content_image/303025.jpg content_image/303026.jpg content_image/303027.jpg content_image/303028.jpg
for every n and every a$_{0}$, a ′ $_{0}$, . . . , a$_{n}$$_{−}$$_{1}$ , a ′ n − 1 ∈ A . The existence of random variables α$_{n}$, α ′ n with the prescribed conditional distribution follows from a standard coupling argument [5, Theorem 5.2]. From (4) it follows that 

<img src='content_image/303025.jpg'>

for every n and every a$_{0}$, . . . , a$_{n}$$_{−}$$_{1}$ ∈ A , i.e., that α$_{0}$, α$_{1}$, . . . is an ( x, y )-random play of Γ( W ). Similarly, from (5) it follows that α ′ $_{0}$, α ′ $_{1}$, . . . is a ( x $^{′}$, y $^{′}$)-random play of Γ( W ). From (6) it follows that 

Therefore, 

<img src='content_image/303026.jpg'>

<img src='content_image/303027.jpg'>

The assertion follows from the last inequality and the fact that µ$_{x,y}$ and µ$_{x}$ ′ ,y ′ are the distri- butions of α$_{0}$, α$_{1}$, . . . and α ′ $_{0}$, α ′ $_{1}$, . . . , respectively. □ 

4.3. Corollary. Let Δ$_{ϵ,n}$ be a finite set which is ϵ/ 2 $^{n}$-dense in Δ( A ) endowed with ‖‖$_{1}$ , i.e., such that the ϵ/ 2 $^{n}$-balls around elements of Δ$_{ϵ,n}$ cover Δ( A ) . Then there exists an ϵ -optimal strategy y 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$_{n}$ . 

Proof. Let y ′ be an ϵ/ 2-optimal strategy of player 2 in Γ( W ) and let y be a strategy of player 2 such that ‖ y$_{n}$ ( p ) − y ′ $_{n}$( p ) ‖$_{1}$ < ϵ/ 2 n and y$_{n}$ ( p ) ∈ Δ$_{ϵ,n}$( A ) for every odd n and every atom p of P$_{n}$ . Then d ( y, y $^{′}$) < ϵ , and therefore, 

<img src='content_image/303028.jpg'>

