Wrong Match Equity Table
Douglas Zare, 2008
Gammon Village Magazine, © 2008 Douglas Zare and GammonVillage

This article originally appeared in Gammon Village Magazine.
Thank you to Douglas Zare and Gammon Village
for their kind permission to reproduce it here.

## Match Equity Tables

One of the key advances in backgammon theory was the introduction of match equity tables by theorists such as Zadeh and Kleinmann. Experts use match equity tables to determine the true risks versus rewards in match play instead of the nominal points won or lost. Using any reasonable match equity table is better than not using anything, but now there are multiple tables in use. Woolsey has a commonly used MET, but Snowie uses its own MET, and GNU uses a different table by default. What happens when the tables conflict?

There can be abstract disagreements. If you are discussing how to treat a match score such as 4-away 5-away, you might disagree with the take points and gammon prices suggested by someone else, only to find that the source of the disagreement is that you are using a different match equity table. Different METs lead to concrete disagreements, particularly in close decisions in some match scores.

Diagram 1
Black trails 4-away 5-away.
Cube action?

According to the Woolsey-Heinrich MET, this is a double. According to Snowie’s MET, doubling would be a 0.067 error. According to Snowie, the difference between being tied at 4-away and leading 3-away 4-away is not as large as the difference between trailing 3-away 5-away and 2-away 5-away.

Diagram 2
Black trails 3-away 5-away.
Pip count 65–74.
Cube action?

White wins 20.7%. According to Snowie’s table, this is a double and a 0.933 take. According to the Woolsey-Heinrich table, this is a pass. While most double/no double decisions are less clear, it’s possible for a bot to mark a series of decisions not to double as errors, costing a large amount of equity, then to reverse itself and agree with you if you use a different match equity table.

Do we have to spend time studying which match equity table to use, in addition to the time spend studying checker plays and cube decisions? While choosing a decent match equity table matters, choosing a good one is good enough. We’ll see that you will rarely give up much equity if you use a good, but not quite perfect match equity table.

## Differing Tables

Woolsey’s MET
 Crawf 2 away 3 away 4 away 5 away 6 away 7 away Crawf 50 70 75 83 85 90 91 2 away 30 50 60 68 75 81 85 3 away 25 40 50 59 66 71 76 4 away 17 32 41 50 58 64 70 5 away 15 25 34 42 50 57 63 6 away 10 19 29 36 43 50 56 7 away 9 15 24 30 37 44 50

The above is also called the Woolsey-Heinrich table, and it was popularized in their book, How to Play Tournament Backgammon.

Snowie’s MET
 Crawf 2 away 3 away 4 away 5 away 6 away 7 away Crawf 50.00 68.50 74.78 81.94 84.30 89.08 90.71 2 away 31.50 50.00 59.42 66.42 73.48 79.07 83.16 3 away 25.22 40.58 50.00 57.16 64.46 70.65 75.45 4 away 18.06 33.58 42.84 50.00 57.33 63.72 69.01 5 away 15.70 26.52 35.54 42.67 50.00 56.48 62.09 6 away 10.92 20.93 29.35 36.28 43.52 50.00 55.85 7 away 9.29 16.84 24.55 30.99 37.91 44.15 50.00

The same table has been used by Snowie for the past few versions.

Snowie’s table is not perfect. I disagree with a lot of the entries, starting with Crawford 2-away. However, I believe it is more accurate than Woolsey’s table. Snowie’s table is based on a higher gammon rate, and the more modern, aggressive play now results in more gammons. This tends to favor the trailer.

Here are the differences between Snowie’s table and Woolsey’s table:

Snowie Minus Woolsey
 Crawf 2 away 3 away 4 away 5 away 6 away 7 away Crawf 0 −1.50 −0.22 −1.06 −0.70 −0.92 −0.29 2 away +1.50 0 −0.58 −1.58 −1.52 −1.93 −1.84 3 away +0.22 0.58 0 −1.84 −1.54 −0.35 −0.55 4 away +1.06 +1.58 +1.84 0 −0.67 −0.28 −0.99 5 away +0.70 +1.52 +1.54 +0.67 0 −0.52 −0.91 6 away +0.92 +1.93 +0.35 +0.28 +0.52 0 −0.15 7 away +0.29 +1.84 +0.55 +0.99 +0.91 +0.15 0

Another modern match equity table is the table Ian Dunstan obtained through Gnu 2-ply rollouts. It’s much closer to Snowie’s table than to Woolsey’s, but there are some important differences. For example, the equity trailing Crawford 2-away is over 32%, which I believe is correct, and the equity is greater for the trailer at Crawford 4-away, too.

I believe it is not that Snowie’s table underestimates the gammon rate, but that the trailer can adjust to the gammon-go scores more than the leader can adjust to gammon-save. Snowie’s table is based on the assumption that the backgammon/gammon/win rates are the same at all match scores as they are for money play.

## Theory

Common sense says that if you do everything else right, it shouldn’t hurt much if you use a match equity table which is slightly off. For example, it shouldn’t matter much if you use a match equity table which is rounded to the nearest percent. Then again, our intuition might say the match leader should be conservative accepting racing doubles, but upon closer inspection, that’s not true.

Suppose you play perfectly, but follow the wrong match equity table for one game only. Suppose the true equity at the match score is 50% and the true equities of the scores you can reach are 70%, 40%, and 30%. We will call the true equities the green equities. However, your table says 68%, 41%, and 29%, which we will call the red equities. We want to know how much green equity you give up while using the red equities. How close to 50% green equity do you get?

Let A be the maximum of the (green − red) equities at the achievable scores, 2% here for 70% − 68%. If we add A to all red equities, they will become greater than the green equities. When you play perfectly in a game with greater payoffs, your average result is at least as good, so

average(A + red) ≥ 50% = green initial equity

A + average(red) ≥ green initial equity

We don’t want to know the average of (A + red) equity. We want to know the average green equity you give up, so we need to relate the (A + red) equity with the green equity. For example, at the middle score the (A + red) equity is 43%, but your green payoff is 40%, 3% lower. Let B be the maximum of (red − green) equities, here 1% for 41% − 40%.

B + green ≥ red

B + average(green) ≥ average(red)

A + B + average(green) ≥ A + average(red) ≥ green initial equity

average(green) ≥ green initial equity − (A + B)

So, the average green equity given up is at most A + B. The cost of using the wrong match equity table is at most A + B in real equity.

Example: Suppose you play perfectly, but use a match equity table which is perfect except it is rounded to the nearest tenth of a percent. Then at each match score, A ≤ 0.05%, and B ≤ 0.05%, so at each match score you give up at most 0.1% MWC.

Of course, a match is made of many games. If you give up 0.1% in each game, you may give up more than 1% over the course of a long match, according to this estimate. In practice, you will give up less than that. You will not always hit the match scores at which you might give up the most equity, and you will often face decisions which do not depend on the match equity table you use.

## Experiments

Several people, notably Joseph Heled, have let Gnu Backgammon play against itself using different match equity tables. In Heled’s trials, he found that a more modern table outperformed the Woolsey’s table by about 50.1%49.9% over 100,000 7-point matches. He used some variance reduction, so the results are statistically significant in two ways.

First, we can conclude Gnu Backgammon was indeed better off using another table than the Woolsey-Heinrich table. Second, we can conclude that it didn’t make much of a difference, no more than about 2 Elo points. This was confirmed by other, shorter experiments.

Other match lengths might lead to greater or smaller differences. Longer matches might emphasize more lopsided match scores more. However, rest assured that as long as you choose a modern table, choosing the wrong one will not cost you much equity on average.

## Summary

Modern match equity tables disagree significantly at a few match scores. The analysis of some positions depends on the MET assumed.

If you use a match equity table which is slightly wrong, this will not cause you to lose a large amount of equity on average. You may occasionally have a string of decisions marked as errors by a bot using a different MET.