For nonatomic games, other than populationgames, the first approach seems to be unexplored, while the second was studiedby Azrieli [4]. Using the same setting as Schmeidler, that is, assuming that theplayers’ space is the unit interval with the Lebesgue measure, Azrieli shows thatself-confirming equilibria exist (that is, when ε = 0), but when utility dependson the entire profile of strategies and the message feedback is the distributionof actions. Moreover, in trying to obtain the nonatomic games of Schmeidleras a limit of finite-players games which become arbitrarily large, he shows thatself-confirming ε -equilibria eventually exist. Finally, Azrieli limits his analysisto the case where there is nonmanipulable information also known as own-actionindependence of feedback. Loosely speaking, this is the case when the feedbackeach player receives does not depend on the action taken by the player. Thisrules out several interesting cases.In our work, we opt for a definition of ε -SCE which requires rational optimiza-tion on the players’ side, but allows them to entertain ε -consistent beliefs. Wedo not assume own-action independence. The assumption of ε being strictlypositive is due to two reasons: one mathematical and one conceptual. Mathe-matically, by considering players’ spaces which involve finitely additive measures λ , one can show that self-confirming equilibria might fail to exist (Example 1).Conceptually, we take the point of view of Kalai and Lehrer [17] and [18]: weimpose rational behavior on players, but allow for slightly inconsistent beliefs.The latter assumption can be justified by interpreting the belief of each playeras the belief entertained after many rounds of play, so that learning yields ap-proximately correct predictions about observables. At the limit, beliefs wouldbe perfectly consistent with observations, but before that they might be just ε -consistent.(ii) The issue of measurability in nonatomic economies and games has been raised In our specification, this would collaps to a Nash equilibrium. For a related concept and result see also Section 5 of Fudenberg and Kamada [13]. σ -algebra with the power set. This comes at acost: the loss of countable additivity of λ . This not only complicates the technicalanalysis, but generates a conceptual loss. In fact, in an independent paper, Khan,Qiao, Rath, and Sun [19] show that the existence of Nash equilibria for any gamewith players’ space ( T, T , λ ) is equivalent to the countable additivity of λ . Sincethe existence of Nash equilibria cannot be guaranteed with mere finite additivity,they study the existence of ε -Nash equilibria, thus overlapping our Corollary 2.(iii) The issue of modelling the players’ space as a continuum or as a discrete spacehas also been discussed by Al-Najjar [2], who considers as competing modelsthe continuum space [0 ,
1] versus a dense countable grid of [0 , ε -equilibriumfor those discrete nonatomic games that arise as limits of proper sequences offinite-players games. Example 2 shows that for our more general class of gamesthese ε -equilibria are not always guaranteed to exist.We conclude by mentioning one more work. One of the important papers onnonatomic games which introduces a novel approach is Mas-Colell [26]. His approach isbased on distributions of strategies, which allows for not considering strategy profiles.In this way, issues of measurability can be partially overridden in the proofs. It is analternative framework which permits the discussion of players’ negligibility. In this This reformulation is connected to the distributional approach for Bayesian games with a con-tinuum of types (see [26, Remarks 3 and 4]).
Roadmap.
In Sections 2 and 3, we formally introduce nonatomic players’ spaces,nonatomic games with estimation feedback, and the definition of ε -estimated equilib-rium whose existence is proven in Theorem 1. In Sections 3.1, 3.2, and 3.3, as aby-product, we obtain the existence of self-confirming, Nash, peer-confirming, andBerk–Nash ε -equilibria. Proofs are relegated to the appendices. In particular, in Ap-pendix A.1, Lemma 1 generalizes Theorem 7 of Khan and Sun [21] which deals withthe set of distributions induced by all the selections of a correspondence. In AppendixA.2, we provide a brief summary of how the main proofs are carried out and prove allthe results contained in the main text. A players’ space is a pair ( T, λ ) where T is a set of players and λ is a (finitely additive) probability on the power set of T . When T = N , a fundamental class of probabilitiesthat are not countably additive are natural densities , that is, probabilities λ such that λ ( E ) = lim k →∞ | E ∩ { , ..., k }| k whenever the limit exists. As is well known, there are many natural densities and allof them satisfy the following property: Strong continuity (Savage’s nonatomicity)
For each ε > there exists a finitepartition { F , F , ..., F k } of T such that λ ( F i ) < ε for all i = 1 , ..., k . Under strong continuity, any singleton (i.e., any single player) has measure 0 andfor each F ⊆ T and β ∈ (0 ,
1) there exists E ⊆ F such that λ ( E ) = βλ ( F ). Thisis the class of probabilities introduced by Savage [31] when he solved De Finetti’sopen problem on the representation of qualitative probabilities (see also Samet andSchmeidler [30]). Recall that λ is a finitely additive probability if and only if λ is a positive finitely additive setfunction such that λ ( T ) = 1. See Maharam [25, Example 2.1 and Theorem 2] and Bhaskara Rao and Bhaskara Rao [7, Theorem5.1.6 and Remark 5.1.7]. In this literature, natural densities are called density measures or densitycharges. Nonatomic games and their equilibria
Nonatomic games are games where each single player has no influence on the strategicinteraction, but only the aggregate behavior of “large” sets of players can changethe players’ payoffs. Formally, a nonatomic (anonymous) game is a triplet G =(( T, λ ) , A, u ) where ( T, λ ) is the players’ space, A is the space of players’ actions/strategiesand u is their profile of utilities. Below, we discuss in detail these mathematical ob-jects and their interpretations. • A = { , ..., n } is the set of pure strategies/actions . • ∆ = (cid:8) x ∈ R n + | P ni =1 x i = 1 (cid:9) is the n − d ∆ the distance on ∆ induced by the Euclidean norm. This set represents all possible distributions of players’ strategies . Note that an element in ∆ can actually taketwo possible interpretations. In fact, given a player t and an element of ∆, thiselement can either be interpreted as a subjective belief of player t (in this case,we often denote it by β t ) or be interpreted as an objective distribution of players’strategies (in this case, we typically denote it by x ). • u = ( u t ) t ∈ T is a profile of functions u t : A × ∆ → R . For each t in T , u t ( a, β t )represents the ex-ante utility of player t , when he chooses strategy a , if his beliefabout the distribution of opponents’ strategies is β t .As mentioned in the Introduction, nonatomic games were first studied by Schmei-dler [32]. In this paper, we consider a class of games which we term nonatomic gameswith estimation feedback . It has a richer structure and nonatomic games can be seenas a specific parametrization.Formally, a nonatomic game with estimation feedback is a quintet G = (( T, λ ) , A, u, (Π , π ) , f )where (( T, λ ) , A, u ) is a nonatomic game defined as above, (Π , π ) is a neighborhoodstructure, and f is a profile of estimation feedback functions which discipline the be-liefs’ formation of agents in equilibrium. Formally, we have that: • (Π , π ) is a neighborhood structure if and only if Π = { T j } mj =1 is a finite cover of T whose elements have strictly positive measure and π is a function from T to { , ..., m } . In particular, each T j is a nonempty subset of T such that λ ( T j ) > In the paper, given a generic set B , we use the term profile to refer to a function from the set ofplayers T to B . We will denote a profile by either b : T → B or by b = ( b t ) t ∈ T . The latter notationwill allow us, with a small abuse, to treat ( b t ) t ∈ T also as a set. ∪ mj =1 T j = T . An important example of finite covers are finite partitions ofthe players’ space. We interpret an element of Π, T j , as the j -th subpopulationof T and for each t ∈ T the value π ( t ) will denote which subpopulation player t observes. • f = ( f t ) t ∈ T is a profile of (estimation) feedback functions f t : A × ∆ × ∆ → [0 , ∞ ).Each f t is assumed to be such that for each y ∈ ∆ there exists x y ∈ ∆ for whichit holds that f t ( a, x y , y ) = 0 ∀ a ∈ A (1)For each t in T , f t ( a, β t , x ) represents a measure of consistency between the belief β t (entertained by player t ) about the players’ actions within the subpopulationobserved by t and the actual distribution of players’ strategies x within thatsame subpopulation, with the idea that the larger f t ( a, β t , x ) is the greater isthe discrepancy between the player’s belief and the subpopulation actions’ distri-bution. In line with this interpretation, property (1) says that for each possibletrue model x there exists a belief β t such that this discrepancy is minimal, nomatter what action a is chosen by player t . To better understand (1), we nextstate a stronger property which implies (1) and has a more immediate interpre-tation. In all our specifications, with the exception of (11), it will be satisfied:for each t ∈ T and for each a ∈ Ax = y = ⇒ f t ( a, x, y ) = 0 (2)In words, this latter property says that discrepancy is minimal provided the belief β t is indeed correct, that is β t = x . Under (1) or (2), we deliberately allow forthe possibility that f t ( a, β t , x ) = 0, but β t = x : a belief might be consistent withevidence, but still incorrect.Finally, we need three extra mathematical objects: • Σ = A T is the set of all functions σ from T to A . Each σ ∈ Σ represents a strategy profile in which the generic player t chooses strategy σ ( t ). Despite being a natural requirement, we can dispense with the assumption that t ∈ T π ( t ) . Inother words, we do not need to assume that any player t belongs to the subpopulation he observes. Note that (2) implies (1). Fix t ∈ T . For each y ∈ ∆, set x y = y . By (2), it follows that f t ( a, x y , y ) = 0 for all a ∈ A . Given j ∈ { , ..., m } , λ j denotes the probability on the power set of T definedby λ j ( E ) = λ ( E ∩ T j ) λ ( T j ) ∀ E ⊆ T In other words, λ j is the players’ conditional measure in the subpopulation j .Note that if λ is strongly continuous, so is each λ j . • Given σ ∈ Σ and j ∈ { , ..., m } , λ jσ ∈ ∆ is the distribution of σ on A in the j -thsubpopulation, that is, λ jσ = (cid:0) λ j ( { t ∈ T j | σ ( t ) = a } ) (cid:1) a ∈ A The vector λ jσ represents the true distribution of players’ pure strategies in the j -th subpopulation, when they all play according to σ . When Π is trivial, thatis, Π = { T } , then Π contains only one element and λ = λ . In this case, wewrite λ σ in place of λ σ . Similarly, the vector λ σ represents the true distributionof players’ pure strategies in the entire population.We can now introduce our most general concept of equilibrium. It provides aunifying structure for the notions of equilibrium that feature players best responding tobeliefs that are possibly wrong, but are nonetheless consistent with their probabilisticinformation. In the next three sections, we discuss three particular and importantspecifications (see also the Introduction). Definition 1
Let ε ≥ . An ε -estimated equilibrium (in pure strategies) for thenonatomic game with estimation feedback G = (( T, λ ) , A, u, (Π , π ) , f ) is a strategyprofile σ ∈ Σ such that there exists a profile of beliefs β ∈ ∆ T satisfying λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Af t (cid:16) σ ( t ) , β ( t ) , λ π ( t ) σ (cid:17) ≤ ε )! = 1 (3)We are ready to state our main result. By definition of λ jσ , note that λ jσ = (cid:0) λ j ( { t ∈ T | σ ( t ) = a } ) (cid:1) a ∈ A for all j ∈ { , ..., m } . heorem 1 Let G = (( T, λ ) , A, u, (Π , π ) , f ) be a nonatomic game with estimationfeedback and ε > . If λ is strongly continuous and f = ( f t ) t ∈ T is a family of functionswhich is equicontinuous with respect to the third argument, then G has an ε -estimatedequilibrium. Remark 1
Three observations are in order: In proving Theorem 1, we actually show that there exists an ε -estimated equi-librium in which each player best-responds to his ε -discrepant belief (cf. alsoRemark 4 and Lemma 3), that is, the set in (3) coincides with T and, in partic-ular, has measure 1. As just mentioned, in an ε -estimated equilibrium players best-respond to their ε -discrepant beliefs. Mimicking the notion of rationalizable self-confirming equi-librium of Rubinstein and Wolinsky [29], we could also require that this is cor-rectly and commonly believed by all players. This will turn out to be useful indiscussing peer-confirming equilibrium (see Section 3.2). In order to do so, wefirst introduce some notation and then propose a recursive definition. Given anonempty subset S ⊆ Σ, we denote by ∆ ( S ) the set of all probabilities over thepower set of S . Consider a player t ∈ T . An element ˜ β t ∈ ∆ ( S ) represents thebelief of the player about which strategy profile in S will realize. At the sametime, given our assumption of anonymity and the neighborhood structure, whatis relevant for t is merely the distribution of players’ strategies ¯ β t , induced by˜ β t , within the subpopulation observed. With this, given δ, ε ≥
0, we can definerecursively the following sequence of sets { S k } k ∈ N : S = Σ and S k +1 = ( σ ∈ S k |∃ ˜ β ∈ ∆ ( S k ) T s.t. ∀ t ∈ T u t (cid:0) σ ( t ) , ¯ β ( t ) (cid:1) ≥ u t (cid:0) a, ¯ β ( t ) (cid:1) − δ ∀ a ∈ Af t (cid:16) σ ( t ) , ¯ β ( t ) , λ π ( t ) σ (cid:17) ≤ ε ) We say that f = ( f t ) t ∈ T is a family of functions which is equicontinuous with respect to the thirdargument if and only if for each ε > δ ε > d ∆ ( x, y ) < δ ε = ⇒ | f t ( a, γ, x ) − f t ( a, γ, y ) | < ε ∀ t ∈ T, ∀ a ∈ A, ∀ γ ∈ ∆In other words, the family of functions { f t ( a, γ, · ) } t ∈ T,a ∈ A,γ ∈ ∆ from ∆ to [0 , ∞ ) is equicontinuous. Formally, we have that¯ β t = (cid:18)Z S λ π ( t ) (cid:0)(cid:8) t ∈ T π ( t ) | σ ( t ) = a (cid:9)(cid:1) d ˜ β t (cid:19) a ∈ A or, more succinctly, ¯ β t = R S λ π ( t ) σ d ˜ β t .
12e say that σ ∈ Σ is a rationalizable ( δ, ε ) -estimated equilibrium (in pure strate-gies) for the nonatomic game with estimation feedback G = (( T, λ ) , A, u, (Π , π ) , f )if and only if σ ∈ ∩ k ∈ N S k . In words, in a rationalizable ( δ, ε )-estimated equi-librium, players δ -best-respond to their ε -discrepant beliefs and this is correctlyand commonly believed by all players. By setting ε = δ = 0, our definitionreduces to a version for nonatomic anonymous games of the equilibrium notionof Rubinstein and Wolinsky [29]. We discuss existence in the next point. Let G = (( T, λ ) , A, u, (Π , π ) , f ) be a nonatomic game with estimation feedback, δ >
0, and ε ≥
0. If λ is strongly continuous, u = ( u t ) t ∈ T is a family of functionswhich is equicontinuous with respect to the second argument, and each f t satisfies condition (2) , then G has a rationalizable ( δ, ε )-estimated equilibrium. N An interesting class of nonatomic games with estimation feedback arises when thefeedback function of player t is generated by a message function m t : A × ∆ → M ,where M is a metric space with distance d . For each t in T , m t ( a, x ) represents the message player t receives when he chooses strategy a and the distribution of players’strategies is x . In games with finitely many players, typically the message function We say that u = ( u t ) t ∈ T is a family of functions which is equicontinuous with respect to thesecond argument if and only if for each ε > δ ε > d ∆ ( x, y ) < δ ε = ⇒ | u t ( a, x ) − u t ( a, y ) | < ε ∀ t ∈ T, ∀ a ∈ A In light of point 1 and, given the equicontinuity of u (cf. the proof of Corollary 2), we can provethat there exists a strategy profile σ ∈ Σ such that u t (cid:16) σ ( t ) , λ π ( t ) σ (cid:17) ≥ u t (cid:16) a, λ π ( t ) σ (cid:17) − δ ∀ a ∈ A, ∀ t ∈ T Set ˜ β ∈ ∆ ( S ) T = ∆ (Σ) T to be such that ˜ β ( t ) coincides with the Dirac at σ for all t ∈ T . It followsthat ¯ β ( t ) = λ π ( t ) σ for all t ∈ T . Given that each f t satisfies (2), we have that f t (cid:16) σ ( t ) , ¯ β ( t ) , λ π ( t ) σ (cid:17) =0 ≤ ε for all t ∈ T . This yields that σ ∈ S . By induction, we can conclude that σ ∈ S k for all k ∈ N ,proving that σ is a rationalizable ( δ, ε )-estimated equilibrium. The complete proof is available uponrequest. To simplify notation, we assume that the message space is the same for all players. This iswithout loss of generality. We could have equivalently assumed that each player has his own messagespace M t , and in the proofs embed this set into a larger common message space M . Our assumptionsof equicontinuity on the message functions m t (cf. Corollary 1) would seamlessly pass through theembedding as well. t depends on the action chosen by the player and the profile of actions chosen bythe opponents. Nevertheless, given our underlying assumption of anonymity, it seemsnatural to replace the latter with the actions’ distribution in the population.With this in mind, the next type of equilibrium models a situation in which thebelief β t adopted by each agent t in equilibrium is consistent/confirmed with/by themessage received. More formally, β t is such that the expected message m t ( σ ( t ) , β t )is ε -close to the received message m t ( σ ( t ) , λ σ ).We define a nonatomic game with message feedback to be a quartet G = (( T, λ ) , A, u, m )where (( T, λ ) , A, u ) is a nonatomic game and m = ( m t ) t ∈ T is a profile of messagefunctions. Note that a nonatomic game with message feedback can be mapped into anonatomic game with estimation feedback. In fact, it is enough to consider (Π , π ) tobe trivial, that is Π = { T } , and set the profile of feedback functions to be such that: f t ( a, x, y ) = d ( m t ( a, x ) , m t ( a, y )) ∀ t ∈ T, ∀ a ∈ A, ∀ x, y ∈ ∆ (4)It can be seen immediately that each f t satisfies (2), and thus (1). We can define ourconcept of self-confirming ε -equilibrium which we discuss below. Definition 2
Let ε ≥ . A self-confirming ε -equilibrium (in pure strategies) for thenonatomic game with message feedback G = (( T, λ ) , A, u, m ) is a strategy profile σ ∈ Σ such that there exists a profile of beliefs β ∈ ∆ T satisfying λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ( m t ( σ ( t ) , β ( t )) , m t ( σ ( t ) , λ σ )) ≤ ε )! = 1 (5)In words, a strategy profile σ ∈ Σ is a self-confirming ε -equilibrium ( ε -SCE) if andonly if1. Almost all players best-respond to their beliefs (optimality);2. Beliefs are not significantly refuted by what they can observe ( ε -confirmation).As noted in the Introduction, self-confirming equilibria were introduced for gameswith finitely many players by Battigalli [6] and Fudenberg and Levine [14], and also ε -confirmation was introduced by Battigalli [6] and Kalai and Lehrer [17] and [18]. Tothe best of our knowledge, the above definition of ε -equilibrium seems to be novel fornonatomic games and also natural (cf. the related literature section). Furthermore, In this case, note that π can only take one value.
14t encompasses the notions of self-confirming equilibrium and ε -Nash equilibrium (afortiori, Nash equilibrium). To see this, we begin by observing that if ε = 0 and m t : A × ∆ → ∆ is such that m t ( a, x ) = x ∀ t ∈ T, ∀ a ∈ A, ∀ x ∈ ∆ (6)that is, ( M, d ) = (∆ , d ∆ ) and feedback is (statistically) perfect , then (5) becomes λ ( { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) ∀ a ∈ A } ) = 1which means that σ is a Nash equilibrium. In this case, beliefs are not only per-fectly consistent with observations but also correct. Maintaining the perfect feedbackassumption (6), but allowing for ε >
0, (5) becomes λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ∆ ( β ( t ) , λ σ ) ≤ ε )! = 1Under a suitable assumption of continuity of u (see Corollary 2 and its proof), we canshow that σ is an ε -Nash equilibrium for some suitable ˆ ε >
0, that is, λ ( { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) − ˆ ε ∀ a ∈ A } ) = 1The intuition is simple: if beliefs are “close” to the true distribution, players are notfar from objective maximization.Finally, if we remove perfect feedback but maintain ε = 0, (5) becomes λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Am t ( σ ( t ) , β ( t )) = m t ( σ ( t ) , λ σ ) )! = 1which is arguably the nonatomic anonymous games counterpart of the definition ofself-confirming equilibrium (SCE).Starting with ε -estimated equilibria, most of our analysis deals with the case inwhich ε >
0. There are two reasons why we do so. First, conceptually, ε > ε -versions do. Ex-15mple 2 provides an instance where ε -uniform Nash equilibria `a la Al-Najjar [2] donot exist, but standard ε -Nash equilibria do. Example 1
The next example builds on Khan, Qiao, Rath, and Sun [19]. Consider T = N and let λ be a natural density. Consider two strategies, that is, A = { , } .Assume that for each t ∈ Tu t ( a, x ) = ( t − x a = 1 x − t a = 2 ∀ x ∈ ∆Let m t = u t for all t ∈ T . This amounts to the standard assumption of mere payoffobservability. Assume that σ ∈ Σ is an SCE, that is, there exists β ∈ ∆ T such that λ ( { t ∈ T | u t ( σ ( t ) , λ σ ) = u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ A } ) = 1For ease of notation, set λ σ = x and define the set of “optimizing” players by O = { t ∈ T | u t ( σ ( t ) , λ σ ) = u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ A } We have two cases: x >
0. Since λ is a natural density and O has mass 1, then O is infinite. Thus,there exists ¯ t ∈ N such that t − x < t ∈ O ∩ { , ..., ¯ t } c . Consider t ∈ O ∩ { , ..., ¯ t } c = ∅ . By contradiction, assume that σ ( t ) = 1. The SCEconditions imply that 1 t − x = 1 t − β ( t ) ≥ β ( t ) − t yielding that 0 ≤ β ( t ) ≤ t − x <
0, a contradiction. Since t was arbitrarilychosen in O ∩{ , ..., ¯ t } c , it follows that σ ( t ) = 2 for all t ∈ O ∩{ , ..., ¯ t } c . Since λ isa natural density and O and O ∩{ , ..., ¯ t } c differ by a finite set λ ( O ∩ { , ..., ¯ t } c ) =1, we have that λ σ = x is such that x = 1, a contradiction with 0 = 1 − x = x > x = 0. Consider t ∈ O . By contradiction, assume that σ ( t ) = 2. The SCE The example of Khan, Qiao, Rath, and Sun [19] seems to be the first one in the literature toexhibit a well-behaved nonatomic game which does not have any Nash equilibrium, be it pure ormixed. x − t = β ( t ) − t ≥ t − β ( t ) yielding that 0 = x = β ( t ) and 0 ≥ t >
0, a contradiction. Since t wasarbitrarily chosen in O , σ ( t ) = 1 for all t ∈ O , yielding that λ σ = x is such that x = 1, a contradiction with x = 0.To sum up, we have just shown that the nonatomic game with message feedbackabove does not have any self-confirming equilibrium and, in particular, any Nashequilibrium. This happens despite the fact that the profile of message functions isextremely well-behaved being m = ( m t ) t ∈ T equicontinuous with respect to the secondargument (cf. Corollary 1). At the same time, consider ε >
0. Let ¯ t ∈ N be suchthat min { , ε } > t for all t ∈ N such that t > ¯ t . Set ¯ ε = min { , ε } . Consider astrategy profile σ ∈ Σ and a belief profile β ∈ ∆ T such that σ ( t ) = 2 and β ( t ) = ¯ ε + t ∈ (cid:0) t , ¯ ε (cid:1) ⊆ (0 ,
1) for all t ∈ N such that t > ¯ t . Since { , ..., ¯ t } is finite and λ is anatural density, we have that λ σ = x is such that x = 1, that is, x = 0. It followsthat for each t ∈ { , ..., ¯ t } c | m t ( σ ( t ) , β ( t )) − m t ( σ ( t ) , λ σ ) | = (cid:12)(cid:12)(cid:12)(cid:12) ¯ ε + t − t − x + 1 t (cid:12)(cid:12)(cid:12)(cid:12) = ¯ ε + t < ¯ ε ≤ ε and u t ( σ ( t ) , β ( t )) = β ( t ) − t = ¯ ε − t > > t − β ( t ) = u t (1 , β ( t )) Two extra observations are in order:a. In the nonatomic game above, SCE equilibria and Nash equilibria coincide. This is by chance,as the next point shows.b. Khan, Qiao, Rath, and Sun [19] consider T = N and let λ be a natural density. They assume A = { , } and ˆ u to be such that for each t ∈ T ˆ u t ( a, x ) = (cid:26) t − x a = 10 a = 2 ∀ x ∈ ∆With similar arguments, they prove that the nonatomic game (( T, λ ) , A, ˆ u ) does not have any Nashequilibrium. At the same time, if we consider the augmented nonatomic game with message feedback(( T, λ ) , A, ˆ u, m ) where m t = ˆ u t for all t ∈ T , then we can show that there exists an SCE equilibrium.In fact, if σ ∈ Σ is such that σ ( t ) = 2 for all t ∈ T , by setting β ∈ ∆ T such that β ( t ) = 1 for all t ∈ T , we obtain the result. Indeed, note that for each ε > δ ε = ε and get d ∆ ( x, y ) < ε = ⇒ | m t ( a, x ) − m t ( a, y ) | = | x − y | ≤ d ∆ ( x, y ) < ε ∀ t ∈ T, ∀ a ∈ A { , ..., ¯ t } c has mass 1, we can conclude that σ ∈ Σ is an ε -SCE. N Example 2
Al-Najjar [2] (cf. the Introduction) also deals with the lack of countableadditivity and studies the following equilibrium: a strategy σ ∈ Σ is an Al-Najjarequilibrium (in pure strategies) if and only if for each ε > λ ( { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) − ε ∀ a ∈ A } ) > − ε (7)We next show that also these equilibria might fail to exist. In what follows, it willoften be useful to set O ε = { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) − ε ∀ a ∈ A } Two observations are in order. First, compared to the ε -Nash equilibria we study(Corollary 2), the key difference is that, in our case, σ might depend on the given ε ,while in Al-Najjar’s case, σ must work with any ε . In particular, one can easily showthat: σ ∈ Σ is an Al-Najjar equilibrium if and only if for each ε > λ ( { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) − ε ∀ a ∈ A } ) = 1Second, by taking the intersection of the sets O /n , this allows us to conclude easilythat an Al-Najjar equilibrium is a Nash equilibrium, provided λ is countably additive.We consider the nonatomic game (( T, λ ) , A, ˜ u ) where ( T, λ ) and A are as in Example1 and for each t ∈ T ˜ u t ( a, x ) = t − x a = 1 and x > x − t a = 2 and x > a = 1 and x = 0 t a = 2 and x = 0 ∀ x ∈ ∆Assume that σ ∈ Σ satisfies (7). For ease of notation, set λ σ = x . As before, we havetwo cases: x >
0. Fix ε >
0. Since λ is a natural density, the set O ε has mass 1, and λ ( { t ∈ T | σ ( t ) = 1 } ) >
0, we have that O ε ∩{ t ∈ T | σ ( t ) = 1 } is infinite. Since It is easy to see that if 0 < ε < ε ′ , then O ε ⊆ O ε ′ , thus λ ( O ε ′ ) ≥ λ ( O ε ) > − ε ∀ ε ′ > , ∀ ε ∈ (0 , ε ′ )yielding that λ ( O ε ′ ) = 1 for all ε ′ > was arbitrarily chosen, this implies that we can construct a strictly increasingsequence { t k } k ∈ N ⊆ N such that t k ∈ O /k ∩ { t ∈ T | σ ( t ) = 1 } for all k ∈ N .Since t k ∈ O /k , σ ( t k ) = 1, and x >
0, we have that for each k ∈ N t k − x ≥ x − t k − k = ⇒ < x ≤ t k + 12 k By passing to the limit, we obtain that 0 < x ≤
0, a contradiction. x = 0. Fix ε >
0. Since λ is a natural density, the set O ε has mass 1, and λ ( { t ∈ T | σ ( t ) = 2 } ) >
0, we have that O ε ∩{ t ∈ T | σ ( t ) = 2 } is infinite. Since ε was arbitrarily chosen, this implies that we can construct a strictly increasingsequence { t k } k ∈ N ⊆ N such that t k ∈ O /k ∩ { t ∈ T | σ ( t ) = 2 } for all k ∈ N .Since t k ∈ O /k , σ ( t k ) = 2, and x = 0, we have that for each k ∈ N t k ≥ − k By passing to the limit, we obtain that 0 ≥
1, a contradiction.To sum up, we have just shown that the nonatomic game ((
T, λ ) , A, ˜ u ) does not haveany equilibrium as defined in (7). At the same time, it is not hard to see that thisgame admits an ε -Nash equilibrium for every ε >
0. One way to observe this isto consider the augmented game ((
T, λ ) , A, ˜ u, m ) in which each player has perfectstatistical feedback: that is m t ( a, x ) = x ∀ t ∈ T, ∀ a ∈ A, ∀ x ∈ ∆Since m = ( m t ) t ∈ T is equicontinuous with respect to the second argument (cf. Corol-lary 1), we have that for each ε > ε -SCE. Given ε ∈ (0 , σ is an ε -SCE if and only if λ ( { t ∈ T | σ ( t ) = 1 } ) ∈ (cid:2) , ε/ √ (cid:3) . Given our choice of m , following the intuitionthat “if beliefs are close to the true distribution, players are not far from objec-tive maximization”, we can prove that, given ε ∈ (0 , σ is an ε √ -SCE and λ ( { t ∈ T | σ ( t ) = 1 } ) >
0, then σ is an ε -Nash equilibrium. In other words, (( T, λ ) , A, ˜ u )does not have any equilibrium as defined in (7), but for each ε > ε -Nashequilibrium. N Since λ is strongly continuous, note that, given ε ∈ (0 , σ ∈ Σ such that λ ( { t ∈ T | σ ( t ) = 1 } ) ∈ (0 , ε/ σ which is an ε √ -SCE such that λ ( { t ∈ T | σ ( t ) = 1 } ) >
19e are ready to state the main results of this section.
Corollary 1
Let G = (( T, λ ) , A, u, m ) be a nonatomic game with message feedbackand ε > . If λ is strongly continuous and m = ( m t ) t ∈ T is a family of functions whichis equicontinuous with respect to the second argument, then G has an ε -SCE. In particular, under the assumption of payoff observability, that is, m t ( a, x ) = u t ( a, x ) for all t ∈ T, a ∈ A, x ∈ ∆, Corollary 1 yields that if λ is strongly continuousand u = ( u t ) t ∈ T is a family of functions which is equicontinuous with respect to thesecond argument, then there exists an ε -SCE strategy profile σ ∈ Σ such that λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ A | u t ( σ ( t ) , β ( t )) − u t ( σ ( t ) , λ σ ) | ≤ ε )! = 1where β ∈ ∆ T . In this case, the objective observed payoff substantially matches theexpected one. Building on Corollary 1 and following a similar intuition, we also obtainthe existence of ε -Nash equilibria. Corollary 2
Let G = (( T, λ ) , A, u ) be a nonatomic game and ε > . If λ is stronglycontinuous and u = ( u t ) t ∈ T is a family of functions which is equicontinuous withrespect to the second argument, then G has an ε -Nash equilibrium, that is, thereexists a strategy profile σ ∈ Σ such that λ ( { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) − ε ∀ a ∈ A } ) = 1At this point, the reader might wonder how restrictive are our assumptions ofequicontinuity. At first sight, it might appear that the degree of similarity amongplayers, imposed by a measurable structure as in the original framework of Schmeidler,is here replaced by equicontinuity. The following example should clarify that this isfar from being the case. We say that m = ( m t ) t ∈ T is a family of functions which is equicontinuous with respect to thesecond argument if and only if for each ε > δ ε > d ∆ ( x, y ) < δ ε = ⇒ d ( m t ( a, x ) , m t ( a, y )) < ε ∀ t ∈ T, ∀ a ∈ A See Footnote 17. See also the discussion following Corollary 3. xample 3 Assume that players have expected-utility like preferences, namely, foreach t ∈ T u t ( a, x ) = X b ∈ A v t ( a, b ) x b ∀ a ∈ A, ∀ x ∈ ∆where v t : A × A → R . As is well known, each v t can be normalized to be takingvalues in [0 , | u t ( a, x ) − u t ( a, y ) | ≤ √ nd ∆ ( x, y ) ∀ t ∈ T, ∀ a ∈ A proving that u = ( u t ) t ∈ T is a family of functions which is equicontinuous with respectto the second argument. Thus, preferences can be extremely different within the aboveclass and yet satisfy equicontinuity. N As mentioned in the Introduction, Khan, Qiao, Rath, and Sun [19] showed that inthe absence of countable additivity the existence of Nash equilibria is not guaranteed.They also reported an independent result of existence of an ε -Nash equilibrium. Theirdefinition is weaker than ours. In their case, a strategy profile σ ∈ Σ is an ε -equilibriumif and only if λ ( { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) − ε ∀ a ∈ A } ) ≥ − ε Lipnowski and Sadler [24] propose a notion of equilibrium in which players best-respond to beliefs which are required to be correct only when it comes to the behaviorof opponents who belong to the same neighborhood. Moreover, they further requirethat this is correctly and commonly believed by players. Formally, the collection ofneighborhoods is a partition of the players in terms of the connected components ofan underlying undirected network. They study games with finitely many players. Forsimultaneous-move games, peer-confirming equilibrium is an example of rationalizableself-confirming equilibrium (see also Rubinstein and Wolinsky [29] as well as Fudenbergand Kamada [13]). In what follows, we dispense with the assumption of correct andcommon belief. This seems reasonable since nonatomic anonymous games model ex-actly situations where individuals are negligible and are not fully aware of the strategicenvironment surrounding them, rendering sophisticated strategic reasoning less plausi-ble. At the same time, our notion of rationalizable ( δ, ε )-estimated equilibrium allows21s to offer a more faithful translation to our setting of peer-confirming equilibrium(see Remark 2). Moreover, given anonymity we require that players’ observations areonly about the actions’ distributions in the subpopulation they face.We define a nonatomic game with a neighborhood structure to be a quartet G =(( T, λ ) , A, u, (Π , π )) where (( T, λ ) , A, u ) is a nonatomic game and (Π , π ) is a neigh-borhood structure. Note that a nonatomic game with a neighborhood structure can bemapped into a nonatomic game with estimation feedback . In fact, it is enough to setthe profile of feedback functions to be such that f t ( a, x, y ) = d ∆ ( x, y ) ∀ t ∈ T, ∀ a ∈ A, ∀ x, y ∈ ∆ (8)It can be seen immediately that each f t satisfies (2), and thus (1). We can define theversion of peer-confirming ε -equilibrium that we analyze below. Definition 3
Let ε ≥ . A peer-confirming ε -equilibrium (in pure strategies) for thenonatomic game with a neighborhood structure G = (( T, λ ) , A, u, (Π , π )) is a strategyprofile σ ∈ Σ such that there exists a profile of beliefs β ∈ ∆ T satisfying λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ∆ (cid:16) β ( t ) , λ π ( t ) σ (cid:17) ≤ ε )! = 1 (9)In words, a strategy profile σ ∈ Σ is a peer-confirming ε -equilibrium ( ε -PCE) ifand only if1. Almost all players best-respond to their beliefs (optimality);2. Beliefs are almost correct in terms of the subpopulation observed ( ε -neighborhoodconfirmation). Corollary 3
Let G = (( T, λ ) , A, u, (Π , π )) be a nonatomic game with a neighborhoodstructure and ε > . If λ is strongly continuous, then G has an ε -PCE. It is important to note how the corollary above does not require any extra prop-erty of continuity. For, in such a case feedback is perfect, when restricted to eachsubpopulation, and action independent, automatically satisfying the requirement ofequicontinuity in Theorem 1. Conceptually, this confirms that, in contrast with mea-surability assumptions, our properties of equicontinuity do not impose automaticallythat “close players” have similar preferences/behavior (cf. Example 3).22 emark 2
Two observations are in order: In the definition of ε -PCE, we could allow for the possibility that each player t has a belief ˜ β t over the entire space of players’ strategy profiles Σ (cf. point 2 ofRemark 1) and require that only the restriction to the subpopulation observed,in terms of actions’ distribution, that is ¯ β t , is ε -confirmed. This would allow formodelling explicitly the possibility that players, in equilibrium, possibly entertainwrong beliefs about players not in their neighborhood. Given our nonatomicstructure and Corollary 3, we could obtain an existence result also for this moregeneral notion. Consider a rationalizable ( δ, ε )-estimated equilibrium as defined in point 2 of Re-mark 1 where the profile of feedback functions is set to be as in (8). Given thisspecification, in a rationalizable ( δ, ε )-estimated equilibrium, all players δ -best-respond to their beliefs which are almost correct in terms of the subpopulationobserved. Moreover, this is correctly and commonly believed by all players. Bysetting ε = δ = 0, our definition provides a more faithful formal translationto nonatomic anonymous games of the equilibrium notion studied by Lipnowskiand Sadler [24]. By point 3 of Remark 1, given a nonatomic game with a neigh-borhood structure G = (( T, λ ) , A, u, (Π , π )) as well as δ > ε ≥
0, if λ isstrongly continuous and u = ( u t ) t ∈ T is a family of functions which is equicon-tinuous with respect to the second argument , then G has a rationalizable ( δ, ε )-estimated equilibrium. N Esponda and Pouzo [12] propose a notion of equilibrium that allows for players’ beliefsto be possibly misspecified (see also Remark 3 below). It is a different way, comparedto self-confirming equilibria, to allow for potentially incorrect beliefs in equilibrium.They term their notion of equilibrium Berk–Nash. Berk–Nash equilibria are basedon the assumption that each player has a set of probabilistic models over the payoff-relevant features, in our case { Q t } t ∈ T ⊆ ∆ o , and:1. All players best-respond to their beliefs (optimality); As usual, ∆ o denotes the set { x ∈ ∆ | x i > ∀ i ∈ { , ..., n }} In other words, ∆ o is the relative interior of ∆.
23. Each player’s belief is restricted to be the best fit (in terms of Kullback–Leiblerdistance) among the set of beliefs he considers possible.In our setup, this would mean that each player t has a (possibly misspecified) setof models Q t ⊆ ∆ o . A strategy profile σ ∈ Σ is a Berk–Nash equilibrium if and only ifthere exists a profile of beliefs β ∈ ∆ T such that the set of all players that satisfy thefollowing two conditions has full measure: u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) for all a ∈ A ;2. β ( t ) ∈ argmin z ∈ Q t K ( λ σ || z ) (where K is the Kullback–Leibler distance).In what follows, we offer a more general version for nonatomic games of the aboveequilibrium. In order to do so, we define a nonatomic game with model misspecification to be a quintet G = (( T, λ ) , A, u, Q , D ) wherea. (( T, λ ) , A, u ) is a nonatomic game;b. Q = ( Q t ) t ∈ T is a profile of sets of actions’ distributions, that is, Q t is a nonempty,compact, and convex subset of ∆ o for all t ∈ T ;c. D : ∆ × ∆ o → [0 , ∞ ) is a statistical divergence, that is, a jointly convex andcontinuous function such that for each x, y ∈ ∆ o x = y ⇐⇒ D ( x || y ) = 0 . (10)The next example describes a class of widely used statistical divergences. Example 4
The most classic statistical divergences are φ -divergences which have theform D φ ( x || z ) = n X i =1 z i φ (cid:18) x i z i (cid:19) where φ is a positive, continuous, strictly convex function on R + such that φ (1) = 0.For example, for φ ( s ) = s log s − s + 1, D φ is the Kullback–Leibler distance, for φ ( s ) = ( s − / D φ is the χ -distance, and for φ ( s ) = ( √ s − , D φ is the Hellingerdistance. In all these specifications, D φ satisfies (10) and it is jointly convex andcontinuous. N Compared to Esponda and Pouzo [12], we do not assume that players’ are expected utility andhave a prior µ over argmin z ∈ Q t K ( λ σ || z ). In other words, players are only allowed to considerdegenerate priors. A priori, this makes it more difficult to obtain an existence result. Moreover, weare also not considering any extra form of feedback (see point 1 of Remark 3 below). Here, it is assumed implicitly that φ (0) = 1 which is obtained by taking the limit for s → model misspecification can be mapped into anonatomic game with estimation feedback . In fact, it is enough to consider (Π , π ) tobe trivial, that is Π = { T } , and set the profile of feedback functions to be such that: f t ( a, x, y ) = d ∆ (cid:0) x, argmin z ∈ Q t D ( y || z ) (cid:1) ∀ t ∈ T, ∀ a ∈ A, ∀ x, y ∈ ∆ (11)It is not hard to show that each f t satisfies (1), but might fail to satisfy (2). We candefine our version of Berk–Nash ε -equilibrium which we discuss below. Definition 4
Let ε ≥ . A Berk–Nash ε -equilibrium (in pure strategies) for thenonatomic game with model misspecification G = (( T, λ ) , A, u, Q , D ) is a strategyprofile σ ∈ Σ such that there exists a profile of beliefs β ∈ ∆ T satisfying λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ∆ (cid:0) β ( t ) , argmin z ∈ Q t D ( λ σ || z ) (cid:1) ≤ ε )! = 1 (12)Note that a strategy profile σ ∈ Σ is a Berk–Nash ε -equilibrium ( ε -BNE) if andonly if1. Almost all players best-respond to their beliefs (optimality);2. Beliefs are ε -close to the set of probabilistic models which are the best fit in theprimitive set Q t of the realized distribution ( ε -fit).Although prima facie they might appear similar, the notion of ε -BNE is concep-tually and formally very different from that of ε -SCE. The next result proves that,under suitable conditions, the former type of equilibria always exists. To do so, weneed a last piece of notation. Given δ >
0, we denote∆ δ = { x ∈ ∆ | x i ≥ δ ∀ i ∈ { , .., n }} In words, ∆ δ is the set of all actions’ distributions which are uniformly bounded awayfrom zero by δ . In this case, note that π can only take one value. Moreover, when x ∈ ∆ and Y is a nonemptysubset of ∆, d ∆ ( x, Y ) denotes the distance of x from the set Y , that is, d ∆ ( x, Y ) = inf y ∈ Y d ∆ ( x, y )In our case, Y = argmin z ∈ Q t D ( y || z ). The two equilibrium notions are distinct, but share some overlap (see Esponda and Pouzo [12]). orollary 4 Let G = (( T, λ ) , A, u, Q , D ) be a nonatomic game with model misspec-ification and ε > . If λ is strongly continuous, D is strictly convex in the secondargument, and there exists δ > such that Q t ⊆ ∆ δ for all t ∈ T , then G has an ε -BNE. Remark 3
Four observations are in order: Unlike Esponda and Pouzo [12] original formulation, in our definition eachplayer’s set of actions’ distributions Q t does not depend on the action played.Conceptually, this amounts to assume that there is perfect statistical feedback. Ifwe were to impose that each Q t was also function of the action, that is a Q t ( a ),the feedback function in (11) would fail to satisfy property (1). In Definition 4, we allow each player’s equilibrium belief β ( t ) to be possiblyoutside the set Q t . This could be interpreted as allowing for the possibility thateach player fears model misspecification and willingly considers probability mod-els that are outside his posited set Q t (see Cerreia-Vioglio, Hansen, Maccheroni,and Marinacci [10]). At the same time, we could have considered the followingmore stringent definition of ε -BNE where this is not allowed. In this case, σ ∈ Σwould be an ε -BNE if and only if there exists a profile of beliefs β ∈ ∆ T satisfying λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ∆ (cid:0) β ( t ) , argmin z ∈ Q t D ( λ σ || z ) (cid:1) ≤ ε and β ( t ) ∈ Q t )! = 1Under the same exact assumptions of Corollary 4, we can show that also these ε -equilibria exist. Our results do not directly apply to the case in which D is the Kullback–Leiblerdistance K . In fact, in this case, K ( x ||· ) might fail to be strictly convex. Atthe same time, any perturbation κ > D , that is D + κd , is a statistical divergence and satisfies the condition of strict convexityin Corollary 4. The assumption “there exists δ > Q t ⊆ ∆ δ for all t ∈ T ” is equivalentto the condition “each Q that belongs to the Hausdorff distance closure of Q isa subset of ∆ o ”. In other words, it is an assumption of relative compactness. N In fact, K ( x ||· ) is strictly convex if x ∈ ∆ o , but it might fail to be so if x ∈ ∆ \ ∆ o . Appendix
In what follows, we first provide the proofs of the results in the main text and thenconclude with one of the authors explaining the origin of nonatomic games. We beginwith Appendix A.1 where we discuss a result which is key in proving Theorem 1.Appendix A.2 contains the remaining proofs. In a nutshell, this latter section is dividedinto two parts. First, we deal with the proof of existence of ε -estimated equilibria.Second, we prove the existence of ε -SCE, ε -NE, ε -PCE, and ε -BNE by showing thatthey are all particular cases or consequence of the existence of ε -estimated equilibria.In the appendix, the vector space we use is the Cartesian product of m copies of R n , that is ( R n ) m , where n is given by the cardinality of the space of actions A and m isgiven by the cardinality of the neighborhood structure (Π , π ). We denote the elementsof ( R n ) m by bold letters, that is x and y , while x j will be the vector in R n which isthe j -th component of x . If m = 1, then we denote x and y simply by x and y . Weendow ( R n ) m with the topology induced by the norm k x k = max j ∈{ ,...,m } k x j k where k k is the Euclidean norm. Finally, we denote the Cartesian product of m copies of∆ by ∆ m . Note that ∆ m is a nonempty, convex, and compact subset of ( R n ) m and weendow it with the distance induced by k k . A.1 A key general result
The next lemma uses the terminology of Bhaskara Rao and Bhaskara Rao [7]. Beforediscussing it, we need a piece of notation which will turn out to be useful in our lateranalysis. If T and A are two generic nonempty sets and Γ : T ⇒ A is a (nonemptyvalued) correspondence, we denote by Sel (Γ) the set of all selections of Γ, that is, theset of all functions γ : T → A such that γ ( t ) ∈ Γ ( t ) for all t ∈ T . Just for thissection, T is an arbitrary σ -algebra of subsets of T . Finally, given a T -measurablemap γ : T → A and a probability µ : T → [0 , µ γ = ( µ ( { t ∈ T | γ ( t ) = a } )) a ∈ A Lemma 1
Let ( T, T ) be a measurable space, A a finite set with n elements, and λ = ( λ , ..., λ m ) a vector of strongly continuous probabilities on T . If Γ : T ⇒ A is acorrespondence, then (cid:8) λ γ = (cid:0) λ γ , ..., λ mγ (cid:1) | γ ∈ Sel (Γ) and γ is T -measurable (cid:9) In the rest of the paper, T is the power set. s a convex subset of ∆ m . Proof. If φ, γ ∈ Sel (Γ) and are T -measurable, for each α ∈ (0 , ψ ∈ Sel (Γ) which is T -measurable and such that λ ψ = α λ φ + (1 − α ) λ γ .Set S ij = φ − ( i ) ∩ γ − ( j ) for all i, j ∈ A . Then { S ij } i,j ∈ A forms a partition of T (with possibly some empty elements) and all its elements belong to T , because φ − ( i ) , γ − ( j ) ∈ T for all i, j ∈ A .Since λ , ..., λ m are strongly continuous and T is a σ -algebra, for any i, j ∈ A ,there are T ij , U ij ∈ T such that S ij = T ij ⊔ U ij , λ ( T ij ) = α λ ( S ij ), and λ ( U ij ) =(1 − α ) λ ( S ij ). This is trivial if S ij is empty, else set T ij = T ∩ S ij λ kij ( S ) = λ k ( S ) ∀ S ∈ T ij , ∀ k = 1 , ..., m and notice that λ ij , ..., λ mij are strongly continuous, positive, and bounded charges onthe σ -algebra T ij . By Bhaskara Rao and Bhaskara Rao [7, Theorem 11.4.9], the set R ( λ ij ) = (cid:8)(cid:0) λ ij ( S ) , ..., λ mij ( S ) (cid:1) | S ∈ T ij (cid:9) is convex in R m . Moreover, both = (cid:0) λ ij ( ∅ ) , ..., λ mij ( ∅ ) (cid:1) and λ ij ( S ij ) = (cid:0) λ ij ( S ij ) , ..., λ mij ( S ij ) (cid:1) belong to R ( λ ij ). By convexity of the latter, there exists T ij ∈ T ij such that λ ij ( T ij ) = α λ ij ( S ij ). But then T ij , U ij = S ij \ T ij ∈ T , S ij = T ij ⊔ U ij , λ ( T ij ) = λ ij ( T ij ) = α λ ij ( S ij ) = α λ ( S ij ), and λ ( U ij ) = (1 − α ) λ ( S ij ) by additivity of λ .The function ψ : T → A defined by ψ ( t ) = ( φ ( t ) = i if t ∈ T ij γ ( t ) = j if t ∈ U ij is well defined and ψ ( t ) ∈ { φ ( t ) , γ ( t ) } ⊆ Γ ( t ) for all t ∈ T , so that ψ ∈ Sel (Γ). For ⊔ denotes the disjoint union. k ∈ A , ψ − ( k ) = { t ∈ T | ψ ( t ) = k } = ( t ∈ G i,j ∈ A T ij ! ⊔ G i,j ∈ A U ij ! | ψ ( t ) = k ) = G i,j ∈ A { t ∈ T ij | ψ ( t ) = k } ! ⊔ G i,j ∈ A { t ∈ U ij | ψ ( t ) = k } ! = G i,j ∈ A { t ∈ T ij | φ ( t ) = k } ! ⊔ G i,j ∈ A { t ∈ U ij | γ ( t ) = k } ! but, for all t ∈ T ij , φ ( t ) = i , then • if i = k , { t ∈ T ij | φ ( t ) = k } = T ij , • else i = k and { t ∈ T ij | φ ( t ) = k } = ∅ ,thus G i,j ∈ A { t ∈ T ij | φ ( t ) = k } = G i,j ∈ A | i = k T ij = G j ∈ A T kj ; analogously, for all t ∈ U ij , γ ( t ) = j ; then • if j = k , { t ∈ U ij | γ ( t ) = k } = U ij , • else j = k and { t ∈ U ij | γ ( t ) = k } = ∅ ,thus G i,j ∈ A { t ∈ U ij | γ ( t ) = k } = G i,j ∈ A | j = k U ij = G i ∈ A U ik ; therefore, ψ − ( k ) = G j ∈ A T kj ! ⊔ G i ∈ A U ik ! ∈ T
29s a consequence, ψ is T -measurable and, for each k ∈ A , and for each l = 1 , ..., m , λ l (cid:0) ψ − ( k ) (cid:1) = X j ∈ A λ l ( T kj ) + X i ∈ A λ l ( U ik ) = X j ∈ A αλ l ( S kj ) + X i ∈ A (1 − α ) λ l ( S ik )= αλ l G j ∈ A S kj ! + (1 − α ) λ l G i ∈ A S ik ! = αλ l G j ∈ A (cid:0) φ − ( k ) ∩ γ − ( j ) (cid:1)! + (1 − α ) λ l G i ∈ A (cid:0) φ − ( i ) ∩ γ − ( k ) (cid:1)! = αλ l φ − ( k ) ∩ G j ∈ A γ − ( j ) ! + (1 − α ) λ l γ − ( k ) ∩ G i ∈ A φ − ( i ) ! = αλ l (cid:0) φ − ( k ) ∩ T (cid:1) + (1 − α ) λ l (cid:0) γ − ( k ) ∩ T (cid:1) = αλ l (cid:0) φ − ( k ) (cid:1) + (1 − α ) λ l (cid:0) γ − ( k ) (cid:1) thus λ lψ = αλ lφ + (1 − α ) λ lγ . Since this is true for each l = 1 , ..., m , then λ ψ = α λ φ + (1 − α ) λ γ , as wanted. (cid:4) Building on this lemma, Gilboa, Maccheroni, Marinacci, and Schmeidler [15] provethat, when m = 1, { λ γ | γ ∈ Sel (Γ) and γ is T -measurable } is indeed the core of thebelief function Bel ( I ) = λ ( { t ∈ T | Γ ( t ) ⊆ I } ) ∀ I ⊆ A and they characterize its extreme points `a la Shapley [33].
A.2 Proofs and related material
In what follows and up to the proof of Corollary 1, we consider a nonatomic gamewith estimation feedback G = (( T, λ ) , A, u, (Π , π ) , f ). Recall that Π is a collectionof nonempty subsets of T , { T j } mj =1 , such that λ ( T j ) > j ∈ { , ..., m } and T = ∪ mj =1 T j . The proof of existence of ε -estimated equilibria rests on two key ideaswhich we formally develop below:1. We first consider different correspondences and study their properties. Thisstudy culminates with the correspondence g BR f,ε : ∆ m ⇒ ∆ m defined in (14)below. All of these correspondences are basically ε -consistent/confirmed best-reply correspondences. To fix ideas, for the case Π = { T } and m = 1, in words,given x ∈ ∆ and y ∈ g BR f,ε ( x ), y is a possible distribution of strategies in the30opulation, which arises if the players’ distribution of actions was x and playersbest-responded to it using a belief which was ε -consistent with respect to x .2. We then show that g BR f,ε has a fixed point by using Browder’s Fixed PointTheorem. This will give us the equilibrium in pure strategies that we are after.Consider ε >
0. First, let BR f,ε : T × ∆ m ⇒ A be defined by BR f,ε ( t, x )= (cid:8) b ∈ A | ∃ β t ∈ ∆ s.t. f t (cid:0) b, β t , x π ( t ) (cid:1) < ε and u t ( b, β t ) ≥ u t ( a, β t ) ∀ a ∈ A (cid:9) for all ( t, x ) ∈ T × ∆ m . Clearly, BR f,ε ( t, x ) is the set of all pure strategies whichare a best-reply of player t to some belief β t where β t is ε -consistent when assumingthe true distribution restricted to the subpopulation T π ( t ) is x π ( t ) . One can deriveseveral related “ ε -consistent best-reply” correspondences from this basic one. Foreach x ∈ ∆ m , denote the x -section BR f,ε ( · , x ) : T ⇒ A of BR f,ε by BR x f,ε . Next, letΦ f,ε : ∆ m ⇒ Σ be defined as Φ f,ε ( x ) = Sel (cid:0) BR x f,ε (cid:1) for all x ∈ ∆ m where Sel (cid:0) BR x f,ε (cid:1) is the set of all selections of BR x f,ε . Thus, for a strategy profile σ ∈ Σ, we have that[ σ ∈ Φ f,ε ( x )] ⇐⇒ (cid:2) ∀ t ∈ T, σ ( t ) ∈ BR x f,ε ( t ) (cid:3) ⇐⇒ [ ∀ t ∈ T, σ ( t ) ∈ BR f,ε ( t, x )] ⇐⇒ (cid:2) ∀ t ∈ T ∃ β t ∈ ∆ s.t. f t (cid:0) σ ( t ) , β t , x π ( t ) (cid:1) < ε and u t ( σ ( t ) , β t ) ≥ u t ( a, β t ) ∀ a ∈ A (cid:3) Remark 4
If there exists x ∈ ∆ m such that σ ∈ Φ f,ε ( x ) and x π ( t ) = λ π ( t ) σ for all t ∈ T , then σ is an ε -estimated equilibrium. In fact, we have For each t ∈ T there exists β t ∈ ∆ such that f t (cid:0) σ ( t ) , β t , x π ( t ) (cid:1) < ε and u t ( σ ( t ) , β t ) ≥ u t ( a, β t ) for all a ∈ A ; We can define β : T → ∆ by β ( t ) = β t for all t ∈ T .This implies that for each t ∈ T a) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) for all a ∈ A (optimality); b) f t (cid:16) σ ( t ) , β ( t ) , λ π ( t ) σ (cid:17) < ε (strict ε -consistency),31hat is, ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Af t (cid:16) σ ( t ) , β ( t ) , λ π ( t ) σ (cid:17) < ε ) = T . In particular, itholds that λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Af t (cid:16) σ ( t ) , β ( t ) , λ π ( t ) σ (cid:17) ≤ ε )! = 1 N Next, consider the correspondence B f,ε : ∆ m ⇒ Σ defined by B f,ε ( x ) = ( σ ∈ Σ | ∃ β ∈ ∆ T s.t. u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ A, ∀ t ∈ T sup t ∈ T f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) < ε ) ∀ x ∈ ∆ m Lemma 2 B f,ε ( x ) = S η ∈ (0 ,ε ) Φ f,η ( x ) ⊆ Φ f,ε ( x ) for all x ∈ ∆ m . Proof.
Fix x ∈ ∆ m . Consider σ ∈ S η ∈ (0 ,ε ) Φ f,η ( x ). It follows that σ ∈ Φ f,η ( x )for some η ∈ (0 , ε ). This implies that σ ∈ Σ and σ ( t ) ∈ BR f,η ( t, x ) for all t ∈ T ,that is, for each t ∈ T there exists β t ∈ ∆ such that f t (cid:0) σ ( t ) , β t , x π ( t ) (cid:1) < η and u t ( σ ( t ) , β t ) ≥ u t ( a, β t ) for all a ∈ A . In particular, if we define β ∈ ∆ T as β ( t ) = β t for all t ∈ T , we have that u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) for all a ∈ A and for all t ∈ T ,and sup t ∈ T f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) ≤ η < ε yielding that σ ∈ B f,ε ( x ). Conversely, if σ ∈ B f,ε ( x ), then there exists β ∈ ∆ T suchthat sup t ∈ T f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) < ε (13)and u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) for all a ∈ A and for all t ∈ T . It follows thatthere exists ¯ η ∈ (0 , ε ) such that (13) holds with ¯ η in place of ε . This implies that σ ∈ Φ f, ¯ η ( x ) ⊆ S η ∈ (0 ,ε ) Φ f,η ( x ).Obviously, if 0 < η < η ′ , then BR f,η ( t, x ) ⊆ BR f,η ′ ( t, x ) for all t ∈ T and for all x ∈ ∆ m and, in particular, Φ f,η ( x ) ⊆ Φ f,η ′ ( x ). This implies that S η ∈ (0 ,ε ) Φ f,η ( x ) ⊆ Φ f,ε ( x ). (cid:4) Remark 4 above will be useful to justify the following last correspondence: g BR f,ε :32 m ⇒ ∆ m defined by g BR f,ε ( x ) = (cid:8) y ∈ ∆ m | ∃ σ ∈ B f,ε ( x ) s.t. λ jσ = y j ∀ j ∈ { , ..., m } (cid:9) ∀ x ∈ ∆ m (14)In other words, g BR f,ε ( x ) is the collection of actions’ distributions y = ( y j ) mj =1 on thesubpopulations of players, which can be induced by the β -optimal choice of strategies σ where beliefs β = ( β t ) t ∈ T are close enough in terms of feedback to x = ( x j ) mj =1 . Notethat g BR f,ε ( x ) = n(cid:0) λ jσ (cid:1) mj =1 | σ ∈ B f,ε ( x ) o = (cid:0) λ jσ (cid:1) mj =1 | σ ∈ [ η ∈ (0 ,ε ) Φ f,η ( x ) (15)= [ η ∈ (0 ,ε ) n(cid:0) λ jσ (cid:1) mj =1 | σ ∈ Φ f,η ( x ) o = [ η ∈ (0 ,ε ) n(cid:0) λ jσ (cid:1) mj =1 | σ ∈ Sel (cid:0) BR x f,η (cid:1)o (16)An immediate implication of the definition in (14) is the next result. Lemma 3 If x ∈ g BR f,ε ( x ) , then there exists an ε -estimated equilibrium σ such that λ jσ = x j for all j ∈ { , .., m } . Proof.
By Lemma 2 and the definition of g BR f,ε , if x ∈ g BR f,ε ( x ), then there exists σ ∈ B f,ε ( x ) ⊆ Φ f,ε ( x ) such that λ jσ = x j for all j ∈ { , ..., m } . Remark 4 yields that σ is an ε -estimated equilibrium. (cid:4) Lemma 4 If λ is strongly continuous, then g BR f,ε ( x ) is nonempty and convex for all x ∈ ∆ m . Proof.
Fix x ∈ ∆ m and η ∈ (0 , ε ). Since f satisfies (1), recall that ∀ t ∈ T, ∀ z ∈ ∆ , ∃ γ t,z ∈ ∆ s.t. ∀ a ∈ A f t ( a, γ t,z , z ) = 0 (17)Since x is given, define β ∈ ∆ T to be such that β ( t ) = γ t,x π ( t ) for all t ∈ T . Note that β ( t ) ∈ ∆ satisfies f t (cid:0) a, β ( t ) , x π ( t ) (cid:1) = 0 < η for all a ∈ A and for all t ∈ T . Since A is finite, for each t ∈ T choose σ ( t ) ∈ A such that u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) forall a ∈ A . This defines a function σ : T → A , that is σ ∈ Σ, such that σ ∈ Φ f,η ( x ).By Lemma 2, we have that σ ∈ B f,ε ( x ) and ( λ jσ ) mj =1 ∈ g BR f,ε ( x ). Convexity is aconsequence of the following two observations:33. By Lemma 1 and since each λ j is strongly continuous, recall that { ( λ jσ ) mj =1 | σ ∈ Sel (cid:0) BR x f,η (cid:1) } is a convex subset of ∆ m for all η ∈ (0 , ε ).2. By (16), we have that g BR f,ε ( x ) = [ η ∈ (0 ,ε ) n(cid:0) λ jσ (cid:1) mj =1 | σ ∈ Sel (cid:0) BR x f,η (cid:1)o It follows that g BR f,ε ( x ) is the union of a chain of convex sets, proving convexity. (cid:4) For the next result recall that a) d ∆ is the distance on ∆ induced by the Euclideannorm; b) we say that f = ( f t ) t ∈ T is a family of functions which is equicontinuous withrespect to the third argument if and only if for each ε > δ ε > d ∆ ( x, y ) < δ ε = ⇒ | f t ( a, γ, x ) − f t ( a, γ, y ) | < ε ∀ t ∈ T, ∀ a ∈ A, ∀ γ ∈ ∆The intuition behind the proof of the next lemma is that if a strategy σ was β -optimaland β was ε -consistent, given x , small perturbations of x do not disrupt optimalityand ε -consistency. Lemma 5 If f = ( f t ) t ∈ T is a family of functions which is equicontinuous with respectto the third argument, then g BR − f,ε ( y ) is open for all y ∈ ∆ m . Proof.
Fix y ∈ ∆ m . Recall that g BR − f,ε ( y ) = n x ∈ ∆ m | y ∈ g BR f,ε ( x ) o . Note that x ∈ g BR − f,ε ( y ) ⇐⇒ y ∈ g BR f,ε ( x )and g BR − f,ε ( y ) is open if and only if “for each ¯x such that y ∈ g BR f,ε ( ¯x ), there existsa ball in ∆ m of radius δ and center ¯x such that y ∈ g BR f,ε ( x ) for all x ∈ B δ ( ¯x )”.Now arbitrarily choose ¯x such that y ∈ g BR f,ε ( ¯x ). By definition of g BR f,ε ( ¯x ), thereexist σ ∈ B f,ε ( ¯x ) ⊆ Σ and β ∈ ∆ T such that1. λ jσ = y j for all j ∈ { , ..., m } ; Recall that if 0 < η < η ′ , then BR f,η ( t, x ) ⊆ BR f,η ′ ( t, x ) ∀ t ∈ T, ∀ x ∈ ∆ m This implies that Sel (cid:0) BR x f,η (cid:1) = Φ f,η ( x ) ⊆ Φ f,η ′ ( x ) = Sel (cid:0) BR x f,η ′ (cid:1) for all x ∈ ∆ m .
34. sup t ∈ T f t (cid:0) σ ( t ) , β ( t ) , ¯ x π ( t ) (cid:1) < ε ;3. u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) for all a ∈ A and for all t ∈ T .By point 2, there exists ¯ ε ∈ (0 , ε ) such thatsup t ∈ T f t (cid:0) σ ( t ) , β ( t ) , ¯ x π ( t ) (cid:1) < ¯ ε < ε Let ˆ ε ∈ (cid:0) , ε − ¯ ε (cid:1) . Since f = ( f t ) t ∈ T is a family of functions which is equicontinuouswith respect to the third argument, there exists δ ˆ ε > d ∆ ( x, y ) < δ ˆ ε = ⇒ | f t ( a, γ, x ) − f t ( a, γ, y ) | < ˆ ε ∀ t ∈ T, ∀ a ∈ A, ∀ γ ∈ ∆For each x ∈ B δ ˆ ε ( ¯x ) note that d ∆ ( x j , ¯ x j ) < δ ˆ ε for all j ∈ { , ..., m } . This implies thatfor each t ∈ T and for each x ∈ B δ ˆ ε ( ¯x ) (cid:12)(cid:12) f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) − f t (cid:0) σ ( t ) , β ( t ) , ¯ x π ( t ) (cid:1)(cid:12)(cid:12) < ˆ ε Since f t ≥ t ∈ T , it follows that for each t ∈ T and for each x ∈ B δ ˆ ε ( ¯x ) f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) = (cid:12)(cid:12) f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1)(cid:12)(cid:12) ≤ (cid:12)(cid:12) f t (cid:0) σ ( t ) , β ( t ) , ¯ x π ( t ) (cid:1)(cid:12)(cid:12) + (cid:12)(cid:12) f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) − f t (cid:0) σ ( t ) , β ( t ) , ¯ x π ( t ) (cid:1)(cid:12)(cid:12) = f t (cid:0) σ ( t ) , β ( t ) , ¯ x π ( t ) (cid:1) + (cid:12)(cid:12) f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) − f t (cid:0) σ ( t ) , β ( t ) , ¯ x π ( t ) (cid:1)(cid:12)(cid:12) < ¯ ε + ˆ ε This implies thatsup t ∈ T f t (cid:0) σ ( t ) , β ( t ) , x π ( t ) (cid:1) ≤ ¯ ε + ˆ ε < ¯ ε + ε < ε ∀ x ∈ B δ ˆ ε ( ¯x )In other words, for each x ∈ B δ ˆ ε ( ¯x ) we have that σ ∈ Σ is such that the same β ∈ ∆ T of above satisfies points 2 and 3, but with x in place of ¯x . This yields that σ ∈ B f,ε ( x )for all x ∈ B δ ˆ ε ( ¯x ). Since y = ( y j ) mj =1 = ( λ jσ ) mj =1 , we obtain that y ∈ g BR f,ε ( x ) for all x ∈ B δ ˆ ε ( ¯x ), proving the statement. (cid:4) Proof of Theorem 1 . By Lemma 3, it is enough to show that g BR f,ε : ∆ m ⇒ ∆ m hasa fixed point. Clearly, ∆ m ⊆ ( R n ) m is nonempty, compact, and convex. By Lemmas 4and 5, g BR f,ε has nonempty and convex values and g BR − f,ε ( y ) is open for all y ∈ ∆ m .35y Browder’s Fixed Point Theorem for correspondences (see Theorem 1 of Browder[8]), g BR f,ε has a fixed point. (cid:4) We next prove the remaining results of the main text.
Proof of Corollary 1 . It is enough to observe that a nonatomic game with message feedback can be mapped into a nonatomic game with estimation feedback where f isdefined as in (4) and Π = { T } . With this identification, an ε -estimated equilibriumis a self-confirming ε -equilibrium. By Theorem 1, it is then enough to show that f = ( f t ) t ∈ T is a family of functions which is equicontinuous with respect to the thirdargument. Since m = ( m t ) t ∈ T is a family of functions which is equicontinuous withrespect to the second argument, we have that for each ε > δ ε > d ∆ ( x, y ) < δ ε = ⇒ d ( m t ( a, x ) , m t ( a, y )) < ε ∀ t ∈ T, ∀ a ∈ A (18)Since for each t ∈ T we have that f t ( a, x, y ) = d ( m t ( a, x ) , m t ( a, y )) for all a ∈ A andfor all x, y ∈ ∆, observe that | f t ( a, γ, x ) − f t ( a, γ, y ) | = | d ( m t ( a, γ ) , m t ( a, x )) − d ( m t ( a, γ ) , m t ( a, y )) |≤ d ( m t ( a, x ) , m t ( a, y )) ∀ t ∈ T, ∀ a ∈ A, ∀ x, y, γ ∈ ∆By (18), we can conclude that for each ε > δ ε > d ∆ ( x, y ) < δ ε = ⇒| f t ( a, γ, x ) − f t ( a, γ, y ) | ≤ d ( m t ( a, x ) , m t ( a, y )) < ε ∀ t ∈ T, ∀ a ∈ A, ∀ γ ∈ ∆proving equicontinuity with respect to the third argument of f . (cid:4) Proof of Corollary 2 . Consider the nonatomic game G = (( T, λ ) , A, u ) and ε > u = ( u t ) t ∈ T is a family of functions which is equicontinuous with respect to thesecond argument, we have that for each ˆ ε > δ ˆ ε > d ∆ ( x, y ) < δ ˆ ε = ⇒ | u t ( a, x ) − u t ( a, y ) | < ˆ ε ∀ t ∈ T, ∀ a ∈ A (19)Consider the profile m = ( m t ) t ∈ T of message functions such that each m t : A × ∆ → ∆is defined to be such that m t ( a, x ) = x ∀ a ∈ A, ∀ x ∈ ∆ Thus, m = 1, T = T , and π ( t ) = 1 for all t ∈ T . M, d ) = (∆ , d ∆ ). Clearly, m = ( m t ) t ∈ T is a family of functionswhich is equicontinuous with respect to the second argument. Given ε >
0, consider δ ε/ > δ ε/ / σ ∈ Σ, that is, there exists β ∈ ∆ T such that1 = λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ( m t ( σ ( t ) , β ( t )) , m t ( σ ( t ) , λ σ )) ≤ δ ε/ / )! = λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ∆ ( β ( t ) , λ σ ) ≤ δ ε/ / )! Define by O the set of “optimizing” players O = ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ Ad ∆ ( β ( t ) , λ σ ) ≤ δ ε/ / ) Since u satisfies (19), note that if t ∈ O , then we have that d ∆ ( β ( t ) , λ σ ) ≤ δ ε/ / <δ ε/ which implies that for each a ∈ A | u t ( σ ( t ) , β ( t )) − u t ( σ ( t ) , λ σ ) | < ε | u t ( a, β ( t )) − u t ( a, λ σ ) | < ε t ∈ O , we can conclude that u t ( σ ( t ) , λ σ ) > u t ( σ ( t ) , β ( t )) − ε ≥ u t ( a, β ( t )) − ε > u t ( a, λ σ ) − ε − ε u t ( a, λ σ ) − ε ∀ a ∈ A Since t was arbitrarily chosen in O , we have that O ⊆ { t ∈ T | u t ( σ ( t ) , λ σ ) ≥ u t ( a, λ σ ) − ε ∀ a ∈ A } Since O has mass 1, it follows that σ ∈ Σ is an ε -Nash equilibrium. (cid:4) Proof of Corollary 3 . It is enough to observe that a nonatomic game with a neighbor-hood structure can be mapped into a nonatomic game with estimation feedback where f is defined as in (8). With this identification, an ε -estimated equilibrium is a peer-confirming ε -equilibrium. By Theorem 1, it is then enough to show that f = ( f t ) t ∈ T is a family of functions which is equicontinuous with respect to the third argument.37ut, note that | f t ( a, γ, x ) − f t ( a, γ, y ) | = | d ∆ ( γ, x ) − d ∆ ( γ, y ) | ≤ d ∆ ( x, y ) ∀ t ∈ T, ∀ a ∈ A, ∀ γ ∈ ∆trivially proving equicontinuity with respect to the third argument of f . (cid:4) We conclude by proving Corollary 4. But, before doing so, we need to makean intermediate observation. Consider a statistical divergence D . Recall that D :∆ × ∆ o → [0 , ∞ ) is a jointly convex and continuous function. Denote by K thecollection of all nonempty compact sets of ∆. We endow K with the Hausdorff distance(see, e.g., Aliprantis and Border [1, Chapter 3, Sections 16 and 17]). We denote by ¯ Q a compact set of K such that each Q ∈ ¯ Q is a nonempty, convex, and compact subsetof ∆ o . Given x ∈ ∆ and Q ∈ ¯ Q , consider the minimization problemmin D ( x || y ) sub to y ∈ Q Define µ : ∆ × ¯ Q ⇒ ∆ to be the solution correspondence of this minimization problem,that is, for each x ∈ ∆ and for each Q ∈ ¯ Q , µ ( x, Q ) = (cid:26) z ∈ ∆ : z ∈ Q and D ( x || z ) = min y ∈ Q D ( x || y ) (cid:27) By Berge’s maximum theorem, note that µ is upper hemicontinuous when ∆ × ¯ Q isendowed with the product topology. In particular, if D is strictly convex with respectto the second argument, µ is single-valued, that is, µ is a continuous function. Finally,define the map g : ∆ × ∆ × ¯ Q → [0 , ∞ ) by g ( β, x, Q ) = d ∆ ( β, µ ( x, Q )) ∀ β, x ∈ ∆ , ∀ Q ∈ ¯ Q Since µ is a continuous function, it follows that g is continuous when ∆ × ∆ × ¯ Q isendowed with the product topology. By Aliprantis and Border [1, Corollary 3.31] andsince ∆ × ∆ × ¯ Q is a compact metric space, g is uniformly continuous. Proof of Corollary 4 . Set ¯ Q = cl Q . By point 4 of Remark 3, note that ¯ Q is acompact subset of K such that each Q ∈ ¯ Q is a nonempty, convex, and compact subsetof ∆ o . For each t ∈ T define f t : A × ∆ × ∆ → [0 , ∞ ) by f t ( a, γ, x ) = g ( γ, x, Q t ) ∀ a ∈ A, ∀ γ, x ∈ ∆ (20)38t is then enough to observe that a nonatomic game with model misspecification can bemapped into a nonatomic game with estimation feedback where f is defined as in (20)and Π = { T } . With this identification, an ε -estimated equilibrium is an ε -BNE. ByTheorem 1, it is then enough to show that f = ( f t ) t ∈ T is a family of functions whichis equicontinuous with respect to the third argument. Since g is uniformly continuous,the statement is trivially true. (cid:4) The proof of the last two points of Remark 3 is routine. Thus, we conclude by onlyproving point 2.
Proof of point 2 of Remark 3 . Set ¯ Q = cl Q . Note that ¯ Q is a compact subset of K such that each Q ∈ ¯ Q is a nonempty, convex, and compact subset of ∆ o . For each t ∈ T define f t : A × ∆ × ∆ → [0 , ∞ ) as in the proof of Corollary 4, that is, f t ( a, γ, x ) = g ( γ, x, Q t ) ∀ t ∈ T, ∀ a ∈ A, ∀ γ, x ∈ ∆Since g is continuous and ∆ × ∆ × ¯ Q is compact, observe that g ≥ M ≥
0. Define the profile of feedback functions ˜ f to be such that for each t ∈ T ˜ f t ( a, γ, x ) = ( f t ( a, γ, x ) γ ∈ Q t M + 1 γ Q t ∀ a ∈ A, ∀ γ, x ∈ ∆Note that each ˜ f t satisfies (1). By the proof of Corollary 4, f = ( f t ) t ∈ T is a family offunctions which is equicontinuous with respect to the third argument. It follows thatfor each ε > δ ε > d ∆ ( x, y ) < δ ε = ⇒ | f t ( a, γ, x ) − f t ( a, γ, y ) | < ε ∀ t ∈ T, ∀ a ∈ A, ∀ γ ∈ ∆Consider x, y ∈ ∆ such that d ∆ ( x, y ) < δ ε and consider t ∈ T , a ∈ A , and γ ∈ ∆. Wehave two cases, either γ ∈ Q t or γ Q t . In the first case, (cid:12)(cid:12)(cid:12) ˜ f t ( a, γ, x ) − ˜ f t ( a, γ, y ) (cid:12)(cid:12)(cid:12) = | f t ( a, γ, x ) − f t ( a, γ, y ) | < ε , while in the second case (cid:12)(cid:12)(cid:12) ˜ f t ( a, γ, x ) − ˜ f t ( a, γ, y ) (cid:12)(cid:12)(cid:12) = | M + 1 − ( M + 1) | = 0 < ε . Since t , a , and γ were chosen arbitrarily, it followsthat ˜ f = (cid:16) ˜ f t (cid:17) t ∈ T is a family of functions which is equicontinuous with respect to thethird argument. Next, we can consider the nonatomic game with estimation feedback (cid:16) ( T, λ ) , A, u, (Π , π ) , ˜ f (cid:17) , where Π = { T } . By Theorem 1, we have that for each Thus, m = 1, T = T , and π ( t ) = 1 for all t ∈ T . Thus, m = 1, T = T , and π ( t ) = 1 for all t ∈ T . ε > ε -estimated equilibrium σ for this game, that is, there exists β ∈ ∆ T such that λ ( t ∈ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t ( σ ( t ) , β ( t )) ≥ u t ( a, β ( t )) ∀ a ∈ A ˜ f t ( σ ( t ) , β ( t ) , λ σ ) ≤ ˜ ε )! = 1If given ε > ε = min { M +1 ,ε } >
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