Equilibrium behaviors of the players in an infinitely repeated $2\times2$ $\varepsilon$-best response game

A stochastic infinitely repeated $\varepsilon$-best response game is analyzed, in which a $2\times2$ bimatrix game is played sequentially in an infinite number of rounds. The limits of the players' expected average gains in the first $n$ rounds of the game as $n\to\infty$ are calculated. These limits are taken as the players' expected average gains in the infinitely repeated $\varepsilon$-best response game. The players' Nash-equilibrium behaviors are described. It is shown that the players' equilibrium gains exceed their gains in the deterministic best-response game.