A $2\times2$ $\varepsilon$-best response stochastic two-step game

A $2\times2$ $\varepsilon$-best response repeated game, in which each player in each subsequent round chooses a pure strategy based on the result of a random test, is analyzed. The random test is generated by the player's arbitrary mixed strategy prescribing the player to choose his/her best response to his/her partner's previously chosen pure strategy with a high probability. The so defined decision making patterns (called $\varepsilon$-best response functions) are interpreted as the players' behavioral strategies. These strategies define a stochastic game, in which the expected benefits averaged over all the rounds act as the players' benefits. The game is analyzed in the two-step case. A classification of the Nash equilibrium points is provided, and the equilibrium values are compared with the average benefits gained through the deterministic usage of the players' best response pure strategies.