Completely Mixed Discounted Bistochastic Games

The study of completely mixed strategies was initiated by Kaplansky, who gave a set of necessary and sufficient conditions for a matrix game to be completely mixed. In this paper, we study completely mixed strategies in two-player nonzero-sum discounted stochastic games (also referred to as bistochastic games), and show two results. Our first result shows that when all the reward matrices are skew-symmetric, the bistochastic game has a unique completely mixed equilibrium if and only if all the individual matrix games are completely mixed. We provide an example to show that this result does not hold true when the matrices are symmetric (in contrast with bimatrix games where an analogous result does hold). Our second result shows that for single player controlled games (controlled by Player 2, say) with symmetric reward matrices for the other player (Player 1), if the bistochastic game has a unique completely mixed equilibrium, then the individual matrix games of the noncontrolling player (Player 1) are completely mixed. We show that this implication does not hold on relaxing any of the assumptions, showing the sharpness of our result.