by a measure on S, a measure on99×△(g9), a measure on9"×△(9)x△(920×△(99) and so on. Let Eu= llk>o a(s)denote the set of such descriptions: the reason for the superscript will be clear momentarily You may immediately see a problem with this representation of beliefs. Fix e=(p,p2,.)E E. Observe that, for every k≥2,p∈△(2-1)=△(92k-2×△(92k-2).Thus, the marginal of p on @2k-2 conveys information about a player's beliefs about @2k-2. However, by definition so does p-1∈△(9k-2 We obviously want the information conveyed by p and p- about our player's beliefs concerning 9-2 to be consistent. We thus concentrate on the subset of Eo defined by E marggk-2P =p Mertens and Zamir prove that, under regularity conditions on Q, the set E is homeo morphic to the set A(Q20 x E): that is, there exists a one-to-one, onto function f: E- △(9°×E0)( continuous, and with a continuous inverse) such that(1) every sequence e∈E corresponds to a unique measure f(e) on the product space Q20 x EO, and(2) every point Moreover, as you would expect, if e=(,p ,.), the to a unique e=f-w,e)EEl (u°,c0)=(u°,p2,yp2…)∈g∈E° can be mapped back marggk-1f(e)=p (recall the definition of Eo) In plain English, this says that, if we are interested in describing a player's beliefs about (1)Q and (2) her opponents full hierarchy of beliefs about S, then we should look no further than E. That is, we can regard elements as E as our player's types But. there is still something wrong with the construction so far. The reason is that hile each sequence e EE satisfies the consistency requirement, its elements may assign positive probability to inconsistent(sub)sequences of measures. That is, a player may hold consistent beliefs, but may believe that her opponent does not On the other hand, it is easy to see that, in any state w of a model of payoff-uncertainty as we have defined it, players' beliefs are consistent. Hence, at any state, players necessarily believe that their opponents hold consistent beliefs, that their opponent believe that their beliefs are consistent, and so on. That is, there is common certainty of consistency. We wish to impose the same restriction on the sequences of probability measures we are considering It is easy to do so inductively. Assume we have defined Ek.Then E+1={e∈E1:f(e)(9×E)=1} This makes sense: f(e)is a measure on 20 x E, so we can use it to read off the probability of the event Q20 x E. Note well what we are doing: the restriction is stated in terms of the probability measure on Q20 x EU induced by eE E, but(via the function f) this entails a restriction on the elements of the sequence e=(p, p,...)by a measure on Ω 0 , a measure on Ω 0 × ∆(Ω0 ), a measure on Ω0 × ∆(Ω0 ) × ∆(Ω0 × ∆(Ω0 )), and so on. Let E 0 = Q k≥0 ∆(Ωk ) denote the set of such descriptions: the reason for the superscript will be clear momentarily. You may immediately see a problem with this representation of beliefs. Fix e = (p 1 , p2 , . . .) ∈ E. Observe that, for every k ≥ 2, p k ∈ ∆(Ωk−1 ) = ∆(Ωk−2×∆(Ωk−2 )). Thus, the marginal of p k on Ωk−2 conveys information about a player’s beliefs about Ω k−2 . However, by definition, so does p k−1 ∈ ∆(Ωk−2 )! We obviously want the information conveyed by p k and p k−1 about our player’s beliefs concerning Ω k−2 to be consistent. We thus concentrate on the subset of E 0 defined by E 1 = {e ∈ E 0 : ∀k ≥ 2, margΩk−2 p k = p k−1 } Mertens and Zamir prove that, under regularity conditions on Ω, the set E 1 is homeo￾morphic to the set ∆(Ω0 × E 0 ): that is, there exists a one-to-one, onto function f : E 1 → ∆(Ω0 ×E 0 ) (continuous, and with a continuous inverse) such that (1) every sequence e ∈ E 1 corresponds to a unique measure f(e) on the product space Ω 0 × E 0 , and (2) every point (ω 0 , e0 ) = (ω 0 , p1 , p2 ...) ∈ Ω 0 ∈ E 0 can be mapped back to a unique e = f −1 (ω 0 , e0 ) ∈ E 1 . Moreover, as you would expect, if e = (p 1 , p2 , . . .), then margΩk−1 f(e) = p k (recall the definition of E 0 ). In plain English, this says that, if we are interested in describing a player’s beliefs about (1) Ω 0 and (2) her opponent’s full hierarchy of beliefs about Ω0 , then we should look no further than E 1 . That is, we can regard elements as E 1 as our player’s types. But, there is still something wrong with the construction so far. The reason is that, while each sequence e ∈ E 1 satisfies the consistency requirement, its elements may assign positive probability to inconsistent (sub)sequences of measures. That is, a player may hold consistent beliefs, but may believe that her opponent does not. On the other hand, it is easy to see that, in any state ω of a model of payoff-uncertainty as we have defined it, players’ beliefs are consistent. Hence, at any state, players necessarily believe that their opponents hold consistent beliefs, that their opponent believe that their beliefs are consistent, and so on. That is, there is common certainty of consistency. We wish to impose the same restriction on the sequences of probability measures we are considering. It is easy to do so inductively. Assume we have defined E k . Then E k+1 = {e ∈ E 1 : f(e)(Ω0 × E k ) = 1} This makes sense: f(e) is a measure on Ω0 × E 0 , so we can use it to read off the probability of the event Ω 0 × E k . Note well what we are doing: the restriction is stated in terms of the probability measure on Ω 0 × E 0 induced by e ∈ E 1 , but (via the function f) this entails a restriction on the elements of the sequence e = (p 1 , p2 , . . .). 8