Match Equities

Introduction

By Rick Janowski

The Rockwell-Kazaross match equity table (MET) was a major achievement, obtaining all the match equities for a 15-point match by direct rollout. It was carried out piecemeal so that all possible resulting scores from any point had already been rolled out, with the MET file in use by the bot suitably updated. Thus, and unlike earler mathematical based METs, the potential for error through invalid assumption was removed.

The tabulated values were subsequently extended out to all scores in a 25 point match by extrapolation. The Gnu 11-point match MET, had followed a similar approach but only rolling out to 11-point scores and with weaker rollout settings. (The eXtremeGammon MET which was previously XG’s default MET also utilised rollout results, though I don’t know the details.) Comparison of the Rockwell-Kazaross MET with the Gnu MET over an 11-point match shows differences of up to 0.5%, demonstrating that the new MET had made a measurable improvement.

The Kazaross-XG2 MET was subsequently constructed in a similar manner but using XG2 exclusively as the bot rollout source rather than Gnu. This was rolled out for all scores in a 9-point match, showing remarkable consistency with the previous Rockwell-Kazaross results. The maximum difference is 0.07% with average difference of 0.02% (treating each difference as a positive result).

More than anything else, I think this exercise demonstrated the validity of the Rockwell-Kazaross MET and that future significant improvement is unlikely. Sensibly it seems, the rolling out process didn’t continue beyond 9-point match scores. Instead, the Kazaross-XG2 MET directly used the previous results from the Rockwell-Kazaross MET for higher scores, including the previous extrapolation from 15 to 25 point match scores.

Which is best? There is obviously hardly anything to distinguish the preference of one over the other for practical purposes. On the one hand, there might be some tiny improvement in accuracy with the Kazaross-XG2 MET, but this could also just be statistical scatter/noise. On the other hand, the Rockwell-Kazaross MET has greater consistency, there being only one joint (15 to 16) rather than two with the Kazaross-XG2 MET (9 to 10 and 15 to 16). However, the consistency of results between the two METs indicates very negligible potential effects.

Creating the Kazaross-XG2 Match Equity Table

By Neil Kazaross

Kazaross-XG2 is the default match equity table for XG2 because it was rolled out using XG2 for all scores in a 9 pt match. I feel that it is more accurate from a technical standpoint since each score was rolled out 100,000 times rather than 46,000 times, resulting in a lower standard deviation. And, since XG2 is clearly a stronger bot than GNU was then, hopefully the stronger level of play results in getting to closer to reality of perfection playing. How that is applies when Player A plays Player B, or when Giant A plays Giant B is, of course, unclear. But that was not the point of making these METs.

XGR cube in XG2 is a tiny bit stronger than 4 ply cube so I used that, since it didn’t add too much to overall rollout time and since I want to minimize the effects of critical cube action errors in score based positions. I used XGR+ cube for 2-away post Crawford since it did a better job of judging when it was a play-on after a great opening roll sequence.

Therefore since Kazaross-XG2 was rolled out with XG2, the strongest bot, and for at least twice as many trials, and with clearly stronger cube settings than Rockwell-Kazaross, I’d use Kazaross-XG2 as the MET for XG2.

Why did I stop after 9 pts when rolling out Kaz-XG2? I stopped because there was very little difference in the scores for matches of this length or slightly shorter between the two METs so, while I originally intended to go to at least 11 pts with my rollouts, I stopped and just used the Rockwell-Kazaross results for longer lengths.

To extrapolate Rockwell-Kazaross from 16 to 25 points, I spent a couple of days work on this, starting with a calculated table that for many scores in the 11 to 15 point range was very close to the Rockwell-Kazaross rollouts, but for lopsided scores needed some adjustments. I used my best mathematical judgements for those adjustments and and then checked take points to make sure that nothing stood as a big difference from what was expected or the expected progression of such. I’ve since tested a few of these match scores where the scores cross over from rolled out Rockwell-Kazaross to extrapolated Rockwell-Kazaross and I am happy with my work.

We all should thank David Rockwell for starting the project to rollout Rockwell-Kazaross with GNU 2 ply. This was necessary since bot rollouts at gammon go using the initial ply (GNU 0 ply, XG 1 ply) doesn’t seem to be able to look ahead far enough in play to take best advantage of the fact that the gammon value is 1 while the gammon cost is 0. I became aware of David’s rollout work and asked if I could help speed it up, having access to more and faster computers than he did.

I did not involve anyone else in creating Kazaross-XG2 since I did it prior to public release of XG so it could be included as default with XG2. Yes, I waited until Xavier locked down the neural nets for XG2 prior to release before starting the rollouts so we could be certain that it was done with the same neural nets we all enjoy in XG2 today.

If there is an XG3, I will probably rollout a MET again for it and will again use 3-ply play XGR cube, but with something longer than 100,000 trials to start with. Perhaps I’ll see nothing more than a 0.02% or so difference from Kazaross-XG2 and give up, or perhaps not.

Another default I sometimes see questioned is our choice of 3-ply play XGR cube as the default for XG2 rollouts. This is a good compromise between speed and strength. XGR cube is considerably stronger than 3-ply cube and while considerably slower, its use doesn’t cause overall rollout time to be greatly longer. We feel that this is worth the improvement from using 3-ply both.

Kazaross-XG2 Match Equity Table

Here is the Kazaross-XG2 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.)

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).

References

Janowski:  Post to bgonline.org

Kazaross:  Post to bgonline.org

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