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The Cut-and-Play Algorithm: Computing Nash Equilibria via Outer Approximations

Margarida Carvalho, Gabriele Dragotto, Andrea Lodi, Sriram Sankaranarayanan

TL;DR

This work addresses the computation of Nash equilibria in broad nonconvex, noncooperative games where each player solves a nonconvex optimization with separable payoffs. It introduces Cut-and-Play, a practical outer-approximation algorithm that builds a sequence of polyhedral convexifications via cuts and branching, leveraging a convex reformulation that relates equilibria in the original game to those in the convexified game. A key component is the Enhanced Separation Oracle, which generates value cuts and verifies membership of candidate equilibria in the convex hulls, enabling finite termination under the assumption of polyhedral representability. The authors validate the method on reciprocally-bilinear and Stackelberg-influenced Nash games, showing significant performance gains and the ability to certify nonexistence in challenging nonconvex settings, often outperforming specialized baselines. This framework offers a scalable, general-purpose tool for equilibrium analysis in complex economic and engineering models, with potential extensions beyond polyhedral representations and unbounded strategy sets.

Abstract

We introduce Cut-and-Play, a practically-efficient algorithm for computing Nash equilibria in simultaneous non-cooperative games where players decide via nonconvex and possibly unbounded optimization problems with separable payoff functions. Our algorithm exploits an intrinsic relationship between the equilibria of the original nonconvex game and the ones of a convexified counterpart. In practice, Cut-and-Play formulates a series of convex approximations of the game and iteratively refines them with cutting planes and branching operations. Our algorithm does not require convexity or continuity of the player's optimization problems and can be integrated with existing optimization software. We test Cut-and-Play on two families of challenging nonconvex games involving discrete decisions and bilevel problems, and we empirically demonstrate that it efficiently computes equilibria while outperforming existing game-specific algorithms.

The Cut-and-Play Algorithm: Computing Nash Equilibria via Outer Approximations

TL;DR

This work addresses the computation of Nash equilibria in broad nonconvex, noncooperative games where each player solves a nonconvex optimization with separable payoffs. It introduces Cut-and-Play, a practical outer-approximation algorithm that builds a sequence of polyhedral convexifications via cuts and branching, leveraging a convex reformulation that relates equilibria in the original game to those in the convexified game. A key component is the Enhanced Separation Oracle, which generates value cuts and verifies membership of candidate equilibria in the convex hulls, enabling finite termination under the assumption of polyhedral representability. The authors validate the method on reciprocally-bilinear and Stackelberg-influenced Nash games, showing significant performance gains and the ability to certify nonexistence in challenging nonconvex settings, often outperforming specialized baselines. This framework offers a scalable, general-purpose tool for equilibrium analysis in complex economic and engineering models, with potential extensions beyond polyhedral representations and unbounded strategy sets.

Abstract

We introduce Cut-and-Play, a practically-efficient algorithm for computing Nash equilibria in simultaneous non-cooperative games where players decide via nonconvex and possibly unbounded optimization problems with separable payoff functions. Our algorithm exploits an intrinsic relationship between the equilibria of the original nonconvex game and the ones of a convexified counterpart. In practice, Cut-and-Play formulates a series of convex approximations of the game and iteratively refines them with cutting planes and branching operations. Our algorithm does not require convexity or continuity of the player's optimization problems and can be integrated with existing optimization software. We test Cut-and-Play on two families of challenging nonconvex games involving discrete decisions and bilevel problems, and we empirically demonstrate that it efficiently computes equilibria while outperforming existing game-specific algorithms.

Paper Structure

This paper contains 49 sections, 5 theorems, 11 equations, 3 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions.
  3. Outline.
  4. Literature Review
  5. Finite Games.
  6. Continuous Games.
  7. Integer Nonconvexities.
  8. The Problem and Our Assumptions
  9. Separable-Payoff Games
  10. Mixed Strategies and Equilibria.
  11. Polyhedral Representability
  12. Algorithmic Scheme
  13. Notation.
  14. Convex Reformulation
  15. Building Convex Approximations
  16. ...and 34 more sections

Key Result

Theorem 1

Let $G$ be an where each player $i$ solves $\min_{x^i}\{f^i(x^i;x^{-i}): x^i \in \mathcal{X}^i\}$. Let $\overline G$ be a convexified version of $G$ where each player $i$ solves $\min_{x^i}\{f^i(x^i;x^{-i}): x^i \in \operatorname{conv} (\mathcal{X}^i) \}$. For any $\overline x$ of $\overline G$, the

Figures (3)

  • Figure 1: A graphical representation of .
  • Figure 2: A 2-dimensional example of \ref{['Alg:EO']} separating $\overline x$ from $\operatorname{conv}(\mathcal{X})$. Here, $\mathcal{X}= \{\operatorname{conv}(\{v^2,\nu\})\} \bigcup \{\operatorname{conv}(\{v^1,v^3\})+\operatorname{cone}(\{r^1\})\}$. The set $\operatorname{conv}(\mathcal{X})$ is the light-blue region, whereas its inner approximation $\mathcal{W}=\operatorname{conv}(\{v^1,v^2,v^3\})$ is in purple.
  • Figure 3: At step $t$, branching might not exclude an infeasible from $\widetilde{\mathcal{X}}^i_{t+1}$. Given $\widetilde{\sigma}^i$ in the interior of $\widetilde{\mathcal{X}}^i_{t}$, the branching operation results in a refined $\widetilde{\mathcal{X}}^i_{t+1}=\operatorname{cl} \operatorname{conv}(Y^i_{t+1} \cup Z^i_{t+1})$ that does not exclude $\widetilde{\sigma}^i$.

Theorems & Definitions (20)

  • Definition 1: Separable-Payoff Game
  • Remark 1: Linear Form
  • Theorem 1
  • proof : Proof of \ref{['thm:convexification']}.
  • Definition 2: Approximate Game
  • Example 1
  • Remark 2
  • Example 2
  • Theorem 2
  • proof
  • ...and 10 more