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Author Akian, M.; Gaubert, S.; Hochart, A. doi  openurl
  Title A Game Theory Approach To The Existence And Uniqueness Of Nonlinear Perron-Frobenius Eigenvectors Type
  Year 2020 Publication Discrete And Continuous Dynamical Systems Abbreviated Journal Discret. Contin. Dyn. Syst.  
  Volume 40 Issue 1 Pages 207-231  
  Keywords Nonlinear eigenproblem; nonexpansive map; Hilbert's projective metric; hypergraph; zero-sum stochastic game  
  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 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.  
  Address [Akian, Marianne; Gaubert, Stephane] Ecole Polytech, INRIA, CNRS, Inst Polytech Paris, F-91128 Palaiseau, France, Email:;  
  Corporate Author Thesis  
  Publisher Amer Inst Mathematical Sciences-Aims Place of Publication Editor  
  Language English Summary Language Original Title  
  Series Editor Series Title Abbreviated Series Title  
  Series Volume Series Issue Edition  
  ISSN 1078-0947 ISBN Medium  
  Area Expedition Conference  
  Notes WOS:000496748500009 Approved  
  Call Number UAI @ eduardo.moreno @ Serial 1075  
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