Scenario-Game ADMM: A Parallelized Scenario-Based Solver for Stochastic Noncooperative Games

TL;DR

This work proposes a new sample-based approximation to a class of stochastic, general-sum, pure Nash games, where each player has an expected-value objective and a set of chance constraints and proposes a decentralized, consensus-based ADMM algorithm to efficiently compute a generalized Nash equilibrium of the approximated game.

Abstract

Decision-making in multi-player games can be extremely challenging, particularly under uncertainty. In this work, we propose a new sample-based approximation to a class of stochastic, general-sum, pure Nash games, where each player has an expected-value objective and a set of chance constraints. This new approximation scheme inherits the accuracy of objective approximation from the established sample average approximation (SAA) method and enjoys a feasibility guarantee derived from the scenario optimization literature. We characterize the sample complexity of this new game-theoretic approximation scheme, and observe that high accuracy usually requires a large number of samples, which results in a large number of sampled constraints. To accommodate this, we decompose the approximated game into a set of smaller games with few constraints for each sampled scenario, and propose a decentralized, consensus-based ADMM algorithm to efficiently compute a generalized Nash equilibrium (GNE) of the approximated game. We prove the convergence of our algorithm to a GNE and empirically demonstrate superior performance relative to a recent baseline algorithm based on ADMM and interior point method.

Figures (2)

• Figure 1: The convergence of Scenario-Game ADMM under different numbers of sampled scenarios in running example \ref{['eq: running example']}. With only 10 samples, we have no binding constraint, and we converge exponentially fast. With 50 and 100 samples, we suffer binding constraints, and the primal residual $\rho\|M(\mathbf{x}(k)-\mathbf{x}^*)\|^2$ oscillates. However, the Lyapunov function, which is defined as the sum of primal residual and dual residual $\frac{1}{\rho}\|\boldsymbol{\lambda}(k)-\boldsymbol{\lambda}^*\|^2$, decays monotonically.
• Figure 2: Comparison of the CPU time under different numbers of sampled scenarios. The solid blue curves represent the implementation of Scenario-Game ADMM which solves step \ref{['alg:parallel solving each scenario']} of Algorithm \ref{['alg:scenario-admm']} sequentially, i.e., one scenario by one scenario. The dashed blue curves represent the expected computation time when we implement step \ref{['alg:parallel solving each scenario']} of Algorithm \ref{['alg:scenario-admm']} in parallel. This expected computation time is derived by dividing the computation time of the blue solid curves by the number of scenarios. In both cases, Scenario-Game ADMM converges faster than ACVI. For each sampled scenario, we have 20 dimensional decision variables, and 35 constraints. When the number of sampled scenarios is 1000, there are $1000\times 35=35000$ constraints. ACVI fails to compile due to the scale of problem. With 1000 samples, our algorithm converges even when we replace the linear dynamics in \ref{['eq: running example']} with the nonlinear unicycle dynamics in laine2021computation, as shown in Fig.\ref{['subfig:size 1000']}.

Theorems & Definitions (25)

• Definition 1: facchinei2010generalized
• Proposition 1
• proof
• Proposition 2
• proof
• Remark 1
• Lemma 1
• proof
• Theorem 1
• proof