A Game Theory Approach To The Existence And Uniqueness Of Nonlinear Perron-Frobenius Eigenvectors
Akian
M
author
Gaubert
S
author
Hochart
A
author
2020
English
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 R->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" S-alpha(beta) :={x is an element of R->0(n) vertical bar alpha x <= f(x) <= 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.
Nonlinear eigenproblem
nonexpansive map
Hilbert's projective metric
hypergraph
zero-sum stochastic game
WOS:000496748500009
exported from refbase (show.php?record=1075), last updated on Mon, 06 Jan 2020 12:32:55 -0300
text
10.3934/dcds.2020009
Akian_etal2020
Discrete And Continuous Dynamical Systems
Discret. Contin. Dyn. Syst.
2020
Amer Inst Mathematical Sciences-Aims
continuing
periodical
academic journal
40
1
207
231
1078-0947