Characterization of values of deterministic absorbing games

Characterize the values realized by deterministic absorbing games, and in particular by deterministic recursive games in which payoff accrues only upon absorption.

Background

The paper’s main construction realizes every real algebraic number as the undiscounted value of a rational halting recursive game, but it relies on genuinely stochastic absorption probabilities. For deterministic transitions, absorption probabilities are restricted to 0 or 1; under strict absorption this forces absorption after a single stage and yields only rational values, while non-strict deterministic absorbing games can already exhibit irrational values in dimension 3 by prior work.

The authors state that the general deterministic case is not understood and specifically leave open the characterization of values attainable by deterministic absorbing games, with deterministic recursive games identified as a particularly relevant subclass.

References

The deterministic case is more delicate and remains largely open. By \citet[Theorem~1]{OV23}, deterministic absorbing games with $\min(m,n)<3$ obey the orderfield property, yet \citet{OV23} also exhibit a deterministic $3\times3$ game whose value is irrational, so determinism neither forces rationality nor is understood in general. Characterizing the values realized by deterministic absorbing games, and in particular by deterministic recursive games, in which payoff accrues only upon absorption, is a natural question left open by our work.

— Values of Absorbing Recursive Games Express All Real Algebraic Numbers  (2609.11583 - Asadi et al., 10 Sep 2026) in Section 7, Discussion, paragraph “Stochastic versus deterministic transitions”