Finally, let E=nk Ek(how do we know the intersection is nonempty? ) The e main esult is as follows Theorem 0.1(Mertens and Zamir, 1985 Under regularity conditions, there exists a home- amorphism g:E→△(90×E) such that, for all k1ande=(p2,p2,)∈E, marg gk-1g(e)=p This is really what we were after. Define &= SxexE(we treat the first E as referring to Player 1's hierarchical beliefs, and the second as referring to Player 2's). Next, for all e1, e t(u)={u′=(如0,21,2):1=e} for each i= 1, 2; that is, in state w, Player i"learns"only her own beliefs. Finally, let T={=(0,1,l2):E=e}:e∈E} i.e. the set of possible types corresponds to E. This is just a big huge model of payoff uncertainty, but it is a model according to our definition. The key points are (1) Every possible" reasonable"hierarchical belief is represented in this big huge (2)Type partitions arise naturally: they correspond to "reasonable"hierarchical Thus, both difficulties with models of payoff uncertainty can be overcomeFinally, let E = T k≥1 E k (how do we know the intersection is nonempty?). The main result is as follows: Theorem 0.1 (Mertens and Zamir, 1985). Under regularity conditions, there exists a home￾omorphism g : E → ∆(Ω0 × E) such that, for all k ≥ 1 and e = (p 1 , p2 , . . .) ∈ E, marg Ωk−1 g(e) = p k This is really what we were after. Define Ω = Ω0×E×E (we treat the first E as referring to Player 1’s hierarchical beliefs, and the second as referring to Player 2’s). Next, for all ω = (ω 0 , e1, e2), let ti(ω) = {ω 0 = (¯ω 0 , e¯1, e¯2) : ¯ei = ei} for each i = 1, 2; that is, in state ω, Player i “learns” only her own beliefs. Finally, let Ti = {{ω 0 = (¯ω 0 , e¯1, e¯2) : ¯ei = ei} : ei ∈ E} i.e. the set of possible types corresponds to E. This is just a big huge model of payoff uncertainty, but it is a model according to our definition. The key points are: (1) Every possible “reasonable” hierarchical belief is represented in this big huge model. (2) Type partitions arise naturally: they correspond to “reasonable” hierarchical beliefs. Thus, both difficulties with models of payoff uncertainty can be overcome. 9