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Minimax representation of nonexpansive functions and application to zero-sum recursive games

Marianne Akian 1, 2 Stéphane Gaubert 1, 2 Antoine Hochart 2, 1
1 TROPICAL - TROPICAL
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
Abstract : 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 extend 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).
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https://hal.inria.fr/hal-01425551
Contributor : Marianne Akian <>
Submitted on : Tuesday, January 3, 2017 - 4:04:40 PM
Last modification on : Friday, April 30, 2021 - 9:56:47 AM

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  • HAL Id : hal-01425551, version 1
  • ARXIV : 1605.04518

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Marianne Akian, Stéphane Gaubert, Antoine Hochart. Minimax representation of nonexpansive functions and application to zero-sum recursive games. Journal of Convex Analysis, Heldermann, 2018, 25 (1). ⟨hal-01425551⟩

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