Stochastic Generalized Dynamic Games with Coupled Chance Constraints

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.

💡 Research Summary

This paper addresses stochastic generalized dynamic games (SGDGs) in which multiple agents interact over a finite horizon, the system dynamics are linear but time‑varying, and the agents are subject to shared safety constraints expressed as chance constraints. Each agent i selects a sequence of control actions (u_i) from a compact convex set, while the global state evolves according to
(s_{t+1}=A_t s_t+\sum_{j=1}^N B_{j t} u_{j t}+w_t),
with an unknown disturbance (w_t). The agents’ individual cost functions consist of an expected state‑dependent term and a deterministic interaction term. The safety requirements are of the form
(\mathbb{P}{\bar\xi_j(s,u,w)\le 0}\ge 1-\gamma_j,; j\in\mathcal S),
where (\bar\xi_j) may depend on the whole trajectory and (\gamma_j) is a prescribed violation tolerance.

The main difficulty is twofold: (i) the chance constraints are non‑convex and render the problem NP‑hard; (ii) the probability distribution of the disturbance is unknown, making the evaluation of expectations intractable. To overcome these challenges the authors propose a two‑step approach.

1. Convex under‑approximation of chance constraints
Assuming the disturbance satisfies a concentration of measure (CoM) property—i.e., for any Lipschitz function (\phi) we have (\mathbb{P}{|\phi(w)-\mathbb{E}