Backgammon Openings

Rollouts of Opening Moves

Rollout Settings
The rollouts on this page where performed using Gnu Backgammon, version 0.14, which is the latest and strongest version of Gnu BG as of early 2006. The rollouts were performed using 2-ply play, which means that Gnu BG looked ahead two rolls as it made each play during the rollouts, averaging the resulting 1296 future positions for each candidate to determine the best play as it played through each trial. This is the highest setting at which Gnu BG can perform rollouts in any reasonable time. In total, the rollouts on this page took over seven months to complete.

Each play was rolled out 46,656 times, which is 36 × 36 × 36, thus guaranteeing that each possible combination of the first three rolls of each play is equally represented. Variance reduction was used to reduce the effect of random error from lucky sequences of dice. The standard error when performing this number of trials using variance reduction is less than 0.001. Standard errors this small are usually (but not always!) enough to separate the best play from the others. A few candidates were rolled out less often than this (between 5,000 and 20,000 times each) when it was clear they would be nowhere near the best in any type of match situation.

Rollouts were performed using cubeless money play. Cubeless means that games are played without a doubling cube and always played to the end of the game. Gammons and backgammons count as usual.

Reading the Charts
The EQUITY column of each chart shows the estimated cubeless equity of each candidate play. The best play is displayed with a white background and the other plays are displayed with progressively darker backgrounds to give a rough idea of the amount of equity lost. Plays that give up more than 0.05 in cubeless equity are not listed.

The check marks denote the best play in each of the following situations:

  • Cubeless (CL): Cubeless money play.
  • Double Match Point (DMP): In a match which has come down to one final game.
  • Gammon-Go (GG): When aggressively going for a gammon win.
  • Gammon-Save (GS): When aggressively avoiding a gammon loss.

The DMP, GG, and GS values are inferred from the probabilities of the cubeless rollout. It is possible that a true rollout (according to score) would give different results than those shown for these special scores.

Rollouts
 
   PLAY W        G       BG     EQUITY CL DMP GG GS
  13/11, 6/5   + .5058, .1415, .0059
  − .4942, .1418, .0084

+.0088 (.0074)
  24/23, 13/11   + .4996, .1369, .0056
  − .5004, .1319, .0053

+.0045 (.0043) (.0124) (.0216)
  24/21   + .4918, .1254, .0056
  − .5082, .1315, .0057

−.0226 (.0314) (.0280) (.0602) (.0148)
  13/10   + .4897, .1378, .0059
  − .5103, .1419, .0068

−.0256 (.0344) (.0322) (.0396) (.0398)
  24/23, 24/22   + .4868, .1213, .0051
  − .5132, .1329, .0053

−.0382 (.0470) (.0380) (.0784) (.0276)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  8/5, 6/5   + .5546, .1708, .0071
  − .4454, .1150, .0051

+.1670

   PLAY W        G       BG     EQUITY CL DMP GG GS
  13/11, 13/10   + .4977, .1493, .0065
  − .5023, .1387, .0072

+.0053 (.0080) (.0172)
  24/21, 13/11   + .5017, .1355, .0064
  − .4983, .1341, .0060

+.0052 (.0001) (.0196)
  24/22, 13/10   + .4985, .1366, .0059
  − .5015, .1345, .0056

−.0006 (.0059) (.0064) (.0238) (.0072)
  13/10, 6/4   + .4918, .1369, .0061
  − .5082, .1455, .0090

−.0279 (.0332) (.0198) (.0366) (.0426)
  24/22, 24/21   + .4924, .1223, .0054
  − .5076, .1350, .0055

−.0280 (.0333) (.0186) (.0646) (.0204)
  13/8   + .4887, .1344, .0067
  − .5113, .1469, .0064

−.0348 (.0401) (.0260) (.0478) (.0516)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  24/23, 13/9   + .4984, .1405, .0058
  − .5016, .1348, .0059

+.0024 (.0032) (.0050)
  13/9, 6/5   + .5000, .1414, .0062
  − .5000, .1448, .0091

−.0063 (.0087) (.0168)
  24/23, 24/20   + .4886, .1223, .0052
  − .5114, .1320, .0045

−.0318 (.0342) (.0228) (.0610) (.0140)
  13/8   + .4887, .1344, .0067
  − .5113, .1469, .0064

−.0348 (.0372) (.0226) (.0366) (.0436)
  24/20, 6/5   + .4879, .1270, .0059
  − .5121, .1429, .0064

−.0406 (.0430) (.0242) (.0530) (.0372)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  8/4, 6/4   + .5380, .1697, .0074
  − .4620, .1242, .0055

+.1234

   PLAY W        G       BG     EQUITY CL DMP GG GS
  13/10, 13/9   + .4984, .1529, .0075
  − .5016, .1374, .0073

+.0125 (.0024) (.0118)
  24/21, 13/9   + .4996, .1404, .0066
  − .5004, .1342, .0060

+.0060 (.0065) (.0226) (.0030)
  24/20, 13/10   + .4989, .1355, .0061
  − .5011, .1330, .0056

+.0008 (.0117) (.0014) (.0338) (.0020)
  24/21, 24/20   + .4980, .1223, .0056
  − .5020, .1311, .0048

−.0120 (.0245) (.0032) (.0620)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  24/23, 13/8   + .5006, .1350, .0054
  − .4994, .1311, .0049

+.0056
  13/8, 6/5   + .4976, .1374, .0058
  − .5024, .1348, .0067

−.0031 (.0087) (.0060) (.0012) (.0134)
  24/18   + .4930, .1179, .0048
  − .5070, .1355, .0053

−.0321 (.0377) (.0152) (.0494) (.0240)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  24/22, 13/8   + .5007, .1334, .0058
  − .4993, .1327, .0047

+.0032 (.0142)
  13/11, 13/8   + .4970, .1442, .0074
  − .5030, .1414, .0065

−.0023 (.0055) (.0074) (.0248)
  13/8, 6/4   + .4899, .1335, .0057
  − .5101, .1403, .0063

−.0276 (.0308) (.0216) (.0356) (.0368)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  8/3, 6/3   + .5173, .1617, .0076
  − .4827, .1339, .0062

+.0638
  24/21, 13/8   + .5061, .1386, .0071
  − .4939, .1338, .0045

+.0196 (.0442) (.0224) (.0686) (.0222)
  13/10, 13/8   + .4970, .1498, .0073
  − .5030, .1406, .0062

+.0042 (.0596) (.0407) (.0645) (.0540)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  24/20, 13/8   + .5051, .1362, .0072
  − .4949, .1331, .0050

+.0155 (.0170) (.0002)
  13/9, 13/8   + .4999, .1499, .0077
  − .5001, .1365, .0064

+.0145 (.0010) (.0104) (.0174)
  24/15   + .5034, .1168, .0052
  − .4966, .1313, .0046

−.0071 (.0226) (.0034) (.0592)
  13/4   + .4852, .1382, .0066
  − .5148, .1426, .0070

−.0344 (.0499) (.0398) (.0528) (.0590)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  13/7, 8/7   + .5353, .1516, .0063
  − .4647, .1201, .0049

+.1035

   PLAY W        G       BG     EQUITY CL DMP GG GS
  24/18, 13/11   + .5068, .1270, .0058
  − .4932, .1346, .0052

+.0066 (.0028)
  13/5   + .4931, .1421, .0067
  − .5069, .1390, .0069

−.0109 (.0175) (.0274) (.0362)
  24/16   + .5028, .1187, .0053
  − .4972, .1363, .0050

−.0117 (.0183) (.0080) (.0274) (.0114)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  24/18, 13/10   + .5053, .1299, .0062
  − .4947, .1339, .0056

+.0072 (.0014)
  24/15   + .5034, .1168, .0052
  − .4966, .1313, .0046

−.0071 (.0143) (.0038) (.0300)
  24/21, 13/7   + .4863, .1346, .0062
  − .5137, .1403, .0063

−.0332 (.0404) (.0380) (.0286) (.0522)
  13/4   + .4852, .1382, .0066
  − .5148, .1426, .0070

−.0344 (.0416) (.0402) (.0236) (.0590)
  13/10, 13/7   + .4816, .1405, .0077
  − .5184, .1489, .0087

−.0462 (.0534) (.0474) (.0262) (.0788)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  8/2, 6/2   + .4981, .1540, .0077
  − .5019, .1417, .0060

+.0102 (.0168) (.0376)
  24/18, 13/9   + .5034, .1348, .0061
  − .4966, .1350, .0058

+.0069 (.0033) (.0062) (.0278) (.0136)
  24/14   + .5065, .1179, .0053
  − .4935, .1313, .0046

+.0003 (.0099) (.0554)

   PLAY W        G       BG     EQUITY CL DMP GG GS
  24/13   + .5282, .1164, .0053
  − .4718, .1200, .0036

+.0545
  24/18, 13/8   + .5099, .1287, .0068
  − .4901, .1350, .0045

+.0158 (.0387) (.0366) (.0120) (.0666)
 

Summary
Here are all the opening rolls ranked from best to worst. The best roll is    with an equity of  +.1670. The worst roll is    with an equity of  +.0024.

Your average equity if you win the opening roll is +.0393.

PLAY EQUITY       W        G       BG       L        G       BG
    8/5, 6/5 +.1670   + .5546, .1708, .0071   − .4454, .1150, .0051
    8/4, 6/4 +.1234   + .5380, .1697, .0074   − .4620, .1242, .0055
    13/7, 8/7 +.1035   + .5353, .1516, .0063   − .4647, .1201, .0049
    8/3, 6/3 +.0638   + .5173, .1617, .0076   − .4827, .1339, .0062
    24/13 +.0545   + .5282, .1164, .0053   − .4718, .1200, .0036
    24/20, 13/8 +.0155   + .5051, .1362, .0072   − .4949, .1331, .0050
    13/10, 13/9 +.0125   + .4984, .1529, .0075   − .5016, .1374, .0073
    8/2, 6/2 +.0102   + .4981, .1540, .0077   − .5019, .1417, .0060
    13/11, 6/5 +.0088   + .5058, .1415, .0059   − .4942, .1418, .0084
    24/18, 13/10 +.0072   + .5053, .1299, .0062   − .4947, .1339, .0056
    24/18, 13/11 +.0066   + .5068, .1270, .0058   − .4932, .1346, .0052
    24/23, 13/8 +.0056   + .5006, .1350, .0054   − .4994, .1311, .0049
    13/11, 13/10 +.0053   + .4977, .1493, .0065   − .5023, .1387, .0072
    24/22, 13/8 +.0032   + .5007, .1334, .0058   − .4993, .1327, .0047
    24/23, 13/9 +.0024   + .4984, .1405, .0058   − .5016, .1348, .0059
Average +.0393   + .5127, .1447, .0065   − .4873, .1315, .0057

Gammon Rate and Backgammon Rate

The totals in this table allow us to compute overall gammon and backgammon rates. We see that the gammon rate when Gnu Backgammon plays itself is .1447 + .1315 = .2762. That is, 27.62% of all backgammon games end in a gammon if played to completion.

And the backgammon rate is .0065 + .0057 = .0122. That is, 1.22% of all backgammon games end in a backgammon if played to completion.

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