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Nash Equilibria in Games with Playerwise Concave Coupling Constraints: Existence and Computation

Philip Jordan, Maryam Kamgarpour

Abstract

We study the existence and computation of Nash equilibria in continuous static games where the players' admissible strategies are subject to shared coupling constraints, i.e., constraints that depend on their \emph{joint} strategies. Specifically, we focus on a class of games characterized by playerwise concave utilities and playerwise concave constraints. Prior results on the existence of Nash equilibria are not applicable to this class, as they rely on strong assumptions such as joint convexity of the feasible set. By leveraging topological fixed point theory and novel structural insights into the contractibility of feasible sets under playerwise concave constraints, we give an existence proof for Nash equilibria under weaker conditions. Having established existence, we then focus on the computation of Nash equilibria via independent gradient methods under the additional assumption that the utilities admit a potential function. To account for the possibly nonconvex feasible region, we employ a log barrier regularized gradient ascent with adaptive stepsizes. Starting from an initial feasible strategy profile and under exact gradient feedback, the proposed method converges to an $ε$-approximate constrained Nash equilibrium within $\mathcal{O}(ε^{-3})$ iterations.

Nash Equilibria in Games with Playerwise Concave Coupling Constraints: Existence and Computation

Abstract

We study the existence and computation of Nash equilibria in continuous static games where the players' admissible strategies are subject to shared coupling constraints, i.e., constraints that depend on their \emph{joint} strategies. Specifically, we focus on a class of games characterized by playerwise concave utilities and playerwise concave constraints. Prior results on the existence of Nash equilibria are not applicable to this class, as they rely on strong assumptions such as joint convexity of the feasible set. By leveraging topological fixed point theory and novel structural insights into the contractibility of feasible sets under playerwise concave constraints, we give an existence proof for Nash equilibria under weaker conditions. Having established existence, we then focus on the computation of Nash equilibria via independent gradient methods under the additional assumption that the utilities admit a potential function. To account for the possibly nonconvex feasible region, we employ a log barrier regularized gradient ascent with adaptive stepsizes. Starting from an initial feasible strategy profile and under exact gradient feedback, the proposed method converges to an -approximate constrained Nash equilibrium within iterations.

Paper Structure

This paper contains 46 sections, 22 theorems, 58 equations, 6 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. Existence
  3. Computation
  4. Contributions
  5. Paper outline
  6. PRELIMINARIES
  7. Notation
  8. Game setting
  9. Nash equilibria
  10. EXISTENCE OF CONSTRAINED NASH EQUILIBRIA
  11. Structure of the feasible region
  12. Existence result
  13. Proof overview for Lemma \ref{['lem:contract']}
  14. Proof overview for Theorem \ref{['thm:existence']}
  15. LEARNING CONSTRAINED NASH EQUILIBRIA
  16. ...and 31 more sections

Key Result

Theorem 1

Let Assumptions ass:player-convex and ass:non-degenerate hold and suppose $\mathcal{C} \not= \emptyset$. Then there exists a constrained Nash equilibrium $x^{\star} \in \mathcal{X}$.

Figures (6)

  • Figure 1: Illustration of the feasible sets $\mathcal{C}^{(1)}$ and $\mathcal{C}^{(2)}$ described in Example \ref{['ex:degenerate']} and Example \ref{['ex:simple']}, respectively.
  • Figure 2: Proof sketch for contractibility (Lemma \ref{['lem:contract']}).
  • Figure 3: Strategy trajectory in the utility landscape of the cooperative game; colors indicate the log barrier value; grey is infeasible.
  • Figure 4: Nash gaps and constraint values over iterations $t$ for the cooperative game.
  • Figure 5: Routing game (tatarenko2017independent, Ex. 2.2).
  • ...and 1 more figures

Theorems & Definitions (49)

  • Example 1
  • Example 2
  • Theorem 1
  • Definition 1
  • Lemma 1
  • Definition 2
  • Example 3
  • Definition 3
  • Theorem 2
  • Proposition 1
  • ...and 39 more