When do Mixed-Integer Games Admit Rational Equilibria?
For game theorists and optimization researchers, this work provides a complete characterization of when rational equilibria exist in mixed-integer games, revealing fundamental limitations.
The paper studies mixed-integer linear-quadratic generalized Nash equilibrium problems and determines that only player-linear games without quadratic terms and without rival-dependent constraints admit rational equilibria; all other subclasses contain instances with only irrational equilibria.
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.