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subset of W and val( S , η − $^{1}$( W$_{0}$ )) ≥ val( S , C ) > val( S , η − $^{1}$( W )) − ϵ , since η − $^{1}$( W$_{0}$ ) ⊇ C . The assertion follows from the fact that the games ( S , η − $^{1}$( W$_{0}$ )) and ( S , η − $^{1}$( W )) are Γ ∗ $_{ϵ}$( W$_{0}$ ) and Γ ∗ $_{ϵ}$( W ), respectively. □ 

The following lemma says that, up to ϵ , player 2 can guarantee in Γ ∗ ϵ the same amount he can guarantee in Γ. 

4.5. Lemma. For every Borel set W of A $^{N}$, 

<img src='content_image/295889.jpg'>

Proof. Note first that by definition of K$_{n}$ and from the perfect recall assumption, there exist functions g$_{n,k}$ : P$_{n}$ → S$_{k}$ for every n and every k ≤ n such that 

<img src='content_image/295890.jpg'>

for every h = ( a$_{0}$, . . . , a$_{n}$$_{−}$$_{1}$ ) ∈ A n and where π$_{n}$ : A n → P$_{n}$ is the natural projection. 

Let y be an ϵ -optimal behavioral strategy for player 2 in Γ( W ) such that y$_{n}$ ( p ) ∈ Δ$_{ϵ,n}$( A ) for every odd n and every atom p of P$_{n}$ . The existence of such a strategy y follows from Corollary 4.3. Consider a pure strategy y ∗ of player 2 in Γ ∗ $_{ϵ}$( W ) that is given by y ∗ $_{n}$( s$_{0}$, b$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$, b$_{n}$$_{−}$$_{1}$, s$_{n}$ ) = y$_{n}$ for every odd n and every partial history ( s$_{0}$, b$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$, b$_{n}$$_{−}$$_{1}$, s$_{n}$ ) of Γ ∗ $_{ϵ}$( W ). (Thus, in every odd stage n , player 2’s action is y$_{n}$ , regardless of the history.) Let x ∗ be any strategy of player 1 in Γ ∗ $_{ϵ}$. Let x be the behavioral strategy of player 1 in Γ( W ) that is given by 

<img src='content_image/295891.jpg'>

where ( s$_{0}$, b$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$, b$_{n}$$_{−}$$_{1}$, s$_{n}$ ) is the finite history of Γ ∗ $_{ϵ}$( W ) defined inductively by b$_{k}$ = x ∗ k ( s$_{0}$, b$_{0}$, . . . , s$_{k}$$_{−}$$_{1}$, b$_{k}$$_{−}$$_{1}$, s$_{k}$ ) for even k , b$_{k}$ = y$_{k}$ for odd k , and s$_{k}$ = g$_{n,k}$ ( p ). 

I am going to join an ( x, y )-random play of Γ( W ) and an ( x $^{∗}$, y$^{∗}$)-random play of Γ∗ $_{ϵ}$( W ) with equal payoffs. Let Π$_{0}$ , ζ$_{0}$, β$_{0}$, α$_{0}$, Π$_{1}$ , ζ$_{1}$, β$_{1}$, α$_{1}$, . . . be a sequence of random variables such that the values of ζ$_{n}$ are in S$_{n}$ , the values of β$_{n}$ are in B$_{n}$ , and the values of α$_{n}$ are in 

