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.

翻译：在先前一篇论文中，我们研究了赌场游戏“百家乐铁路牌戏”的若干模型，包括模型B2（一个$2\times2^{484}$矩阵博弈）和模型B3（一个$2^5\times2^{484}$矩阵博弈），两者均依赖于正整数参数$d$（牌副数）。求解模型B2下的博弈关键在于我们称之为Foster算法的方法，它适用于可加性$2\times2^n$矩阵博弈。其中“可加性”指收益在构成玩家II纯策略的$n$个二元选择中具有可加性。本文中，我们考虑赌场游戏“百家乐铁路牌戏”的类似模型，加入对庄家（玩家II）胜局征收$100\,\alpha\%$的佣金，其中$0\le\alpha\le1/10$。因此，该博弈现在不仅依赖于离散参数$d$，还依赖于连续参数$\alpha$。此外，该博弈不再是零和博弈。为求解模型B2下的所有纳什均衡，我们将Foster算法推广至可加性$2\times2^n$双矩阵博弈。我们发现，除少数例外情况，纳什均衡是唯一的。我们还基于模型B2的结果获得了模型B3下的一个纳什均衡，但在此无法证明其唯一性。