Open-loop potential difference games with inequality constraints

TL;DR

This paper investigates potential difference games with inequality constraints, establishing conditions for Nash equilibria via optimal control and potential functions, especially in linear-quadratic cases, with applications to smart grid energy storage.

Contribution

It introduces a framework linking potential difference games with inequality constraints to optimal control problems, providing construction methods for potential functions and equilibrium computation techniques.

Findings

Conditions for existence of Nash equilibria in constrained potential games.

Method to construct potential functions from optimal control formulations.

Application to energy storage incentives in smart grids.

Abstract

Static potential games are non-cooperative games which admit a fictitious function, also referred to as a potential function, such that the minimizers of this function constitute a subset (or a refinement) of the Nash equilibrium strategies of the associated non-cooperative game. In this paper, we study a class $N$-player non-zero sum difference games with inequality constraints which admit a potential game structure. In particular, we provide conditions for the existence of an optimal control problem (with inequality constraints) such that the solution of this problem yields an open-loop Nash equilibrium strategy of the corresponding dynamic non-cooperative game (with inequality constraints). Further, we provide a way to construct potential functions associated with this optimal control problem. We specialize our general results to a linear-quadratic setting and provide a linear complementarity problem-based approach for computing the refinements of the open-loop Nash equilibria. We illustrate our results with an example inspired by energy storage incentives in a smart grid.