A game-theoretic analysis of baccara chemin de fer, II

TL;DR

This paper extends the analysis of baccara chemin de fer to include commission and non-zero-sum aspects, generalizing Foster's algorithm to find Nash equilibria in complex, parameter-dependent game models.

Contribution

It introduces a generalized Foster's algorithm for additive bimatrix games with continuous parameters and applies it to analyze Nash equilibria in more realistic baccara models.

Findings

Nash equilibrium is typically unique under the new models.

The game depends on both discrete and continuous parameters.

The generalized algorithm effectively finds equilibria in complex game settings.

Abstract

In a previous paper, we considered several models of the parlor game baccara chemin de fer, including Model B2 (a $2\times2^{484}$ matrix game) and Model B3 (a $2^5\times2^{484}$ 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 $2\times2^n$ 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 $100\,\alpha$ percent commission on Banker (player II) wins, where $0\le\alpha\le1/10$. Thus, the game now depends not just on the discrete parameter $d$ but also on a continuous parameter $\alpha$. Moreover, the game is no longer zero sum. To find all Nash equilibria under Model B2, we generalize Foster's algorithm to additive $2\times2^n$ 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.