Stochastic Generalized Dynamic Games with Coupled Chance Constraints

TL;DR

This paper introduces a convex approximation approach for stochastic generalized dynamic games with coupled chance constraints, proving equilibrium existence and developing a sampling-based algorithm with convergence guarantees.

Contribution

It proposes a novel convex under-approximation method for non-convex chance-constrained games and a semi-decentralized algorithm that converges without prior distribution knowledge.

Findings

Proved existence of a stochastic generalized Nash equilibrium (SGNE).

Developed a sampling-based algorithm with almost sure convergence.

Showed the approximation yields an $oldsymbol{oldsymbol{ ext{ε}}}$-SGNE for the original game.

Abstract

This paper investigates stochastic generalized dynamic games with coupling chance constraints, where agents have incomplete information about uncertainties satisfying a concentration of measure property. This problem, in general, is non-convex and NP-hard. To address this, we propose a convex under-approximation by replacing chance constraints with tightened expected-value constraints, yielding a tractable game. We prove the existence of a stochastic generalized Nash equilibrium (SGNE) in this new game and show that its variational SGNE is an $\boldsymbol{\varepsilon}$-SGNE for the original game, with $\boldsymbol{\varepsilon}$ expressed via the approximation errors and Lagrange multipliers. A semi-decentralized, sampling-based algorithm with time-varying step sizes is developed, requiring no prior knowledge of the uncertainty distribution or expectation evaluations. Unlike existing methods, it avoids step-size tuning based on Lipschitz constants or adaptive rules. Under standard assumptions on the pseudo-gradient, the algorithm converges almost surely to an SGNE.