This paper is concerned with three typical issues about two-player repeated games. The first question regards the convergence of applied strategy to Nash equilibrium (NE) and corresponding rate of convergence. The second question asks about the optimal exit time for each player. In the third question, it is asked if the pure NE maximizes the mixed NE joint distribution. A Gibbs type sampling is proposed and the above mentioned issues are answered using the sampling method. Finally, concluding remarks are given.