BATH ECONOMICS RESEARCH PAPERS

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1 Distributional Perfect Equilibrium in Bayesian Games with Applications to Auctions Elnaz Bajoori Dries Vermeulen No. 15 /13R BATH ECONOMICS RESEARCH PAPERS Department of Economics

2 Distributional Perfect Equilibrium in Bayesian Games with Applications to Auctions ELNAZ BAJOORI DRIES VERMEULEN December 27, 2017 Abstract In second-price auctions with interdependent values, bidders do not necessarily have dominant strategies. Moreover, such auctions may have many equilibria. In order to rule out the less intuitive equilibria, we define the notions of distributional perfect and strong distributional perfect equilibria for Bayesian games with infinite type and action spaces. We prove that every Bayesian game has a distributional perfect equilibrium provided that the information structure of the game is absolutely continuous and the payoffs are continuous in actions for every type. We apply strong distributional perfection to a class of symmetric second-price auctions with interdependent values and show that the efficient equilibrium defined by Milgrom [22] is strongly distributionally perfect, while a class of less intuitive, inefficient, equilibria introduced by Birulin [11] is not. JEL Codes. C72. Keywords. Trembling hand perfect equilibrium, Bayesian games with infinite type and action spaces, Second-price auctions with interdependent values. The authors would like to thank Philip J. Reny and János Flesch for their very helpful and constructive comments. Corresponding author, e.bajoori@bath.ac.uk. University of Bath, Dept. of Economics, Bath, BA2 7AY, United Kingdom. d.vermeulen@maastrichtuniversity.nl. Maastricht University, School of Business and Economics, Dept. of Quantitative Economics, Maastricht, The Netherlands., P.O. Box 616, 6200 MD Maastricht, The Netherlands.

3 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 1 1 Introduction In private value second-price auctions each bidder has a dominant strategy in which he bids his own valuation of the object. However, in second-price auctions with interdependent values bidders may not have dominant strategies while there may exist multiple equilibria in undominated strategies. For example Birulin [11] shows that any auction with an efficient ex-post equilibrium has a continuum of inefficient undominated ex-post equilibria. Therefore, a selection tool is needed in such Bayesian games to rule out the less intuitive equilibria. We develop a concept of trembling hand perfect equilibrium for Bayesian games with infinite type and action spaces, called distributional perfect equilibrium. We prove two main results for distributional perfection. First, we prove existence of distributionally perfect BNE in Bayesian games for which the type space of each player is a separable metric space, the action space of each player is a compact metric space, and player types are drawn from a prior probability measure on the product of the type spaces that is absolutely continuous with respect to the product measure of its marginal probabilities. 1 Second, we apply a stronger notion of distributional perfection, called strong distributional perfect equilibrium, to second price auctions with interdependent values in order to select among equilibria. We show that the efficient equilibrium introduced by Milgrom [22] is strongly distributionally perfect. Moreover, our refinement rules out the inefficient equilibria introduced by Birulin [11]. Related literature. The literature of perfect equilibrium for infinite games started by Simon and Stinchcombe [26], who proposed two different approaches to extend perfect equilibrium to games with infinite action spaces. The first approach is a generalization of the definition of Selten, based on the notion of completely mixed strategy. The second approach, referred to as the finitistic approach, uses finite approximations of the original game to define perfect equilibrium. Bajoori et al. [4] examine these two approaches in further detail and provide an improved version of the finitistic definition. Bajoori et al. [5] define perfect equilibrium for Bayesian games, based on behavioral strategies. 1 This property is called absolute continuity of information in Milgrom and Weber [23].

4 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 2 For further studies on the existence of equilibrium in Bayesian games, we refer to Athey [2], Mcadams [18], Reny [24], Jackson et al. [15],[16], Mertens [21], Meirowitz [19], Milgrom and Weber [23], Mallozzi et al. [20], Kim and Yannelis [27], Zandt and Vives [29], and Balder [6]. The paper is structured as follows. In Section 2 we describe the model. In section 3 we introduce the concept of distributionally perfect BNE and strong distributionally perfect BNE, and show existence of distributionally perfect BNE. In section 4 the concept of strong distributional perfection is applied to a class of second-price auctions with interdependent values as a tool to select the efficient equilibrium. 2 The model A Bayesian game Γ is defined as follows. There are n players. For each player i, the set T i of types is a complete separable metric space, and the set A i of actions is a compact metric space. Let d Ti and d Ai denote the respective metrics, and let T i and A i denote the induced Borel σ-fields on T i and A i respectively. Moreover, let T = n i=1t i and A = n i=1a i. Let µ be a probability measure on the product σ-field T = n i=1t i with marginal probability µ i on T i for every player i. We assume that each µ i is a completely mixed probability measure. 2 We also assume that the measure µ is absolutely continuous 3 with respect to ˆµ = n i=1µ i. 4 Player i s payoff function π i : T A IR is assumed to be bounded and jointly measurable. Further, payoffs are assumed to be continuous in actions for every type, that is, π i (t, ) : A IR is continuous for every t T. 2 A probability measure is completely mixed if it assigns strictly positive weight to every nonempty open set. 3 Let µ and ν be two measures on a σ-field Σ. The measure µ is absolutely continuous with respect to ν if for every G Σ, ν(g) = 0 implies µ(g) = 0. 4 In Milgrom and Weber [23] this property is called Absolutely Continuous Information. Absolutely continuous information is a rather weak assumption. It is satisfied when the type spaces are finite, or when each player s type is drawn independently from his type space.

5 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES Strategies and equilibrium In this section, we define two classes of strategies: behavioral strategies and distributional strategies. We also define Bayesian Nash Equilibrium (BNE). A pure strategy for player i in Γ is a measurable function p i : T i A i. A behavioral strategy for player i is a function β i : T i A i [0, 1] such that [1] the section function β i (t i, ) : A i [0, 1] is a probability measure for every t i T i, and [2] the section function β i (, B) : T i [0, 1] is measurable for every B A i. When player i plays according to the behavioral strategy β i, for each type t i T i, he chooses his action according to the probability measure β i (t i, ). A behavioral strategy β i for player i is called pure if there is a pure strategy p i for player i such that β i (t i, ) = D pi (t i )( ) for every type t i T i, where D denotes Dirac measure. For simplicity, we denote this pure behavioral strategy by p i. Following Milgrom and Weber [23], a distributional strategy for player i in a Bayesian game is a probability measure γ i on T i A i such that the marginal distribution of γ i on T i equals the probability measure µ i. That is, γ i (U A i ) = µ i (U) for every U T i. Given a behavioral strategy β i, the induced distributional strategy γ i is uniquely determined by γ i (U B) = β i (t i, B) µ i (dt i ) U for every rectangle U B T i A i. This induces a many-to-one mapping from behavioral strategies to distributional strategies that preserves the players ex ante expected payoffs. The vector β = (β 1, β 2,..., β n ), where β i is player i s behavioral strategy, is called a behavioral strategy profile. For every player i we write β i = (β j ) n i j=1 to denote the behavioral strategy profile for his opponents. The definitions are similar for distributional strategies. Let i be the set of all distributional strategies for player i and = n i=1 i. Let Π i : IR be the expected payoff of player i playing distributional strategy γ i

6 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 4 against a distributional strategy profile γ i. So, Π i (γ i, γ i ) = π i (t, a) γ i (da i t i )γ i (da i t i ) µ(dt), where t = (t 1,..., t n ) T, a = (a 1,..., a n ) A, and γ i (da i t i ) is a version of conditional probability on A i given t i. By integrals with respect to γ i (da i t i ) we mean the iterated integrals with respect to γ j (da j t j ) for all j i. As the measure µ is absolutely continuous with respect to ˆµ, by the Radon- Nikodym theorem there is a measurable function f : T IR such that for every measurable set G T we have µ(g) = fdˆµ. Therefore, the expected payoff Π i can be expressed in an easier form as Π i (γ i, γ i ) = π i (t, a)f(t) dγ i dγ i. Let β i be a behavioral strategy corresponding to the distributional strategy γ i, for every player i. Then, by Theorem in Dudley [12] 5 we conclude that Π i (γ i, γ i ) = π i (t, a)f(t) β i (t i, da i ) µ i (dt i ) β i (t i, da i ) µ i (dt i ). By Π b i(β i, β i t i ) we denote player i s expected payoff given his behavioral strategy β i and his type t i against a behavioral strategy profile β i. Thus, Π b i(β i, β i t i ) = π i (t, a)f(t) β i (t i, da i ) β i (t i, da i ) µ i (dt i ). G Furthermore, notice that Π i (γ 1,..., γ n ) = Π b i(β i, β i t i ) µ i (dt i ). In the special case where player i chooses a pure behavioral strategy p i, his expected payoff is Π b i(p i, β i t i ) = π i (t, (p i (t i ), a i )) f(t) β i (t i, da i ) µ i (dt i ). 5 Dudley s notation and terminology in [12] differs from ours. Part (II) of Theorem in Dudley [12] with our terminology can be expressed as follows: For every measurable function g : T i A i IR we have g dγ i = T i A i g β i (t i, da i ) µ i (dt i ). A i T i

7 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 5 One can also define Π b i(σ i, β i t i ) and Π b i(a i, β i t i ) respectively for every probability measure σ i on A i and every action a i A i. A probability measure σ i on A i is called a best response of player i for type t i T i against a behavioral strategy profile τ i, if for every probability measure σ i on A i we have Π b i(σ i, τ i t i ) Π b i(σ i, τ i t i ). The set of such best responses is denoted by BR i (t i, τ). A behavioral strategy β i is called a best response of player i against the behavioral strategy profile τ i, if β i (t i, ) BR i (t i, τ) for every t i T i. A behavioral strategy profile β = (β 1,..., β n ) is a Bayesian Nash Equilibrium (BNE) if β i is a best response to β i for every player i. A distributional strategy γ i of player i is a best response against a distributional strategy profile η i, if Π i (γ i, η i ) Π i ( γ i, η i ), for every distributional strategy γ i of player i. Let BR i : i denote the best response correspondence for player i on the set of distributional strategies. Moreover, define BR :, for every γ, as BR(γ) = n i=1br i (γ). We say that a behavioral strategy β i (weakly) dominates behavioral strategy ˆβ i if, for their respective induced distributional strategies γ i and ˆγ i, it holds that Π i (γ i, γ i ) Π i (ˆγ i, γ i ) for all γ i i, and with a strict inequality for at least one γ i i. A behavioral strategy β i is called (weakly) dominant if it dominates every other behavioral strategy of player i. A behavioral strategy β i is undominated if there is no other behavioral strategy that dominates β i. 3 Distributional perfection In this section, we define perfect equilibrium for Bayesian games in terms of behavioral strategies. Since only the induced distribution on opponents actions matter for computation of expected payoffs, we use distributional strategies as a tool to verify whether the behavioral strategy profile satisfies the conditions of trembling hand perfect equilibrium.

8 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 6 Let C b (T i A i ) be the set of all continuous and bounded functions with domain T i A i. The weak topology on i is the coarsest topology for which all the maps γ i h dγ i, h C b (T i A i ) are continuous. Since T i A i is a separable metric space, the weak topology on i is separable and metrizable with the weak (Prohorov) metric 6, denoted by ρ w. Definition 3.1 A behavioral strategy profile β = (β 1,..., β n ) is called distributional perfect if for the induced distributional strategy profile γ = (γ 1,..., γ n ) there exists a sequence (γ k ) k=1 = (γk 1,..., γn) k k=1 of completely mixed distributional strategy profiles such that for every player i it holds that (i) lim ρ w (γi k, γ i ) = 0, k (ii) lim k ρ w (γ k i, BR i (γ k )) = 0. A behavioral strategy profile is a distributional perfect BNE if it is both distributional perfect and a BNE. In the next theorem we show the existence of distributional perfect BNE. The proof can be found in the appendix and it is based on the existence proof in Simon and Stinchcombe [26] and Milgrom and Weber [23]. Theorem 3.2 The set of distributional perfect BNEs is nonempty. We also introduce a stronger notion of distributional perfection, called strong distributional perfection. Definition 3.3 A behavioral strategy profile β = (β 1,..., β n ) is called strongly distributionally perfect if, given the induced profile γ = (γ 1,..., γ n ) of distributional strategies, for every completely mixed strategy profile ν = (ν 1,..., ν n ) and every sequence (ε k i ) k=1 of real numbers converging to zero, the sequence (γi k ) k=1 defined by γk i = (1 ε i k ) γ i + ε k i ν i for every i is such that lim k ρw (γi k, BR i (γ k )) = 0, 6 See Aliprantis and Border [1]. Let µ and ν be two probability measures on a σ-field Σ. Weak (Prohorov) metric is defined for µ and ν by ρ w (µ, ν) = inf{ε > 0 B Σ : µ(b) < ν(b ε ) + ε and ν(b) < µ(b ε ) + ε}.

9 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 7 in which γ k = (γ1 k,..., γn). k A behavioral strategy profile is a strong distributional perfect BNE if it is both strongly distributionally perfect and a BNE. Clearly strong distributional perfection implies distributional perfection. 4 Second-price auctions In this section we study symmetric second-price auctions with interdependent values for two bidders and apply strong distributional perfection in this context. Let Λ be the following sealed-bid second-price auction for a single indivisible object. There are 2 bidders. Prior to bidding, each bidder i receives a private signal t i T i = [t, t], which is called the type of bidder i. Signals are drawn independently according to a distribution µ i. Bidder i s valuation of the object may depend on both types and is denoted by v i (t 1, t 2 ). Each bidder i, after observing his own type, submits a bid from a set A i independently of his opponent, where A i is sufficiently large in the sense that it contains the range of v i. Given the bids submitted by the bidders, the highest bid wins the auction for the price equal to the second highest bid. 7 Suppose the following assumptions in this auction: [1] v 1 (t 1, t 2 ) = v 2 (t 2, t 1 ). [2] v i is continuously differentiable. [3] v i t i (t 1, t 2 ) > v i t j (t 1, t 2) 0 for all t 1, t 2, t 1, t 2. Condition [3], without the non-negativity requirement, is a strong form of the single crossing condition (SCC). It is for example satisfied by linear valuation functions v 1 (t 1, t 2 ) = a t 1 + b t 2 for non-negative numbers a, b with a > b. Condition [3] has several important consequences, of which we mention three. First, it implies that v i is strictly increasing in t i and increasing in t j for j i. Second, in combination with condition [1] it ensures that the bidder with the higher type has the higher ex post valuation. 8 Third, the pair of strategies 7 In the event of a tie for the highest bid, the winner is chosen according to a probability distribution which may depend on the identity of the highest bidder and his type as well. In our setting, a tie will only occur with probability 0, even given the type of one bidder, which makes the specification of this tie-breaking rule irrelevant for the expected profits. 8 Thus, common value auctions do not satisfy SSC.

10 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 8 (b 1, b 2 ) in which b i = v i (t i, t i ) for every bidder i, is an efficient equilibrium. 9 More information on SCC can be found in for example Krishna [17]. In the auction Λ the set of available bids for bidder i is an interval A i = [a, a], where v i (t, t) = a and v i (t, t) = a, for i = 1, 2. In this auction bidders do not have any dominant strategy (disregarding the classic case of private values for a moment), and there are many equilibria. In the next two subsections we show how strong distributional perfection can be used to refine among these equilibria. We show that the efficient equilibrium introduced in Milgrom [22] is a strong distributionally perfect BNE. In contrast with this, we show that within the class of inefficient equilibria introduced in Birulin [11] none of them are strongly distibutionally perfect. 4.1 Equilibrium refinement I: equilibrium selection Milgrom [22] introduced the following class of strategy pairs. Let φ : [t, t] [t, t] be an increasing bijection. Define the strategy pair (b 1, b 2 ) by b 1 (t 1 ) = v 1 (t 1, φ(t 1 )) and b 2 (t 2 ) = v 2 (φ 1 (t 2 ), t 2 ). In Proposition 4.1 we show that within this class of continuous strategy pairs only one focal strategy pair is a BNE. In Proposition 4.2 we show that this focal BNE is strongly distributionally perfect. We also argue that the BNE is not in dominant strategies (unless we are considering the special case of pure private values). Thus, dominance cannot be used as a means to refine among equilibria in this context. Proposition 4.1 Let (b 1, b 2 ) be the strategy pair defined above. Then, (b 1, b 2 ) is a BNE if and only if φ(t) = t. Proof. A. Suppose that φ(t) = t for every t [t, t]. We show that (b 1, b 2 ) is a BNE. Define w(t) = v 1 (t, t) for every t [t, t]. Clearly w is an increasing bijection from [t, t] to [a, a]. Suppose that bidder 1 uses bid function b 1 (t 1 ) = v 1 (t 1, t 1 ). The expected payoff of bidder 2 having type t 2 and bidding p 2 is: Π b 2(b 1, p 2 t 2 ) = w 1 (p 2 ) (v 2 (t 1, t 2 ) v 1 (t 1, t 1 )) µ 1 (dt 1 ). t 9 An outcome in the auction is called efficient if the winner is the one with the highest ex post valuation.

11 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 9 As v 2 is strictly increasing in the second argument, and moreover v 2 (t 1, t 2 ) v 1 (t 1, t 1 ) = v 2 (t 1, t 2 ) v 2 (t 1, t 1 ), it is clear that v 2 (t 1, t 2 ) v 1 (t 1, t 1 ) 0 precisely when t 1 t 2. So, the maximum of Π b i(b 1, p 2 t 2 ) is obtained by choosing p 2 such that w 1 (p 2 ) = t 2. Hence, the optimal bidding strategy is b 2 (t 2 ) = w(t 2 ) = v 1 (t 2, t 2 ) = v 2 (t 2, t 2 ). B. Conversely, suppose that φ(t) t for some t [t, t]. We show that (b 1, b 2 ) is not a BNE. If φ(t) > t. Then v 2 (t, φ(t)) > v 2 (φ(t), t) by assumption [3] in our auction model. So, v 2 (t, φ(t)) v 2 (φ(t), t). A similar argument applies for the case φ(t) < t. Thus, there is a type ˆt 2 with v 2 (ˆt 2, φ(ˆt 2 )) v 2 (φ(ˆt 2 ), ˆt 2 ). Write t 2 = φ(ˆt 2 ). We first argue that t 2 φ 1 (t 2). Suppose that t 2 = φ 1 (t 2). Then φ(t 2) = t 2 = φ(ˆt 2 ), so that ˆt 2 = t 2 = φ(ˆt 2 ). Then v 2 (ˆt 2, φ(ˆt 2 )) = v 2 (φ(ˆt 2 ), ˆt 2 ). This contradicts our assumption. Now suppose that bidder 1 uses bid function b 1 (t 1 ) = v 1 (t 1, φ(t 1 )). The expected payoff of bidder 2 having type t 2 and bidding p 2 is: Π b 2(b 1, p 2 t 2) = w 1 (p 2 ) Define f(t 1 ) = v 2 (t 1, t 2) v 1 (t 1, φ(t 1 )). Since it follows that t (v 2 (t 1, t 2) v 1 (t 1, φ(t 1 ))) µ 1 (dt 1 ). f(t 1 ) = v 2 (t 1, t 2) v 1 (t 1, φ(t 1 )) = v 2 (t 1, t 2) v 2 (φ(t 1 ), t 1 ) f (t 1 ) = v 2 t 1 v 2 t 1 φ (t 1 ) v 2 t 2 = v 2 t 1 v 2 t 2 v 2 t 1 φ (t 1 ). So, by assumption [3], f is strictly decreasing. Further, f(φ 1 (t 2)) = v 2 (φ 1 (t 2), t 2) v 2 (φ(φ 1 (t 2)), φ 1 (t 2)) = v 2 (φ 1 (t 2), t 2) v 2 (t 2, φ 1 (t 2))

12 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 10 Now recall that t 2 φ 1 (t 2). If t 2 > φ 1 (t 2). Then f(φ 1 (t 2)) > 0. As the function f is strictly decreasing, we can conclude that Π b 2(b 1, p 2 t 2) is maximized at a type t 1 > φ 1 (t 2). If t 2 < φ 1 (t 2), then Π b 2(b 1, p 2 t 2) is maximized at a type t 1 < φ 1 (t 2). In both cases, b 2 (t 2) v 2 (φ 1 (t 2), t 2). Thus, this class of strategy pairs contains only one BNE, namely the pair for which b i = v i (t i, t i ) for every i = 1, 2. We show that the BNE is also strongly distributionally perfect. Proposition 4.2 The strategy pair (b 1, b 2 ), where b i = v i (t i, t i ) for i = 1, 2 is a strong distributional perfect BNE. Further, b i is undominated for every bidder i. Proof. We prove that the BNE (b 1, b 2 ) is strongly distributionally perfect. Let γ i be the distributional strategy induced by b i. For every k IN define γ k 1 = (1 ε k ) γ 1 + ε k ν 1, where ε k (0, 1), ε k 0 as k, and ν 1 is any completely mixed distributional strategy on the product space T 1 A It suffices to show that for each k there is a strategy η k 2 BR 2 (γ k 1 ) such that lim k ρw (γ 2, η2) k = 0. The proof is in two steps. Let t 2 be a type of bidder 2. A. We show that p 2 = b 2 is the unique maximizer of F (p 2 ) = Π b 2(b 1, p 2 t 2 ) and F (p 2 ) is continuous. Recall that F (p 2 ) = Π b 2(b 1, p 2 t 2 ) = w 1 (p 2 ) t (v 2 (t 1, t 2 ) v 1 (t 1, t 1 )) µ 1 (dt 1 ), where w denotes the strictly increasing function w(t) = v 1 (t, t). Since the integrand v 2 (t 1, t 2 ) v 1 (t 1, t 1 ) = v 2 (t 1, t 2 ) v 2 (t 1, t 1 ) is continuous and strictly decreasing in t 1, and w 1 (p 2 ) is continuous and strictly increasing in p 2, it is clear that F (p 2 ) is continuous. Further, since the integrand is zero at t 1 = t 2, it is also clear that F (p 2 ) has a unique maximum location at value p 2 with w 1 (p 2 ) = t 2. Thus, p 2 = w(t 2 ) = v 1 (t 2, t 2 ) = v 2 (t 2, t 2 ) = b 2 (t 2 ) 10 That is, η 1 is a completely mixed measure on T 1 A 1 and η 1 (S 1 A 1 ) = µ 1 (S 1 ) for every S 1 T 1.

13 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 11 is the unique maximizer of F (p 2 ). B. We prove that for each k there is a strategy η k 2 BR 2 (γ k 1 ) such that lim k ρw (γ 2, η2) k = 0. Let p 2 be a pure behavioral strategy for bidder 2 and σ 2 be the distributional strategy induced by p 2. Then, the expected payoff of bidder 2 choosing σ 2 against γ k 1 is: Π 2 (γ k 1, σ 2 ) = = = T 2 (v 2 (t 1, t 2 ) a 1 ) dγ1 k dσ 2 p 2 >a 1 p 2 >a 1 (v 2 (t 1, t 2 ) a 1 ) dγ k 1 µ 2 (dt 2 ) T 2 Π b 2(β k 1, p 2 t 2 ) µ 2 (dt 2 ), where β1 k is a behavioral strategy corresponding to γ1 k for every k IN. Let b k 2 be any maximizer of Π b 2(β1 k, p 2 t 2 ) for every k, where Π b 2(β1 k, p 2 t 2 ) = (1 ε k ) (v 2 (t 1, t 2 ) a 1 ) dγ 1 p 2 >a 1 + ε k p 2 >a 1 (v 2 (t 1, t 2 ) a 1 ) dη 1 = (1 ε k ) w 1 (p 2 ) t + ε k p 2 >a 1 (v 2 (t 1, t 2 ) a 1 ) dη 1 (v 2 (t 1, t 2 ) v 1 (t 1, φ(t 1 ))) µ 1 (dt 1 ) = (1 ε k ) Π b 2(b 1, p 2 t 2 ) + ε k p 2 >a 1 (v 2 (t 1, t 2 ) a 1 ) dη 1. Note that the maximizer b k 2 exists, because Π b 2(β k 1, p 2 t 2 ) is a continuous function over the compact set T 2. We know that p 2 = b 2 is the unique maximizer of Π b 2(b 1, p 2 t 2 ). Thus, according to Lemma 6.6, we have that b k 2 converges to b 2 when k. Now, let η k 2 be the induced distributional strategy by b k 2. As b k 2 BR 2 (t 2, β k 1 ), then we have η k 2 BR 2 (γ k 1 ). Furthermore, since b k 2(t 2 ) converges to b 2 (t 2 ) for every t 2 T 2, we conclude ρ w (γ 2, η k 2) 0 when k. With a similar argument for bidder 2, we can conclude that (b 1, b 2 ) is strongly distributionally perfect.

14 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 12 Finally, as bidding b 2 is the unique best response for bidder 2 against b 1, we conclude that b 2 is undominated. A similar argument holds for b 1. In the next proposition we prove that this strategy pair is not in dominant strategies, unless we are in the pure private values case. Proposition 4.3 Suppose that the valuation function v i is not private values. Then the strategy b i = v i (t i, t i ) is not dominant. Proof. Assume that v 1 is not private values. We prove that b 1 = v 1 (t 1, t 1 ) is not a dominant strategy. In particular, we construct a bid function d 2 for bidder 2 and an open set I 1 of types for bidder 1, such that there is a bid a A 1 for bidder 1 that yields a strictly higher payoff against d 2 than b 1 (t 1 ) = v 1 (t 1, t 1 ) for any type t 1 I 1,. This then is a sufficient argument. Since v 1 is not private values, by condition [3] there is an s 1 in T 1 and an open set B 2 in T 2 such that the map v 1 (s 1, ): T 2 A 2 is strictly increasing on B 2. Suppose that B 2 [t, s 1 ) is not empty. Then we may assume that t 2 < s 1 for all t 2 B 2. Then, for every t 2 B 2 we have v 1 (s 1, t 2 ) < v 1 (s 1, s 1 ). Define y(t 1 ) = t1 t v 1 (t 1, t 2 ) µ 2 (dt 2 ). Since v 1 (s 1, t 2 ) v 1 (s 1, s 1 ) for every t 2 < s 1, and v 1 (s 1, t 2 ) < v 1 (s 1, s 1 ) for every t 2 B 2, it follows that y(s 1 ) = s1 t v 1 (s 1, t 2 ) µ 2 (dt 2 ) < v 1 (s 1, s 1 ) = b 1 (s 1 ). So, y(s 1 ) < b 1 (s 1 ). Since t < s 1 and b 1 is strictly increasing and continuous, there is u 1 < s 1 with b 1 (u 1 ) > y(s 1 ). Take I 1 = (u 1, s 1 ). Then, since y and b 1 are strictly increasing functions, for every t 1 I 1 we have y(t 1 ) < y(s 1 ) < b 1 (u 1 ) < b 1 (t 1 ). Take the (constant) bid function d 2 (t 2 ) = b 1 (u 1 ) for every t 2 T 2 for bidder 2. Take t 1 I 1 and any t 2 T 2. Since bidder 1 bids according to b 1 (t 1 ), and d 2 (t 2 ) = b 1 (u 1 ) < b 1 (t 1 ), bidder 1 wins the auction. Consequently, his expected payoff against d 2 is T 2 (v 1 (t 1, t 2 ) b 1 (u 1 )) µ 2 (dt 2 ) = y(t 1 ) b 1 (u 1 ) < 0.

15 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 13 A similar argument applies to the case where B 2 (s 1, t] is not empty. This concludes the proof. 4.2 Equilibrium refinement II: discarding equilibria Birulin in [11] on page 678 introduces the following class of discontinuous asymmetric strategy profiles. 11 For given s 1, s 2 [t, t] with s 1 < s 2, define the strategy pair (ˆb 1, ˆb 2 ) by { v 1 (s 2, t 1 ) if t 1 [s 1, s 2 ] ˆb1 (t 1 ) = v 1 (t 1, t 1 ) otherwise, ˆb2 (t 2 ) = { v 2 (t 2, s 1 ) if t 2 [s 1, s 2 ] v 2 (t 2, t 2 ) otherwise. Birulin [11] shows that, under SCC, the strategy pair (ˆb 1, ˆb 2 ) is an ex post equilibrium that allocates the object inefficiently. Moreover, this equilibrium may be in undominated strategies. In the equilibrium above, bidder 1 overbids and bidder 2 underbids for types between s 1 and s 2. In the proposition below we prove that strong distributional perfection rules out these equilibria. 12 The intuition behind this result is that if there is a small chance that one bidder bids b i = v i (t i, t i ) for t i [s 1, s 2 ] instead of ˆb i, then ˆb j will no longer be a best response for bidder j, where j i. In other words, underbidding (overbidding) would not be a best response against overbidding (underbidding) if even with a tiny probability one bidder bids according to the efficient bidding strategy b i = v i (t i, t i ). As an extreme case, we see that the ex post equilibrium 13 in which one bidder always bids a and the other bidder always bids a is not strongly distributionally perfect. Proposition 4.4 The equilibrium (ˆb 1, ˆb 2 ) is not strongly distributionally perfect. Proof. A. First we show that if bidder 1 bids according to a behaviorial completely mixed strategy that converges to ˆb 1, then bidding ˆb 2 for bidder 2 is not 11 Our presentation of the bid functions is slightly different from Birulin [11]. He presents the bid functions in terms of the opponents valuations. We do this for each bidder in terms of his own valuations. 12 In finite games, an equilibrium is perfect if and only if it is in undominated strategies. This is in general not true in games with infinite action spaces, even if undominatedness is replaced by limit undominatedness. Bajoori et. al in [4] give a detailed analysis on this issue. They prove that, in normal form games under appropriate assumptions, every weak perfect equilibrium is in limit undominated strategies, but not vice versa. Example 6 in Bajoori et. al [4] provides the relevant counter-example. 13 This kind of equilibrium is often called a wolf and sheep equilibrium.

16 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 14 optimal on a set of types with strictly positive measure. Assume bidder 1 bids according to β1 k strategy: β1 k (t 1 ) = (1 2δ k )ˆb 1 (t 1 ) + δ k b 1 (t 1 ) + δ k η 1 (t 1 ), where b 1 (t 1 ) = v 1 (t 1, t 1 ), η 1 is any completely mixed strategy for bidder 1, and δ k 0 when k. Clearly, β1 k converges to ˆb 1 when k pointwisely. We show that bidding b 2 (t 2 ) = v 2 (s 1, s 2 ) is strictly better than bidding ˆb 2 for t 2 (s 1, s 2 ) for bidder 2. Note that ˆb 2 = v 2 (t 2, s 1 ) for t 2 (s 1, s 2 ). Define w(t) = v 1 (t, t). Bidder 2 s expected payoff by bidding ˆb 2 = v 2 (t 2, s 1 ) against β k 1 for every t 2 (s 1, s 2 ) is Π b 2(β k 1, ˆb 2 t 2 ) = (1 2δ k ) s1 t (v 2 v 1 (t 1, t 1 )) dµ(t 1 ) w 1 (ˆb 2 ) + δ k (v 2 v 1 (t 1, t 1 )) dµ(t 1 ) t + δ k ˆb2 >a 1 (v 2 a 1 ) dη 1 (t 1 ). Moreover, bidder 2 s expected payoff by bidding b 2 = v 2 (s 1, s 2 ) against β k 1 for every t 2 (s 1, s 2 ) is Π b 2(β k 1, b 2 t 2 ) = (1 2δ k ) s1 t (v 2 v 1 (t 1, t 1 )) dµ(t 1 ) w 1 ( b 2 ) + δ k (v 2 v 1 (t 1, t 1 )) dµ(t 1 ) t + δ k b2 >a 1 (v 2 a 1 ) dη 1 (t 1 ). By condition [3] in our auction model, we have ˆb 2 (t 2 ) = v 2 (t 2, s 1 ) < v 2 (s 1, s 2 ) = b2 (t 2 ). Because the function w is a strictly increasing bijection, we can have w 1 (ˆb 2 ) < w 1 ( b 2 ). Hence, Π b 2(β1 k, ˆb 2 t 2 ) < Π b 2(β1 k, b 2 t 2 ). Therefore, ˆb2 (t 2 ) / BR 2 (t 2, β1 k ) for t 2 (s 1, s 2 ) and every k. B. To show that (ˆb 1, ˆb 2 ) is not strongly distributionally perfect, we show that for the induced distributional strategy profile (ˆγ 1, ˆγ 2 ) ν = (ν 1, ν 2 ) and (ε k ) k=1 = (ε 1 k, ε 2 k) k=1

17 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 15 in which ν 1, ν 2 are completely mixed distributional strategies and ε 1 k, ε2 k 0 when k such that for γ1 k = (1 ε 1 k )ˆγ 1 + ε 1 k ν 1 and γ2 k = (1 ε 2 k )ˆγ 2 + ε 2 k ν 2 we have ρ w (γ2 k, BR 2 (γ1 k )) > ε. for an ε > 0. Let ε 2 k = 0 for every k, then we have γk 2 = ˆγ 2 for every k. Therefore, it is enough to prove that ρ w (ˆγ 2, BR 2 (γ1 k )) > ε. for an ε > 0. Let β 2 k be such that β 2 k (t 2, ) BR 2 (t 2, β1 k ) for every t 2 T 2 and every k. In part A we show that ˆb 2 (t 2 ) / BR 2 (t 2, β1 k ) for t 2 (s 1, s 2 ). As the function Π b 2(β1 k, p 2 t 2 ) is continuous in p 2, then any point in a neighborhood of ˆb 2 (t 2 ) is not a best response against β1 k either. Therefore, there are ε > 0 and λ > 0 such that if for every t 2 (s 1, s 2 ) we define B t2 = (ˆb2 (t 2 ) λ, ˆb ) 2 (t 2 ) + λ, then we have 1. s 1 + ε < s 2 ε, 2. µ 2 (s 1 + ε, s 2 ε) > ε, 3. βk 2 (t 2, (B t2 ) ε ) = 0. Now, let B = {(t 2, a 2 ) t 2 (s 1 + ε, s 2 ε), a 2 B t2 }. Clearly, B is measurable. Also, let γ 2 k and γ1 k be the distributional strategies induced by β 2 k, and β1 k respectively. One can write γ1 k = (1 ε 1 k)ˆγ 1 + ε 1 kν 1 in which ε 1 k = 2δ k and ν 1 is the completely mixed distributional strategy induced by behaviorial strategies b 1 and η 1. Moreover, we have γ 2 k BR 2 (γ1 k ). As ˆb2 (t 2 )(B t2 ) = 1 14 and β 2 k (t 2, (B t2 ) ε ) = 0, we have ˆγ 2 (B) = s2 ε ˆb2 (t 2 )(B t2 )µ 2 (dt 2 ) = µ 2 (s 1 + ε, s 2 ε) s 1 +ε 14 Notice that ˆb 2 (t 2 ) is interpreted as a behavioral strategy that assigns probability 1 on ˆb 2 (t 2 ) for every t 2 T 2.

18 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 16 and γ k 2 (B ε ) = Consequently, s2 ˆγ 2 (B) > γ k 2 (B ε ) + ε, s 1 βk 2 (t 2, (B t2 ) ε )µ 2 (dt 2 ) = 0. then ρ w (ˆγ 2, γ k 2 ) > ε for every k. Hence, we have ρ w (ˆγ 2, BR 2 (γ k 1 )) > ε for every k. The proof is complete. To illustrate our results, we discuss the following example. Example 4.5 Consider a 2-bidder second-price auction with interdependent values v 1 = t t 2 and v 2 = 1 2 t 1 + t 2. Types t 1, t 2 for bidder 1 and 2 are drawn independently from [0, 1] according to the uniform distribution. Each bidder simultaneously submits a bid from the set [0, 3 ]. The efficient BNE introduced by 2 Milgrom is the strategy pair (b 1, b 2 ) in which b i = 3t 2 i for every bidder i = 1, 2. By Proposition 4.2 this BNE is strongly distributionally perfect. On the other hand, the class of inefficient ex post equilibria introduced by Birulin looks as follows. For fixed s 1, s 2 [0, 1] with s 1 < s 2, define { { s 2 + 1t 2 1 if t 1 [s 1, s 2 ] s 1 + 1t 2 2 if t 2 [s 1, s 2 ] ˆb1 (t 1 ) = 3 t 2 1 otherwise, ˆb2 (t 2 ) = 3 t 2 2 otherwise. An example of such an equilibrium is depicted in Figure 1. In each such an equilibrium, when the realizations of both types are within the set [s 1, s 2 ], bidder 1 wins the auction and pays the amount ˆb 2 (t 2 ) = s t 2, which is his minimum ex post valuation. Thus, in this case bidder 1 overbids, while bidder 2 underbids. So, since such equilibria lead to inefficient outcomes, we would like to be able to rule them out. Proposition 4.4 shows that the equilibria (ˆb 1, ˆb 2 ) are not strongly distributionally perfect. Intuitively, the reason is that bidder 1, when his type is between s 1 and s 2, bids higher than his maximum valuation. Now, if bidder i makes mistakes, any bid from the set [0, 3 ] becomes possible. This prevents the 2 other bidder j from overbidding in a perfect equilibrium, because overbidding strictly decreases his payoff.

19 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 17 ˆb1 ˆb2 s t 1 s t 2 s t 1 s 2 1 s t 1 s 2 2 Figure 1 5 Conclusion We introduce the concept of distributional perfect equilibrium as a refinement of Bayesian Nash Equilibrium. We prove the existence of distributional perfect equilibrium in a class of games where the information structure of the game is absolutely continuous and the payoffs are continuous in actions for every type. We also define a stronger notion of distributional perfection, strong distributional perfect BNE. We apply this concept in symmetric second-price auctions with interdependent values for two bidders to select the efficient BNE, and to rule out the less intuitive equilibria. In particular, we show that the efficient BNE discussed in Milgrom [22] is strongly distributionally perfect. We also show that strong distributional perfection rules out all the discontinuous, inefficient, equilibria introduced by Birulin [11]. These equilibria feature overbidding and underbidding. Intuitively, we show that underbidding (overbidding) cannot be a best response against overbidding (underbidding), if one bidder bids according to the efficient equilibrium with a small probability. 6 Appendix Lemma 6.1 Consider a Bayesian game Γ. Let β = (β 1,..., β n ) be a behavioral strategy profile and γ = (γ 1,..., γ n ) be the induced distributional strategy profile. If γ i BR i (γ), then β i (t i, ) BR i (t i, β) for µ i -a.e. t i T i. Proof. Suppose for every player i that γ i BR i (γ). Then, we have Π i (γ i, γ i ) Π i ( γ i, γ i ),

20 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 18 for every player i s distributional strategy γ i. Therefore, for every player i s behavioral strategy β i we have Π b i(β i, β i t i ) µ i (dt i ) T i Π b i( β i, β i t i ) µ i (dt i ). T i (1) Theorem 9.1 in [28] 15 implies that for every player i there is a behavioral strategy ζ i such that ζ i (t i, ) BR i (t i, β) for every t i T i, which means having type t i T i Π b i(ζ i, β i t i ) Π b i(a i, β i t i ), for every a i A i. To prove that β i (t i, ) for µ i -a.e. t i T i is a best response for every player i against β i, we show that Π b i(ζ i, β i t i ) = Π b i(β i, β i t i ) for µ i -a.e. t i T i. Suppose the opposite. First, let S i = { t i T i Π b i(ζ i, β i t i ) < Π b i(β i, β i t i ) }. Clearly, S i is µ i -measurable. Suppose µ i (S i ) > 0. Notice that given t i T i, for β i (t i, )-a.e. a i A i we have Π b i(β i, β i t i ) = Π b i(a i, β i t i ). Then, given t i S i, for β i (t i, )-a.e. a i A i we conclude that Π b i(ζ i, β i t i ) < Π b i(a i, β i t i ), which is a contradiction with optimality of ζ i for every type. Second, let S i = { t i T i Π b i(ζ i, β i t i ) > Π b i(β i, β i t i ) } and suppose µ i (S i ) > 0. Therefore, we have Π b i(ζ i, β i t i ) µ i (dt i ) > Π b i(β i, β i t i ) µ i (dt i ). S i S i 15 Suppose (T, M) is a measurable space, X is a topological space with the Borel σ-algebra B, and F is a correspondence from T to X. Denote GrF = {(t, x) : x F (t)}. Suppose u : GrF IR is M B measurable and u(t, ) is upper semicontinuous on F (t) for t T, and let ν(t) = sup{u(x, t) : x F (t)} for t T. Theorem 9.1 in [28] states that if F is compactvalued and measurable, X is separable metric, and u is the limit of a decreasing sequence of Caratheodory maps, then ν is measurable and there exits f : T X measurable such that u(, f( )) = ν.

21 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 19 Since ζ i (t i, ) is a best response for every type t i a i A i we have T i against β i, for every Π b i(ζ i, β i t i ) Π b i(a i, β i t i ), thus Π b i(ζ i, β i t i ) µ i (dt i ) Π b i(β i, β i t i ) µ i (dt i ). T i \S i T i \S i Consequently, Π b i(ζ i, β i t i ) µ i (dt i ) > Π b i(β i, β i t i ) µ i (dt i ). T i T i This is a contradiction with (1). Overall, we showed that if there is a subset of T i on which the expected payoff of player i, playing according to ζ i is different from playing according to β i, the measure of that subset is zero. In other words, for every i there exists a set S i T i with µ i (S i ) = 0 such that for every t i T i \S i we have β i (t i, ) BR i (t i, β). This completes the proof. Lemma 6.2 Consider a Bayesian game Γ. For every player i and every ε > 0, there is a set E T such that µ(e c ) < ε and the family {π i (t, ) t E} is equicontinuous. 16 Proof. Let C(A) be the set of all continuous real functions on A. For every player i consider the mapping t π i (t, ) which is a measurable mapping from T to C(A). Therefore, the probability measure µ on T induces a probability measure on C(A), say η. Since A is compact metric space, then C(A) is a compact separable metric space that implies that η is tight. That means, for every ε > 0 there is a compact set G C(A) with η(g c ) < ε. Let E be the inverse image of G under the above mapping. Then we have µ(e c ) < ε. Since G is compact, by Ascoli-Arzela theorem one concludes that G is equicontinuous. This completes the proof. 16 A family F of real functions on the metric space X is equicontinuous if to every ε > 0 corresponds a δ > 0 such that for every x, y X with d(x, y) < ε we have f(x) f(y) < δ, for every f F.

22 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 20 In the following lemma, we use the following metrics: For every t, s T and a, b A, define ( n d T (t, s) = d 2 T i (t i, s i ) i=1 ) 1 2 ( n, d A (a, b) = d 2 A i (a i, b i ) Moreover, let d T A ((t, a), (s, b)) = ( d 2 T (t, s) + d 2 A(a, b) ) 1 2. Lemma 6.3 Consider a Bayesian game Γ. For every player i and every ε > 0, there is a set K T with µ(k c ) < ε such that π i is continuous on K A for every i. Proof. Take ε > 0 and a A. First we prove that there is a set K T with µ(k c ) < ε such that the function π i (, a) : K IR is continuous. As A is separable, it has a countable dense subset, say B. Hence, there exists a sequence (a k ) k=1 in B such that a k a, for every k, and d A (a k, a) 0 as k. According to Lusin s Theorem for the collection of the functions {π i (, a k ) k = 1, 2,...} there is a compact set E T with µ(e c ) < ε 2 such that π i(, a k ) is continuous on E for every k IN. Moreover, by the lemma 6.2 we know that payoffs are equicontinuous, i.e. there is a set F T with µ(f c ) < ε such that 2 the collection of the functions {π i (t, ) : A IR t F } is equicontinuous. Let K = E F. It is clear that µ(k c ) < ε. Moreover, because π i (t, ) is continuous in actions for every t K, the sequence of the functions (π i (, a k )) k=1 converges pointwise to π i (, a) as k. It is easy to check that equicontinuity of payoffs and the fact that d A (a k, a) 0 as k, implies that the sequence of functions (π i (, a k )) k=1 uniformly converges to π i(, a) on K as k. Hence, π i (, a) : K IR is continuous too. Now, we prove that π i is continuous at (t, a) K A. We know that {π i (t, ) t K} is equicontinuous, then there is a δ 1 > 0 such that if d A (a, b) < δ 1, we have π i (t, a) π i (t, b) < ε 2 for every t K. Also, we know π i(, a) is continuous at t K, so there is a δ 2 > 0 such that if d T (t, s) < δ 2, we have π i (t, a) π i (s, a) < ε. Let δ = min{δ 2 1, δ 2 }. Then, for every (s, b) K A with d T A ((t, a), (s, b)) < δ we have d A (a, b) < δ 1 and d T (s, t) < δ 2. Consequently, we have π i (t, a) π i (s, b) < π i (t, a) π i (s, a) + π i (s, a) π i (s, b) < ε 2 + ε 2 = ε. i=1 ) 1 2.

23 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 21 This proves that π i is continuous on K A. Lemma 6.3 gives a slightly more general result than Lusin s theorem. Indeed, Lusin s theorem would guarantee the existence of a subset of T A such that π i is continuous on that subset, while Lemma 6.3 would do that for a set of the form K A, where K T. The following lemma is the basic continuity result in Milgrom and Weber [23]. Here we provide a detailed proof of that. Lemma 6.4 Consider a Bayesian game Γ. For every player i, the function Π i : IR is continuous with respect to the weak metric on. Proof. To prove that Π i is continuous at a distributional strategy profile γ, let (γ k ) k=1 be a sequence of distributional strategy profiles converging to γ with respect to the weak metric as k. Now, by Lemma 6.3, for every i and every δ > 0, there exist a continuous and bounded function v δ : T A IR and a set K T such that µ(k c ) < δ, and v δ = π i on K A. As µ(k c ) < δ, we have γ k (K c A) < δ, for every k. Similarly, γ(k c A) < δ. Moreover, because v δ and π i are bounded, there is an M > 0 such that π i v δ M. Hence, for every k we have π i v δ f(t) dγ k = T A Similarly, T A π i v δ f(t) dγ = K c A K c A π i v δ f(t) dγ k M δ. π i v δ f(t) dγ M δ. Furthermore, as f is ˆµ-integrable, there is a sequence (f l ) l=1 of bounded and

24 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 22 continuous functions such that T f(t) f l (t) ˆµ(dt) 0, as l. We have Π i (γ k ) Π i (γ) = π i f(t)dγ k π i f(t)dγ π i v δ f(t)dγ k + f(t) f l (t) v δ dγ k + v δ f l (t)dγ k v δ f l (t)dγ + f(t) f l (t) v δ dγ + π i v δ f(t)dγ 2Mδ + f(t) f l (t) v δ dγ k + v δ f l dγ k v δ f l dγ + f(t) f l (t) v δ dγ. Hence, Π i (γ k ) Π i (γ) converges to zero when k, l, and δ 0. This proves that Π i is continuous at γ. We prove the existence of a distributional perfect BNE, with the help of the following fixed point theorem from Glicksberg [14]. Theorem 6.5 Let S be a nonempty compact and convex subset of a locally convex Hausdorff space. Let F : S S be an upper hemicontinuous correspondence with nonempty and convex values. Then, F has a fixed point. Lemma 6.6 Let f and g be two continuous functions on an interval [a, b]. For every k IN, define f k (x) = (1 ε k )f(x) + ε k g(x) in which ε k 0 as k. Let y k be a maximizer of f k for every k. Suppose that function f has a unique maximizer x = y, then y k y when k. Proof. We have f k converges to f uniformly. For every k, y k is any maximizer of f k, which exists since [a, b] is compact. Consider any convergent subsequence {y k l } kl =1 of the sequence {yk } k=1. Assume yk l converges to y [a, b] when k l. Then we have f k l (y k l) f k l(x) for every x [a, b]. Now, by uniform convergence, we have f k l (y k l) f(y ) and f k l (x) f(x), when kl.

25 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 23 Therefore, we have f(y ) f(x) for every x [a, b]. Then, by uniqueness of maximizer we have y = y. Thus, any convergent subsequence has the same limit y. Hence, we conclude that {y k } k=1 is convergent and converges to y when k. Proof of Theorem 3.2. First we prove that the set of distributional strategy profiles satisfying the conditions in the Definition 3.1 is not empty. For every player i, let ν i be a completely mixed distributional strategy. Note that such a ν i exists, because T i A i is separable. Define for every k IN i (k) = {γ i i γ i (B i ) 1 k ν i(b i ), B i T i A i }. Moreover, let (k) = n i=1 i (k) and BR k i : (k) i (k) be the best response correspondence for player i restricted to i (k) 17 and define the correspondence BR k : (k) (k) to be BR k (γ) = n i=1br k i (γ), for every γ (k). We verify the conditions of Theorem 6.5 for the correspondence BR k. Since T is a complete and separable metric space, by Theorem 1.4 in [10] µ is a tight measure. 18 This fact together with the compactness of the set A implies that i is a tight set of probability measures. 19 Now, by Prohorov s Theorem we conclude that i is a compact metric space with respect to the weak metric. It is easy to see that i (k) is a closed subset of i with respect to the strong metric. Consequently, by Theorem V.3.13 in [13], it is closed with respect to the weak metric. Hence, i (k) is compact with respect to the weak metric. Also, one can easily check that i (k) is convex. Moreover, upper hemicontinuity of 17 Distributional strategy γ i of player i is a best response restricted to i (k) against a distributional strategy profile η i i (k), if Π i (γ i, η i ) Π i ( γ i, η i ), for every distributional strategy γ i i (k) of player i. 18 Let X be a metric space with the Borel σ-field Σ. A measure µ on Σ is tight if for every G Σ, µ(g) = sup{µ(k) K G, and K compact}. Moreover, Theorem 1.4 in [10] states that if X is separable and complete, then each probability measure on Σ is tight. 19 A family of probability measures on a metric space is tight if for every ε > 0 there is a compact set K satisfying µ(k) > 1 ε, for every µ in the family.

26 DISTRIBUTIONAL PERFECT EQUILIBRIUM IN BAYESIAN GAMES 24 BR k follows by Lemma 6.4. Now, we apply Theorem 6.5, which leads us to the existence of an equilibrium point γ k (k), for every k. Now, define 20 PBR i (γ) = {(t i, a i ) T i A i (t i, a i ) supp(γ i ) and γ i BR i (γ)}. ( It is clear that γi k PBRi (γ k ) ) 1 1ν k i(t i A i ) = 1 1. This implies that k ρ w (γi k, BR i (γ k )) 1. Furthermore, as k i is compact with respect to the weak metric, without loss of generality we can assume that there is a distributional strategy profile γ such that ρ w (γ k, γ) 0 when k. Let β be a behavioral strategy profile corresponding to γ. Obviously, β is a distributionally perfect. We show that there is a BNE ˆβ such that for µ i -a.e. t i T i we have β i (t i, ) = ˆβ i (t i, ), for every player i. Then, since β is distributionally perfect, trivially the BNE ˆβ is also distributionally perfect. Hence, the set of distributional perfect BNEs is nonempty, as desired. Since β = (β 1,..., β n ) is distributionally perfect, Definition 3.1 states that, for the induced distributional strategy profile γ = (γ 1,..., γ n ) there exists a sequence (γ k ) k=1 = (γk 1,..., γn) k k=1 of completely mixed distributional strategy profiles such that for every player i we have lim ρ w (γi k, γ i ) = 0 and k lim k ρw (γi k, BR i (γ k )) = 0. Therefore, for every i, there exists a sequence (ˆγ k i ) k=1 such that ˆγk i BR i (γ k ) for every k IN and ρ w (γ k i, ˆγ k i ) 0 as k. By the triangle inequality for ρ w this implies for every i that ρ w (γ i, ˆγ k i ) 0 as k. Now, by Lemma 6.4 we conclude the upper hemicontinuity of the correspondence BR i, for every player i. Hence, γ i BR i (γ). Now, by Lemma 6.1 we have β i (t i, ) BR i (t i, β) for µ i -a.e. t i T i. In other words, for every i there exists a set S i T i with µ i (S i ) = 0 such that for every t i T i \S i we have β i (t i, ) BR i (t i, β). Define ˆβ = ( ˆβ 1,..., ˆβ n ) by letting ˆβ i (t i, ) = β i (t i, ) for every t i T i \S i and selecting any ˆβ i (t i, ) BR i (t i, β) for every t i S i. Note that the result of this selection is measurable. Clearly, ˆβ is a BNE and satisfies the requirement of the proposition. 20 Recall that supp(µ) = {x X for every open set G if x G then µ(g) > 0}.

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