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Computing Nash Equilibria in Potential Games with Private Uncoupled Constraints

Nikolas Patris, Stelios Stavroulakis, Fivos Kalogiannis, Rose Zhang, Ioannis Panageas

TL;DR

This work tackles the computation of Nash equilibria in potential games with private convex constraints by formulating a regularized Lagrangian and solving via independent gradient descent. It introduces Algorithm IGD$_{\bm{\lambda}}$, a distributed method that updates each player's strategy using local cost and constraint information, and proves an $O(\varepsilon)$-approximate feasible generalized Nash equilibrium can be reached in $T=O(1/\varepsilon^6)$ iterations. The analysis establishes smoothness of the max-regularized Lagrangian, strong duality under Slater's condition, and bounded Lagrange multipliers, leading to convergence guarantees. Numerical experiments on a constrained congestion game demonstrate diminishing Nash gaps and shrinking constraint violations, indicating practical viability for decentralized decision-making under private constraints.

Abstract

We consider the problem of computing Nash equilibria in potential games where each player's strategy set is subject to private uncoupled constraints. This scenario is frequently encountered in real-world applications like road network congestion games where individual drivers adhere to personal budget and fuel limitations. Despite the plethora of algorithms that efficiently compute Nash equilibria (NE) in potential games, the domain of constrained potential games remains largely unexplored. We introduce an algorithm that leverages the Lagrangian formulation of NE. The algorithm is implemented independently by each player and runs in polynomial time with respect to the approximation error, the sum of the size of the action-spaces, and the game's inherent parameters.

Computing Nash Equilibria in Potential Games with Private Uncoupled Constraints

TL;DR

This work tackles the computation of Nash equilibria in potential games with private convex constraints by formulating a regularized Lagrangian and solving via independent gradient descent. It introduces Algorithm IGD, a distributed method that updates each player's strategy using local cost and constraint information, and proves an -approximate feasible generalized Nash equilibrium can be reached in iterations. The analysis establishes smoothness of the max-regularized Lagrangian, strong duality under Slater's condition, and bounded Lagrange multipliers, leading to convergence guarantees. Numerical experiments on a constrained congestion game demonstrate diminishing Nash gaps and shrinking constraint violations, indicating practical viability for decentralized decision-making under private constraints.

Abstract

We consider the problem of computing Nash equilibria in potential games where each player's strategy set is subject to private uncoupled constraints. This scenario is frequently encountered in real-world applications like road network congestion games where individual drivers adhere to personal budget and fuel limitations. Despite the plethora of algorithms that efficiently compute Nash equilibria (NE) in potential games, the domain of constrained potential games remains largely unexplored. We introduce an algorithm that leverages the Lagrangian formulation of NE. The algorithm is implemented independently by each player and runs in polynomial time with respect to the approximation error, the sum of the size of the action-spaces, and the game's inherent parameters.
Paper Structure (33 sections, 28 theorems, 42 equations, 4 figures, 1 algorithm)

This paper contains 33 sections, 28 theorems, 42 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our contributions
  3. Technical overview
  4. Related work
  5. Preliminaries
  6. Notation and Definitions
  7. Notation
  8. Normal-form Games
  9. Potential Games
  10. Problem Statement
  11. Lagrangian dual function
  12. Dual problem
  13. Nonconvex Constrained Games and Lagrangian Functions
  14. Nonconvex Games
  15. Solution Concept
  16. ...and 18 more sections

Key Result

Lemma 3.4

For any player $i \in [n]$ and for any joint strategy of the other players ${\bm{x}}_{-i}$ along with their Lagrange multipliers $\boldsymbol{\lambda} _{-i}$, strong duality holds. where $\mathcal{L}_i$ is the player-wise Lagrangian function defined in Appendix.

Figures (4)

  • Figure 1: Nonconvex feasibility set
  • Figure 2: Players can pick out of four options: R1, R2, R3, HW by consuming {2,3,4,10} from their gas allowance.
  • Figure 3: Spider chart capturing the final distributions over paths of all players.
  • Figure 4: Collection of plots that are used to identify an approximate feasible approximate Nash Equilibrium.

Theorems & Definitions (56)

  • Definition 2.1: Approximate Nash equilibrium
  • Remark 2.2
  • Definition 2.3: Potential function
  • Definition 2.4: Primal problem
  • Definition 2.5: Lagrangian function
  • Definition 2.6: Lagrangian dual function
  • Definition 2.7: Weak duality
  • Definition 2.8: Dual problem
  • Definition 3.1: Approximate First Order Stationary Point
  • Definition 3.2: Approximate feasible Approximate Nash equilibrium
  • ...and 46 more