The Asymptotic Theory of Stochastic Games

We study two person, zero sum stochastic games. We prove that lim_n→∞{V_n/n} = lim_r→0rV(r), where V_n is the value of the n-stage game and V(r) is the value of the infinite-stage game with payoffs discounted at interest rate r > 0. We also show that V(r) may be expanded as a Laurent series in a fractional power of r. This expansion is valid for small positive r. A similar expansion exists for optimal strategies. Our main proof is an application of Tarski's principle for real closed fields.