On the three-person game baccara banque

TL;DR

This paper analyzes three different equilibrium solutions in the three-person zero-sum game baccara banque, providing explicit derivations for each across all parameter values under specific assumptions.

Contribution

It introduces and derives explicit forms of the independent, correlated, and Nash equilibria for baccara banque for all parameter values, clarifying their existence conditions.

Findings

Nash equilibrium exists only for certain parameters

Correlated cooperative equilibrium always exists

Explicit solutions are derived under specific assumptions

Abstract

Baccara banque is a three-person zero-sum game parameterized by $\theta\in(0,1)$. A study of the game by Downton and Lockwood claimed that the Nash equilibrium is of only academic interest. Their preferred alternative is what we call the independent cooperative equilibrium. But this solution exists only for certain $\theta$. A third solution, which we call the correlated cooperative equilibrium, always exists. Under a "with replacement" assumption as well as a simplifying assumption concerning the information available to one of the players, we derive each of the three solutions for all $\theta$.