Definition 1 Fix a finite normal-form game G=(N, (Ai, Lien), a player i E I and a probability measure a_i E A(A-i. An action ai E Ai is a best reply to the belief a-i iff ∑a-(a-)u(an,a-)≥∑a-(a-)u1,a-)1∈A In finite games, expected payoffs are always well-defined. Not so for games with infinite action spaces: even assuming a nice measure-theoretic structure, it is not hard to construct (mildly pathological) examples in which payoff functions are not integrable Thus, a better definition for infinite games is the following. Assume that, for every player i E N, the set of actions(Ai, Ai) is a measure space. Endow the product sets A and A-i with the natural product sigma-algebras, denoted A and A_i. Finally, assume that ui is A- measurable. The sigma-algebras on the set of actions will be indicated explicitly for infinite Definition 2 Fix a normal-form game G=(N, (Ai, Ai, uiieN), a player i E N and a belief a-i E A(A-i, A-i). An action a; E Ai is a best reply to the belief a-i iff the Lebesgue integralS ui(ai, a-i)a-i(da )exists, and moreover u(4,a-)a-()2/a(0,a-)a-()∈A whenever the right-hand side integral exists I emphasize that integrals are taken(here and in the following) in the Lebesgue sense Even if expected payoffs are well-defined, the very structure of the game might lead to the non-existence of best replies. An obvious example arises in the Bertrand price competition game. Example. Two firms produce and sell the same good, facing a market of size Q, and zero fixed and marginal cost; finally, the two firms compete by posting unit prices pi, i=1, 2. Assume that consumers have unit demand and buy from the cheapest producer; if P1= P2 buy from each firm with probability 2 Suppose that Firm 1 expects Firm 2 to post a price equal to p2= 2; this corresponds to a degenerate belief, concentrated on 2. Then any price p1 >2 yields 0, any price p1 2 yields Qpl, and P1=P2=2 yields 22=Q. Clearly no price pi satisfies our definition of a best reply The conditions for the existence of best replies are related to those which guarante the existence of an optimum. We need to specify a topology Ti on every action space A 2Definition 1 Fix a finite normal-form game G = (N,(Ai , ui)i∈N ), a player i ∈ I and a probability measure α−i ∈ ∆(A−i). An action ai ∈ Ai is a best reply to the belief α−i iff X A−i α−i(a−i)ui(ai , a−i) ≥ X A−i α−i(a−i)ui(a 0 i , a−i) ∀a 0 i ∈ Ai . In finite games, expected payoffs are always well-defined. Not so for games with infinite action spaces: even assuming a nice measure-theoretic structure, it is not hard to construct (mildly pathological) examples in which payoff functions are not integrable. Thus, a better definition for infinite games is the following. Assume that, for every player i ∈ N, the set of actions (Ai , Ai) is a measure space. Endow the product sets A and A−i with the natural product sigma-algebras, denoted A and A−i . Finally, assume that ui is A￾measurable. The sigma-algebras on the set of actions will be indicated explicitly for infinite games. Definition 2 Fix a normal-form game G = (N,(Ai , Ai , ui)i∈N ), a player i ∈ N and a belief α−i ∈ ∆(A−i , A−i). An action ai ∈ Ai is a best reply to the belief α−i iff the Lebesgue integral R A−i ui(ai , a−i)α−i(dai ) exists, and moreover Z A−i ui(ai , a−i)α−i(dai ) ≥ Z A−i ui(a 0 i , a−i)α−i(dai ) ∀a 0 i ∈ Ai whenever the right-hand side integral exists. I emphasize that integrals are taken (here and in the following) in the Lebesgue sense. Even if expected payoffs are well-defined, the very structure of the game might lead to the non-existence of best replies. An obvious example arises in the Bertrand price competition game. Example. Two firms produce and sell the same good, facing a market of size Q, and zero fixed and marginal cost; finally, the two firms compete by posting unit prices pi , i = 1, 2. Assume that consumers have unit demand and buy from the cheapest producer; if p1 = p2, consumers buy from each firm with probability 1 2 . Suppose that Firm 1 expects Firm 2 to post a price equal to p2 = 2; this corresponds to a degenerate belief, concentrated on 2. Then any price p1 > 2 yields 0, any price p1 < 2 yields Qp1, and p1 = p2 = 2 yields 2Q 2 = Q. Clearly no price p1 satisfies our definition of a best reply. The conditions for the existence of best replies are related to those which guarantee the existence of an optimum. We need to specify a topology Ti on every action space Ai ; 2