Generalized open-loop Nash equilibria in linear-quadratic difference games with coupled-affine inequality constraints

TL;DR

This paper analyzes linear-quadratic difference games with coupled affine inequality constraints, deriving conditions for generalized open-loop Nash equilibria via coupled complementarity systems and providing a numerical solution approach.

Contribution

It introduces a novel framework linking Nash equilibria to coupled complementarity systems and offers a computational method for these equilibria in constrained dynamic games.

Findings

Necessary conditions involve coupled complementarity systems.

Sufficient conditions relate to convexity of objectives.

A numerical method for computing equilibria is proposed.

Abstract

In this note, we study a class of deterministic finite-horizon linear-quadratic difference games with coupled affine inequality constraints involving both state and control variables. We show that the necessary conditions for the existence of generalized open-loop Nash equilibria in this game class lead to two strongly coupled discrete-time linear complementarity systems. Subsequently, we derive sufficient conditions by establishing an equivalence between the solutions of these systems and convexity of the players' objective functions. These conditions are then reformulated as a solution to a linear complementarity problem, providing a numerical method to compute these equilibria. We illustrate our results using a network flow game with constraints.