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Explanation of new XG rollout statistics

Posted By: Timothy Chow
Date: Monday, 16 February 2015, at 8:45 p.m.

Ray Kershaw asked for an explanation of XG's new rollout statistics. In this post I will give a bare-bones explanation. I won't, at the moment, go into the topic of what these statistics are good for. That's for another time, hopefully soon. For now, let's just make sure we understand how to parse the information.

It should be pointed out that these statistics have long been available in GNU Backgammon, along with some other statistics (hitting percentage, etc.) that are still not available in XG. However, apart from the fact that XG rollouts are much faster than GNU rollouts, the GNU statistics have two serious limitations: (1) they are available only for cube decisions, and (2) the statistics are not saved in the .sgf file. XG remedies both of these defects.

The first step is to generate a rollout in the usual way. As I said before, the feature applies to both cube decisions and checker-play decisions. The example below is a cube decision. Note: I have rolled this out with no truncation, so all the numbers below are whole numbers. If you enable truncation then you will get decimals for some of the numbers.

White is Player 2

score: 0
pip: 57
Unlimited Game
pip: 54
score: 0

Blue is Player 1
Blue on roll, cube action?

Analyzed in Rollout No double Double/Take
Player Winning Chances: 69.77% (G:0.00% B:0.00%) 69.74% (G:0.00% B:0.00%)
Opponent Winning Chances: 30.23% (G:0.00% B:0.00%) 30.26% (G:0.00% B:0.00%)
Cubeless Equities +0.395 +0.789
Cubeful Equities
No double:+0.622±0.004 (+0.619..+0.626)
Double/Take:+0.614 (-0.009)±0.004 (+0.609..+0.618)
Double/Pass:+1.000 (+0.378)
Best Cube action: No double / Take
Percentage of wrong pass needed to make the double decision right: 2.2%
Rollout details
1296 Games rolled with Variance Reduction.
Dice Seed: 271828
Moves: 3-ply, cube decisions: XG Roller
Double Decision confidence:99.8%
Take Decision confidence:100.0%
Duration: 2 minutes 44 seconds

eXtreme Gammon Version: 2.19.208.pre-release

After generating the rollout, select the play in the box on the left, and right-click to get a menu, where you can ask for the rollout statistics to be displayed. You'll get a pop-up window with information that looks something like this (which I cut and pasted using Ctrl-C):


No double Non VR Equity: +0.639 (Cost: +0.639)
Cube Win BG Win G Win S Cash Pass Lose BG Lose G Lose S D/T D/P Pass % D/T D/P Pass %
1 370 83 1 749 370 66.93% 93 83 52.84%
2 515 10 128 104 9 10 47.37% 76 128 37.25%
4 13 18 3 47 4 18 18.18% 3 0.00%
8 4

The key information is displayed in the table on the left. The main point is that each of the 1296 trials in the rollout ended either with a dropped double, or in a single/gammon/backgammon win for someone with the cube on some level. This table displays all the possible categories and tells you how many of the 1296 trials fell into each category. (There is a typo, which I'm sure Xavier will fix in the release version, which is that the "Lose BG" and "Lose S" column headings have been inadvertently interchanged.) In this particular position, gammons are not possible, so the Win BG/Win G/Lose BG/Lose G columns are empty, but in general there will be numbers in these columns as well.

So for example, the "370" under the "Cash" column and in the "1" row means that 370 of the 1296 trials ended with the player on roll offering a double to 2 that was passed. The "83" under "Pass" means that 83 of the 1296 trials ended with the opponent offering a double to 2 that the player on roll passed. And so on. We see from the bottom row that in 4 of the 1296 trials, the player on roll won with the cube on 8.

Regarding that last fact, you might wonder, in those 4 trials, did the player own the 8-cube or did the opponent own the 8-cube? To determine this, look at the check boxes above the table (not shown here, but visible when you actually do this in XG). By selecting or unselecting these boxes, you can choose whether to display only the trials that ended with the player having cube access, or only the trials that ended with the player not having cube access, or both.

At the very top, there is a number labeled "non VR equity". This is the estimate of the equity that you get by simply adding up all the trials, weighting each game by its value (e.g., a backgammon with the cube on 4 is worth 12 points), and of course using a plus sign for wins and a minus sign for losses. More precisely, this calculation gives you the "Cost", and the "Equity" is the cost divided by the cube value in the position that you originally rolled out. If you do enough trials, the non VR equity should closely match the equity displayed in the rollout itself (which will be a variance-reduced equity except in the unlikely event that you unchecked the variance reduction box when doing the rollout). Here, for example, the non VR equity is 0.639 while the (variance-reduced) equity displayed for the rollout is 0.622. The variance-reduced equity is more accurate. The non VR equity is useful because if it is close to the variance-reduced equity, then you can feel confident that conclusions drawn from the table (such as, "370/1296 ~ 28.5% of the time, this game will end in Blue cashing a single point") are quantitatively close to the truth.

On the right are two other tables, displaying statistics about the number of times that each player offered a double at each cube level and the number of times that the double was taken or passed. The "Pass %" indicates the percentage of times the double was accepted (as Maik noted, this is a typo, which again I'm sure Xavier will fix); so for example 749/(749+370) ~ 66.93%. If you click on the "Show equity" box, the numbers will change to show you the average EMG equity of the trials in that box at the moment that the cube was offered. For example, the "749" doubles to 2 by the player that were taken will change to a number that is the average of the 749 D/T equities tallied in that box. For the D/P column, the average is of the equities if the cube were incorrectly taken.

One final remark that will not come up much in practice but that could be confusing if it does. Suppose you roll out a checker play that is D/P after the play is made, so that the (EMG) equity is -1.000. What XG does in such cases is to roll out the position assuming D/T after the play, and then at the end it reports -1.000 if the D/T equity it comes up with is less than -1.000. The main table will display the statistics for D/T, but the tables on the right will indicate that all the trials ended in D/P.

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