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

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).