INTERIM CORRELATED RATIONALIZABILITY IN INFINITE GAMES JONATHAN WEINSTEIN AND MUHAMET YILDIZ Abstract. In a Bayesian game, assume that the type space is a complete, separable metric space, the action space is a compact metric space, and the payoff functions are continuous. We show that the iterative and fixed-point definitions of interim correlated rationalizability (ICR) coincide, and ICR is non-empty-valued and upper hemicontinuous. This extends the finite-game results of Dekel, Fudenberg and Morris (2007), who introduced ICR. Our result applies, for instance, to discounted infinite-horizon dynamic games. JEL Numbers: C72, C73. 1. Introduction Interim correlated rationalizability (henceforth ICR) has emerged as the main notion of rationalizability for Bayesian games. Among other reasons, it has the following two desirable properties: Firstly, it is upper hemicontinuous in types, i.e., one cannot obtain a substantially new ICR solution by perturbing a type. Secondly, two distinct definitions of ICR coincide: The fixed-point definition states that ICR is the weakest solution concept with a specific best-response property, and this is the definition that gives ICR its epistemological meaning, as a characterization of actions that are consistent with common knowledge of rationality. Under an alternate definition, ICR is computed by iteratively eliminating the actions that are never a best response (type by type), and this iterative definition is often more amenable for analysis. The above properties were originally proven for games with finite action spaces and finite sets of payoff parameters (by Dekel, Fudenberg, and Morris (2006, 2007), who introduced ICR). However, in many important games these sets are quite large. For example, in the infinitely repeated prisoners’dilemma game, the set of outcomes is uncountable. Hence, the In earlier versions of the paper, the results were confined to compactly metrizable type spaces. We thank an anonymous referee for detailed comments and for providing the main arguments that extended our proofs to the type spaces that are completely metrizable and separable. 1 2 JONATHANWEINSTEINANDMUHAMETYILDIZ set of all possible payoff functions (with payoffs in [0, 1]) is the unit cube with uncountably many dimensions. Therefore, to analyze the infinitely repeated prisoners’ dilemma game under payoff uncertainty without restricting the payoffs, we would need to consider a space of underlying payoff parameters larger than the continuum. The action space is also un- countably infinite, because the action space consists of all mappings that specify whether one cooperates or defects at each of infinitely many histories. In this note, we establish the aforementioned two properties of ICR in greater general- ity so that it can be applied to commonly-used games with large action and type spaces. Specifically, we establish that ICR is upper hemicontinuous and its iterative and fixed-point definitions coincide in games where (i) the spaces of payoff parameters and types are com- plete, separable metric spaces, (ii) the space of actions is a compact metric space, and (iii) the belief and payoff functions are continuous. An immediate consequence of our result is that ICR is upper hemicontinuous when types are endowed with the product topology of belief hierarchies. That is, one cannot obtain a substantially distinct ICR solution for a type by perturbing the type’s higher-order beliefs (possibly by considering another type space). In that sense, ICR predictions are robust to higher-order uncertainty. Interestingly, ICR is unique in that regard: any strict refine- ment of ICR, such as Bayesian Nash equilibrium, fails to be upper hemicontinuous as a correspondence from belief hierarchies (Weinstein-Yildiz, 2007). When the action space is a compact metric space, the set of all continuous payoff func- tions is a complete separable metric space under the uniform metric. Then, the universal type space is also complete and separable metric space. Therefore, we obtain an upper- hemicontinuity result for ICR on universal type space under the same conditions for individ- ual best response: the action space is a complete separable metric space and it is common knowledge that the utility functions are continuous. In Section 6, we apply our results to infinite-horizon games with finitely many moves at every stage, where the relevant notion of continuity is continuity at infinity. As examples, we show that ICR is upper hemicontinuous in discounted repeated games with uniformly bounded stage-game payoffs, even if discount factors are unknown and stage payoffs are unknown, history-dependent or even stochastic. We applied our results here to general infinite-horizon games, including repeated games, in Weinstein and Yildiz (2013). INTERIMCORRELATEDRATIONALIZABILITY 3 Our note relates to the existing literature as follows. The usual solution concepts, includ- ing Nash equilibrium and rationalizability under complete information (Bernheim (1984) and Pearce (1984)), are upper hemicontinuous with respect to the parameters of the game. However, as we mentioned above, most incomplete-information versions of these concepts are not upper hemicontinuous with respect to higher-order beliefs, and we are not aware of any positive result on this matter other than the result of Dekel, Fudenberg and Morris that we extend here. On the other hand, there is a sizeable literature on the equivalence of the iterative and fixed-point definitions of rationalizability under complete information. The equivalence holds when the action spaces are compact metric spaces and the payoff functions are continuous (Tan and Werlang, 1988), and we extend this result to Bayesian games (using ICR as the notion of rationalizability).1 Under complete information, Lipman (1994) shows that the equivalence may fail in games with countably infinite, discrete action spaces, because reaching the fixed point may require iterations through a transfinite ordinal; Arieli (2010) discusses the same issue in continuous games, in which the payoff functions are continuous and the strategy spaces are complete separable metric spaces, finding that the iterations may need to reach the first uncountable ordinal (but not further). These results show that our continuity and compactness assumptions are not superfluous for the equivalence result.2 2. Basic Definitions Given any topological space X, we will write (X) for the space of Borel probability measures on X and endow it with the weak topology. We will also endow all product spaces with their product topology. We write f : X Y to mean that f is a correspondence from X to Y , i.e. an arbitrary subset of X Y with the convention that f(x) denotes y :(x, y) f . f 2 g We consider the defining subset of X Y to be the graph of the correspondence and denote this G(f) for clarity. 1The equivalence was also discussed, for a more narrow class of games, in Bernheim (1984), and in Dufwenberg-Stegeman (2002). This extension to Bayesian games was also covered by Ely-Peski (2006), using interim independent rationalizability as the solution concept. 2Also related is Chen-Long-Luo (2007) who form an iterative definition which is independent of the order of elimination, by allowing dominance by previously eliminated strategies; standard concepts may be order- dependent without the presence of regularity assumptions, as noted by Dufwenberg-Stegeman (2002). 4 JONATHANWEINSTEINANDMUHAMETYILDIZ The correspondence f is said to have the closed-graph property if G (f) is closed. In general, the closed-graph property is weaker than upperhemicontinuity, but the two concepts coincide when Y is compact. When Y is compact, a correspondence f : X Y is said to be upper hemicontinuous (UHC) if f has the closed-graph property. Consider a Bayesian game (N, A, u, , T, ) in normal form where N is a set of players, • 3 A = i N Ai is a compact metric space of action profiles, • 2 is a complete, separable metric space of payoff parameters, • ui : A [0, 1] is a continuous payoff function for each player i N, • ! 2 T = i N Ti is a complete, separable metric space of type profiles, and • 2 ti ( T i) is the interim belief of type ti on the payoff functions and the other • 2 players’types. The map ti t is assumed to be continuous. 7! i In the definition of a Bayesian game above, we made some topological assumptions that we will maintain throughout. These assumptions are satisfied in many applications, including discounted dynamic games, as discussed in Section 6. For each player i, define the best-response correspondence BRi : ( A i) Ai by setting BRi () = arg max ui (, ai, a i) d (, a i) ai Ai 2 Z for each ( A i), where arg max is the set of maximizers. The best-response 2 correspondence is extended to ( T i A i) by BRi () = BRi marg A i , where margX takes the marginal with respect to X. Since Ai is compact and ui is continuous and bounded, BRi is always non-empty. Moreover: Fact 1. BRi is UHC. This fact is a version of Berge’s Maximum Theorem suitable for our purposes; see the appendix for a straightforward proof. 3 For any family (Xi)i N of sets and any family of functions fi : Xi Yi, i N, we write X = i N Xi 2 ! 2 2 and X i = j N i Xj and define functions f : X Y and f i : X i Y i by setting f i (x i) = 2 nf g ! ! (fj (xj))j N i . 2 nf g INTERIMCORRELATEDRATIONALIZABILITY 5 We next present the two definitions of interim correlated rationalizability (ICR). The definitions may differ in general infinite games, but will coincide under the present conditions. The first definition is given by iterated elimination of strictly dominated actions that are never a weak best response, as follows: m Define the family of correspondences Si : Ti Ai, i N, m N, iteratively, by setting 0 2 m 2 S (ti) = Ai and for each m > 0 and ti Ti, let ai S (ti) if and only if there exists i 2 2 i ( T i A i) such that 2 (2.1) ai BRi () , 2 (2.2) ti = marg T i µ, m 1 (2.3) G S i = 1.