Solving Multiparametric Generalized Nash Equilibrium Problems and Explicit Game-Theoretic Model Predictive Control

TL;DR

This paper introduces a method for explicitly solving parametric Generalized Nash Equilibrium problems with quadratic costs and linear constraints, enabling fast, interpretable solutions and applications in game-theoretic Model Predictive Control.

Contribution

The paper provides an exact multiparametric solution for GNE problems with specific assumptions, facilitating real-time computation and zero-shot updates in game-theoretic MPC.

Findings

Exact multiparametric GNE solutions enable minimal real-time computation.

The method allows for interpretability and enumeration of multiple equilibria.

Application to game-theoretic MPC shows improved performance over centralized solvers.

Abstract

We present a method to compute explicit solutions of parametric Generalized Nash Equilibrium (GNE) problems with convex quadratic cost functions and linear coupling and local constraints. Assuming the parameters only enter the linear terms of the cost functions and constraint right-hand sides, we provide the exact multiparametric solution of the GNE problem. Such a solution enables (i) minimal real-time computation, (ii) inherent interpretability, explainability, and exact enumeration of all multiple equilibria, (iii) determine desired GNE solution types in the case of infinitely-many equilibria, and (iv) zero-shot updates of the GNE solution due to changes of constraint right-hand sides and/or linear costs. In line with explicit Model Predictive Control (MPC) approaches, we apply our method to solve game-theoretic MPC (Receding Horizon Games) explicitly, comparing performance against centralized solvers in a battery charging game and in a toy two-mass spring system control problem. A Python implementation of the algorithms presented in this paper is available on https://github.com/bemporad/nash_mpqp.