Example: Consider a two-firm version of the Cournot quantity competition game, with action spaces A;=0, 1] and payoffs ui( ai, a_i)=(2-ai-a-i)ai: that is, marginal costs are zero, and inverse demand is given by P(a)=2-q Notice that Player i's expected payoff depends linearly on Player -i's expected quantity thus, our analysis can be restricted to degenerate (point)beliefs without loss of generality For any a_i E A-i, Player i's best reply is found by solving Inax so Tila Then notice that no a; E[0, 2)is ever a best reply: after all, even if a-i=l, there would still be residual demand in the market, and consumers would be willing to pay 1 dollar for an dditional marginal unit of the good. That is, Player i is actually facing the residual inverse demand curve P(ai)=1-ai, so setting ai=,(the corresponding monopoly quantity)is optima This is true for i= 1, 2. Thus, assume that Player i understands that a-iE 1]: that is, Player i believes that, since her opponent is rational, he will never produce less than units of the good. But then, producing ai E(, 1] units cannot be a best reply against a belief satisfying this restriction. Since, moreover, we have already concluded that quantities ai E0, 5)must be discarded on the basis of rationality considerations alone, the assumptions Bayesian rationality Belief in the opponent,s Bayesian rationality already imply that only action profiles(a1, a2) with ai E [, i should be expected to Once we are willing to make this assumption about the players beliefs, it becomes natural at least conceive the possibility that the players themselves might contemplate it. This leads to a hierarchy of hypotheses about mutual belief in rationality. That is, we might think that players are Bayesian rational (2) players believe that their opponents are also Bayesian rational (3)players believe that their opponents believe that their own opponents are Bayesian rational: and so on The following definition captures the implications of these assumptions Definition 4 Fix a game G=(N, (Ai, Ti, ui)iEN ) For every player iE N, let A9=Aj.For k≥1 and for every i∈I, let ai∈ A' iff there exists a-∈△(A-,B(T-) such thatExample: Consider a two-firm version of the Cournot quantity competition game, with action spaces Ai = [0, 1] and payoffs ui(ai , a−i) = (2−ai −a−i)ai : that is, marginal costs are zero, and inverse demand is given by P(q) = 2 − q. Notice that Player i’s expected payoff depends linearly on Player −i’s expected quantity; thus, our analysis can be restricted to degenerate (point) beliefs without loss of generality. For any a−i ∈ A−i , Player i’s best reply is found by solving max ai∈[0,1] (2 − ai − a−i)ai , so ri(a−i) = 1 − 1 2 a−i Then notice that no ai ∈ [0, 1 2 ) is ever a best reply: after all, even if a−i = 1, there would still be residual demand in the market, and consumers would be willing to pay 1 dollar for an additional marginal unit of the good. That is, Player i is actually facing the residual inverse demand curve P R(ai) = 1 − ai , so setting ai = 1 2 (the corresponding monopoly quantity) is optimal. This is true for i = 1, 2. Thus, assume that Player i understands that a−i ∈ [ 1 2 , 1]: that is, Player i believes that, since her opponent is rational, he will never produce less than 1 2 units of the good. But then, producing ai ∈ ( 3 4 , 1] units cannot be a best reply against a belief satisfying this restriction. Since, moreover, we have already concluded that quantities ai ∈ [0, 1 2 ) must be discarded on the basis of rationality considerations alone, the assumptions of Bayesian rationality + Belief in the opponent’s Bayesian rationality already imply that only action profiles (a1, a2) with ai ∈ [ 1 2 , 3 4 ] should be expected. Once we are willing to make this assumption about the players’ beliefs, it becomes natural to at least conceive the possibility that the players themselves might contemplate it. This leads to a hierarchy of hypotheses about mutual belief in rationality. That is, we might think that: (1) players are Bayesian rational; (2) players believe that their opponents are also Bayesian rational; (3) players believe that their opponents believe that their own opponents are Bayesian rational; and so on. The following definition captures the implications of these assumptions. Definition 4 Fix a game G = (N,(Ai , Ti , ui)i∈N ). For every player i ∈ N, let A0 i = Ai . For k ≥ 1 and for every i ∈ I, let ai ∈ Ak i iff there exists α−i ∈ ∆(A−i , B(T−i)) such that 5