The Inverse Problem of Linear-Quadratic Differential Games: When is a Control Strategies Profile Nash?

TL;DR

This paper addresses the inverse problem in linear quadratic differential games by establishing analytical conditions under which a control strategy profile is Nash, using frequency-domain techniques and the Kalman equation to simplify the analysis.

Contribution

It introduces a frequency-domain necessary and sufficient condition for Nash profiles, including the circle criterion and a novel Kalman equation, reducing computational complexity.

Findings

Derived a frequency-domain condition for Nash control profiles.

Developed the Kalman equation to characterize cost parameters.

Showed leader can enforce Nash profiles via shared state penalties.

Abstract

This paper aims to formulate and study the inverse problem of non-cooperative linear quadratic games: Given a profile of control strategies, find cost parameters for which this profile of control strategies is Nash. We formulate the problem as a leader-followers problem, where a leader aims to implant a desired profile of control strategies among selfish players. In this paper, we leverage frequency-domain techniques to develop a necessary and sufficient condition on the existence of cost parameters for a given profile of stabilizing control strategies to be Nash under a given linear system. The necessary and sufficient condition includes the circle criterion for each player and a rank condition related to the transfer function of each player. The condition provides an analytical method to check the existence of such cost parameters, while previous studies need to solve a convex feasibility problem numerically to answer the same question. We develop an identity in frequency-domain representation to characterize the cost parameters, which we refer to as the Kalman equation. The Kalman equation reduces redundancy in the time-domain analysis that involves solving a convex feasibility problem. Using the Kalman equation, we also show the leader can enforce the same Nash profile by applying penalties on the shared state instead of penalizing the player for other players' actions to avoid the impression of unfairness.