Backgammon Theory

Theory

Double/take/drop rates

From:   Gary Wong
Address:   gary@cs.arizona.edu
Date:   20 June 1999
Subject:   Re: cubeful distribution
Forum:   rec.games.backgammon
Google:   wt7loyxm5b.fsf@brigantine.CS.Arizona.EDU

(B@ckgammon.com) writes: > I rolled out the starting position 50.000 times using Jellyfish > Level 5 to produce the table below to show cubeful distribution. > > [table edited] > Cube 1 2 4 8 16 32 Total > > Cashes 37.83 13.54 1.78 0.28 0.03 0.01 53.47 > Single 0.00 26.63 5.03 0.59 0.06 32.31 > Gammon 1.45 10.90 1.17 0.10 0.01 13.63 > Backgammon 0.07 0.46 0.04 0.01 0.01 0.59 > ----- ----- ---- ---- ---- ---- ------ > 39.35 51.53 8.02 0.98 0.11 0.01 100.00 Thanks very much for that data! > I'm not technically minded enough to make use of these numbers, but > would they maybe affect match equity tables etc or could they shed > any sort of light on recube vig? Unfortunately the results are probably inapplicable to match play (where cube behaviour is likely to vary drastically from money play depending on the score) but they ought to help us model certain features of money games. I have attempted to produce a simple Markov process which would produce a distribution similar to that above. It is: If the cube is centred: - it is offered and dropped with probability 0.378 (terminal) - it is offered and taken with probability 0.606 (go to the state below) - a player wins a gammon with probability 0.015 (terminal) - a player wins a backgammon with probability 0.07 (terminal) If the cube is owned: - it is offered and dropped with probability 0.220 (terminal) - it is offered and taken with probability 0.152 (remain in this state) - a player wins a single game with probability 0.455 (terminal) (These wins can reasonably expected to belong to the player who last turned the cube; we presume the cube holder can always reach a point where she has a correct double before she wins a single game). - a player wins a gammon with probability 0.172 (terminal) - a player wins a backgammon with probability 0.001 (terminal) (It's hard to determine who the gammon and backgammon wins belong to. Certainly the vast majority will be the player who turned the cube, but there will be occasions where the cube owner becomes too good to double and goes on to win a gammon or backgammon without having had a correct double.) If two players happen to play with cube behaviour described by the above model, then we can make the following observations about the results of their games: - Most (62%) initial doubles are takes. This value agrees reasonably closely to an earlier result at: http://x24.deja.com/=dnc/getdoc.xp?AN=376082872 - Most (59%) redoubles are drops. - The final cube value follows the following (geometric-like) distribution: p(X=1) = 0.394 p(X=2^n) = 0.514 x 0.152^{n-1} for n = 1, 2, 3... - The cube is turned and accepted 0.71 times per game on average; the average final value of the cube is 1.87. - A single game is worth 2.2 points on average; the standard deviation is 3.0. I have previously assumed 2 and 3 respectively for these parameters in my own games. Those statistics do not change a significant amount if the Jacoby rule is used in interpreting the score (naturally, the parameters of the model might change depending on the use of the rule). - Assuming all gammon and backgammon wins belong to the player who last cubed, a double/take leaves the taker with an average normalised cubeful equity of -0.446. (If all doubles were perfectly efficient, this figure would be -0.5.) With a few more assumptions it may be possible to come up with other statistics on cube efficiencies, recube vig, etc. I don't have time to think about it right now, though -- more another day perhaps. Cheers, Gary. -- Gary Wong, Department of Computer Science, University of Arizona gary@cs.arizona.edu http://www.cs.arizona.edu/~gary/

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