(1) P ( α$_{n}$ = a | α$_{0}$, . . . , α$_{n}$$_{−}$$_{1}$ ) =     $^{x$_{n}$}$$^{(}$$^{α$_{0}$}$$^{, . . . , α$_{n}$$_{−}$$_{1}$}$$^{)[}$$^{a}$$^{]}$$^{,}$ if n is even , y$_{n}$ ( α$_{0}$, . . . , α$_{n}$$_{−}$$_{1}$ )[ a ] , if n is odd . I call such a sequence of random variables an ( x, y )- random play . If the players play according to the strategy profile ( x, y ), then the expected payoff for player 1 is given by randomizes his action according to x$_{n}$ ( π$_{n}$ ( h )), where π$_{n}$ ( h ) is the atom of P$_{n}$ that contains h . Abusing notations, I sometimes write x$_{n}$ ( h ) instead of x$_{n}$ ( π$_{n}$ ( h )). Behavioral strategies y of player 2 are defined analogously. Every pair x, y of strategies induces a probability distribution µ$_{x,y}$ over the set A N of infinite histories or plays : µ$_{x,y}$ is the joint distribution of a sequence α$_{0}$, α$_{1}$, . . . . . . of A - valued random variables such that where α$_{0}$, α$_{1}$, . . . is an ( x, y )-random play. The lower value val Γ( W ) and upper value val Γ( W ) of the game Γ( W ) are defined by: where the suprema are taken over all strategies x of player 1 and the infima over all strategies y of player 2. The game is determined if the lower and upper values are equal, val Γ( W ) = val Γ( W ), in which case their common value is called the value of the game. For ϵ ≥ 0, a strategy x of player 1 is ϵ -optimal if µ$_{x,y}$ ( W ) ≥ val Γ( W ) − ϵ for every strategy y of player 2. We also say that player 1 can guarantee payoff of at least val Γ( W ) − ϵ by playing such a strategy x . ϵ -optimal strategies of player 2 are defined analogously. Let ∼$_{n}$ be the equivalence relation over infinite histories such that u ∼$_{n}$ u ′ whenever u |$_{n}$ and u $^{′}$|$_{n}$ belong to the same atom of P$_{n}$ , where u |$_{n}$ and u $^{′}$|$_{n}$ are the initial segments of u and u ′ of length n . The interpretation is that if u, u ′ ∈ A N and u ∼$_{n}$ u $^{′}$, then at stage n the player cannot distinguish between u and u $^{′}$. Say that at stage n the player observes the 2 (2) µ$_{x,y}$ ( W ) = P (( α$_{0}$, α$_{1}$, . . . ) ∈ W ) , val Γ( W ) = sup x inf y µ$_{x,y}$ ( W ) , and val Γ( W ) = inf y sup x µ$_{x,y}$ ( W ) ,