We consider mixed-integer linear-quadratic generalized Nash equilibrium problems, i.e., games in which each player solves a mixed-integer program subject to linear constraints in her own and rivals' strategies as well as an objective which is quadratic in her own strategies and bilinear in her own and rivals' strategies. For this class of games, we study the question of the existence of rational equilibria assuming rational input data. We distinguish four subclasses according to the presence of player-quadratic terms in the objective and rival-dependent constraints. As our main result, we completely settle the rationality question for all four subclasses, i.e., we show that only player-linear games without player-quadratic terms and without rival-dependent constraints admit rational equilibria -- if the game admits equilibria at all. All other three classes contain instances with irrational equilibria only.

翻译：我们考虑混合整数线性二次广义纳什均衡问题，即每个玩家求解一个受自身及对手策略线性约束且目标函数为自身策略二次项及自身与对手策略双线性项的混合整数规划博弈。针对此类博弈，我们研究在输入数据为有理数的假设下理性均衡的存在性问题。根据目标函数中玩家二次项及对手依赖约束的存在性，我们将博弈划分为四个子类。作为主要结论，我们完全解决了所有四个子类的理性问题：仅当博弈中既无玩家二次项亦无对手依赖约束时（即玩家线性博弈），若博弈存在均衡，其必为理性均衡；其余三个子类均存在仅允许无理数均衡的实例。