Proposition 0.1 Fix a game=(N, (Ai, lilieN) and a profile of actions a=(alieN 1)If there exists a model M=(Q, (Ti, ai, piie) for G and a state w in the model such that u∈Rn∩ CFBi and a1(u)= ai for all i∈N, then a is a Nash equilibrium of G 2)If a E A is a Nash equilibrium of G, there exists a model M=(Q, (Ti, ai, piie)for G and a state w in the model such that w E RnneN CFB; and ai(w)=ai for all N Proof: (1)At w, ai=ai(w)Eri(a-i(w))=ri(sa-i)for all i E N; hence, a is a Nash equilibrium.(2)Consider a model with a single state w(so that Q= w= ti(w)for all i E N) such that ai( w)=ai. Since a is a Nash equilibrium, W E R; since necessarily D2(u)=1 for all i∈N,w∈CSB; for all i∈N.■ You are authorized to feel cheated. This characterization does not add much to the definition of Nash equilibrium. However, it does emphasize what is entailed by correctness of beliefs I mentioned this in the informal language we had to make do with a while ago but i can be more precise now. Player i's beliefs at w are correct if a-i(w)=d(a (w ieN which is a condition relating one player's beliefs with her opponents choices at a state. If at w this is true for all players, and if players are rational, then the profile(ai ( w))ien they play at w must be a Nash equilibrium On the other hand. if a profile a is a Nash equilibrium. the assumption that there exists a state w where players are rational and a-(u)=6(a)≠ (ai(w))ien that is actually played at w may fail to be a Nash equilibrium. ,and the profile for all iE N, but still it may be the case that ai( w)t ai for In some sense, the most interesting part of Proposition 0.1 is(2), because it states that the strong notion of correctness of first-order beliefs is necessary if we are to interpret Nash equilibrium as yielding behavioral predictions Equilibrium beliefs The"standard"approach, usually attributed to Aumann and Brandenburger, is intellectually more satisfying. Instead of asking what conditions imply that players' behavior will be consistent with Nash equilibrium, Aumann and Brandenburger ask what conditions imply that their beliefs will b The result is most transparent in two-player games. The idea is to consider a pair(o1, 2) of (possibly degenerate) mixed actions as a pair of beliefs. Thus, 1, which is conventionallyProposition 0.1 Fix a game G = (N,(Ai , ui)i∈N ) and a profile of actions a = (ai)i∈N . (1) If there exists a model M = (Ω,(Ti , ai , pi)i∈N ) for G and a state ω in the model such that ω ∈ R ∩ T i∈N CFBi and ai(ω) = ai for all i ∈ N, then a is a Nash equilibrium of G. (2) If a ∈ A is a Nash equilibrium of G, there exists a model M = (Ω,(Ti , ai , pi)i∈N ) for G and a state ω in the model such that ω ∈ R ∩ T i∈N CFBi and ai(ω) = ai for all i ∈ N. Proof: (1) At ω, ai = ai(ω) ∈ ri(α−i(ω)) = ri(δa−i ) for all i ∈ N; hence, a is a Nash equilibrium. (2) Consider a model with a single state ω (so that Ω = {ω} = ti(ω) for all i ∈ N) such that ai(ω) = ai . Since a is a Nash equilibrium, ω ∈ R; since necessarily pi(ω) = 1 for all i ∈ N, ω ∈ CSBi for all i ∈ N. You are authorized to feel cheated. This characterization does not add much to the definition of Nash equilibrium. However, it does emphasize what is entailed by correctness of beliefs. I mentioned this in the informal language we had to make do with a while ago, but I can be more precise now. Player i’s beliefs at ω are correct if α−i(ω) = δ(ai(ω))i∈N which is a condition relating one player’s beliefs with her opponents’ choices at a state. If at ω this is true for all players, and if players are rational, then the profile (ai(ω))i∈N they play at ω must be a Nash equilibrium. On the other hand, if a profile a is a Nash equilibrium, the assumption that there exists a state ω where players are rational and α−i(ω) = δ(aj )j6=i for all i ∈ N, but still it may be the case that ai(ω) 6= ai for some i ∈ N, and the profile (ai(ω))i∈N that is actually played at ω may fail to be a Nash equilibrium. In some sense, the most interesting part of Proposition 0.1 is (2), because it states that the strong notion of correctness of first-order beliefs is necessary if we are to interpret Nash equilibrium as yielding behavioral predictions. Equilibrium beliefs The “standard” approach, usually attributed to Aumann and Brandenburger, is intellectually more satisfying. Instead of asking what conditions imply that players’ behavior will be consistent with Nash equilibrium, Aumann and Brandenburger ask what conditions imply that their beliefs will be. The result is most transparent in two-player games. The idea is to consider a pair (φ1, φ2) of (possibly degenerate) mixed actions as a pair of beliefs. Thus, φ1, which is conventionally 3