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.

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.


UPDATE, September 2026. I asked ChatGPT to see if it could close the gap in my argument above. It did not, but there were two important outcomes of the chat. First, it pointed me to Douglas Zare’s article on the topic, which I think I was vaguely aware of before, but had failed to appreciate properly. In particular, Zare’s position has one clear advantage over my position, because it is reachable from the standard starting position, whereas my position is not. Moreover, Zare’s position is symmetric, which is what I need for my argument to go through.

But the more important point is that there is a major fallacy in my argument. Let us assume for the moment that the equity is not zero. I argued that the assumption that D/P is the perfect-play cube action leads to a contradiction, and the assumption that D/T is the perfect-play cube action leads to undefined equities. I then jumped to the incorrect conclusion that this proves that the equity is undefined. But this is fallacious. All I have really shown is that if equities are always finite, then the perfect play is not to double. This is a much weaker conclusion than I thought I had proved. Therefore, the assertion that there are backgammon positions with undefined equity is even less well-supported than I thought.


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