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Undefined equities

Posted By: Timothy Chow
Date: Tuesday, 20 October 2026, at 4:18 a.m.

In Response To: Undefined equities (ah_clem)

You're on the right track with your thinking. This basic line of reasoning had already been identified back in 1982. However, by itself, it does not prove that the equity of the Paradox Position is undefined. The skeptic could say, "XG is just confused about the cube action. The correct cube action is ND/T, and the cube does not in fact get turned repeatedly in a game between perfect players."

Ruling out the possibility that the correct cube action is ND/T is the trickiest part of the argument. Zare admitted he couldn't rule it out, and strictly speaking I can't rule it out either. However, I've taken one step further than Zare. Suppose we roll a 6 from the bar. It seems plausible that XG's estimate of the equity after you roll a 6 should be fairly trustworthy, since then there's no more repeated dancing by both sides to worry about. However, what I can prove is that the XG rollouts cannot in fact be giving us reliable estimates of the equity after entering with a 6, if by "equity" we mean that the cube is allowed to increase without limit. This last observation is the main new theoretical contribution of this article.

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