A pure strategy of player 1 in Γ ∗ $_{ϵ}$( W ) is a sequence { x ∗ n : S$_{0}$ × B$_{0}$ ×· · ·× S$_{n}$$_{−}$$_{1}$ × B$_{n}$$_{−}$$_{1}$ × S$_{n}$ → B$_{n}$ }$_{n}$$_{=0}$$_{,}$$_{2}$$_{,...}$ of functions: at stage n , after observing the finite history ( s$_{0}$, b$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$, b$_{n}$$_{−}$$_{1}$, s$_{n}$ ), player 1 plays x $^{∗}$( s$_{0}$, b$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$, b$_{n}$$_{−}$$_{1}$, s$_{n}$ ). Pure strategies y ∗ of Player 2 are defined anal- ogously. Let X ∗ and Y ∗ be the sets of pure strategies of players 1 and 2 respectively. The expected payoff for player 1 in the game Γ ∗ $_{ϵ}$( W ) when the players play according to ( x $^{∗}$, y $^{∗}$) is given by R ( x $^{∗}$, y $^{∗}$) = P ( F ( ζ$_{0}$, ζ$_{1}$, . . . ) ∈ W ) where ζ$_{0}$, β$_{0}$, ζ$_{1}$, β$_{1}$, . . . is a sequence of random variables, where the values of β$_{n}$ are in B$_{n}$ and the values of ζ$_{n}$ are in S$_{n}$ such that I call such a sequence ζ$_{0}$, β$_{0}$, ζ$_{1}$, β$_{1}$, . . . of random variables an ( x $^{∗}$, y $^{∗}$) -random play of Γ $_{ϵ}$( W ). Identifying the game Γ ∗ $_{ϵ}$( W ) with its normal form, say that Γ ∗ $_{ϵ}$( W ) is determined if P ( ζ$_{n}$ = s | ζ$_{0}$, β$_{0}$, . . . , ζ$_{n}$$_{−}$$_{1}$, β$_{n}$$_{−}$$_{1}$ ) = z ( ζ$_{0}$, β$_{0}$, . . . , ζ$_{n}$$_{−}$$_{1}$, β$_{n}$$_{−}$$_{1}$ ) [ s ] , β$_{n}$ = x ∗ $_{n}$( ζ$_{0}$, β$_{0}$, . . . , ζ$_{n}$$_{−}$$_{1}$, β$_{n}$$_{−}$$_{1}$, ζ$_{n}$ ) for even n, and β$_{n}$ = y ∗ $_{n}$( ζ$_{0}$, β$_{0}$, . . . , ζ$_{n}$$_{−}$$_{1}$, β$_{n}$$_{−}$$_{1}$, ζ$_{n}$ ) for odd n. In this case the common value of the two sides of the last equations is called the value of the game, and is denoted by val Γ ∗ $_{ϵ}$( W ). 4.4. Lemma. Let W ⊆ A N be a Borel set. Then the game Γ ∗ $_{ϵ}$( W ) is determined, and val Γ ∗ $_{ϵ}$( W$_{0}$ ) > val Γ ∗ $_{ϵ}$( W ) − ϵ for some compact subset W$_{0}$ of W . Proof. In the terminology of appendix A, the game Γ ∗ $_{ϵ}$( W ) is the stochastic game with stochastic setup S = (( S$_{n}$, B$_{n}$ )$_{n}$$_{∈}$ N , z ) and the winning set η − $^{1}$( W ), where η : S$_{0}$ × B$_{0}$ × S$_{1}$ × B$_{1}$ × · · · → A N is the continuous map given by Thus η − $^{1}$( W ) is a Borel set and therefore by Proposition A.1 the game ( S , η − $^{1}$( W )) is determined. Moreover, there exists a compact set C ⊆ S$_{0}$ × B$_{0}$ × S$_{1}$ × B$_{1}$ × . . . such that C ⊆ η − $^{1}$( W ) and val( S , C ) > val( S , η − $^{1}$( W )) − ϵ . Let W$_{0}$ = η ( C ). Then W$_{0}$ is a compact 13 sup ξ ∈ Δ( X ∗ ) inf y $^{∗}$∈ Y ∗ ∫ R ( x $^{∗}$, y $^{∗}$) ξ (d x $^{∗}$) = inf τ ∈ Δ( Y ∗ ) sup x $^{∗}$∈ X ∗ ∫ R ( x $^{∗}$, y $^{∗}$) τ (d y $^{∗}$) . (10) η ( s$_{0}$, b$_{0}$, s$_{1}$, b$_{1}$, . . . ) = F ( s$_{0}$, s$_{1}$, . . . ) .