Linear Programming for Computing Equilibria Under Truncation Selection and Designing Defensive Strategies Against Malicious Opponents

Abstract

Linear programming and polyhedral representation conversion methods have been widely applied to game theory to compute equilibria. Here, we introduce new applications of these methods to two game-theoretic scenarios in which players aim to secure sufficiently large payoffs rather than maximum payoffs. The first scenario concerns truncation selection, a variant of the replicator equation in evolutionary game theory where players with fitnesses above a threshold survive and reproduce while the remainder are culled. We use linear programming to find the sets of equilibria of this dynamical system and show how they change as the threshold varies. The second scenario considers opponents who are not fully rational but display partial malice: they require a minimum guaranteed payoff before acting to minimize their opponent’s payoff. For such cases, we show how generalized maximin procedures can be computed with linear programming to yield improved defensive strategies against such players beyond the classical maximin approach. For both scenarios, we provide detailed computational procedures and illustrate the results with numerical examples.