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Journal of Convex Analysis 25 (2018), No. 1, [final page numbers not yet available]
Copyright Heldermann Verlag 2018



Minimax Representation of Nonexpansive Functions and Application to Zero-Sum Recursive Games

Marianne Akian
INRIA Saclay-Ile-de-France, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France
marianne.akian@inria.fr

Stéphane Gaubert
INRIA Saclay-Ile-de-France, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France
stephane.gaubert@inria.fr

Antoine Hochart
INRIA Saclay-Ile-de-France, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France
antoine.hochart@polytechnique.edu



We show that a real-valued function on a topological vector space is positively homogeneous of degree one and nonexpansive with respect to a weak Minkowski norm if and only if it can be written as a minimax of linear forms that are nonexpansive with respect to the same norm. We derive a representation of monotone, additively and positively homogeneous functions on L spaces and on Rn, which extends results of Kolokoltsov, Rubinov, Singer, and others. We apply this representation to nonconvex risk measures and to zero-sum games. We derive in particular results of representation and polyhedral approximation for the class of Shapley operators arising from games without instantaneous payments (Everett's recursive games).

Keywords: Nonexpansive maps, weak Minkowski norms, zero-sum games, recursive games, Shapley operators, risk measures, minimax representation.

MSC: 49J35, 91A15, 26B25

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