furthermore, we assume that the relevant sigma-algebra in the definition of a best reply is the corresponding Borel sigma-algebra B(Ti)(hence, ui ()is Borel-measurable if it is continuous). Finally, we assume product sets are endowed with the product topology and the induced Borel product sigma-algebra Proposition 0.1 Consider a game G=(N, (Ai, Ii, uiieN) and fix a player i E N. If ui ( is bounded and continuous, and Ai is a compact metrizable space, then for every measure a-i∈△(A-a,B(T) there exists a best reply a;∈A Proof: We need to show that the map p: A,HJA ui(ai, a-i)a-i(da-i) is continuous Since A; is a metric space, sequences characterize continuity. Thus, fix a sequence ai-ai in A;. By assumption, ui(ah, a-i)ui(ai, a-i)for every a_i E A-i, and ui is bounded Thus, by Lebesgue's Dominated Convergence theorem, p(ai-p(ai).B Observe that, in the above proof, it is essential that expectations be defined in the Lebesgue sense: otherwise, stronger conditions are required on the payoff functions and action spaces There are two reasons why we are interested in infinite games. First, for Nash equilibria to exist one must enlarge the players' action spaces to include all mixtures of strategies thus, effectively, one is looking at an infinite game, in which every strategy set is a simplex, hence a compact subset of Euclidean space, and every payoff function is continuous(because the expected payoff from a mixed strategy profile is linear in the probabilities). Thus, the preceding result applies Second. several interesting economic models lend themselves to a formulation involving infinite strategy spaces. Thus, getting some exposure to the issues that arise in such settings can be useful The Best Reply Correspondence Recall that a correspondence is a set-valued function. Since, in general, Player i may have more than one best reply to a belief a-i E A(A-i, A-i), we use correspondences to describe the mapping from beliefs to best replies Definition 3 Fix a game G=(N, (Ai, AieN) and a player i E N. The best-reply corre- spondence for player i, ri: A(A-i, A-i)= Ai, is defined by va-∈△(A-,A-),a1∈ra(a-)台a; is a best reply given a- The preceding proposition then gives conditions under which the best-reply correspon- dence is nonemptyfurthermore, we assume that the relevant sigma-algebra in the definition of a best reply is the corresponding Borel sigma-algebra B(Ti) (hence, ui(·) is Borel-measurable if it is continuous). Finally, we assume product sets are endowed with the product topology and the induced Borel product sigma-algebra. Proposition 0.1 Consider a game G = (N,(Ai , Ti , ui)i∈N ) and fix a player i ∈ N. If ui(·) is bounded and continuous, and Ai is a compact metrizable space, then for every measure α−i ∈ ∆(A−i , B(Ti)) there exists a best reply ai ∈ Ai . Proof: We need to show that the map ϕ : Ai 7→ R A−i ui(ai , a−i)α−i(da−i) is continuous. Since Ai is a metric space, sequences characterize continuity. Thus, fix a sequence a k i → ai in Ai . By assumption, ui(a k i , a−i) → ui(ai , a−i) for every a−i ∈ A−i , and ui(·) is bounded. Thus, by Lebesgue’s Dominated Convergence theorem, ϕ(a k i ) → ϕ(ai). Observe that, in the above proof, it is essential that expectations be defined in the Lebesgue sense: otherwise, stronger conditions are required on the payoff functions and action spaces. There are two reasons why we are interested in infinite games. First, for Nash equilibria to exist one must enlarge the players’ action spaces to include all mixtures of strategies; thus, effectively, one is looking at an infinite game, in which every strategy set is a simplex, hence a compact subset of Euclidean space, and every payoff function is continuous (because the expected payoff from a mixed strategy profile is linear in the probabilities). Thus, the preceding result applies. Second, several interesting economic models lend themselves to a formulation involving infinite strategy spaces. Thus, getting some exposure to the issues that arise in such settings can be useful. The Best Reply Correspondence Recall that a correspondence is a set-valued function. Since, in general, Player i may have more than one best reply to a belief α−i ∈ ∆(A−i , A−i), we use correspondences to describe the mapping from beliefs to best replies. Definition 3 Fix a game G = (N,(Ai , Ai)i∈N ) and a player i ∈ N. The best-reply corre￾spondence for player i, ri : ∆(A−i , A−i) ⇒ Ai , is defined by ∀α−i ∈ ∆(A−i , A−i), ai ∈ ri(α−i) ⇔ ai is a best reply given α−i The preceding proposition then gives conditions under which the best-reply correspon￾dence is nonempty. 3