Approximative Methods for Monotone Systems of Min-Max-Polynomial Equations

A monotone system of min-max-polynomial equations (min-maxMSPE) over the variables X1, . . . , Xn has for every i exactly one equation of the form Xi = fi(X1, . . . , Xn) where each fi(X1, . . . , Xn) is an expression built up from polynomials with non-negative coefficients, minimumand maximum-operators. The question of computing least solutions of min-maxMSPEs arises naturally in the analysis of recursive stochastic games [4, 5, 13]. Min-max-MSPEs generalize MSPEs for which convergence speed results of Newton’s method are established in [10, 2]. We present the first methods for approximatively computing least solutions of min-max-MSPEs which converge at least linearly. Whereas the first one converges faster, a single step of the second method is cheaper. Furthermore, we compute ǫ-optimal positional strategies for the player who wants to maximize the outcome in a recursive stochastic game.