# (Min Max)2=Min Max

A repeated game with absorbing states is played over the infinite future. A fixed one-shot game is played over and over again. However, for each action combination there is a probability that once it has occurred all future payoffs for the players are constant (that depends on the action combination that caused the "termination"), whatever the players play in the future. Given such a game, we define a modified game, by changing the payoff function. The new daily payoff for each player is the minimum between his expected payoff given the mixed-actions the players play in this stage, and his min-max value of the original game. Clearly the min-max value of the modified game, when the players are restricted to pure strategies (i.e. they cannot lotter between mixed-actions) cannot exceed the min-max value of the original game. We prove that the two values are equal.