A Game-Theoretic Analysis of Baccara Chemin de Fer , II

In a previous paper, we considered several models of the parlor game baccara chemin de fer , including Model B2 (a 2 × 2 484 matrix game) and Model B3 (a 2 5 × 2 484 matrix game), both of which depend on a positive-integer parameter d , the number of decks. The key to solving the game under Model B2 was what we called Foster’s algorithm, which applies to additive 2 × 2 n matrix games. Here “additive” means that the payoffs are additive in the n binary choices that comprise a player II pure strategy. In the present paper, we consider analogous models of the casino game baccara chemin de fer that take into account the 100 α percent commission on Banker (player II) wins, where 0 ≤ α ≤ 1 / 10 . Thus, the game now depends not just on the discrete parameter d but also on a continuous parameter α . Moreover, the game is no longer zero sum. To find all Nash equilibria under Model B2, we generalize Foster’s algorithm to additive 2 × 2 n bimatrix games. We find that, with rare exceptions, the Nash equilibrium is unique. We also obtain a Nash equilibrium under Model B3, based on Model B2 results, but here we are unable to prove uniqueness.