The Competive Spectral Radius of Families of Nonexpansive Mappings

TL;DR

This paper introduces a new class of two-player zero-sum games based on the escape rate of switched nonexpansive dynamical systems, generalizing the joint spectral radius concept to non-linear, nonexpansive mappings.

Contribution

It characterizes the game's value through an infinite-dimensional non-linear eigenproblem, extending spectral radius concepts to a two-player setting with nonexpansive operators.

Findings

Existence of the game value is proven.

Characterization of the value via a non-linear eigenproblem.

Extension of spectral radius concepts to non-linear, two-player games.

Abstract

We consider a new class of repeated zero-sum games in which the payoff is the escape rate of a switched dynamical system, where at every stage, the transition is given by a nonexpansive operator depending on the actions of both players. This generalizes to the two-player (and non-linear) case the notion of joint spectral radius of a family of matrices. We show that the value of this game does exist, and we characterize it in terms of an infinite dimensional non-linear eigenproblem. This provides a two-player analogue of Ma\~ne's lemma from ergodic control. This also extends to the two-player case results of Kohlberg and Neyman (1981), Karlsson (2001), and Vigeral and the second author (2012), concerning the asymptotic behavior of nonexpansive mappings. We discuss two special cases of this game: order preserving and positively homogeneous self-maps of a cone equipped with Funk's and Thompson's metrics, and groups of translations.