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BGonline.org Forums
Undefined equities
Posted By: ah_clem In Response To: Undefined equities (ah_clem)
Date: Saturday, 17 October 2026, at 6:31 p.m.
The equity value for when the cube is turned n times would be the limit as n -> ∞. Since the growth is exponential (order is 2ⁿ) the limit must be infinite . But each turn there is some finite probability that the on roll player will roll a six and the cube. Is this probability low enough that the exponential growth out-races it? My math skills are pretty rusty, and probability was never my field, but I'm not seeing it.
Ok, I think I'm seeing it now. Once either player enters it's basically game over and the cube remains where it was. The probability of entering after n rolls grows slowly while the cube value grows as 2ⁿ so it swamps the probability of the game ending after n rolls.
An analogy to the Zare posiition, which I think simplifies things while leaving the basic idea intact is as follows:
It's a game with two players who alternate spinning a wheel with N possible outcomes, one of which wins the game and the other N-1 allow the other player to take the next turn. The stakes double with each failed attempt.
If N==2, it's like a coin flip and the probability of winning is .5 each time. If N==100 it's like a giant roulette wheel with the probability of winning equal to 1% each time. Extend the analogy to have some other random process and one can imagine the game taking place with a one-trial probability, p, of anything between 0 and 1.
Conjecture: The equity is undefined regardless of the value of p, since the value of the stakes grows faster than the probability of the game ending.
It would seem that this would be easier to prove mathematically than the Zare position. But, perhaps the result does depend on p. In a backgammon example, p needs to be sufficiently high to prompt the voluntary cube, while in my example the stakes double regardless.
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