3.1. Lemma. An infinite game with a finite action set, perfect recall, and a compact winning set is determined. Proof. Let Γ = ( A, { P$_{n}$ }$_{n}$$_{∈}$ N ,W ) be an infinite game with a finite action set A , perfect recall, and a compact winning set W . By Kuhn’s Theorem it is sufficient to prove that the normal form game N (Γ) = (Σ , Θ ,R ) of Γ is determined. This follows from the minimax theorem. Indeed, the set Σ of pure strategies of player 1 is compact in the product topology, and the payoff function R is upper semicontinuous as a composition of the continuous function ( σ,θ ) ↦→ u ( σ,θ ) and the function 1$_{W}$ , which is upper semi continuous because W is closed. □ Throughout this section, we fixe a game Γ that satisfies the conditions of Theorem 1.3. Overview of the proof. Eventual perfect monitoring entails that the action of stage m is known to the opponent at infinity. Lemma 4.1 below shows that in fact more is true: for every m there exists some finite stage n > m at which, regardless of the play path, the opponent knows the action of stage m . This means that at such a stage n the action taken by the player at stage m becomes common knowledge : The opponent knows the action played at stage m , the player that played the action knows that the opponent knows, the opponent knows that the player that play the action knows that the opponent knows, ad infinitum. Roughly speaking, I am going to construct a stochastic game Γ ∗ with perfect information that mimics the original game Γ. In Γ $^{∗}$, at every stage m , the player announces a mixture over the set A of actions contingent on his information at that stage. So in Γ $^{∗}$, instead of choosing an action which is not revealed to his opponent (as in Γ), the player announces how he intends to randomize his action. The actual randomization is performed by Chance at a future stage k ( m ), in which the action should become commonly known, and the realization of that randomization is immediately made public. So in the game Γ $^{∗}$, Chance performs the randomization for the player. The determinacy of Γ ∗ follows from Martin’s Theorem, and I 8 4. Proof of Theorem 1.3