TITLE: What equity is necessary to offer the doubling cube in Backgammon (dice game)
QUESTION [6 upvotes]: Edit: This question is a lot shorter than it is. Don't get intimidated. If you know backgammon, just skip to question 2.
In Backgammon, each game is played for one point (or one dollar) between two players. There is a die, called the doubling cube, which has the numbers $2, 4, 8, ...$, in the middle of the board (it is not 'owned' by anyone). The players take turns rolling two regular dice (not the doubling cube) and moving. But before each roll, a player can 'offer the cube' (or 'offer to double', 'double', etc.) which is basically saying "Hey, why don't we play this game for 2 points". The other player can drop, or refuse the cube, and lose one point, or may accept, or take, the cube, in which case the game continues for twice as many points as before.
A player's equity in the game is the probability that he'll win a cubeless game, where cubeless means neither player can offer the cube (or a one point game). (For backgammon players who know the rules, I'm ignoring gammons and backgammons, so the equity equals the probability of winning. Edit: As @Henning Makholm's first answer indicates, I also do not want to include the equity of owning the doubling cube.
I have two questions, but I know the answer to the first one and I think I'm calculating it right.
1) What equity does the player receiving the cube require in order to accept it? Answer, I'm told, is $.25$?
The receiving player will accept the double when the expected value of taking it is greater than the expected value of dropping (which automatically loses him $1$ point). $p$ is the probability (equals equity) of the receiving player winning (are you guys following all this?).
$E(take) \ge E(drop) \\
2p-2(1-p) \ge -1 \\p > .25$
I'm nearly certain that's correct, so
2) What equity is required for a player to offer the cube (offer to double the game's stakes)?
How is that calculated? We don't know whether the cube's recipient will accept or not.
I start the same as question 1: The giver will double when his expected value of doubling is greater than his EV of not doubing (duh!). If $EV(rolling)$ is $p-(1-p)$, and $EV(doubling)$ is $2p-2(1-p)$, then
$E(doubling) > E(rolling) \\
2p-2(1-p) > p-(1-p) \\
p > .5$
Which can't possibly be correct. While I'm not BG expert, I did used to play for (small amounts) of money in NYC. There is no way in heck that I would double with 51% chances.
OK, that's all I got. How do we figure this out?
Thanks.

REPLY [5 votes]: You can actually do a fair bit better than the 20%/80% approximation.
Rick Janowski wrote a paper about it here:
http://www.bkgm.com/articles/Janowski/cubeformulae.pdf
Two big innovations in that paper:

You want to account for the possibility of gammons and backgammons. You do that by assuming that the proportion of wins that are gammon wins (and the proportion that are backgammon wins) stays constant as the probability of win changes. Ditto for gammon/backgammon losses.
He noted that the 20%/80% thing above is a "live cube" limit. It assumes that the probability of win diffuses continuously, when really it jumps from turn to turn. Because of that you can wait right until before the cash point to offer the double, capturing all the theoretical value. Actually, because you might jump across the cash point, you want to double earlier than right before the cash point. So the "live cube" limit is an upper limit to the proper cubeful equity. The "dead cube" limit, where you assign no value to owning the cube, is a lower limit. Janowski proposed a linear interpolation in equity between those two limits; the linear interpolation parameter is a proxy for all the complicated stuff about jumps in probability, size of doubling window, etc.

With this you can get relatively simple expressions for take and cash points, as well as initial double and redouble points. It's all in that paper, which is a real classic.
I also did a bit of work on this, modeling the changes in probability as jumps instead of a continuous diffusion. Still a bit rough but if you're interested you can check it out here:
http://compgammon.blogspot.com/2012/03/new-cube-decision-model-posted-on.html?showComment=1333785363926