On noncooperative $n$-player principal eigenvalue games

TL;DR

This paper studies a noncooperative game involving multiple players controlling a stochastic dynamical system, focusing on the principal eigenvalue related to system exit rates and establishing the existence of a Nash equilibrium linked to invariant sets.

Contribution

It introduces a novel framework connecting noncooperative eigenvalue games with invariant set analysis and proves the existence of Nash equilibria in this context.

Findings

Existence of Nash equilibrium in the eigenvalue game.

Relation between equilibrium strategies and invariant sets.

Characterization of the principal eigenvalue in stochastic control.

Abstract

We consider a noncooperative $n$-player principal eigenvalue game which is associated with an infinitesimal generator of a stochastically perturbed multi-channel dynamical system -- where, in the course of such a game, each player attempts to minimize the asymptotic rate with which the controlled state trajectory of the system exits from a given bounded open domain. In particular, we show the existence of a Nash-equilibrium point (i.e., an $n$-tuple of equilibrium linear feedback operators) that is distinctly related to a unique maximum closed invariant set of the corresponding deterministic multi-channel dynamical system, when the latter is composed with this $n$-tuple of equilibrium linear feedback operators.