Profitable Proffers and Pineapple Bluffers
by Jake Jacobs
19 September 2016
Like Joe Cocker I get by with a little help from my friends. The friends helping this month are Phil Simborg and Sam Pottle. Thanks guys!
Phil likes to tinker, and is constantly experimenting with variations, or new games. In the process he often seeks help to answer questions that arise during these games.
The first to come up was about a variation that is part of standard money backgammon, albeit one that most players have never encountered: the proffer. Typically the player making the proffer offers to beaver if his opponent doubles or redoubles. Phil asked, assuming the choices were No Redouble, or Redouble-Beaver, was there a formula that would tell you when it was right to accept a proffer?
I read most email in the morning, and this one caught me hurrying to wrap it up and call a taxi for an appointment. In lieu of thought, I turned to XG. It didn't take long to set up the position above. Black leads by one pip, 89 to 90. His cubeless winning chances are 61.81%. That gives a cubeless equity of +.236, or +.472 doubled. The cubeful numbers are .454 for No Redouble, and +.211 for Redouble-Take. Thus Redouble-Beaver would be +.422, less than +.454, but tantalizingly close. I replied to Phil that my hunch was that 62.5% was the sweet spot, while (based upon a different position) 72.5% was about the point where one would redouble without a proffer. That I would think about why this was so, and get back to him.
Of course I was now busily trying to come up with a formula that fit those numbers, rather than finding the correct formula. Luckily, Sam came to my rescue. I won't go into detail about the wrong way to approach the matter, and instead will quote Sam on the right way:"Better not to use separate variables X and Y. If your cubeless winning chance is X, your equity on a 1 cube is X - (1-X) = 2X - 1. On a 2 cube it's 4X - 2.
In the continuous model, holding a 2 cube, it's 4*(1.25X) - 2 = 5X - 2. This can also be written (4X - 2) + X, so if you call your "cubeless 2-cube" equity E, then your continuous model "owns 2-cube" equity is just E + X, which is kind of neat.
It turns out that the breakeven between no-double and double/beaver is at X = 2/3. Then the no-double equity is 5*(2/3) - 2 = 4/3. From the beaver side, equity on a 2 cube would be 5*(1/3) - 2 = -1/3, so on an 8 cube would be -4/3."
If you try to reconcile the formula with XG's numbers for the position above, you may be confused (if you are not confused already). Here is what is going on. If the cubeless winning chances are 61.81% for the roller, then 5 * .6181 – 2 = 1.0905, while if you do 5 * .3819 – 2 it equals -.0905, so four times that (redoubled and beavered) equals -.362. That's a long way from the values XG gives.
By the way, XG's numbers are values on a 1-cube, so when it says No Redouble is .454, you could double that to .908, your equity on a 2-cube, which is a bit less than one point, while Redoubling would give you .422, or a bit less than half a point on your opponent's 4-cube, and .844 if he held an 8-cube.
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