Absorbing games with irrational values

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 $3\times 3$ absorbing game $$A = \begin{bmatrix} 1^* & 1^* & 2^* \\ 1^* & 2^* & 0\phantom{^*} \\ 2^* & 0\phantom{^*} & 1^* \end{bmatrix},$$ and a sequence of $2\times 2$ absorbing games whose limit values are $\sqrt{k}$, for all integer $k$. Finally, we conjecture that any algebraic number can be represented as the limit value of an absorbing game.