content_image/311711.jpg content_image/311712.jpg content_image/311713.jpg content_image/311714.jpg content_image/311715.jpg
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 

<img src='content_image/311711.jpg'>

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 $^{N}$. 

<img src='content_image/311712.jpg'>

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 

<img src='content_image/311713.jpg'>

<img src='content_image/311714.jpg'>

<img src='content_image/311715.jpg'>

