Ordered field property for zero-sum stochastic games

TL;DR

This paper proves that the value vectors of finite zero-sum stochastic games with algebraic number data also lie within the same algebraic number field, resolving an open problem since 1991.

Contribution

It establishes that the value vectors in finite zero-sum stochastic games with algebraic data are themselves algebraic numbers, for both discounted and average cases.

Findings

Value vectors are algebraic numbers in the same field as game data.

Results apply to both discounted and average payoff versions.

Settles an open problem from 1991.

Abstract

We consider a finite state, finite action, zero-sum stochastic games with data defining the game lying in the ordered field of algebraic numbers. In both the discounted and the limiting average versions of these games we prove that the value vector also lies in the same field of algebraic numbers. In a prescribed sense, our results settle a problem that has remained open since, at least, 1991.