Computing Equilibria in Stochastic Nonconvex and Non-monotone Games via Gradient-Response Schemes

TL;DR

This work tackles the challenge of computing equilibria in stochastic N-player games with smooth, possibly nonconvex objectives, by introducing single-step gradient-response schemes that achieve almost-sure convergence to a quasi-Nash equilibrium (QNE) under a quadratic growth condition. When a potential structure with strong pseudomonotonicity is present, the methods guarantee convergence to a Nash equilibrium (NE); the deterministic variant further yields a local linear rate under a weak sharpness condition, enabling a two-stage global-to-local scheme. For convex but potentially non-monotone cases, a zeroth-order asynchronous modified gradient-response (ZAMGR) scheme is developed, leveraging a smoothing-based gap function and copositivity to obtain sublinear convergence to a stationary point of the gap, which under copositivity can be an NE. The framework is validated via network congestion, Nash-Cournot, and copositive congestion game applications, with numerics indicating promising performance. Overall, the paper broadens the tractability of equilibrium computation in nonconvex stochastic games through single-step, last-iterate guarantees and practical algorithms.

Abstract

We consider a class of smooth $N$-player noncooperative games, where player objectives are expectation-valued and potentially nonconvex. In such a setting, we consider the largely open question of efficiently computing a suitably defined {\em quasi}-Nash equilibrium (QNE) via a single-step gradient-response framework. First, under a suitably defined quadratic growth property, we prove that the stochastic synchronous gradient-response ({\bf SSGR}) scheme and its asynchronous counterpart ({\bf SAGR}) are characterized by almost sure convergence to a QNE and a sublinear rate guarantee. Notably, when a potentiality requirement is overlaid under a somewhat stronger pseudomonotonicity condition, this claim can be made for NE, rather than QNE. Second, under a weak sharpness property, we show that the deterministic synchronous variant displays a {\em linear} rate of convergence sufficiently close to a QNE by leveraging a geometric decay in steplengths. This suggests the development of a two-stage scheme with global non-asymptotic sublinear rates and a local linear rate. Third, when player problems are convex but the associated concatenated gradient map is potentially non-monotone, we prove that a zeroth-order asynchronous modified gradient-response ({\bf ZAMGR}) scheme can efficiently compute NE under a suitable copositivity requirement. Collectively, our findings represent amongst the first inroads into efficient computation of QNE/NE in nonconvex settings, leading to a set of single-step schemes that are characterized by broader reach while often providing last-iterate rate guarantees. We present applications satisfying the prescribed requirements where preliminary numerics appear promising.

Figures (2)

• Figure 1: SSGR and SAGR on network congestion problem.
• Figure 2: SGR and ZAMGR schemes.

Theorems & Definitions (67)

• Definition 1: Quasi-Nash equilibrium pang-scutari-2011
• Theorem 1: QNE existence
• proof
• Remark 1
• Definition 2: Four properties
• Proposition 1
• proof
• Remark 2
• Lemma 1
• Lemma 2: SSGR recursion