...olish spaces, A is a Borel subset of X, B is a Borel subset of Y , and A and B have the same cardinality. Then there is a Borel bijection from A onto B. For a proof of Fact A.2, see Theorem 13.1.1 in =-=[3]-=-. Corollary A.3 Suppose X and Y are Polish spaces, An, n ∈ N is a family of pairwise disjoint Borel subsets of X, Bn, n ∈ N is a family of pairwise disjoint Borel subsets of Y , and for each n, An has...

...4.6, we see that Proposition 2.3 holds. A Appendix: Borel Sets and Mappings Fact A.1 Every uncountable Borel subset of a Polish space has cardinality 2 ℵ0 For a proof of Fact A.1, see Theorem 13.6 in =-=[5]-=-, . Fact A.2 Suppose X and Y are Polish spaces, A is a Borel subset of X, B is a Borel subset of Y , and A and B have the same cardinality. Then there is a Borel bijection from A onto B. For a proof o...

... of P b m into finitely many pairwise disjoint classes, and elements of the same class behave in a similar way with respect to strategies. We will need the following result, which is Proposition 1 in =-=[1]-=-, . 6Fact 3.1 Let (ν0, . . . , νn) be a sequence of probability measures on S b . Then there is a probability measure µ on S b such that (i) The support of µ is the union of the supports of νi, i ≤ n...

...longs to a state with rationality and common assumption of rationality. 1 Introduction Given a finite game in strategic form with two players, say Ann and Bob, we show that the epistemic framework in =-=[2]-=- allows the following. (a) Each player considers all possibilities. (b) Each player is rational in the sense of [2] (avoids weakly dominated strategies but rules nothing out). (c) Ann assumes that Bob...

...e are complete lexicographic type structures which satisfies condition (1) and in addition has other completeness properties, such as containing all hierarchies of beliefs in the sense of Friedenberg =-=[4]-=-. The author thanks Adam Brandenburger and Amanda Friedenberg for helpful discussions related to this paper. 22 The Underlying Framework In this section we give a brief review of the concepts we will...