When your opponent wins the opening roll, there are fifteen possible rolls. Those fifteen rolls can be played in perhaps a total 33 reasonable ways. After that, you can roll 21 different numbers in reply. That makes a total of 33 × 21 = 693 different positions. Rollouts for all 693 positions appear on the pages that follow (a total 2218 plays are rolled out). A rollout determines the best way to play a roll by simulating the game on computer. Whichever play wins more games is deemed to be correct according to the rollout. As long enough games are rolled out, the luck of the dice plays only a small part in determining the winning play in a rollout. You can find more information here on the rollout settings used.

Rollout Settings

Gnu Backgammon was used to rollout the candidate plays. Each play was rolled out 2592 times using cubeless 0-ply play to a truncation depth of 16.5 plies. "Cubeless" means that games were played without a cube in play. Gammons and backgammons still count. "0-ply play" means that Gnu Backgammon did no lookahead as it played the games. "Truncation depth" refers to how many rolls into the game each trial of the rollout is performed. These rollouts were truncated at an average depth of 16.5 plies. (Half the trials were truncated at 16 plies, the other half at 17 plies.) The advantage of these settings is (1) speed and (2) low random error. Stronger settings, such as using cubeful play, 2-ply evaluation, and untruncated rollouts, would have needed considerably more time to achieve the same low random error. (Perhaps a thousand times more.) Such advanced settings were simply not feasible with so many plays to be rolled out. Nevertheless, I believe the rollout results will hold up quite well. Cubeless play will do just about as well as cubeful play when the starting position is even. Errors in 0-ply play (compared to 2-ply) will tend to be distributed evenly between the sides. Bias introduced by truncation will also tend to be evenly distributed between the sides. And the truncation bias affects relative errors much less than absolute errors. As long as the relative errors are small, the plays will still usually be ranked correctly. The rollouts have a standard error of about .002 cubeless equity. That means two plays that differ by less than .002 could easily be ranked out of order. Plays that differ by more than .002 cubeless equity are probably ordered correctly as far as random error is concerned. (Though the order could still be wrong due to systematic errors in play or bias from truncation.) The DMP, gammon-go, and gammon-save values shown in the tables were derived from the listed cubeless probabilities. They were not performed according to score, so you can't depend on them as much.

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