# THE DETERMINACY OF INFINITE GAMES WITH EVENTUAL PERFECT MONITORING 

## ERAN SHMAYA 

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. 

## 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 

