发表机构
Institute of Science and Technology Austria (ISTA)（奥地利科学技术研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明每个实数代数数都可作为有理吸收递归博弈的无折扣值，并给出严格吸收与递归构造，从而完整刻画了有理吸收博弈的值集合。
AI 中文摘要
许多类别的两人零和随机博弈具有序域性质：如果所有收益和转移概率位于 $\mathbb{R}$ 的一个子域中，那么无折扣值也位于该子域中。吸收博弈不满足这一性质，Oliu-Barton 和 Vigeral [\emph{具有无理值的吸收博弈}，Oper.\\ Res.\\ Lett.\\ 51 (2023) 555--559] 猜想这一失效的精确程度：每个在 $\mathbb{Q}$ 上次数 $m\geq 1$ 的实数代数数 $\alpha$ 都是某个有理 $m\times m$ 吸收博弈的无折扣值。我们证明了这个猜想，并且实际上是在吸收博弈的一个特殊子类中证明的：对于每个这样的 $\alpha$，实现它的博弈是\emph{严格吸收}且\emph{递归}的，即每个行动对以正概率吸收且所有非吸收阶段收益为零；当 $\alpha>0$ 时，它还可以被取为\emph{正递归}的，具有正吸收收益。作为推论，有理 $m\times m$ 吸收博弈的无折扣值的集合恰好是次数至多为 $m$ 的实数代数数的集合。
英文摘要
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 [Absorbing games with irrational values, Oper. Res. Lett. 51 (2023) 555--559] conjectured the precise extent of the failure: every real algebraic number $α$ 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 $α$, the game realizing it is strictly absorbing and recursive, i.e., every action pair is absorbing with positive probability and all non-absorbing stage payoffs are zero; when $α>0$ it can moreover be taken 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 $m$.
