Computing the smallest fixed point of nonexpansive mappings arising in game theory and static analysis of programs

The problem of computing the smallest fixed point of a monotone map arises classically in the study of zero-sum repeated games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative.