Absorbing games with irrational values

Many classes of two-player zero-sum stochastic games have the orderfield property; that is, if all payoffs and transitions belong to some field, so does the limit value. Is it also the case for absorbing games? No: In this note, we exhibit m x m absorbing games with rational data whose limit values are algebraic of degree m, for each m E N*. Furthermore, we provide maximal conditions for the orderfield property to hold, namely if transitions are deterministic and one player has at most two actions. Last, we prove that any algebraic number of degree 2 is the limit value of a 2 x 2 absorbing game, which leads to the conjecture that any algebraic number of degree m is the limit value of an m x m absorbing game.(c) 2023 Elsevier B.V. All rights reserved.