The Rockwell-Kazaross Match Equity Table
Neil Kazaross, 2010

This is the new match equity table (MET) created by David Rockwell and myself. Rockwell-Kazaross was developed by rolling out every score in a 15-point match almost 39,000 times using GNU 2-ply Supremo. I then carefully extrapolated out to 25 points.

## How the Table Was Created

Every score in a 15-point match was rolled out 38,880 times using GNU 2-ply Supremo. That is 19,440 (15 × 1,296) for each side on roll, of course, with GNU set to roll out as initial position (i.e., you can’t start the game with doubles).

This will certainly suffice for we Americans who almost never play anything longer than 15 points, but for those interested in longer matches I extrapolated to 25 points using a calculated MET as a template (was a very close match to the rolled out MET in 12 to 15 point matches as close scores, and was a little low for the trailer’s chances to come back at lopsided scores). Therefore, I used very slight adjustments to improve the trailer’s chances to come back from big deficits in long matches in the extrapolation.

To check to make sure of my extrapolation, we looked very carefully at cubeless take points for 2- and 4-cubes and all follow the numerical trends. There are a lot fewer issues (I didn’t find any) when crossing over from scores that were rolled-out to scores extrapolated with Rockwell-Kazaross than with g11.

I am convinced this MET works well when a high-level bot plays a high-level bot. I sent the .xls with the MET to Xavier and he checked it versus his rolled out XG MET (1 ply, and close to g11 (0-ply GNU)) and quickly said that he loved the new MET and it was stronger by 0.4 Elo. That may seem to be an insignificant amount, but using XG’s Elo calculator it means that Rockwell-Kazaross wins 50.04% of all 11 pt matches vs XG’s MET.

My recollection of lots of matches someone ran with a program to compare g11 to Woolsey MET ended up with g11 winning 50.05% of the matches. So in real life, the gain from switching MET is very small. However, more and more, the top players play like the top bots. And most serious students of backgammon study match play using XG or GNU and do rollouts with them. So why not use a MET that best reflects their play on high levels? Now we have a MET that is based on how the top bots play in rollouts so our rollout results should be a bit more accurate, IMHO.

The key figure for any MET is -2,-1 Crawford. The differences in Rockwell-Kazaross vs g11 or XG or calculated tables is that our rollouts showed 32.31% chances for the trailer at that score vs 31.85% for g11. This may seem to be an unbelievable difference between GNU 0 ply and 2 ply Supremo, but I have verified this a couple of times, as have others.

I also rolled out -2,-1 Crawford using XG 3-ply, over 3000 trials for each of the 30 opening plays according to score. When I tabulated the data, I got 32.32%. Additionally a quick look at Stick’s rollouts on his site for gammon save and gammon go will also convince one that the resulting match equity for -2,-1C is clearly over 32%.

When can a new MET come out and replace our’s? Well, someone could redo it using XG 3-ply (assuming XG can be set to not roll doubles when starting a rollout from the opening position). But unless he did several times more trials, I doubt it would be much more accurate.

My guess is that in a few years someone will train a very high level bot to play better at gammon go and gammon save than our current bots do. The gains will be slight, and perhaps balance out, but if someone does a very long rollout using a very high strength (like XGR+) and shows that -2,-1C is clearly different than 32.3%, it may make sense for them to attempt the many-month-long project to roll out a new match equity table.

But for now and the foreseeable future, I will be using Rockwell-Kazaross since I think it best reflects how both the top bots and top players play and it will make bot evaluations and rollouts slightly more likely to return the correct result for match analysis.

## Rockwell-Kazaross Match Equity Table

Here is the Rockwell-Kazaross match equity table up to 15-away. The numbers represent percent probability of winning the match for the player whose “away score” is listed down the left side of the table. Row 1 and Column 1 represent the Crawford game. (Post-Crawford equities are not shown in this abridged table.)

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 50 67.7 75.1 81.4 84.2 88.7 90.7 93.2 94.4 95.9 96.6 97.6 98 98.5 98.8 2 32.3 50 59.9 66.9 74.3 79.9 84.2 87.5 90.2 92.3 93.9 95.2 96.2 97.1 97.7 3 24.9 40.1 50 57.1 64.8 71.2 76.3 80.5 84 87.1 89.4 91.5 93.1 94.4 95.5 4 18.6 33.1 42.9 50 57.7 64.3 70 74.6 78.8 82.4 85.4 87.9 90 91.8 93.3 5 15.8 25.7 35.2 42.3 50 56.7 62.7 67.8 72.6 76.7 80.3 83.4 86 88.3 90.2 6 11.3 20.1 28.8 35.7 43.3 50 56.3 61.7 66.8 71.3 75.3 78.9 82 84.7 87 7 9.3 15.8 23.7 30 37.3 43.7 50 55.5 60.9 65.6 70 73.9 77.4 80.5 83.3 8 6.8 12.5 19.5 25.4 32.2 38.3 44.5 50 55.4 60.4 65 69.1 72.9 76.4 79.4 9 5.6 9.8 16 21.2 27.4 33.2 39.1 44.6 50 55 59.8 64.1 68.2 71.9 75.3 10 4.1 7.7 12.9 17.6 23.3 28.7 34.4 39.6 45 50 54.9 59.3 63.6 67.5 71.1 11 3.4 6.1 10.6 14.6 19.7 24.7 30 35 40.2 45.1 50 54.6 58.9 63 66.8 12 2.4 4.8 8.5 12.1 16.6 21.1 26.1 30.9 35.9 40.7 45.4 50 54.4 58.6 62.5 13 2 3.8 6.9 10 14 18 22.6 27.1 31.8 36.4 41.1 45.6 50 54.2 58.3 14 1.5 2.9 5.6 8.2 11.7 15.3 19.5 23.6 28.1 32.5 37 41.4 45.8 50 54.1 15 1.2 2.3 4.5 6.7 9.8 13 16.7 20.6 24.7 28.9 33.2 37.5 41.7 45.9 50

## Unabridged Table

The raw, unabridged table goes out to 25-away. Here is the full table, with all digits of precision and post-Crawford equities.

The column and row labeled “1” represent the Crawford game, that is, the first game after the leading player reaches 1-away.

The column and row labeled “PC” represent any game after the Crawford game (a.k.a., the “post-Crawford” games).

 PC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 PC .500000 0.512323 0.676888 0.689701 0.809204 0.818838 0.884079 0.89094 0.93027 0.934839 0.957931 0.96094 0.974629 0.976572 0.984696 0.98595 0.99076 0.99158 0.99444 0.99495 0.99664 0.99697 0.99797 0.99818 0.99877 1 .500000 0.676888 0.751179 0.813772 0.841941 0.886867 0.907188 0.932313 0.943975 0.959275 0.966442 0.975534 0.979845 0.985273 0.987893 0.99114 0.99273 0.99467 0.99563 0.99679 0.99737 0.99807 0.99842 0.99884 0.99905 2 .487677 .323112 0.5 0.598994 0.668586 0.743447 0.798991 0.842141 0.875198 0.90172 0.923034 0.939311 0.95247 0.962495 0.970701 0.976887 0.98196 0.9858 0.98893 0.99129 0.99322 0.99466 0.99585 0.99675 0.99746 0.99802 3 .323112 .248821 0.401006 0.5 0.571438 0.647713 0.711608 0.762548 0.80485 0.840196 0.870638 0.894417 0.914831 0.930702 0.944426 0.954931 0.96399 0.97093 0.97687 0.98139 0.98522 0.98814 0.99062 0.99248 0.99407 0.99527 4 .310299 .186228 0.331414 0.428562 0.5 0.577415 0.643063 0.699664 0.746157 0.78829 0.824059 0.853955 0.879141 0.900233 0.91804 0.932657 0.94495 0.95499 0.96341 0.97021 0.97589 0.98044 0.98422 0.98726 0.98975 0.99174 5 .190796 .158059 0.256553 0.352287 0.422585 0.5 0.566621 0.626658 0.678181 0.725507 0.767055 0.802732 0.833654 0.859934 0.882866 0.902013 0.91847 0.93223 0.94397 0.95367 0.96189 0.96864 0.97432 0.97896 0.98283 0.986 6 .181162 .113133 0.201009 0.288392 0.356937 0.433379 0.5 0.562783 0.616561 0.667856 0.713057 0.753427 0.788634 0.819569 0.846648 0.869999 0.89021 0.90756 0.92246 0.93508 0.94583 0.95488 0.96254 0.96894 0.97432 0.97879 7 .115921 .092812 0.157859 0.237452 0.300336 0.373342 0.437217 0.5 0.554919 0.608614 0.656283 0.700209 0.739054 0.774121 0.805203 0.832566 0.85659 0.87761 0.89591 0.91171 0.92535 0.93702 0.94703 0.95553 0.96276 0.96887 8 .109060 .067687 0.124802 0.19515 0.253843 0.321819 0.383439 0.445081 0.5 0.554384 0.603718 0.649899 0.691356 0.729447 0.763593 0.794397 0.82158 0.84578 0.86714 0.88589 0.9023 0.91658 0.92898 0.93968 0.94891 0.95682 9 .069730 .056025 0.09828 0.159804 0.21171 0.274493 0.332144 0.391386 0.445616 0.5 0.550196 0.597926 0.641481 0.682119 0.718927 0.752814 0.78301 0.81037 0.83483 0.85662 0.87591 0.89294 0.90791 0.92098 0.9324 0.9423 10 .065161 .040725 0.076966 0.129362 0.175941 0.232945 0.286943 0.343717 0.396282 0.449804 0.5 0.548547 0.593459 0.63588 0.67483 0.711113 0.74371 0.77375 0.80093 0.82543 0.84741 0.86703 0.88448 0.89991 0.91353 0.9255 11 .042069 .033558 0.060689 0.105583 0.146045 0.197268 0.246573 0.299791 0.350101 0.402074 0.451453 0.5 0.545552 0.589242 0.629736 0.667927 0.70303 0.7353 0.76494 0.79198 0.81648 0.83862 0.85849 0.87629 0.89214 0.90622 12 .039060 .024466 0.04753 0.085169 0.120859 0.166346 0.211366 0.260946 0.308644 0.358519 0.406541 0.454448 0.5 0.544068 0.585701 0.625259 0.66178 0.6961 0.72778 0.75703 0.78381 0.80826 0.83044 0.85051 0.86856 0.88476 13 .025371 .020155 0.037505 0.069298 0.099767 0.140066 0.180431 0.225879 0.270553 0.317881 0.36412 0.410758 0.455932 0.5 0.541943 0.582545 0.62036 0.65619 0.68966 0.72081 0.74963 0.77619 0.80054 0.82276 0.84295 0.86123 14 .023428 .014727 0.029299 0.055574 0.08196 0.117134 0.153352 0.194797 0.236407 0.281073 0.32517 0.370264 0.414299 0.458057 0.5 0.54075 0.57942 0.61634 0.65117 0.68391 0.71448 0.7429 0.76917 0.79339 0.81559 0.83586 15 .015304 .012107 0.023113 0.045069 0.067343 0.097987 0.130001 0.167434 0.205603 0.247186 0.288887 0.332073 0.374741 0.417455 0.45925 0.5 0.53916 0.57679 0.61261 0.64659 0.67859 0.70862 0.73664 0.76265 0.78669 0.80883 16 .01405 .00886 0.01804 0.03601 0.05505 0.08153 0.10979 0.14341 0.17842 0.21699 0.25629 0.29697 0.33822 0.37964 0.42058 0.46084 0.5 0.53796 0.57441 0.60929 0.64241 0.67376 0.70323 0.73084 0.75657 0.78046 17 .00924 .00727 0.0142 0.02907 0.04501 0.06777 0.09244 0.12239 0.15422 0.18963 0.22625 0.2647 0.3039 0.34381 0.38366 0.42321 0.46204 0.5 0.53676 0.57222 0.60618 0.63856 0.66925 0.69822 0.72542 0.75087 18 .00842 .00533 0.01107 0.02313 0.03659 0.05603 0.07754 0.10409 0.13286 0.16517 0.19907 0.23506 0.27222 0.31034 0.34883 0.38739 0.42559 0.46324 0.5 0.53574 0.57023 0.60336 0.63501 0.6651 0.69356 0.72038 19 .00556 .00437 0.00871 0.01861 0.02979 0.04633 0.06492 0.08829 0.11411 0.14338 0.17457 0.20802 0.24297 0.27919 0.31609 0.35341 0.39071 0.42778 0.46426 0.5 0.53475 0.56838 0.60073 0.63171 0.66122 0.68921 20 .00505 .00321 0.00678 0.01478 0.02411 0.03811 0.05417 0.07465 0.0977 0.12409 0.15259 0.18352 0.21619 0.25037 0.28552 0.32141 0.35759 0.39382 0.42977 0.46525 0.5 0.53387 0.56667 0.5983 0.62864 0.6576 21 .00336 .00263 0.00534 0.01186 0.01956 0.03136 0.04512 0.06298 0.08342 0.10706 0.13297 0.16138 0.19174 0.22381 0.2571 0.29138 0.32624 0.36144 0.39664 0.43162 0.46613 0.5 0.53303 0.56508 0.59603 0.62576 22 .00303 .00193 0.00415 0.00938 0.01578 0.02568 0.03746 0.05297 0.07102 0.09209 0.11552 0.14151 0.16956 0.19946 0.23083 0.26336 0.29677 0.33075 0.36499 0.39927 0.43333 0.46697 0.5 0.53226 0.5636 0.59391 23 .00203 .00158 0.00325 0.00752 0.01274 0.02104 0.03106 0.04447 0.06032 0.07902 0.10009 0.12371 0.14949 0.17724 0.20661 0.23735 0.26916 0.30178 0.3349 0.36829 0.4017 0.43492 0.46774 0.5 0.53153 0.56221 24 .00182 .00116 0.00254 0.00593 0.01025 0.01717 0.02568 0.03724 0.05109 0.0676 0.08647 0.10786 0.13144 0.15705 0.18441 0.21331 0.24343 0.27458 0.30644 0.33878 0.37136 0.40397 0.4364 0.46847 0.5 0.53086 25 .00123 .00095 0.00198 0.00473 0.00826 0.014 0.02121 0.03113 0.04318 0.0577 0.0745 0.09378 0.11524 0.13877 0.16414 0.19117 0.21954 0.24913 0.27962 0.31079 0.3424 0.37424 0.40609 0.43779 0.46914 0.5

## References

Kazaross:  Post to bgonline.org