We read Bi (E)"Player i is certain that E is true. Also define the event, "Everybody is certain that e is true”byB(E)=∩eNB2( Beliefs operators satisfy the following properties (1)t=B2(t); (2)E C F implies B(E)C Bi(F); (3)B(E∩F)=B(E)∩B(F); (4)Bi (E)C Bi (B(E))and Q2\B(E)C B(Q\B(E); (5)R2CB2(R2) Finally, a few shorthand definitions vi∈N,q∈△(A-): and vi∈N,a1∈A:[a1=al]={u:a(u)=a} Nash equilibrium I will provide two distinct epistemic characterizations of Nash equilibrium. The first is behavioral, nonstandard, and rather trivial. The second is actually a characterization of equilibrium beliefs, but it is the standard one, and requires a modicum of work Simple-minded characterization of Nash equilibrium You will recall from our past informal discussions that Nash equilibrium incorporates two key assumptions:(1)Players are Bauesian rational;(2) Their beliefs are correct: what they believe their opponents will do is exactly what they in fact do We now have the machinery we need to formalize this statement. The key idea is that correctness of first-order beliefs is easy to define in our model Definition 1 Fix a game G=(N, (Ai, ui)ieN)and a model M=(Q, (Ti, ai, pi)ien) for G For every i E N and w E Q, Player i has correct first-order beliefs at w iff there exists a-i=(a1)≠∈A- i such that (1)≠,a(u)=a; 2)a-(u)({a-})=1 Let CFBi denote the set of states where Player i's first-order beliefs are correct. That is, concisely. CFB1={:a-(u)=6a)≠ At every state in a model, players choose a single action-they do not randomize. If randomization is an actual strategic option, it must be modelled explicitly. The present approach only deals with pure Nash equilibriaWe read Bi(E) “Player i is certain that E is true.” Also define the event, “Everybody is certain that E is true” by B(E) = T i∈N Bi(E). Beliefs operators satisfy the following properties: (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). Finally, a few shorthand definitions: ∀i ∈ N, q ∈ ∆(A−i) : [α−i = q] = {ω : α−i(ω) = q} and ∀i ∈ N, ai ∈ Ai : [ai = ai ] = {ω : ai(ω) = ai} Nash Equilibrium I will provide two distinct epistemic characterizations of Nash equilibrium. The first is behavioral, nonstandard, and rather trivial. The second is actually a characterization of equilibrium beliefs, but it is the standard one, and requires a modicum of work. Simple-minded characterization of Nash equilibrium You will recall from our past informal discussions that Nash equilibrium incorporates two key assumptions: (1) Players are Bauesian rational; (2) Their beliefs are correct: what they believe their opponents will do is exactly what they in fact do. We now have the machinery we need to formalize this statement. The key idea is that correctness of first-order beliefs is easy to define in our model. Definition 1 Fix a game G = (N,(Ai , ui)i∈N ) and a model M = (Ω,(Ti , ai , pi)i∈N ) for G. For every i ∈ N and ω ∈ Ω, Player i has correct first-order beliefs at ω iff there exists a−i = (aj )j6=i ∈ A−i such that: (1) ∀j 6= i, aj (ω) = aj ; (2) α−i(ω)({a−i}) = 1. Let CFBi denote the set of states where Player i’s first-order beliefs are correct. That is, concisely, CFBi = {ω : α−i(ω) = δ(aj (ω))j6=i } At every state in a model, players choose a single action—they do not randomize. If randomization is an actual strategic option, it must be modelled explicitly. The present approach only deals with pure Nash equilibria. 2