Geodesic
Minimax Representation of Nonexpansive Functions and Application to Zero-Sum Recursive Games
Journal of convex analysis, Tome 25 (2018) no. 1, pp. 225-24
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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).
Classification : 49J35, 91A15, 26B25
Mots-clés : Nonexpansive maps, weak Minkowski norms, zero-sum games, recursive games, Shapley operators, risk measures, minimax representation 
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     title = {Minimax {Representation} of {Nonexpansive} {Functions} and {Application} to {Zero-Sum} {Recursive} {Games}},
     journal = {Journal of convex analysis},
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     year = {2018},
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M. Akian; S. Gaubert; A. Hochart. Minimax Representation of Nonexpansive Functions and Application to Zero-Sum Recursive Games. Journal of convex analysis, Tome 25 (2018) no. 1, pp. 225-24. http://geodesic.mathdoc.fr/item/JCA_2018_25_1_JCA_2018_25_1_a13/