INTERIM CORRELATED RATIONALIZABILITY IN INFINITE GAMES

Transcription

1 INTERIM CORRELATED RATIONALIZABILITY IN INFINITE GAMES JONATHAN WEINSTEIN AND MUHAMET YILDIZ A. We show that, under the usual continuity and compactness assumptions, interim correlated rationalizability is well-defined and upper hemicontinuous in infinite games. We then apply this result to infinite-horizon games. JEL Numbers: C72, C I 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 upperhemicontinuous 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 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 1

2 2 JONATHAN WEINSTEIN AND MUHAMET YILDIZ of underlying payoff parameters larger than the continuum. The action space is also uncountably 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 generality so that it can be applied to commonly-used games with large action and type spaces. Specifically, we establish that ICR is upperhemicontinuous and its iterative and fixed-point definitions coincide in games where (i) the spaces of payoff parameters, types and actions are compact metric spaces and (ii) the belief and payoff functions are continuous. An immediate consequence of our result is that ICR is upperhemicontinuous 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 refinement of ICR, such as Bayesian Nash equilibrium, fails to be upperhemicontinuous as a correspondence from belief hierarchies (Weinstein-Yildiz, 2007). Here the key assumption is that the space of payoff functions is compactly metrizable (or, more broadly, sequentially compact). In the applications we have in mind this assumption can be onerous. For example, in the infinitely repeated prisoners dilemma game, the space of all payoff functions cannot be made sequentially compact. We also delineate further broad assumptions on dynamic games under which our suffi cient conditions are satisfied. In particular, we show that our conditions are satisfied whenever the underlying payoff functions are equicontinuous. In multistage games with finitely many moves at every stage, this condition reduces to equicontinuity at infinity, i.e., for every ε > 0, there exists a finite horizon such that the payoff differences due to the variations after that horizon are all less than ε under all payoff functions. For example, in the repeated prisoners dilemma game, our conditions are all satisfied if the payoffs from the repeated game are the discounted sum of the payoffs that are attained in the stage games, provided that the discount factor is bounded away from 1 and the stage-game payoffs are uniformly bounded. (Here, the stage game payoffs may be stochastic and may depend on the history, and the discount factor may be unknown.) We applied our results here to general infinite-horizon games, including repeated games, in Weinstein and Yildiz (2013).

3 INTERIM CORRELATED RATIONALIZABILITY 3 Our note relates to the existing literature as follows. The usual solution concepts, including Nash equilibrium and rationalizability under complete information (Bernheim (1984) and Pearce (1984)), are upperhemicontinuous with respect to the parameters of the game. However, as we mentioned above, most incomplete-information versions of these concepts are not upperhemicontinuous 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). Under complete information, Lipman (1994) shows that the equivalence may fail in discontinuous games because reaching the fixed point may require a transfinite number of iterations; Arieli (2010) further shows that the same issue arises even in continuous games, in which the payoff functions are continuous and the strategy spaces are complete separable metric spaces, but compactness fails. These results show that our continuity and compactness assumptions are not superfluous for the equivalence result. 2. B D 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}. Unless stated otherwise, correspondences are assumed to be non-empty, meaning that f(x) is non-empty for all x. We consider the defining subset of X Y to be the graph of the correspondence and denote this G(f) for clarity. 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 upperhemicontinuous (UHC) if f has the closed-graph property. Consider a Bayesian game (N, A, u, Θ, T, κ) in normal form where

4 4 JONATHAN WEINSTEIN AND MUHAMET YILDIZ N is a set of players, A = i N A i is a compact metric space of action profiles, 1 Θ is a compact metric space of payoff parameters, u i : Θ A [0, 1] is a continuous payoff function for each player i N, T = i N T i is a compact metric space of type profiles, and κ ti (Θ T i ) is the interim belief of type t i on the payoff functions and the other players types. In the definition of a Bayesian game above, we made several continuity and compactness assumptions that we will maintain throughout. For the most part, the assumptions would be satisfied in applications. The assumption that Θ is a compact metric space is the most stringent assumption, as it fails for some function spaces. However, it is needed to to show (via the Maximum Theorem) that the best-reply correspondence is UHC, and this is the most basic single-player version of the continuity property we will prove. In Section 5.2 we will discuss suffi cient conditions for our restriction on Θ. For each player i, define the best-response correspondence BR i : (Θ A i ) A i by setting BR i (µ) = arg max u i (θ, a i, a i ) dµ (θ, a i ) a i A i for each µ (Θ A i ), where arg max is the set of maximizers. The best-response correspondence is extended to (Θ T i A i ) by BR i (µ) = BR i ( margθ A i µ ), where marg X takes the marginal with respect to X. Since A i is compact and u i is continuous and bounded, BR i is always non-empty. Moreover: Fact 1. BR i is UHC. This fact is a consequence of the following general lemma. Note that we need both spaces to be metric in order to apply the Maximum Theorem. Lemma 1. Let (S, d) and (X, d) be compact metric spaces, and u : S X R be continuous. Then, B i (µ) = arg max s S u (s, x) dµ (x) 1 For any family (X i ) i N of sets and any family of functions f i : X i Y i, i N, we write X = i N X i and X i = j N\{i} X j and define functions f : X Y and f i : X i Y i by setting f i (x i ) = (f j (x j )) j N\{i}.

5 is upperhemicontinuous in µ. INTERIM CORRELATED RATIONALIZABILITY 5 Proof. By the Maximum Theorem, it suffi ces to show that U (s, µ) = u (s, x) dµ (x) is continuous. To this end, take any sequence (s n, µ n ) with limit (s, µ) and any ε > 0. First note that since u is continuous and S X is compact, by Heine-Cantor Theorem, u is uniformly continuous. Hence, there exists δ > 0 such that, whenever d (s, s ) < δ, u (s, x) u (s, x) < ε/2, yielding U (s, µ n ) U (s, µ n ) u (s, x) u (s, x) dµ n (x) ε/2 for every n. Since s n s, there then exists n 1 such that U (s, µ n ) U (s n, µ n ) ε/2 for all n > n 1. On the other hand, since u (s, ) is a continuous function of x and µ n µ, u (s, x) dµ n (x) u (s, x) dµ (x). Hence, there exists n > n 1 such that U (s, µ) U (s, µ n ) < ε/2 for all n > n. Combining the last two displayed inequalities, one obtains that U (s, µ) U (s n, µ n ) < ε for all n > n as desired. 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: Define the family of correspondences Si m : T i A i, i N, m N, iteratively, by setting Si 0 (t i ) = A i and for each m > 0 and t i T i, let a i Si m (t i ) if and only if there exists

6 6 JONATHAN WEINSTEIN AND MUHAMET YILDIZ µ (Θ T i A i ) such that (2.1) (2.2) (2.3) a i BR i (µ), µ ( G ( )) S m 1 i = 1. κ ti = marg Θ T i µ, Here, the first condition requires that a i is a best response to belief µ; the second condition requires that µ is a belief of type t i (coherence), and the last condition requires that the other players play according to S m 1 i under µ. Here, the domain of S m 1 i is taken to be Θ T i, where θ Θ does not affect S m 1 i (θ, t i ). We consider such larger domains whenever it is convenient. The limiting correspondence S : T A i is defined by (2.4) S (t) = m 0 S m (t). The correspondence S is our first definition of ICR. The second definition is given as a fixed-point property. A solution concept f : T A is said to have the best-response property if for every t i T i and a i f (t i ), there exists µ (Θ T i A i ) such that (2.5) (2.6) (2.7) a i BR i (µ), κ ti = marg Θ T i µ, µ (G (f i )) = 1. The interim correlated rationalizability is defined as the largest correspondence R : T A with the best-response property. Note that, since the best-response property is closed under coordinate-wise union, R is well-defined, i.e., R i (t i ) = f i (t i ) (t i T i ) f:t A with best-response property has the best-response property. Note also that the only difference from the iterative definition of S is that the definition of R requires that the other players play according to R as well, while S m requires only that they play according to S m 1. Consequently, R is a stronger solution concept. Fact 2. For any t i and m, R i (t i ) Si m (t i ).

7 INTERIM CORRELATED RATIONALIZABILITY 7 Proof. It suffi ces to show this for any finite m. The statement is true for m = 0, by definition. Towards an induction, assume that it is true for m 1. For any a i R i (t i ), there exists µ with (2.5)-(2.7). But since R i S m 1 i by the inductive hypothesis, µ also satisfies (2.1)-(2.3), showing that a i Si m (t i ). When the two definitions differ, the fixed-point definition should be taken as the definition of rationalizability because it characterizes the strategic implications of common knowledge of rationality. However, under the present assumptions, the definitions are equivalent, as we show in the next section. 3. U In this section, we will show that S is UHC, and further that it coincides with the fixed-point definition. Proposition 1. For every m, S m is UHC. Proof. For each finite m N, we will show that S m has the closed-graph property, i.e., G (S m ) is closed. This further implies that G (S ) = G (S m ) m 0 is also closed. Since A is compact, this is indeed the desired result: S m is UHC for each m. Clearly, G (S 0 ) = T A is closed. Towards an induction, assume that G ( ) S m 1 i is closed for some i N and m N. Take a sequence (t i,k, a i,k ) G (Si m ) with limit (t i, a i ). For each k, since (t i,k, a i,k ) G (S m i ), there exists µ k (Θ T i A i ) such that (3.1) (3.2) (3.3) a i,k BR i (µ k ), κ ti,k = marg Θ T i µ k, ( ( )) µ k G S m 1 i = 1. Note that, since Θ T i A i is compact and metrizable, so is (Θ T i A i ) by Prohorov s Theorem. Hence, µ k has a convergent subsequence with a limit µ (Θ T i A i ).

8 8 JONATHAN WEINSTEIN AND MUHAMET YILDIZ Switching to the convergent subsequence, we will show that µ satisfies the conditions (2.1)- (2.3), showing that a i S m i (t i ), as desired. Firstly, since Θ T i A i is endowed with product topology, the projection mapping is continuous and hence κ ti,k = marg Θ T i µ k marg Θ T i µ, where the equality is by (3.2). Since κ ti,k κ ti marg Θ T i µ = κ ti, by continuity of κ, this further implies that proving (2.2). Secondly, since BR i is UHC by Fact 1, (3.1) implies that a i BR i (µ), proving (2.1). Finally, since G ( ) S m 1 i is closed (by the inductive hypothesis) and µk µ, by Portmanteau Theorem, µ ( G ( )) ( ( )) S m 1 i lim sup µk G S m 1 i = 1, where the last equality is by (3.3). This proves (2.3) and completes the proof. By Fact 2, R is always contained in S. The next result shows that the reverse inclusion is also true when the type space can be embedded in a compact metric space. Proposition 2. S = R. Proof. By Fact 2, it suffi ces to show that S R. We will prove this by showing that S has the best-response property. To this end, take any t i T i and a i Si (t i ). Now, since a i Si m (t i ) for each m, there exists a sequence µ m (Θ T i A i ) such that a i BR i (µ m ), ( ( )) µ m G S m 1 i = 1. κ ti = marg Θ T i µ m, As in the proof of Proposition 1, µ m has a limit µ with a i BR i (µ) and κ ti = marg Θ T i µ. We will also show that µ ( G ( S i)) = 1, showing that S has the best-response property. Now, for any m N, since µ k ( G ( S m i )) = 1 for every k > m and G ( S m i ) is closed, by the Portmanteau Theorem, µ ( G ( S m i)) = 1. Since G ( S m i ) G ( S i ), this implies that µ ( G ( S i)) = limm µ ( G ( S m i)) = 1, completing the proof.

9 INTERIM CORRELATED RATIONALIZABILITY 9 Combining the two propositions above, we come to the main conclusion in this section: Proposition 3. R is UHC. 4. U B H In Bayesian games, a type is meant to represent a belief hierarchy in the interim stage. A central issue in game theory is whether the solution is robust to small specification errors in modeling the belief hierarchies using types. Such robustness is formalized by upperhemicontinuity in belief hierarchies. In this section, we show that ICR is UHC in belief hierarchies, by embedding the type spaces in the universal type space and applying Proposition 3 to the universal type space. Fix Θ, A, and utility functions u i : Θ A [0, 1], and vary the type spaces (T, κ). We assume that Θ and A are compact metric spaces and u i and κ are continuous as in the rest of the study, but we allow T to have any topology. A solution concept Σ is defined as a mapping that yields a correspondence Σ ( T, κ) : T A on each type space (T, κ). When we refer to ICR as a solution concept, we use the obvious notation R ( T, κ) and S ( T, κ). The type spaces are continuously embedded in the universal type space (T, κ ) using the following belief-hierarchy mapping (see Mertens and Zamir (1985)). For any type space (T, κ) and type t i T i, belief hierarchy is computed inductively by setting and h i (t i T, κ) = ( h 1 i (t i T, κ), h 2 i (t i T, κ), h 3 i (t i T, κ),... ) h 1 i (t i T, κ) = κ ti ρ 1 Θ h k i (t i T, κ) = κ ti ( ρ Θ, h 1 i ( T, κ),..., h k 1 i ( T, κ) ) 1 where ρ Θ : Θ T i Θ is the projection mapping. Here, h 1 i (t i T, κ) is the first-order belief of type t i about the payoff parameter θ, and h k i (t i T, κ) is the kth-order belief of type t i about the payoff parameter θ and first k 1 orders of beliefs. Each level of beliefs is given the weak-star topology. The universal type space T consists of all belief hierarchies obtained as above and is endowed with product topology. Since Θ is a compact metric space, the universal type space T is also a compact metric space, and κ is a continuous

10 10 JONATHAN WEINSTEIN AND MUHAMET YILDIZ function of belief hierarchies. 2 Hence, all of our assumptions hold for the Bayesian game (N, A, u, Θ, T, κ ), to which we apply our previous results. Our first result establishes that ICR only depends on the belief hierarchies and its iterative and fixed-point definitions coincide (although T need not be compact). Proposition 4. For any type space (T, κ), we have R ( T, κ) = S ( T, κ), and (4.1) Si (t i T, κ) = Si (h i (t i T, κ) T, κ ) ( i N, t i T i ). Proof. Fix any (T, κ). It is straightforward to obtain (4.1) from the definitions following Dekel, Fudenberg, and Morris. Since T is compact, by Proposition 2, we also have (4.2) R ( T, κ ) = S ( T, κ ). This further implies that R ( T, κ) = S ( T, κ). To see this, take any t i T i and any a i S i (t i T, κ). By (4.1), a i S i (h i (t i T, κ) T, κ ). However, by (4.2), S ( T, κ ) has the best-response property. Hence, a i is a best response to a belief µ ( Θ T i A i ) with (i) marg Θ T i µ = κ h i (t i T,κ) and (ii) µ ( a i S i ( t i T, κ )) = 1. Now, consider the belief µ µ ( ι Θ, h i ( T, κ), ι A i ) 1 (Θ T i A i ) where ι Θ and ι A i are the identity mappings on Θ and A i, respectively. Since marg Θ A i µ = marg Θ A i µ, a i is also a best response to belief µ. Moreover, by (i), marg Θ T i µ = κ ti definition of κ ), and by (ii) and (4.1), µ ( a i S i (t i T, κ) ) = 1. Proposition 4 establishes that ICR solution set does not depend the way one models the belief hierarchies. In contrast, many solution concepts, such as Bayesian Nash equilibrium and "interim independent rationalizability", fail (4.1), and the solution set depends on the way one models the belief hierarchies. Indeed, if the original type space has redundant types, some of the equilibria may disappear when the type space is embedded in the universal type space; see for example Ely and Peski (2006). There then exists some type t i in some type space (T, κ) such that Σ (t i T, κ) Σ (h i (t i T, κ) T, κ ). 3 2 See Mertens and Zamir (1985) for details. They assume that the type spaces are compact, but compactness is not needed for continuity of κ and h. 3 These solution concepts may still have selections that are invariant to the way one models the hierarchies (Yildiz (2009)). (by

11 INTERIM CORRELATED RATIONALIZABILITY 11 We now turn to the main concept of this section: upperhemicontinuity in belief hierarchies. We use the following notion of convergence among types. Definition 1. A sequence of types t m i from type spaces (T m, κ m ) is said to converge in belief hierarchies to a type t i from a type space (T, κ), denoted by t m i t i, if (4.3) h k i (t m i T m, κ m ) h k i (t i T, κ) k. Here, we use the product topology on belief hierarchies, which reflects the point of view of an observer who can only access to finite orders of beliefs (see Weinstein and Yildiz (2007) for a detailed discussion). Our notion of convergence among types reflects the same view. Definition 2. A solution concept Σ is said to be upperhemicontinuous in belief hierarchies if [a m i Σ (t m i T m, κ m ) m] = a i Σ (t i T, κ) for every sequence of actions a m i with limit a i and for every sequence of types t m i from type spaces (T m, κ m ) that converges in belief hierarchies to a type t i from a type space (T, κ). Observe that when (4.1) fails, the solution concept cannot be UHC in belief hierarchies, no matter what topology one uses on the belief hierarchies. More broadly, if h i (t i T, κ) = h i (t i T, κ ) while a i Σ (t i T, κ) for some a i Σ (t i T, κ ), then the condition for UHC in belief hierarchies fails for the constant sequence (a m i, t m i ) = (a i, t i). In particular, Bayesian Nash equilibrium and interim independent rationalizability are not UHC in belief hierarchies. Indeed, as we mentioned in the introduction, no strict refinement of ICR can be UHC in belief hierarchies. ICR turns out to be UHC in belief hierarchies, as Dekel, Fudenberg, and Morris show for finite A and Θ. Applying Proposition 3 to the universal type space (T, κ ), the next result shows that ICR remains UHC in belief hierarchies in our more general setup. Proposition 5. R is upperhemicontinuous in belief hierarchies. Proof. Consider a sequence of actions a m i with limit a i and a sequence of types t m i T m i from type spaces (T m, κ m ) with limit t i T i from a type space (T, κ) in the sense of (4.3).

12 12 JONATHAN WEINSTEIN AND MUHAMET YILDIZ Assume that for each m, a m i R i (t m i T m, κ m ). Then, for each m, we have a m i R i (t m i T m, κ m ) = S i (t m i T m, κ m ) = S i (h i (t m i T m, κ m ) T, κ ), where the first equality is by Proposition 2 and the second equality is by (4.1). S ( T, κ ) is UHC (by Proposition 1) and h i (t m i T m, κ m ) h i (t i T, κ), we have Since a i S i (h i (t i T, κ) T, κ ) = S i (t i T, κ) = R i (t i T, κ), where the equalities are by Proposition E In the previous sections we have shown that ICR is UHC and computable via iterated strict dominance under all our assumptions, including the metrizability of the type space. Now, we will illustrate that the assumptions made on the action space, utility functions, and the beliefs are satisfied in canonical cases. Furthermore, we note that metrizability of the type space is the most stringent condition in large games. We will then give general examples of games for which the latter assumption holds or fails Repeated Games with common knowledge of the discount factor. Consider an infinitely repeated game with perfect-monitoring and with a stage game-form (N, S), where N = {1,..., n} is the finite set of players and S = S 1 S n is the finite set of strategy profiles in the stage game. The stage payoff-function g : S [0, 1] n is unknown. It is common knowledge that each player s payoff is the discounted sum of his stage-payoffs with respect to a known discount factor δ (0, 1). We write θ g,δ (h) = (1 δ) δ l g ( s l) for the payoff vector at any history h = (s 0, s 1,...). We also write l=0 Θ δ = { θ g,δ g : S [0, 1] n} for the set of all payoff functions with the repeated-game payoff structure for the fixed δ. Endow Θ δ with the metric d ( ) θ g,δ, θ g,δ = g g

13 INTERIM CORRELATED RATIONALIZABILITY 13 where is a Euclidean metric on [0, 1] N S. Clearly, (Θ δ, d) is a compact and complete metric space. Note that the topology on (Θ δ, d) is both product and uniform topologies, as they are equivalent. Let T δ be the Θ δ-based universal type space. By Mertens and Zamir (1985) and Brandenburger and Dekel (1993), T δ is also a compact and complete metric space. Of course, as we have discussed above, under the product topology, A is compact and metrizable, u i : Θ δ A [0, 1] with u i (θ, a) = θ i (z (a)), where z (a) is the outcome of a, is continuous, and κ is continuous with respect to the types in the universal type space T by Mertens and Zamir. Therefore, the game (N, A, u, Θ δ, T, κ) satisfies all of our assumptions, and ICR is UHC and computable by iterated strict dominance on this game. In other words, our results here show that, in a repeated game, ICR is UHC and computable by iterated strict dominance whenever the repeated game structure and the discount factor (but not the stage payoffs) are common knowledge. This is an instance of a broader class of games that we will present next Suffi cient Conditions Equicontinuity. A family F Y X of functions f : X Y is said to be equicontinuous if for every x and every ɛ, there exists a δ such that for all x with d(x, x) < δ, and all f F, d(f(x), f(x )) < ɛ. That is, the choice of δ can be made independently of f. We will now show that when the set of payoff functions is equicontinuous and compact under the sup norm (or product topology), all of our assumptions hold, and ICR is UHC. This concept applies to both static and dynamic games. The key is that for equicontinuous families, the Arzela-Ascoli theorem tells us that pointwise convergence implies uniform convergence. That is, the uniform and product topology coincide, so that the sup norm actually metrizes the product topology. First consider an n-player static game in which A is compact and Θ is an equicontinuous set of payoff functions θ : A [0, 1] n. Let (T, κ) be the (Θ, d u )-based universal type space where d u is the uniform metric, defined by d u (θ, θ ) = sup θ (a) θ (a). a A If it is also the case that Θ is compact under the product topology, it is compact under the sup norm as well. Note that compactness under the product topology is often easy to verify via Tychonoff s Theorem; for instance, the space of all functions from an arbitrary domain to a compact range is compact. Of course, our utility functions are continuous (by

14 14 JONATHAN WEINSTEIN AND MUHAMET YILDIZ equicontinuity) and the beliefs are continuous. Therefore, all of our assumptions are satisfied, and ICR is UHC on (N, A, u, Θ, T, κ). For dynamic games, equicontinuity is defined as follows. A set Θ of functions θ : Z [0, 1] n is said to be equicontinuous at infinity if for every ε > 0 and z Z, there exists L such that for every θ Θ, θ i (z) θ i (z ) < ε whenever z and z agree in the first L periods. This is the general definition of equicontinuity applied to the metric defined by d(z, z ) = 2 l, where l is the earliest discrepancy between z and z. It strengthens the usual notion of continuity at infinity (introduced in Fudenberg-Levine, 1983) by picking L simultaneously for all θ Θ. Now consider an n-player multistage game as above and assume that Θ, which is now a set of payoff functions θ : Z [0, 1] n, is equicontinuous at infinity. Apply the uniform metric d u. Then, for the same reasons as above, provided that Θ is compact in the product topology, all of our assumptions are satisfied. Therefore, ICR is UHC on (N, A, u, Θ, T, κ). Example 1 (Repeated games with uncertainty about the stage payoffs and discount factor). Let Θ = Θ δ δ D for some closed subset D of (0, 1), where Θ δ is as in the last section, and let (T, κ) be the (Θ, d u )-based universal type space. Then Θ is equicontinuous at infinity. Hence, ICR is UHC on (N, A, u, Θ, T, κ). It is critical that the discount factor is bounded away from 1; see the next section. The stationarity of payoffs is not critical; equicontinuity holds for any class of dynamic games with discounting where stage payoffs are uniformly bounded and δ is bounded away from When the Assumptions Fail. The assumptions that action space A is compact and the utility functions and the beliefs are continuous are not superfluous for the continuity results here. Indeed, it is well-known that the best-response functions could easily fail to be UHC and non-empty without these assumptions, proving that these conditions are not superfluous even in single-player games. We also assume that Θ and T are compact metric spaces; we do not assume compactness of T in Section 4 because T is embedded in the universal type space, which is a compact metric space whenever Θ is so. This assumption is needed

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