Values of Absorbing Recursive Games Express All Real Algebraic Numbers

Abstract

Many classes of two-player zero-sum stochastic games have the orderfield property: if all payoffs and transition probabilities lie in a subfield of $\mathbb{R}$, so does the undiscounted value. Absorbing games fail this property, and Oliu-Barton and Vigeral [\emph{Absorbing games with irrational values}, Oper.\ Res.\ Lett.\ 51 (2023) 555--559] conjectured the precise extent of the failure: every real algebraic number $\alpha$ of degree $m\geq 1$ over $\mathbb{Q}$ is the undiscounted value of a rational $m\times m$ absorbing game. We prove this conjecture, and in fact within a special subclass of absorbing games: for every such $\alpha$, the game realizing it is \emph{strictly absorbing} and \emph{recursive}, i.e., every action pair is absorbing with positive probability and all non-absorbing stage payoffs are zero; when $\alpha>0$ it can moreover be taken \emph{positive recursive}, with positive absorbing payoffs. As a corollary, the set of undiscounted values of rational $m\times m$ absorbing games is exactly the set of real algebraic numbers of degree at most $m$