Then u$_{m}$ = a = u ′ m and therefore u = u $^{′}$, but u ∼$_{n}$ u ′ for every n , in contradiction to the eventual perfect monitoring assumption. By K¨ onig’s Lemma, T$_{a}$ is finite. Let n a be the maximal length of elements of T$_{a}$ , and let n = max { n $^{a}$| a ∈ A } + 1. Then from stage n onwards the opponent observes the action of stage m . □ Approximating strategies. For two strategies x, x ′ of player 1, let d ( x, x $^{′}$), the distance between x and x $^{′}$, be given by where the maximum is taken over all atoms p of P$_{n}$ . The distance d ( y, y $^{′}$) between two behavioral strategies y, y ′ of player 2 is defined analogously. 4.2. Lemma. Let x, x ′ be strategies of player 1 and y, y ′ be strategies of player 2. Then for every Borel subset W of A $^{N}$. Proof. The idea is to join a ( x, y )-random play and a ( x $^{′}$, y $^{′}$)-random play such that the two random plays are equal with high probability. Let z$_{n}$ : P$_{n}$ → A be given by z$_{n}$ = x$_{n}$ for even n ’s and z$_{n}$ = y$_{n}$ for odd n ’s and z ′ n : P$_{n}$ → A be given by z ′ n = x ′ n for even n ’s and z ′ n = y ′ n for odd n ’s. Let α$_{0}$, α ′ $_{0}$, α$_{1}$ , α ′ $_{1}$, . . . be a sequence of A -valued random variables defined inductively such that the conditional joint distribution of the pair ( α$_{n}$, α ′ $_{n}$) given the event { α$_{i}$ = a$_{i}$, α ′ i = a ′ i for 0 ≤ i < n } satisfies 10 d ( x, x $^{′}$) = ∑ n even max p ∈ P n ‖ x$_{n}$ ( p ) − x ′ $_{n}$( p ) ‖$_{1}$ , ‖ µ$_{x,y}$ ( W ) − µ$_{x}$ ′ ,y ′ ( W ) ‖ ≤ ( d ( x, x $^{′}$) + d ( y, y $^{′}$) ) / 2 P ( α$_{n}$ = a | α$_{i}$ = a$_{i}$, α ′ i = a ′ i for 0 ≤ i < n ) = z$_{n}$ ( a$_{0}$, . . . , a$_{n}$$_{−}$$_{1}$ )[ a ] , (4) P ( α ′ n = a ′ | α$_{i}$ = a$_{i}$, α ′ i = a ′ i for 0 ≤ i < n ) = z ′ $_{n}$( a ′ $_{0}$, . . . , a ′ n − $_{1}$)[ a $^{′}$] , and (5) P ( α ′ n = α$_{n}$ | α$_{i}$ = a$_{i}$, α ′ i = a ′ i for 0 ≤ i < n ) ≤ ‖ z$_{n}$ ( a$_{0}$, . . . , a$_{n}$$_{−}$$_{1}$ ) − z ′ $_{n}$( a ′ $_{0}$, . . . , a ′ n − $_{1}$) ‖$_{1}$ / 2 , (6)