Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A game theory approach to the existence and uniqueness of nonlinear
  Perron-Frobenius eigenvectors   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.

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The categories of this paper are: cond-mat
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The categories of this paper are: bayes-an
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The categories of this paper are: physics
15
The categories of this paper are: nucl-ex
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The categories of this paper are: adap-org
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The categories of this paper are: chao-dyn
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The categories of this paper are: supr-con
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The categories of this paper are: astro-ph
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The categories of this paper are: nlin
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-th
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The categories of this paper are: econ
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio