THE DETERMINACY OF INFINITE GAMES WITH EVENTUAL PERFECT MONITORING Abstract. An infinite two-player zero-sum game with a Borel winning set, in which the opponent’s actions are monitored eventually but not necessarily immediately after they are played, is determined. The proof relies on a representation of the game as a stochastic game with perfect information, in which Chance operates as a delegate for the players and performs the randomizations for them, and on Martin’s Theorem about determinacy of such games. ERAN SHMAYA 1. Setup Consider an infinite two-player zero-sum game that is given by a triple ( A, ( P$_{n}$ )$_{n}$$_{∈}$$_{N}$ , W ) where A is a finite set of actions , P$_{n}$ is a partition of A n for every n ∈ N , the information partition of stage n , and W ⊆ A N is a Borel set, the winning set of player 1. The game is played in stages: player 1 chooses an action a$_{0}$ ∈ A ; then player 2 chooses an action a$_{1}$ ∈ A ; then player 1 chooses an action a$_{2}$ ∈ A , and so on, ad infinitum. Before choosing a$_{n}$ , the player who plays at stage n receives some information about the actions of previous stages: Let h = ( a$_{0}$, a$_{1}$, . . ., a$_{n}$$_{−}$$_{1}$ ) be the finite history that consists of the actions played before stage n ; then before choosing a$_{n}$ , the player who plays at stage n observes the atom of P$_{n}$ that contains h . Player 1 wins the game if the infinite history ( a$_{0}$, a$_{1}$, . . . ) is in W . When the action set and information partitions are fixed, I denote the game by Γ( W ). A behavioral strategy x = ( x$_{n}$ )$_{n}$$_{∈}$ N of player 1 is a sequence { x$_{n}$ : P$_{n}$ → Δ( A ) }$_{n}$$_{=0}$$_{,}$$_{2}$$_{,}$$_{4}$$_{,...}$ of functions: At stage n , after observing the finite history h = ( a$_{0}$, a$_{1}$, . . . , a$_{n}$$_{−}$$_{1}$ ), player 1 Date : May 31, 2018. 2000 Mathematics Subject Classification. Primary: 91A15, 03E75. Secondary: 03E15, 91A60. Key words and phrases. Infinite games, determinacy, stochastic games, imperfect monitoring. Thanks to to the anonymous referee for helpful suggestions and comments, and also to Chris Chambers, Tzachi Gilboa, John Levy, Ehud Lehrer, Wojciech Olszewski, Phil Reny, Eilon Solan, Bill Sudderth and Rakesh Vohra. 1 arXiv:0902.2254v2 [math.LO] 5 Jul 2011