Absorbing games with irrational values

TL;DR

This paper demonstrates that absorbing games with rational data can have irrational limit values, providing simple examples and conjecturing that all algebraic numbers can be represented as such limit values.

Contribution

It presents the first simple examples of absorbing games with irrational limit values and conjectures that any algebraic number can be realized as a limit value.

Findings

Provided the simplest examples of irrational limit values in absorbing games

Constructed sequences of 2x2 absorbing games with limit values as square roots of integers

Conjectured that any algebraic number can be a limit value of an absorbing game

Abstract

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.