Can an absorbing game with rational data have an irrational limit value? Yes:
In this note we provide the simplest examples where this phenomenon arises.
That is, the following 3Γ3 absorbing game A=β1β1β2ββ1β2β0ββ2β0β1βββ, and a sequence of 2Γ2 absorbing games whose limit
values are kβ, for all integer k. Finally, we conjecture that any
algebraic number can be represented as the limit value of an absorbing game