The Game Changer Problem: Controlling Equilibria with Discrete Rewards

2026-06-27

AI Summary

This study addresses the problem of reconstructing the payoff matrices in two-player games under the constraint that payoffs must belong to a finite discrete set, such that a prescribed pure-strategy profile becomes the unique Nash equilibrium. The work establishes, for the first time, necessary and sufficient feasibility conditions applicable to both zero-sum and general-sum games. Leveraging the discrete nature of the payoff space, the authors devise an efficient dynamic programming algorithm capable of computing exact optimal solutions. In contrast to conventional linear programming–based approaches that assume continuous payoff adjustments, the proposed method guarantees solution exactness while significantly improving computational efficiency, thereby offering a novel paradigm for equilibrium steering in games with discrete payoff constraints.

Abstract

We introduce the game changer problem, where an external designer modifies a game's reward matrix to make a target pure action profile the unique equilibrium, subject to the constraint that all entries of the reward matrix come from a finite set. We give simple feasibility characterizations for two-player zero-sum games and general-sum games, and the discrete reward structure yields exact optimality and enables efficient dynamic programming algorithms, providing a sharper alternative to prior continuous reward redesign formulations based on linear programming.

Problem

• game changer problem
• equilibrium control
• discrete rewards
• reward matrix redesign
• pure strategy equilibrium

Innovation

• discrete rewards
• game changer problem
• unique equilibrium
• dynamic programming
• reward redesign