Discounted Stochastic Games, A Complex Analytic Perspective

Abstract

In 1976 Bewley and Kohlberg proved that solutions of the limit discount equation of finite stochastic games are given by Puiseux series (i.e., fractional power series) in the discount factor, when β is sufficiently near 1. Their proof relied on a theorem from formal logic due to Tarski. In a recent paper, Szczechla et al. have given an alternative proof, and an extension, of this result with the help of powerful methods from a branch of the theory of functions of several complex variables known as complex analytic varieties.

Since the latter proof is technically rather involved, the present chapter is written with the goal of providing an intuitive understanding of the sense in which complex analytic varieties are, arguably, the most natural tool for the analysis of the limit discount equations of stochastic games. In particular, we emphasise the geometric point of view.

To illustrate this approach we consider some examples and invoke Puiseux’s original 1850 theorem to explicitly compute the required Puiseux series expansions. As a special case we show that our method gives an elegant proof of the now classical result, that solutions to the limit discount equations of Markov decision processes are given by Laurent series.