Backgammon Theory

Theory

Janowski's formulas

From:   Stig Eide
Address:   stigeide@yahoo.com
Date:   2 September 1999
Subject:   Converting cubeless to cubefull
Forum:   rec.games.backgammon
Google:   7qlqvg$cmg$1@nnrp1.deja.com

I'd like to start a discussion of how to convert a cubeless equity- estimate to cubeful. I guess the easiest is to start with moneygame, and then generalize to match later. I suspect that one way to go is to use the Janowski method. That's a set of formulas that gives the needed win% to take and be too good. I'll start by describing the formulas, using an example: Let's say the player on roll has these probabilities (cubeless): Player Opponent ------------ ------------- BG G W W G BG 0 30 60 40 10 0 That is, the player wins 60%, including 30% gammons. While he loses 40%, including 10% gammons. No bg's in this example. The cubeless equity for Player (for the rest of the article I'll refer to 'Player' as the player on roll and 'Opponent' as the other) is +0.4. The cubefull equity will depend upon the position of the cube and its value. I'll assume that the cube always has the value '1'. If Player owns the cube, the equity should be higher than if Opponent owns it. And if it's in the middle, the equity should be somewhere in between. The Janowski formulaes uses the expected points won, for both players, _given_ they win. That is, if Player win he'll win gammon in 30 of the 60%, and normal in 30 of the 60%. That is an average of 1.5 points pr. won game. (60+30)/60 would also do the trick. Opponent will win 1.25 points pr. won game. These figures are cubeless. If we name expected points for Player EP and expected points for Opponent EO, The Janowski takepoint for Opponent, OT, is: EP - 0.5 OT = ------------- EP + EO + c/2 'c' is cube-efficiency, a number between 0 and 1. That is, if the cube is dead, c will be 0. In the continuous model, c is 1 (all doubles are borderline take/pass). This number will depend upon the position. From now on I'll just estimate it to be 3/4. It's more common to estimate it to be 2/3 but that's another discussion. If we use the example, we find that Opponents takepoint is 32%. That is, to take a double he need to win 32% cubeless when he factor in gammons and the value of owning the cube. Note that Player would have a cubefull equity of 1, if he had had 68% wins, because Opponent would then have a borderline take/pass. We also need Players takepoint. I'll assume that the expected points won for Player and Opponent is the best estimate for this position and all of the future ones. This is a crucial assumption. I'm not sure if it is sound to assume that Players gammonrate will be independent of his wins. But, using this assumption, the Janowski takepoint for player, PT, is: EO - 0.5 PT = ------------- EO + EP + c/2 The cubeless figures in the example and a cubeefficiency of 3/4, says that Players takepoint is 24%. We should now be ready to calculate the cubefull equities: If the cube is in the middle, Player can cash if he reach 68% wins, while he has a drop if he reach 24%. He's now at 60%, so the probability that he reach 68% before 24% should be: 60 - 24 36 ------- = -- = 82% 68 - 24 44 So, Player will win one point in 82% and lose one point in the remaining 18%. His cubefull equity is 1*0.82 + (-1)*0.18 = 0.64 If Player owns the cube, he can still cash at 68%, but he has the right to play until he loses, so the probability of Player cashing before he loses is: 60 - 0 60 ------ = -- = 88% 68 - 0 68 So, player will win one point in 88% and lose 1.25 in the remaining 12%. His cubefull equity is 1*0.88 + (-1.25)*0.18 = 0.655 If Opponent owns the cube, Player needs to climb to 100% before he gets his expected 1.5 points while Opponent gets one point if Player gets to 24%. The probability of this happening is: 60 - 24 36 -------- = -- = 47% 100 - 24 76 So, Player will win 1.5 points in 47% and lose one point in the remaining 53%. His cubefull equity is 1.5*0.47 + (-1)*0.53 = 0.175 The remaining part is to figure out cubefull equity when one of the players are too good to double. The Janowski formula for the too good point is in the same spirit as the formula for take point: Player is too good if his probability of winnig is more than EO - 1 ------------- EP + EO + c/2 Using the same method as earlier, we should be able to convert cubeless equity to cubefull in positions where one of the players are too good. Questions, anyone? Stig Eide

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