The Introduction of a Three-Dimensional Payoff Matrix

We introduce the novel concept of a three-dimensional payoff matrix based on a simplified version of a card game. This matrix is necessary to represent the three variables which correspond to the dynamics of the game. We find Sonnaville-Mensink Equilibria (SME) that represent the optimal actions for a game in a match with an infinite number of games. Subsequently, we identify three Nash Equilibria (NE) in both the infinite and finite match. The SME differs from the NE in that the former is applicable in a game with incomplete information where the optimal action of a player depends not only on the actions and corresponding payoffs of both players, but also on a third element that is influenced by the actions of both players. We illustrate that in most cases the NE is unilateral and thus known to one player only. The other player cannot recognize the NE, but will nevertheless act accordingly to the NE