for every strategy x of player 1, where the first inequality follows from Lemma 4.2, and the second inequality from the fact that y ′ is ϵ/ 2-optimal. Therefore y is ϵ -optimal. □ Chance as the players’ randomization delegate. Let Γ = ( A, P$_{n}$, W ) be an infinite game with perfect recall and eventual perfect monitoring. In this section, I define an auxiliary stochastic game Γ ∗ ϵ = Γ ∗ $_{ϵ}$( W ) with perfect information, which mimics the original game Γ. Fix ϵ > 0 and, for every n ∈ N , let Δ$_{ϵ,n}$ be a finite subset of the interior of Δ( A ) which is ϵ/ 2 $^{n}$-dense in Δ( A ) endowed with ‖‖$_{1}$ . For every m ∈ N , fix k ( m ) > m such that m = k ( m ) mod 2, and such that at stage k ( m ) the opponent observes the action of stage m , as in Lemma 4.1. For every n , let B$_{n}$ = { b : P$_{n}$ → Δ$_{ϵ,n}$ } be the set of actions of stage n in Γ ∗ $_{ϵ}$( W ), so that an action is a function from P$_{n}$ (viewed as a collection of atoms) to Δ$_{ϵ,n}$; and let S$_{n}$ = A K n be the set of states of stage n in Γ ∗ $_{ϵ}$( W ), where K$_{n}$ = { m | k ( m ) = n } . Let f$_{n}$ : A n → S$_{n}$ be the projection over the corresponding coordinates m ∈ K$_{n}$ , and let F : S$_{0}$ × S$_{1}$ · · · → A N be such that for every u ∈ A $^{N}$. Γ ∗ $_{ϵ}$( W ) is played as follows: Player 1 plays at even stages and player 2 at odd stages. At every stage n , Chance announces a state s$_{n}$ in S$_{n}$ , and then the player that play at that stage announces an action b$_{n}$ in B$_{n}$ . Chance chooses the state s$_{n}$ of stage n from the distribution z ( s$_{0}$, b$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$, b$_{n}$$_{−}$$_{1}$ ) that is given by where ¯ α$_{0}$, . . . , ¯ α$_{n}$ is a sequence of A -valued random variables such that 12 (7) F ( f$_{0}$ ( u |$_{0}$ ) , f$_{1}$ ( u |$_{1}$ ) , . . . ) = u (8) z ( s$_{0}$, b$_{0}$, . . . , s$_{n}$$_{−}$$_{1}$, b$_{n}$$_{−}$$_{1}$ ) [ s ] = P ( f$_{n}$ (¯ α$_{0}$, . . . , ¯ α$_{n}$$_{−}$$_{1}$ ) = s | f$_{k}$ (¯ α$_{0}$, . . . ¯ α$_{k}$$_{−}$$_{1}$ ) = s$_{k}$ for 0 ≤ k < n − 1) , (9) P (¯ α$_{k}$ = a | ¯ α$_{0}$, . . . , ¯ α$_{k}$$_{−}$$_{1}$ ) = b$_{k}$ ( π$_{k}$ (¯ α$_{0}$, . . . , ¯ α$_{k}$$_{−}$$_{1}$ ))[ a ] .