On the three-person game baccara banque

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 $\theta$.