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BGonline.org Forums
Undefined equities
Posted By: MK In Response To: Undefined equities (Timothy Chow)
Date: Friday, 23 October 2026, at 9:53 p.m.
After 20+ years, Chow Chow puppy is still chasing his tail...
One fundamental fallacy that ruins his useless arguments is the notions of "perfect play", "perfect opponent", etc. and XG's being nearly perfect with its equity calculations being accurate enough, when no such assumptions are necessary for the paradox.
He writes: [[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.]]
This had been discussed at length about 4 years ago in various threads in RGB, with articles referring to "St. Patersburg Paradox", etc. Start by reading this:
https://groups.google.com/g/rec.games.backgammon/c/MZY1nS9b2I4/m/MY4lT0Y5AgAJ
The cube can go infinetely high between non-perfect players also, if both sides are played by the "same consistent player", i.e. a bot, of any strength since both sides' cubeful equities will be calculated exactly the same, thus making all decisions "perfect play" regardless of whether "perfecter plays" may or may not exist.
In fact it can happen between players of different strengths, even if one is claimed to be perfect or near-perfect, (although, of course, there are no such things). If one player defies the so-called "cube skill strategy" by constantly and purposefully wrongly doubling and thus forcing his "superior" opponent to keep doubling also, the cube can go infinetely high with cubeful equities becoming "undefined" and irrelevant since the result will be determined by the game being played out to the end effectively as a cubeless game.
I will leave this short to not be repetitive for the ones who may have read what I had written on the subject in the past, here and in RGB, and perhaps even visited my backgammon page to see the related experiments that I had done.
MK
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