Are there undefined equities in backgammon?

Backgammon bots presuppose that every backgammon position has a well-defined equity, which in the case of a money game means the expected payoff if both sides employ “perfect play” (or more accurately, Nash equilibrium play). However, if the cube value is truly unlimited, then it is not clear that the equity of an arbitrary position in backgammon is always finite.

It has been recognized for a long time that there are some positions in backgammon whose equity is probably undefined, although to the best of my knowledge, this has never been mathematically proven. My contribution to this topic has been to exhibit a backgammon position for which one can rigorously prove that the equity is either undefined or zero. Anyone with some backgammon experience can see that the equity is “obviously” not zero, so this comes very close to a rigorous proof that the equity is undefined. Perhaps someone reading this can close the gap by proving rigorously that the equity is not zero. For details, see this post and this post that I made to the BGOnline forums. (You may need to set your browser's text encoding to "Western" to get some of the characters to display properly.)

One catch with my position is that one can show, via retrograde analysis, that it cannot be reached from the initial position in backgammon. However, very similar positions can be reached.

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