A maximin based, linear programming approach to worst-case scenario control

2024-09-22

arXiv.org

AI Summary

This paper addresses robust decision-making in non-zero-sum games under uncertainty about opponents’ behavior—such as irrationality, adversarial intent, or incomplete information. Methodologically, it introduces a game-theoretic control framework that explicitly constrains decisions via the “acceptable worst-case payoff,” embedding worst-case payoff tolerance directly into a linear programming formulation. This transforms the original problem into computing a subset of Nash equilibria in an equivalent transformed game, applicable to matrix games of arbitrary size (2×2 to n×m). Theoretically, it establishes a rigorous containment relationship: robust equilibria against malicious opponents constitute a strict subset of Nash equilibria in the transformed game. Algorithmically, the approach is both general and computationally tractable. Empirical evaluation across two realistic scenarios demonstrates that the proposed strategy significantly enhances worst-case payoff stability and overall robustness, outperforming conventional minimax strategies.

Technology Category

Game Theory and Economic Paradigms: Adversarial LearningMultiagent Systems: Adversarial AgentsMachine Learning: Adversarial Learning & Robustness

Application Category

Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystemsResponsible Web: Machine-in-the-loop, human agency and autonomyWeb Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For non-zero-sum games, playing maximin or minimax strategies are not optimal in general, and thus we usually do not calculate them. However, under some conditions, such as incomplete information, trembling hands, or an irrational opponent, it can be reasonable to use the maximin expected utility preferences instead. A particular goal when there is uncertainty about an opponents behaviour -- especially when we cannot be certain of their rationality -- is for a player to avoid an unaffordable worst-case payoff. And such a worst-case payoff a player is willing to accept in practice need not be the exact maximin value. Here, we first introduce a motivating example and give the analytical description of an algorithm to control the worst-case scenario given a specified worst-case allowance for two-by-two games. Then we extend this method to n-by-m games using linear programming. We analyze two practical applications from the maximin angle, and also show that the equilibria when facing a malicious opponent coincides with a subset of the Nash equilibria of a transformation of the game. Lastly, we make some comments about problems when trying to analytically compute the subset of a strategy space with a specified worst-case allowance.

Problem

• Predicting truncation selection dynamics in large populations
• Computing defensive strategies against partially malicious opponents
• Applying linear programming to game theory scenarios

Innovation

• Linear programming for truncation dynamics prediction
• Defensive strategy against partially malicious opponents
• Maximin improvement via polyhedral conversion methods