Subgame Perfect Equilibria in Infinite Stage Games: an alternative existence proof

Transcription

1 Subgame Perfect Equilibria in Infinite Stage Games: an alternative existence proof Alejandro M. Manelli Department of Economics Arizona State University Tempe, Az Version: January 2003 Abstract Any stage-game with infinite choice sets can be approximated by finite games obtained as increasingly finer discretizations of the infinite game. The subgame perfect equilibrium outcomes of the finite games converge to a limit distribution. We prove that (i) if the limit distribution is feasible in the limit game, then it is also a subgame perfect equilibrium outcome of the limit game; and (ii) if the limit distribution prescribes sufficiently diffused behavior for first-stage players, then it is a subgame perfect equilibrium outcome of the limit game. These results are potentially useful in determining the existence of subgame perfect equilibria in applications. I am grateful to Kim Border for helpful discussions and for bringing to my attention the Arsenin-Kunugui Selection Theorem. I also benefited from conversations with Andreas Blume, Hector Chade, Arthur Robson, Edward Schlee, and Jeroen Swinkels. Financial support from the National Science Foundation under Grant SBR is gratefully acknowledged. 1

2 1 Introduction Games where players have a continuum of choices (henceforth infinite games) are commonly used to model economic phenomena. Despite their numerous applications, infinite games with incomplete or imperfect information, even if very well behaved, may have no equilibrium. Indeed van Damme (1987) constructs a signaling game with no sequential equilibrium, and Harris, Reny, and Robson (1995) construct a two-stage game with no subgame perfect equilibrium. (We provide a brief summary of the latter example toward the end of the Introduction for the benefit of the reader unfamiliar with the subject.) General existence results that don t rely on particular payoff functions, partial effects, or distributional assumptions are few. Still certain similarities have been identified in seemingly different infinite games: correlation in various forms restores the existence of equilibrium. Cotter (1991) proves the existence of a correlated equilibrium in simultaneous-move games of incomplete information. (Whether those games admit a Bayesian Nash equilibrium is an open question. 1 ) In addition to their non-existence example Harris, Reny, and Robson (1995) prove the existence of a subgame perfect equilibrium in stage-games that have a public, randomization device in each stage. In previous work, we have shown that infinite signaling games have a sequential equilibrium when cheap talk is allowed. In this essay we provide an alternative proof of the existence of a subgame perfect equilibrium in stage games with public randomization devices. Our approach may prove useful in analyzing other classes of infinite games. Throughout the paper, we consider games with two stages. In each stage finitely many players simultaneously and independently select their actions (from a compact set) after observing choices made in previous stages. Payoffs depend continuously on all players choices. Harris, Reny, and Robson s non-existence example is within the class of games that we study. The approach we follow has three main steps. 1. Fix an infinite game and consider a sequence of finite games constructed by taking increasingly finer discretizations of the infinite game. Each finite game has a subgame perfect outcome, the outcome of a subgame perfect equilibrium. Outcomes are distributions and converge in a subsequence to a limit probability distribution. 2. We prove that the limit distribution is a subgame perfect outcome of the infinite game if and only if it is feasible, i.e., if there are some strategies, though not necessarily equi- 1 Milgrom and Weber (1985), Balder (1988) and more recently Al-Najjar and Solan (1999) prove existence of Bayesian Nash equilibrium using restrictions on the distribution of types. Khan and Sun (1995) extend the purification theorems in Radner and Rosenthal (1982) to countable action spaces. 2

3 librium strategies, of the infinite game that generate the limit distribution (Corollary 1 of Theorem 1). 3. We identify conditions on or variations of the infinite game under which either the limit distribution is always feasible (and therefore a subgame perfect equilibrium exists), or a variation of the limit distribution is a subgame perfect equilibrium (Theorem 2). The first step is a straight forward observation that holds generally. 2 The second step contains a result that holds in the stage games we consider as well as in signaling games (Manelli (1996), Theorem 1). We believe that a similar result that limit distributions when feasible are sequentially rational outcomes of the limit game may hold for other types of infinite games. If the limit distribution is infeasible, the reason is typically that some sort of correlation has been added to it through the limiting process. Note that in Harris, Reny, and Robson s non-existence example the limit distribution is infeasible; it requires that second-stage players correlate their actions, and since players must move independently, no strategies of the infinite game can generate the necessary correlation. In van Damme s non-existence example the limit distribution requires that the receiver in the signaling game correlate her actions with the sender s private information; since the sender s signals are uninformative, it is again impossible to attain the necessary correlation. Still a similar phenomenon occurs in some examples due to Jackson, Simon, Swinkels, and Zame (2001) of non-existence of Bayesian Nash equilibrium in auctions. (In their games, however, payoff functions are not continuous on actions.) The third step identifies conditions to transform the second step into an existence result. One may envision three types of conditions. First, one may impose restrictions so that the limit distribution does not exhibit correlation. This is hard to accomplish; spurious correlation is easily generated specially when the approximating games have multiple equilibria. (We expand on this point in Section 3 when presenting our formal results.) Second, one may add features to the infinite game (or identify a class of games) such that strategies can reproduce any potential correlation of the limit distribution. For instance, in signaling games the ability of the sender to transmit cheap-talk messages is one such feature. Public randomization devices in stage games is another example. A similar addition or extension restores existence in the auctions studied by Jackson, Simon, Swinkels, and Zame (2001). Finally, it is sometimes possible to alter the limit distribution to identify an equilibrium distribution. This is accomplished by Theorem 2, the main technical contribution of the 2 Provided action sets are compact metric spaces, any sequence of outcome distributions has a convergent subsequence with respect to the topology of weak* convergence. 3

4 essay, for the games we study. As noted earlier, the limit distribution is infeasible whenever it requires second-stage correlation, i.e., a mixture of second-stage Nash equilibria. Theorem 2 provides, to some extent, a purification of this mixture: if according to the limit distribution first-stage players choose non-atomic strategies, then even if the limit distribution is infeasible there is a subgame perfect equilibrium of the limit game. (Of course the identified equilibrium outcome may be different from the limit distribution.) The combination of Theorems 1, Corollary 1 and Theorem 2 may be useful in determining existence in applications. For instance, it follows from those results that if the limit distribution prescribes non-atomic behavior for first-stage players, then the infinite game has a subgame perfect equilibrium (Corollary 2). Similarly consider extensions of the game that incorporate cheap talk the ability of first-stage players to send payoff-irrelevant messages to second-stage players. Equilibrium outcomes of approximating games can be made into equilibrium outcomes of the cheap-talk extension of those games by letting players send completely uninformative messages say all cheap-talk messages are sent with equal probability. With a sufficiently rich cheap-talk space, the limit outcome will prescribe non-atomic behavior for first-stage players. Hence, infinite games with sufficiently rich cheap talk have a subgame perfect equilibrium (Corollary 3). The existence of subgame perfect equilibria in stage games with cheap talk was first established by Harris, Reny, and Robson (1995) (see their discussion following their Theorem 45). Corollary 2 can be derived directly from Harris, Reny, and Robson s results (Manelli (2001)). Our proofs differ from those in Harris, Reny, and Robson (1995). These authors derive their existence result from a general upper hemi-continuity property. The subgame perfect outcomes of games with correlation devices are distributions on spaces that include payoff-irrelevant variables, the realization of the public randomization devices. Consider the projection of those distributions (i.e., their marginals) on the payoff-relevant variables. Harris, Reny, and Robson prove the upper hemi-continuity of the correspondence that maps games with public randomization devices into the projection of subgame perfect outcomes on payoff-relevant variables. Their proof uses an argument reminiscent of backward induction. Proceeding from the last to the first stage, a backward step identifies for each history the combinations of actions and continuation-payoffs that are optimal in that stage. Proceeding from the first to the last stage, a forward induction step shows that there are equilibrium strategies that realize them. 3 The proof of Theorem 1 and Corollary 1 shows that certain equilibrium properties of the 3 The interested reader should consult Harris, Reny, and Robson s paper for precise statements, important additional results, and an accurate description of their method of proof. 4

5 approximating outcomes are inherited by the limit distribution. Essentially given any history, the expected payoff of the continuation outcome converges to the expected payoff derived from the limit distribution. Equilibrium strategies are then constructed, when possible (i.e. when the limit distribution is feasible), from the marginal and conditional probabilities implicit in the limit distribution. This procedure identifies subgame perfect equilibrium strategies on the equilibrium path. The procedure is completed by constructing strategies off the equilibrium path. The proof of Theorem 2 has a geometric quality. The theorem is, technically, a bangbang result; it permits the purification of the mixing implicit in the limit distribution. The limit distribution implies certain behavior on the part of first-stage players. The implicit continuation, however, may involve correlation. The idea behind Theorem 2 is to identify an alternative continuation that does not involve correlation but that still provides the correct incentives for first-stage players. The set of possible continuations that provide correct incentives for first-stage players is a convex, compact set of functions, and must therefore have extreme points. It is then shown that extreme points of that set do not involve correlation. Therefore, there exists a continuation that provides the correct incentives for first-stage players and does not prescribe correlation. We conclude the introduction by listing some salient features of Harris, Reny, and Robson s non-existence example. First, there are two stages and two players in each stage. Only one player, say player 1 in the first stage, has a continuum of choices. If player 1 s choices are discretized, the game has a subgame perfect equilibrium. Suppose that in the infinite game player 1 s choice set is the interval [ 1, 1]. Imagine finite games indexed by the integer n where player 1 s choice set is replaced by the finite set { 1,..., 2/n, 1/n, 0, 1/n, 2/n,..., 1} but where payoff functions and other players choice sets remain unchanged. Second, the example is constructed so that in the subgame perfect equilibrium of the n th game player 1 randomizes between 1/n and 1/n with equal probability. Second-stage players respond by choosing Up when they observe 1/n and Down when they observe 1/n. Thus, second-stage players use the randomization in the first stage to coordinate their actions between two Nash equilibria of the second-stage game, (Up,Up) and (Down, Down). Third, as n increases, the n th game and the infinite game become closer. The subgameperfect-equilibrium outcomes of the finite games specify, in the limit, that player 1 choose zero with certainty and that second stage players continue to coordinate their actions, playing (Up,Up) and (Down,Down) with equal probability. This, however, is impossible. Secondstage players choose their actions independently and player 1 s choice, always zero, does not 5

6 help them correlate second-stage actions. Were each player independently to select Up and Down with equal probability, the outcome of the second stage would include (Up,Down) and (Down, Up). Finally, payoffs in the example are specified so that no subgame perfect equilibrium exists: in order to induce second-stage correlation which player 1 prefers to no correlation player 1 is willing to mix between some positive and some negative action. Using non-zero actions, however, is costly to player 1 and such cost increases as the selected actions move away from zero. Player 1 would therefore prefer to randomize between two actions, a positive and a negative one, as close to zero as possible. Since player 1 s choice set is an interval, there is no strictly positive or strictly negative action closest to zero. There is no least expensive way to ensure second-stage correlation. 2 Definitions and Notation We consider the following game. In the first stage, each player i in a finite set I selects in isolation an action a i from a set A i. After observing the actions selected in the first stage, each player j in a finite set J selects also in isolation an action b j from a set B j. 4 Action spaces are compact metric spaces. All payoff functions {U l } l I J are continuous, real valued, and have as arguments the actions selected by all players. The game described is summarized by Γ = [I, J, {A i } i I, {B j } j J, {U l } l I J ]. For convenience, A denotes the Cartesian product of the action spaces {A i } i I, a is a typical element of A, and a i is the element obtained by removing the i th component from a. The same notational convention applies to other objects, for instance, B = j J B j, B l = j J\{l} B j, b B, and b j = (b 1,..., b j 1, b j+1,... ). Given any metric space X, M(X) denotes the set of probability distribution on X. Probability measures are defined on the corresponding Borel σ-fields. Unless otherwise specified, all sets and functions mentioned are measurable. For any first-stage player i I a typical (mixed) strategy is a distribution over i s available actions, α i M(A i ). For any second-stage player j J a typical (behavior) strategy is a measurable function that assigns to each first-stage realization a A a distribution over j s actions, β j : A M(B j ). Let α be the collection (α 1, α 2,..., α I ) of first-stage strategies and β the collection 4 For notational simplicity, a player that moves in both stages is split into two players, one per stage, each with the payoff function of the original player. Any subgame perfect equilibrium of the game so obtained will also be a subgame perfect equilibrium of the original game. Thus, we assume that no player moves in both stages. 6

7 (β 1, β 2,..., β J ) of second-stage strategies. Then, β(a) = (β 1 (a), β 2 (a),..., β J (a)) indicates the choices made by second-stage players after observing the realization a. Sometimes, specially when integrating, we will write β j (db j a) and β(db a) = (β 1 (db 1 a) β 2 (db 2 a)... β J (db j a)). Player i s expected payoff when players move according to (α, β) is E[U i α, β]. Similarly, E[U i a i, α i, β] is player i s expected payoff when i chooses a i and the other players move according to (α i, β). The same notational convention will be used to represent expected payoffs given other combinations of play. A profile (α, β) is a subgame perfect equilibrium of Γ if i I, E[U i α, β] E[U i ā i, α i, β], ā i A i, and (1) a A, j J E[U j a, β(a)] E[U j a, b j, β j (a)], bj B j. (2) Condition (1) states that every first-stage player selects a best response. Condition (2) requires that for any realization a in the first stage, β(a) be a Nash equilibrium of the ensuing second-stage game. For any a A, let NE(a) be the set of Nash equilibria of the second-stage game defined after a is realized. Thus, β(a) NE(a) if and only if (2) holds for β(a). The outcomes of approximating finite games will be used to establish equilibrium properties of limit infinite games. Typically, given a strategy profile β the outcome of a game is defined as the distribution on terminal nodes (i.e., an element of M(A B)) generated by the given profile. It is useful to represent the outcome as a distribution on a slightly different space: for any history a A (i.e., any realization of the first stage game), the strategies β determine a continuation path, i.e., an element η of j J M(B j). By considering outcomes as distributions on M[A j J M(B j)] any element of the support specifies a first-stage realization and its continuation path. Formally, define the outcome of playing the profile (α, β) as ( ) λ = α i f 1, where f(a) = (a, β(a)), i I and i I α i is the product distribution obtained from first-stage strategies. 5 Finally, η denotes a typical element of j J M(B j), λ A and λ Ai denote the marginal distributions of λ on A and A i respectively. 5 The symbol is used to indicate products of measures as well as Cartesian products. Abusing notation we will sometimes write β(a) = (β 1(a), β 2(a),..., β J(a)) for j J βj(a) and vice versa. 7

8 3 Results Fix a game Γ = [I, J, {A i } i I, {B j } j J, {U l } l I J ] for the remainder of the paper. Finite games, i.e., games with finite action spaces, that approximate Γ can be constructed by discretizing Γ. We say that a sequence of finite games Γ n = [I, J, {A n i } i I, {B n j } j J, {U l } l I J ], n = 1, 2,..., converges to the limit game Γ if for all players i I, j J, (i) A n i A i and B n j B j n, and (iii) lim inf n A n i = A i and lim inf n B n j = B j. The last condition requires that feasible actions in the limit game be approximated by feasible actions of the approximating games. Thus, finite approximating games are simply increasingly finer discretizations of the infinite game. Since finite games always have a subgame perfect equilibrium, any sequence Γ n of finite approximating games yields a sequence of equilibrium outcomes λ n. These outcomes are distributions (on a compact space) and therefore converge in a subsequence to a limit distribution λ. Our first results are Theorem 1 and Corollary 1. Corollary 1 states that if there are strategies of the limit game that implement the limit distribution λ, then there are subgame perfect equilibrium strategies that also implement λ. Theorem 1 offers somewhat weaker conditions: if there are strategies in Γ that yield to first-stage players the same expected payoffs they would obtain under λ, and the second-stage strategies are sequentially rational for second-stage players, then a subgame perfect equilibrium exists. commentary follows the formal presentation of the theorem. A more informative Theorem 1 Let Γ n, n = 1, 2,..., be a sequence of finite games converging to a limit game Γ, and let λ n be a subgame perfect equilibrium outcome of Γ n. Through a subsequence, λ n converges to a limit distribution λ. Let γ( a) be a version of the regular conditional distribution of η j J M(B j) given a derived from λ. If there are any second-stage strategies β = (β 1, β 2,..., β J ), such that (i) i I, E[U i a i, λ A i, β ] = E[U i a i, λ A i, γ] λ Ai a.e., and (ii) β (a) N(a) λ A a.e., then there exists a subgame perfect equilibrium (α, β) where α i = λ Ai β j (a) = β j (a) λ A-a.e. for all j J. for all i I, and 8

9 The limit distribution λ implicitly specifies certain behavior on the limit game: first-stage players must move according to their corresponding marginal distribution λ Ai ; second-stage players must move according to the conditional distribution γ( a) derived from λ. Thus, E[U i a i, λ A i, γ] represents player i s expected payoff of choosing the action a i when all other players move according to λ. Condition (i) states that there are strategies that give firststage players the same payoffs they would obtain under λ. Condition (ii) simply states that second-stage players choose best responses almost always. The proof of Theorem 1 (and Corollary 1) constructs equilibrium strategies of the limit game (on the equilibrium path) from the marginal and conditional probabilities of the limit distribution λ. The limit distribution λ, however, conveys little or no information about behavior off the equilibrium path. As previously noted, any element in the support of an outcome λ n specifies a first-stage realization a n A n and a continuation η n j J M(B j). Any element in the support of the limit distribution can be approximated by elements in the support of λ n. This property and the continuity of expected payoffs with respect to outcomes are used repeatedly in the proof to show that the equilibrium inequalities of the approximating equilibria hold in the limit. Although the details are different, we used the same ideas elsewhere to prove the analogue of Corollary 1 for signaling games with infinite type and action spaces. Therefore, we relegate the proof to the appendix. We believe that a similar relationship between equilibrium outcomes of finite approximating games and those of infinite games might hold in other classes of infinite games. Our proof of Theorem 1 may help elucidate that relationship. Proof of Theorem 1: provided in the Appendix. It follows as a corollary to Theorem 1 that the limit distribution is a subgame perfect equilibrium outcome of the limit game if and only if it is feasible. Thus, the non-existence example briefly discussed in the introduction captures the essence of the problem. Corollary 1 The limit distribution λ is a subgame perfect equilibrium outcome of the limit game Γ if and only if there are strategies in Γ that generate λ. Proof: If the strategy profile (α, β ) generates λ, α i = λ Ai for all i I. Second-stage strategies β are equivalent to γ: note that γ : A M[ j J M(B j)], and β : A j J M(B j). Both (α, γ) and (α, β ) generate λ. Therefore, γ(a) = δ β (a) α a.e. (where δ η M[ j J M(B j)] represents the degenerate distribution at η j J M(B j)). Hence, (i) is satisfied. The outcomes λ n belong to M[graph(NE n )] where NE n (a) is the set of second-stage Nash equilibria in Γ n after the realization a A. Since lim sup n graph(ne n ) graph(ne) 9

10 (Lemma 3 in the Appendix), λ belongs to M[graph(NE)]. Hence, (ii) is also satisfied. Q.E.D. Corollary 1 implies, albeit for a much smaller class of games, Harris, Reny, and Robson s existence result for games with public, randomization devices: adding such device to Γ allows second-stage players to coordinate their actions, thus making the limit distribution λ feasible. When λ is feasible (without altering the game Γ), then randomization devices are not necessary to obtain a subgame perfect equilibrium. Corollary 1 does not follow immediately from Harris, Reny, and Robson s upper hemicontinuity theorem. To see this, consider a sequence of finite games that converge to Γ, and the corresponding subgame perfect outcomes λ n that converge to a feasible limit distribution λ. Extend Γ with a randomization device whose realizations z come from some set Z. The upper hemi-continuity result in Harris, Reny, and Robson (1995) identifies a subgame perfect equilibrium such that (a) each first-stage player i moves according to λ Ai ; (b) every secondstage player j uses a strategy β (a, z); and (c) the outcome of the profile (λ A, β ) is a distribution over A Z j J M(B j) whose marginal distribution over the payoff-relevant variables A j J M(B j) is λ. Subgame perfection implies that β (a, z) is a second-stage Nash equilibrium for any (a, z). To obtain Corollary 1, one must find second-stage strategies β (a) that depend only on first-stage realizations, that generate λ, and that constitute a second-stage Nash equilibrium, i.e., β (a) NE(a) for all a. Any such strategies β must coincide (almost everywhere) with the strategies obtained by integrating β (a, z) over z (by (c) above). It is not immediate that the strategies β (a), obtained by integrating out z, are a second-stage Nash equilibrium. Even if the limit distribution λ is not feasible, Γ may have a subgame perfect equilibrium. This is our next result. Theorem 2, the main technical contribution in the essay, establishes that whenever the limit distribution exhibits diffused behavior there are strategies in Γ that satisfy the conditions of Theorem 1, and thus an equilibrium exists. Given a correspondence ϕ, a measure λ, and a set of continuous functions, Theorem 2 establishes the existence of a measurable selection β = (β 1,..., β J ) that satisfies certain properties. Theorem 2 Let {A i } i I and {B j } j J be finite families of compact metric spaces, {U i } i I be continuous real-valued functions defined on A B, ϕ : A j J M(B j) be a correspondence with closed graph, λ be any probability measure on graph(ϕ( )) with regular conditional distribution of η given a denoted by γ( a). If for any two i, i I, the marginal distributions λ Ai and λ Ai are non-atomic, then there are measurable functions β j : A M(B j), j J 10

11 such that (i) i I, E[U i a i, λ A i, β ] = E[U i a i, λ A i, γ] λ Ai a.e., and (ii) β (a) ϕ(a) λ A a.e. In our application ϕ(a) is the second-stage Nash correspondence NE(a), λ is the limit distribution, and β is a second-stage strategy-profile. Condition (ii) states that β ( ) is an a.e. selection, i.e. a Nash equilibrium of the second stage almost always; condition (i) states that for almost any action a i, player i receives the same expected payoff under β than under the limit distribution λ. A few comments before the proof of Theorem 2 are in order. The limit distribution λ can be infeasible in two main cases. The first case is illustrated by the example described in the introduction: all through the sequence of approximating games a first-stage player uses a mixed strategy. Second-stage players focus on first-stage realizations to correlate their actions. Although the support of the mixed strategy collapses in the limit rendering secondstage coordination impossible, the limit distribution still requires second-stage players to correlate their moves: the conditional distribution of η given a derived from λ prescribes mixing among different Nash equilibria of the second stage. The second case in which the limit distribution λ is infeasible occurs when second-stage strategies oscillate at an increasing rate along the sequence of approximating games. As a result of such oscillation, the limit outcome prescribes correlation among second-stage players. To see this, imagine for instance a game with two players in each stage. In the n th approximating game, player 1 plays all her available actions ( A n 1 = {k/n, k = 1,..., n}) with equal probability (α1 n(a 1) = 1/n, a 1 A n 1 ). (For simplicity, player 2, a dummy player, uses a similar strategy.) Second stage players coordinate their actions on player 1 s choices: β n 1 (a 1, a 2 ) = β n 2 (a 1, a 2 ) = Up if a 1 = k/n for k even, and β n 1 (a 1, a 2 ) = β n 2 (a 1, a 2 ) = Down if a = k/n for k odd. Thus second-stage strategies jointly oscillate between (Up, Up) and (Down, Down) as the finite games approach the limit game. The limit distribution λ stipulates that for any given (a 1, a 2 ), the second-stage outcome be (Up, Up) or (Down, Down) with equal probability. Hence, the limit distribution is infeasible. 6 A stage-correlation device solves this problem by making the limit distribution feasible. Theorem 2 finds, instead, that the limit distribution can be purified. A limit distribution λ is infeasible only when its conditional distribution γ( a) randomizes among different secondstage Nash equilibria. Theorem 2 establishes the existence of a second-stage strategy profile 6 One may define all payoffs to be identically one for all actions. Then the strategies of the approximating games form a subgame perfect equilibrium but the limit distribution is still infeasible. 11

12 β that avoids the randomization in γ( a), prescribes equilibrium behavior for second-stage players (i.e., it is a selection of the Nash correspondence), and guarantees first-stage players the same expected payoffs they obtained under γ. Theorem 2 extends Liapunov s Theorem in the following sense. Expected payoffs under λ for first-stage players are (E[U 1 λ],..., E[U I λ]) = (E[U 1 λ A, γ],..., E[U I λ A, γ]). This I-tuple belongs to the convex hull of ϕ dλ A. Richter s version of Liapunov s Theorem (see for instance Theorem 3, D.II.4, page 62, Hildenbrand (1974)) implies the existence of a measurable selection β of ϕ such that E[U i λ A, β ] = E[U i λ A, γ], i I. (3) This equivalence, however, is not sufficient to prove our results: even though β yields i s expected payoffs under λ, player i may have an incentive to choose a strategy different from λ Ai. Player i will not mix according to λ Ai if i s expected payoff E[U i a i, λ i, β ] for different actions a i (in the support of λ Ai ) is not the same. To guarantee that player i moves according to λ Ai, β must satisfy a stronger requirement such as condition (i). Integrating both sides of (i) with respect to λ Ai, (3) obtains. The proof of Theorem 2 involves three main parts. The first one uses ideas in Dvoretzky, Wald, and Wolfowitz (1951). The result is first proved for the case where the range of the correspondence NE( ) is finite, and then it is extended to the continuum case by a limiting argument. The second part uses ideas in Lindenstrauss (1966). It is shown that the set W of candidate solutions, i.e., all second-stage continuations f that yield the desired payoffs (E[U i a i, λ Ai, f] = E[U i a i, λ Ai, γ], for most a i and all i ), is convex and compact and therefore has an extreme point. Demonstrating that any member of W that prescribes correlation cannot be an extreme point is the last step of the proof. An illustrative (but not entirely accurate) description of the third part follows. The objective is to find a non-trivial continuation f 0 that yields expected payoff zero (E[U i a i, λ Ai, f ] = 0 for λ Ai -almost all a i and all i). Armed with such a continuation, it is possible to show that any candidate f W that prescribes correlation cannot be an extreme point: the linearity of expected payoffs implies that both f + f and f f belong to W. To see how the existence of such f is established, suppose there are only two first-stage players. Using non-atomicity, their choice spaces are divided into two sets of equal measure, thus partitioning A 1 A 2 into four sets. As indicated in the graph below, we construct a function f that takes values 1 and 1 in alternating sets. 12

13 A A 1 If the payoff functions U i were constant, f would be the desired continuation f. The continuation f and an application of the Banach Inverse Theorem (Lemma 2) are used to prove the existence of the desired f with zero expected payoff. Proof of Theorem 2: Suppose first that the set a A ϕ(a) (which in our application is the set of Nash equilibria of the second-stage game) is finite, and represent its elements by η k, k = 1, 2,..., K. Given a first-stage realization a A, each η k is played with certain probability γ k (a). Thus, γ is represented by measurable functions γ k : A [0, 1], k = 1, 2,..., K, with K k=1 γk (a) = 1, almost everywhere in A. Let E k = {a A : γ k (a) > 0}. For any g = (g 1,..., g K ) (L (A, λ A )) K, define T i g = K k=1 A i 1 E k [ B T g = (T 1 g,..., T I g). U i (a i, a i, b) dη k ] g k (a i, a i ) dλ A i, i I, and The expression T i γ(a i ) (i.e, T i γ evaluated at a i supp[λ Ai ]) represents, in our application, player i s expected payoff when i selects a i and i s opponents play according to γ. The operators T i and T have several continuity properties. Let g = max 1 k K gk and T g = max i I T ig. (4) The operators T i, T are linear and continuous with respect to the the described norms; they are also weak continuous. 7 Let W = T 1 (T γ) {g : K g k (a) = 1 λ A a.e., and 0 g k 1, k = 1,..., K, }. (5) k=1 The set W contains potential continuations of the game that give first-stage players the same payoff as γ; indeed γ belongs to W. The set T 1 (T γ) is weak closed and convex by continuity 7 For n = 1, 2,..., let g n (L (A, λ A)) K. Weak w continuity is understood in the following sense: [g n w w g T g n T g], where g n g if for k = 1,..., K, gn k converges to g k with respect to the weak topology (denoted gn k w g k w w ). Similarly, T g n T g if for all i I, T ig n T ig. 13

14 and linearity. The second set in the definition of W is a weak closed, convex, subset of the unit ball. Hence W is weak compact (Banach-Aloglou Theorem), and since it is convex it has an extreme point (Krein-Milman Theorem). To complete the proof in the case where a A ϕ(a) is finite, it suffices to show that there is an element g W all of whose coordinate functions g k are indicator functions. We now prove that all extreme points of W have this property. Suppose then that g W has some component that is not an indicator function. Relabeling coordinates if necessary, suppose such component is g 1. To demonstrate that g is not an extreme point of W we must identify a function f 0 such that g ± f W. It follows from the definition of W that the candidate function f must satisfy three conditions: T f = 0, (6) 0 f k + g k 1, for k = 1,..., K, and (7) K f k = 0 λ A a.e. (8) k=1 Given (7) and (8), g ± f belong to the second set in (5); (6) implies that they also belong to the first. Lemma 1 below establishes some properties of any g W that are used to show that such an f exists. For any two elements η k, η k of a A ϕ(a), define U i (a, η k, η k ) = U i (a, b) dη k B B U i (a, b) dη k. Lemma 1 Let g = (g 1,..., g K ) be any element of W. Suppose that for some k, λ A ({a : 0 < g k (a) < 1}) > 0. Then there is an ɛ > 0, a coordinate g k, k k, and a set D = i I D i with D i A i i I, such that (a) λ Ai (D i ) > 0 i I; (b) a, a D, U i (a, η k, η k ) U i (a η k, η k ) < ɛ, (c) a D, ɛ < g k (a) < 1 ɛ and ɛ < g k (a) < 1 ɛ. The proof of the lemma is provided in the Appendix; only a brief intuition is offered here. Suppose g is an element of W and g k is not an indicator function. Then there must be some set where g k (a) takes values strictly below one and strictly above zero. Since the different coordinate functions must add up to one, there must be another function g k, and a set of positive measure in which both functions are bounded away from zero and one. This 14

15 reasoning leads to (a) and (c). Uniform continuity of the payoff functions yields condition (b). We proceed with the proof of Theorem 2. Relabeling coordinates if necessary, let g k = g 1 and g k = g 2 (Lemma 1 (c)). Let X be the subspace of (L (A, λ A )) K defined by X = {f : f 2 = f 1, f 1 (a) = f 2 (a) = 0 a / D, and f k = 0 for 3 k K}. Note that X is a closed linear subspace and therefore a Banach space. Define Y = T X. Recall that we must find f X, f 0 with T f = 0. Suppose f = 0 is the unique solution of T f = 0 in X. Then T 1 : Y X is a well-defined function. The space Y with the norm T defined below is a Banach space: y Y, y T = 1 2 y T 1 y. (9) Lemma 2 Let T be a bounded linear operator from a Banach space X onto a Banach space Y. Then, the equation T f = 0 has f = 0 as its unique solution if and only if C > 0 such that f X, f C T f. The proof of the lemma is provided in the Appendix. We now show that and then use Lemma 2. C > 0, f c X such that f c > C T f c T, By hypothesis, at least two marginal distributions λ Ai and λ Ai are non-atomic. Relabeling first-stage agents if necessary, suppose that λ A1 and λ A2 are the non-atomic distributions. For l = 1, 2 partition D l (identified in Lemma 1) into D l and D l so that D l D l =, D l D l = D l, and λ Al (D l ) = λ A l (D l ). (10) Such partition exist because λ Al is non-atomic. Let D + = [(D 1 D 2) (D 1 D 2)] D = [(D 1 D 2) (D 1 D 2)] I D i, i=3 I D i, i=3 15

16 and let f = ((1 D + 1 D ), (1 D + 1 D ), 0,, 0). Then, for any i I T i f = U i (a, b) dη 1 (1 D + 1 D ) dλ A i = = = = D i D i B ( B D i ) U i (a, b) dη 1 U i (a, b) dη 2 (1 D + 1 D ) dλ A i B B U i (a, b) dη 2 (1 D + 1 D ) dλ A i U i (a, η 1, η 2 )(1 D + 1 D ) dλ A i D i U i (a, η 1, η 2 ) 1 D + dλ A i U i (a) 1 D dλ A i D i D ( ) i ( ) sup U i (a, η 1, η 2 ) 1 D + dλ A i inf U i(a, η 1, η 2 ) 1 D dλ A i. a D D i a D D ( ) i sup U i (a, η 1, η 2 ) inf U i(a, η 1, η 2 ) 1 D + dλ A i a D a D D i ɛ 1 2 λ A i (D i ), (11) where the first four equations follow using the definition of T i, rearranging terms, and using the definition of U i (a, η 1, η 2 ). The first inequality follows by taking extreme values over D, which includes the area of integration. The last two lines follow from the fact that 1 D + dλ A i = 1 D dλ A i, i I. D i D i To see this, note that (D 1 D 2 ) (D 1 D 2 ) = and therefore using (10) λ A1 λ A2 [(D 1 D 2) (D 1 D 2)] = λ A1 (D 1)λ A2 (D 2) + λ A1 (D 1)λ A2 (D 2) = (λ A2 (D 1) + λ A1 (D 1))λ A2 (D 2) = λ A1 (D 1 )λ A2 (D 2) = λ A 1 (D 1 )λ A2 (D 2 ). 2 Thus, λ A (D + ) = 1 2 λ A(D). Hence D i 1 D + dλ A i = 1 2 λ A i (D i ). The same argument applies to D. 16

17 For C > 0, let f c = C f. Then, using the definitions of T and ((9) and (4) respectively), the definition of f c, and the inequality (11), T f c T = = max i I 1 = max i I 1 2 T if c f c 2 C T i f C 1 max i I 2 C ɛ 1 2 λ A i (D i ) C < C = f c. By Lemma 2 there exists f X, f 0, such that T f = 0. Let f = ɛf 2 f. It is immediate that f satisfies (6). For k = 1, 2, f k < ɛ/2, and for k 3 f k = 0. Thus f satisfies (7). By construction f also satisfies (8). Then g ± f W and therefore g cannot be an extreme point. This completes the proof when a A ϕ(a) is finite. Using the argument in Dvoretzky, Wald, and Wolfowitz (Section 4, 1951), it follows that the theorem holds when a A ϕ(a) is compact, which is the case here. Roughly a sequence of increasingly finer partitions of B is constructed. Fix a partition. Expected payoffs are defined so as to be constant on each partition element. Our theorem for the finite case then applies because each partition has finitely many elements. As the partitions become increasingly finer, a limiting argument yields the desired result. Q.E.D. Note that the proof of Theorem 2 could be applied directly to the case of public randomization devices; no extreme point involves correlation. To conclude this section, we illustrate one potential application of our results with two corollaries that follow from combining Theorems 1 and 2. According to the limit distribution λ, any first-stage player i must choose the strategy λ Ai. Corollary 2 states that if the limit distribution stipulates sufficiently diffused behavior for at least two first-stage players, then Γ has a subgame perfect equilibrium. We emphasize, however, that λ itself need not be a subgame perfect equilibrium outcome. Corollary 2 Let Γ n, n = 1, 2,..., be a sequence of finite games converging to a limit game Γ, and let λ n be a subgame perfect equilibrium outcome of Γ n. Through a subsequence, λ n converges to a limit distribution λ. If for two first stage agents, i, i I, the marginal distributions λ Ai and λ Ai are non-atomic, then Γ has a subgame perfect equilibrium. 17

18 The diffuse behavior in the first stage is used in place of a public randomization device by second-stage players to coordinate their actions. Having at least two first-stage players with non-atomic behavior guarantees that no first-stage player has, individually, the ability to manipulate to her advantage the continuation of the game. Corollary 2 can be obtained directly from Harris, Reny, and Robson s results (Manelli (2001)). As an existence result, Corollary 2 is not entirely satisfactory because imposes conditions on limit distributions, not on primitives. Also, it might be difficult to check the non-atomicity of the limit distribution in applications. Corollary 3 states that if the infinite game Γ is extended to incorporate cheap talk, i.e., first-stage players ability to send payoff irrelevant messages to second-stage players, then the new game has a subgame perfect equilibrium. Equilibrium outcomes of finite approximating games without cheap talk can be made into equilibrium outcomes of games with cheap talk: players use completely uninformative messages such as randomization with equal probability over all their cheap-talk possibilities. If the cheap-talk spaces are sufficiently rich, the unit interval for instance, then the limit distribution obtained from the approximating finite outcomes prescribes non-atomic behavior for first-stage players who randomize over all their cheap-talk messages. This argument leads to the following corollary. Corollary 3 was first established by Harris, Reny, and Robson (1995) in their discussion following their Theorem 45. Corollary 3 Let Γ n, n = 1, 2,..., be a sequence of finite games converging to a limit game Γ, and let λ n be a subgame perfect equilibrium outcome of Γ n. Through a subsequence, λ n converges to a limit distribution λ. Extend Γ so that first-stage players can send to second-stage players payoff-irrelevant messages from the unit interval or any other compact metric space with a continuum of elements. Then the extended game has a subgame perfect equilibrium. Furthermore, there is a subgame perfect equilibrium of the extended game that generates the distribution λ over the payoff-relevant variables. 4 Conclusions We conclude with a few remarks and comments on related literature. 1. Harris, Reny, and Robson (1995) prove many interesting results that we have not mentioned. They also consider more general games than those studied in this essay, mainly games with arbitrarily many stages. We believe that our approach would extend to those games but the details are by no means trivial. It is likely that the extension 18

19 would be as lengthy as Harris, Reny, and Robson s original treatment. Our proof emphasizes certain regularities observed in different types of infinite games. It may prove useful in analyzing other infinite games with imperfect or incomplete information. 2. Formally the games considered by Simon and Zame (1990) are not stage games. An interpretation that Simon and Zame favor, however, is that their model is a reduced form of a two-stage game where the second-stage players are not explicitly modeled. Instead, second-stage players are replaced by a payoff correspondence specifying feasible first-stage payoffs for any realization in the first stage. (The payoff correspondence is obtained from the second-stage Nash equilibrium correspondence by associating to every Nash equilibrium its corresponding payoffs for first-stage players.) Simon and Zame prove that if the payoff correspondence is upper hemi-continuous and convex-valued, then it has a measurable selection for which an equilibrium exists. Our results complement Simon and Zame s. We replace the assumption of a convex payoff-correspondence with the non-atomicity of first-stage behavior. 3. Reny and Robson (1995) provide an alternative proof of the main existence theorem in Harris, Reny, and Robson (1995) using Simon and Zame s result. Their proof also uses a backward and forward step. It is shorter than the proof in Harris, Reny, and Robson s paper but focus entirely on equilibrium payoffs rather than on equilibrium paths. 4. As noted, the addition of cheap talk solves the non-existence problem in stage games and in signaling games with infinite choice-sets. It would be of interest to identify the class of games in which this property holds. Harris, Stinchcombe, and Zame (1999) report an example of an incomplete-information game in which cheap talk fails to restore existence. They also use non-standard analysis to study in a single framework cheap talk, public randomization devices, and sharing rules. 5. Chakrabarti (1999) proves the existence of epsilon-perfect equilibria for a general class of dynamic games that includes the games we study. 6. Börgers (1991) defines a notion of approximation for a class of games and proves that the map from games to pure subgame perfect equilibrium outcomes is upper hemicontinuous. Börgers result only applies to outcomes generated by pure strategies. Since finite games may not have pure strategy equilibria, his result is not enough to yield existence. Our proof of Theorem 1 and Corollary 1 uses the outcomes of behavior strategies. 19

20 7. There are certain similarities between the games considered in this essay and games of incomplete information. Rosenthal and Radner (1982) and Milgrom and Weber (1985) study games in which players observe some private information (from a continuum of alternatives) and then simultaneously and independently choose an action. 8 Under certain distributional assumptions on information, they show that there exist equilibria in behavior strategies. If in addition action spaces are finite, they prove that pure strategy equilibria exist (or equivalently, that mixed-strategy equilibria can be purified ). This is not the case, however, when there is a continuum of actions. Khan, Rath, and Sun (1999) have constructed a well-behaved game with intervals as choice sets and with no pure strategy equilibrium. 8 A strategy maps a player s types into actions. In stage games, a second-stage strategy maps first-stage realizations into actions. Thus, the vector of first-stage realizations is the analogue of the vector of types. 20

21 5 References Al-Najjar, N. and E. Solan, Equilibrium Existence in Incomplete Information Games with Atomic Posteriors, working paper, MEDS, Northwestern University, May Balder, E.J., Generalized Equilibrium Results for Games of Incomplete Information, Mathematics of Operations Research 13 (1988): Billingsley, P., Convergence of Probability Measures. New York: John Wiley, , Probability and Measure. New York: John Wiley, Börgers, T., The Upper Hemi-Continuity of the Correspondence of Subgame Perfect Equilibrium outcomes, Journal of Mathematical Economics 20 (1991): Chakrabarti, S., Finite and infinite action dynamic games with imperfect information, Journal of Mathematical Economics 32 (1999): Dvoretzky, A., A. Wald, and J. Wolfowitz, Elimination of Randomization in Certain Statistical Decision Procedures and Zero-Sum Two-Person Games, Annals of Mathematical Statistics 22 (1951): Harris, C. J., Reny P. J. and A. J. Robson, The Existence of Subgame Perfect Equilibrium in Continuous Games with Almost Perfect Information: a Case for Extensive-Form Correlation, Econometrica 63 (1995): Harris, C. J., Stinchcombe, M. B., and W. Zame, The Finitistic Theory of Infinite Games, photocopy, (1999). Hildenbrand, W., Core and Equilibria of a Large Economy. Princeton, (1974). Princeton University Press, Jackson, M., L. Simon, J. Swinkels, and W. Zame, Communication and Equilibrium in Discontinuous Games of Incomplete Information, photocopy, (2001). Khan, M. Ali and Y. Sun, Pure Strategies in Games with Private Information, Journal of Mathematical Economics 24 (1995): Khan, M. Ali, Rath K. P., and Y. Sun, On a private information game without pure strategy equilibria, Journal of Mathematical Economics 31 (1999): Kechris, A. S., Classical Descriptive Set Theory. Springer-Verlag, New York, (1995). Lindenstrauss, J., A Short Proof of Liapunov s Convexity Theorem, Journal of Mathematics and Mechanics, 15, 6, (1966): Manelli, A.M., Cheap Talk and Sequential Equilibria in Signaling Games, Econometrica, 66 (1996):

22 Manelli, A.M., Subgame Perfect Equilibria in Stage Games, forthcoming in Journal of Economic Theory. Milgrom, P., and R. J. Weber, Distributional Strategies for Games with Incomplete Information, Mathematics of Operations Research 10 (1985): Parthasarathy, K., Probability Measures on Metric Spaces. (1967). New York: Academic Press, Radner, R., and R. W. Rosenthal, Private Information and Pure Strategy Equilibria, Mathematics of Operations Research 7 (1982): Reny, P. J. and A. J. Robson, A Short Proof of Existence of Subgame Perfect Equilibrium in Continuous Stage Games With Public Randomization, photocopy, (1995). Simon, L., and W. Zame, Discontinuous Games and Endogenous Sharing Rules, Econometrica 58 (1990): Van Damme, E., Equilibria in Non-Cooperative Games, in Surveys in Game Theory and Related Topics (H. Peters and O. Vrieze, Eds.) C.W.I. Tract 39. Amsterdam,

23 6 Appendix Lemma 3 is used in the proof of Corollary 1 and in the proof of Theorem 1 step three. Recall that NE(a) is the set of second-stage Nash Equilibria in Γ after the first-stage realization a. Lemma 3 Let Γ n, n = 1, 2,..., be a sequence of finite games and NE n be the corresponding sequence of second-stage Nash equilibrium correspondences. Suppose Γ n converges to Γ. Then lim sup graph(ne n ) graph(ne). n As a consequence, NE(a) is non-empty a A, and if λ n is a subgame perfect equilibrium outcome of Γ n and λ n λ, then supp[λ] graph(ne). Proof of Lemma 3: If (a, η 1,..., η J ) belongs to lim sup graph(ne n ), then there is a sequence (a n, η n 1,..., ηn J ) graph(nen ) for n = 1, 2,..., such that (a n, η n 1,..., ηn J ) converges to (a, η 1,..., η J ). For any j J, E[U j a n, η n 1,..., η n J ] E[U j a n, b n j, η n j], b n j B n j. (12) For any b j B j, there is b n j Bn such that b n j converges to b j. Taking limits in (12), it follows that (a, η 1,..., η J ) is in graph(ne). Q.E.D. Proof of Theorem 1: Let α i = λ Ai for all i I. The proof constructs second-stage strategies β and shows that (α, β) satisfies (1) and (2). The equilibrium strategies β j for second-stage players are constructed in successive steps by defining strategies that satisfy (2) and then refining them to satisfy (1). λ n. Note for later use that there is a subgame perfect equilibrium (α n, β n ) in Γ n that generates The proof proceeds in four steps. [ 1.] Second-stage equilibrium. N(a). By hypothesis (ii), there is a set A A, λ A (A ) = 1, such that for any a A, β (a) Let β be any measurable everywhere selection from the Nash Equilibrium correspondence NE( ). Such a selection exists because NE( ) is upper hemi-continuous and compact valued (see for instance, Arsenin-Kunugui Theorem, Kechris (1995), Theorem 35.46, page 297). For each j J, define β j (a) = { β j (a) if a A β j (a) 23 otherwise.