Existence and computation of monomial families of near-optimal strategies for recursive games

Abstract

In a finite recursive game in the sense of Everett, both players have stationary epsilon-optimal strategies for every epsilon>0. Frederiksen and Miltersen strengthened this result by showing that the strategies for all sufficiently small epsilon can be encoded by finitely many monomials: at every state, all but possibly one of the action probabilities are constants times integer powers of epsilon. The resulting finite symbolic object specifies a strategy for every sufficiently small accuracy. Their proof uses semialgebraic selection and Puiseux series. We give an alternative elementary proof of this regularity theorem for recursive games. We start with stationary strategies that guarantee vectors approaching the value through Everett's one-sided region. After fixing their support, we express, for each pure stationary reply, all absorption probabilities as quotients of directed-forest polynomials with nonnegative coefficients and a common positive denominator. Each payoff is a fixed signed linear combination of these quotients. We then compress the asymptotic orders of the finitely many forest monomials into one integer weight vector. This proof uses neither semialgebraic selection nor Puiseux series. Furthermore, for rational games with a fixed number N of active states, we present a deterministic polynomial-time algorithm that computes a monomial family exactly. It returns all algebraic coefficients in one ordered real univariate representation. The representation length and running time are at most L^{(N+1)^{O(N)}}, where L is the input length.