When do Mixed-Integer Games Admit Rational Equilibria?
2026-06-15 • Computer Science and Game Theory
Computer Science and Game Theory
AI summaryⓘ
The authors study a type of game where each player has to solve problems involving both integers and real numbers, with their outcomes depending on quadratic relationships involving their own and others' choices. They focus on whether these games can have equilibrium points that are simple rational numbers, given rational inputs. Dividing the games into four categories based on certain mathematical terms, the authors find that only the simplest category guarantees rational equilibria when equilibria exist. The other more complex types can lead to equilibria that are irrational numbers.
Mixed-Integer ProgramGeneralized Nash EquilibriumQuadratic ObjectiveBilinear TermsRational EquilibriumLinear ConstraintsPlayer-Quadratic TermsRival-Dependent Constraints
Authors
Aloïs Duguet, Tobias Harks, Martin Schmidt, Julian Schwarz
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.