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Minimally Modifying a Markov Game to Achieve Any Nash Equilibrium and Value

Young Wu, Jeremy McMahan, Yiding Chen, Yudong Chen, Xiaojin Zhu, Qiaomin Xie

TL;DR

The paper introduces a formal framework for minimally modifying a two-player zero-sum Markov game to install a target MPE with a specified value range, while minimizing a cost of modification. It provides necessary and sufficient conditions (SIISOW and INV) for the target NE to be unique and reformulates the problem into a convex linear program augmented with spectral constraints, then remedies nonconvexity via a Relax and Perturb (RAP) approach that adds a small eRPS-based perturbation to guarantee invertibility with probability one. The authors extend the method from normal-form games to Markov games, using stage-game Q-functions and Bellman consistency, and propose RAP-MG with analogous feasibility and asymptotic optimality guarantees. Through toy experiments and scale benchmarks, they demonstrate the algorithm’s ability to install mixed-strategy equilibria, control game value, and scale to large action spaces and horizons, with public code available for replication. The work advances understanding of strategic game modification, offering practical tools for both benign design and defensive modeling in multi-agent settings, with several directions for future extension to more general settings and constraints.

Abstract

We study the game modification problem, where a benevolent game designer or a malevolent adversary modifies the reward function of a zero-sum Markov game so that a target deterministic or stochastic policy profile becomes the unique Markov perfect Nash equilibrium and has a value within a target range, in a way that minimizes the modification cost. We characterize the set of policy profiles that can be installed as the unique equilibrium of a game and establish sufficient and necessary conditions for successful installation. We propose an efficient algorithm that solves a convex optimization problem with linear constraints and then performs random perturbation to obtain a modification plan with a near-optimal cost. The code for our algorithm is available at https://github.com/YoungWu559/game-modification .

Minimally Modifying a Markov Game to Achieve Any Nash Equilibrium and Value

TL;DR

The paper introduces a formal framework for minimally modifying a two-player zero-sum Markov game to install a target MPE with a specified value range, while minimizing a cost of modification. It provides necessary and sufficient conditions (SIISOW and INV) for the target NE to be unique and reformulates the problem into a convex linear program augmented with spectral constraints, then remedies nonconvexity via a Relax and Perturb (RAP) approach that adds a small eRPS-based perturbation to guarantee invertibility with probability one. The authors extend the method from normal-form games to Markov games, using stage-game Q-functions and Bellman consistency, and propose RAP-MG with analogous feasibility and asymptotic optimality guarantees. Through toy experiments and scale benchmarks, they demonstrate the algorithm’s ability to install mixed-strategy equilibria, control game value, and scale to large action spaces and horizons, with public code available for replication. The work advances understanding of strategic game modification, offering practical tools for both benign design and defensive modeling in multi-agent settings, with several directions for future extension to more general settings and constraints.

Abstract

We study the game modification problem, where a benevolent game designer or a malevolent adversary modifies the reward function of a zero-sum Markov game so that a target deterministic or stochastic policy profile becomes the unique Markov perfect Nash equilibrium and has a value within a target range, in a way that minimizes the modification cost. We characterize the set of policy profiles that can be installed as the unique equilibrium of a game and establish sufficient and necessary conditions for successful installation. We propose an efficient algorithm that solves a convex optimization problem with linear constraints and then performs random perturbation to obtain a modification plan with a near-optimal cost. The code for our algorithm is available at https://github.com/YoungWu559/game-modification .
Paper Structure (33 sections, 7 theorems, 88 equations, 3 figures, 2 tables, 2 algorithms)

This paper contains 33 sections, 7 theorems, 88 equations, 3 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. The Game Modification Problem
  3. Our Contributions
  4. Related Work
  5. Modifying Normal Form Games
  6. Preliminaries
  7. Equivalent Formulation of Game Modification
  8. Feasibility of Game Modification
  9. An Efficient Algorithm for Game Modification in Normal Form Games
  10. Performance Guarantees for RAP
  11. Markov Games Modification
  12. Preliminaries
  13. Reformulation and Feasibility of Markov Game Modification
  14. Efficient Algorithm for Modifying Markov Games
  15. Experiments
  16. ...and 18 more sections

Key Result

Proposition 1

For normal form games and a target policy $\left(\mathbf p, \mathbf q\right)$ with supports $\mathcal{I}, \mathcal{J}$, the game modification problem eq:GM is equivalent to the following optimization problem: where $\sigma_{\min}(\cdot)$ denotes the smallest singular value.

Figures (3)

  • Figure 1: Convergence to Optimal Cost
  • Figure 2: Scale Benchmark for Number of Actions
  • Figure 3: Scale Benchmark for Number of Periods

Theorems & Definitions (20)

  • Definition 1: Game Modification
  • Definition 2: Nash Equilibrium
  • Proposition 1: Reformulation of Normal-Form Game Modification
  • Lemma 2: Uniqueness of NE
  • Theorem 3: Feasibility of Game Modification
  • Definition 3: Extended Rock-Paper-Scissors Game
  • Lemma 4
  • Remark 1
  • Example 1: One-Time Cost
  • Example 2: Forever Cost
  • ...and 10 more