Definition 3 Fix a game G=(N, (Ai, uiieN) and a model M=(Q, (Ti, ai, pilieN) for G A player i E I is deemed rational at state w E Q iff a(w)E_i(w)). Define the event Player i is rational"b R1={u∈9:a(u)∈r(a-l(u)} and the event, "Every player is rational"by R= nien ri This is quite straightforward. Finally, adapting the definition we gave last time Definition 4 Fix a game G=(N, (Ai, ui)ieN) and a model M=(@2, (Ti, ai, piie) for G Player i's belief operator is the map Bi: 23-22 defined by VEc9,B1(E)={∈9:(E|t(u)=1} Also define the event, "Everybody is certain that E is true "by B(e)=nien Bi(e) The following shorthand definitions are also convenient vi∈N,q∈△(A-):[a-i=q={u:a-(u)=q} which extends our previous notation, and vi∈N,a1∈A:[a;=al={u:a(u)=a} We now have a rather powerful and concise language to describe strategic reasoning in games. For instance, the following relations summarize our discussion of Figure 1 u1∈B1(a2=L])∩B2(a1=T); and also, more interestingly ∈R1;w∈R2;∈B1(B2) In fact ∈9\B2(B1);a1∈B1(9\B2(R1) Notice that we are finally able to give formal content to statements such as "Player 1 is certain that Player 2 is rational". These correspond to events in a given model, which in turn represents well-defined hierarchies of beliefs I conclude by noting a few properties of belief operators Proposition 0.1 Fix a game G=(N, (Ai, uiieN) and a model M=(Q2, (Ti, ai, pi)ieN)for G.Then, for every i∈N: (1)t=B2(t) (2)EC F implies Bi(E)C Bi(F); (3)B(EnF)=B(E)∩B(F) (4)Bi (E)C B(B(E)) and 2\ B(E)C Bi(Q\B(E)): (5)R2CB2(R)Definition 3 Fix a game G = (N,(Ai , ui)i∈N ) and a model M = (Ω,(Ti , ai , pi)i∈N ) for G. A player i ∈ I is deemed rational at state ω ∈ Ω iff ai(ω) ∈ ri(α−i(ω)). Define the event, “Player i is rational” by Ri = {ω ∈ Ω : ai(ω) ∈ ri(α−i(ω))} and the event, “Every player is rational” by R = T i∈N Ri . This is quite straightforward. Finally, adapting the definition we gave last time: Definition 4 Fix a game G = (N,(Ai , ui)i∈N ) and a model M = (Ω,(Ti , ai , pi)i∈N ) for G. Player i’s belief operator is the map Bi : 2Ω → 2 Ω defined by ∀E ⊂ Ω, Bi(E) = {ω ∈ Ω : pi(E|ti(ω)) = 1}. Also define the event, “Everybody is certain that E is true” by B(E) = T i∈N Bi(E). The following shorthand definitions are also convenient: ∀i ∈ N, q ∈ ∆(A−i) : [α−i = q] = {ω : α−i(ω) = q} which extends our previous notation, and ∀i ∈ N, ai ∈ Ai : [ai = ai ] = {ω : ai(ω) = ai} We now have a rather powerful and concise language to describe strategic reasoning in games. For instance, the following relations summarize our discussion of Figure 1: ω1 ∈ B1([a2 = L]) ∩ B2([a1 = T]); and also, more interestingly: ω1 ∈ R1; ω2 ∈ R2; ω1 ∈ B1(R2). In fact: ω2 ∈ Ω \ B2(R1); ω1 ∈ B1(Ω \ B2(R1)). Notice that we are finally able to give formal content to statements such as “Player 1 is certain that Player 2 is rational”. These correspond to events in a given model, which in turn represents well-defined hierarchies of beliefs. I conclude by noting a few properties of belief operators. Proposition 0.1 Fix a game G = (N,(Ai , ui)i∈N ) and a model M = (Ω,(Ti , ai , pi)i∈N ) for G. Then, for every i ∈ N: (1) ti = Bi(ti); (2) E ⊂ F implies Bi(E) ⊂ Bi(F); (3) Bi(E ∩ F) = Bi(E) ∩ Bi(F); (4) Bi(E) ⊂ Bi(Bi(E)) and Ω \ Bi(E) ⊂ Bi(Ω \ Bi(E)); (5) Ri ⊂ Bi(Ri). 4