5. The role of eventual perfect monitoring In the auxiliary game that I use in the proof of Theorem 1.3, Chance plays the role of the players ‘randomization delegate’, and all the actions of the players are perfectly mon- itored. What happens if we try to apply the same argument to a game without eventual perfect monitoring ? Consider the following game, which is a modification of the game in Example 2.3 with Chance as player 1’s randomization delegate. 5.1. Example. Let A = { S , L } and consider the game with perfect monitoring that is played in stages n = 0 , 1 , . . . as follows: Player 1 plays at even stages and player 2 plays at odd stages. At every even stage n player 1 declares a mixed action p$_{n}$ ∈ [0 , 1], viewed as a dice over A ( p$_{n}$ is the probability that player 1 will leave the game at stage n if he didn’t stop already). At every odd stage n player 2 declares an action A = { S , L } . At infinity, Chance tosses the dice that player 1 declared one after another, thus obtaining an infinite history h ∈ A $^{∞}$. Player 1 wins the game if h ∈ W , where W is given by (3). Unlike the original game of Example 2.3, the game of Example 5.1 is determined and has value 1 / 2. An optimal strategy for player 2 is to leave the game at the first stage n for which (1 − p$_{0}$ ) · (1 − p$_{2}$ ) · · · · · (1 − p$_{n}$$_{−}$$_{1}$ ) < 1 / 2. However, as we saw, in the game of Example 2.3, where player 1’s action is not monitored by player 2, player 2 cannot guarantee to pay a payoff smaller than 1. So, the transition from imperfect monitoring to Chance acting as the player’s randomiza- tion delegate might fundamentally change the strategic potential of game in the absence of eventual perfect monitoring. On the other hand, in games with eventual perfect monitoring, as with finite games, a player does not lose anything if he declares his mixed contingent plan and let Chance do the randomization for him. The idea that a player declares his contingent mixed plan and let Chance randomize for him can be traced back to von Neumann and Morgenstern’s discussion of mixed actions. They argue that a player who fears his plan will be found out or deduced by the opponent may want to use randomization. When the player let Chance choose his action, the opponent 17