A game theory approach to the existence and uniqueness of nonlinear Perron-Frobenius eigenvectors

TL;DR

This paper generalizes the Perron-Frobenius theorem for nonlinear maps using a game-theoretic approach, establishing conditions for the existence and uniqueness of eigenvectors based on dominions and hypergraph methods.

Contribution

It introduces a combinatorial criterion involving dominions to determine the existence and uniqueness of eigenvectors for nonlinear, order-preserving, positively homogeneous maps.

Findings

Established a generalized Perron-Frobenius theorem for nonlinear maps.

Characterized when the slice space is bounded in Hilbert's projective metric.

Provided hypergraph methods to verify dominion conditions.

Abstract

We establish a generalized Perron-Frobenius theorem, based on a combinatorial criterion which entails the existence of an eigenvector for any nonlinear order-preserving and positively homogeneous map $f$ acting on the open orthant $\mathbb{R}_{\scriptscriptstyle >0}^n$. This criterion involves dominions, i.e., sets of states that can be made invariant by one player in a two-person game that only depends on the behavior of $f$ "at infinity". In this way, we characterize the situation in which for all $\alpha, \beta > 0$, the "slice space" $\mathcal{S}_\alpha^\beta := \{ x \in \mathbb{R}_{\scriptscriptstyle >0}^n \mid \alpha x \leq f(x) \leq \beta x \}$ is bounded in Hilbert's projective metric, or, equivalently, for all uniform perturbations $g$ of $f$, all the orbits of $g$ are bounded in Hilbert's projective metric. This solves a problem raised by Gaubert and Gunawardena (Trans. AMS, 2004). We also show that the uniqueness of an eigenvector is characterized by a dominion condition, involving a different game depending now on the local behavior of $f$ near an eigenvector. We show that the dominion conditions can be verified by directed hypergraph methods. We finally illustrate these results by considering specific classes of nonlinear maps, including Shapley operators, generalized means and nonnegative tensors.