Fixed Point Sets of Payment-Free Shapley Operators and Structural Properties of Mean Payoff Games

Shapley operators are the dynamic programming operators of zero-sum stochastic games, they can be characterized as order-preserving maps commuting with the addition of a constant. We study a subclass of Shapley operators which are characterized by the property of commuting with the multiplication by a positive constant. We call them paymentfree, as they arise in the study of recursive games, in which the payment only occurs when the game stops. They also arise in the study of structural properties of parametric mean payoff games (the transition probabilities are fixed, not the transition payments) with finite action spaces and perfect information: their fixed point set can be shown to be all the possible mean payoff vectors of such games. A basic problem is to check whether the fixed point set of such an operator is trivial (reduced to the multiples of the unit vector), and more precisely to determine its characteristics, for instance decide whether there is a fixed point with a prescribed Arg min. Yang and Zhao showed (in Systems and Control Letters, 2004) that the former problem is co-NP-complete for deterministic games. We show that the latter problem is polynomial, and deduce that the former problem remains in co-NP for stochastic games. The proof relies on the construction of a Galois connection between faces of the hypercube that are invariant by the operator, and on a reduction to a reachability problem in a directed hypergraph.