Problem
Research questions and friction points this paper is trying to address.
Mixed-Integer Games
Rational Equilibria
Generalized Nash Equilibrium
Quadratic Objectives
Bilinear Constraints
When do Mixed-Integer Games Admit Rational Equilibria?
🤖 AI Summary
This study investigates the existence of rational solutions to mixed-integer linear-quadratic generalized Nash equilibrium problems with rational input data. Focusing on four classes of mixed-integer games distinguished by their objective functions and constraint structures, the authors combine techniques from mixed-integer programming, game theory, and algebraic number theory to provide the first complete characterization of necessary and sufficient conditions for the existence of rational equilibria. The main finding reveals that a rational equilibrium is guaranteed to exist only when all players have purely linear objectives—free of quadratic terms—and their constraints are independent of opponents’ strategies. For the remaining three classes, the paper constructs explicit counterexamples admitting only irrational equilibria, thereby demonstrating stringent limitations on the existence of rational solutions in mixed-integer generalized Nash games.
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
mixed-integer games
rational equilibria
generalized Nash equilibrium
linear-quadratic games
computational game theory
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