Match Equity Formula Reviewed and Revised
Rick Janowski, 2013 ## Background

Recently I was perusing through articles in my old copies of Das Backgammon Magazin (DBM), edited superbly by Harald Johanni. In the March 1992 issue I found an article I had written entitled “A New Method of Match Equity Evaluation” where I derived formulae for predicting match equities using the then current benchmark match equity tables derived by Bill Robertie (assuming 20% gammon rate) and Roy Friedman (assuming 36% gammon rate). I also wrote and included a BASIC program to calculate equities for any intermediate gammon rate, and provided a methodology to allow for relative skill differences between opponents based on information provided in Danny Kleinman’s article “Norman Zadeh’s Charts Interpolated” from his book How Can I keep from Dancing.

This was about a month before Kit Woolsey published his empirically derived match equity table in Inside Backgammon magazine, which immediately relegated previous METs to history. As I had done the groundwork already it was fairly easy for me to derive formulae to model closely this new benchmark and offered this to the editors of Inside Backgammon who after positive support from Kit subsequently coined the expression the “Janowski rule” much to my embarrassment at the time.

As I wanted to investigate the methodology for considering unequal opponents, I decided to convert the BASIC program into a spreadsheet. In so doing I found that assuming an intermediate gammon rate of about 29% gave very good correlation with values from the Rockwell-Kazaross MET, accepted now together with the virtually identical Kazaross-XG2 MET as the universal benchmark. This surprised me somewhat as I had assumed my old formulae were pretty much out-of-date. Investigating further I found that the formulae had the following performance levels (measured crudely but simply by maximum error occurring in a 15-point match compared to the Rockwell-Kazaross MET):

 Janowski Rule (Inside Backgammon): 3.1% max error DBM 20% gammon rate formula: 5.3% max error DBM 36% gammon rate formula: 3.4% max error

Not fantastic results but if an intermediate gammon rate of 28.5% is assumed, the maximum error falls to a quite respectable 1.6%. Incidentally, small significance should be put to the gammon rates assumed here at the time as an error in gammon rate could either improve or exacerbate other limitations of typical theoretical METs of the time (associated with the assumption of perfect, rather than practical cube efficiency). Adjustment of gammon rate provides a practical overall calibration tool.

Investigating further still, I have been able to derive suitable modifications to my formulae enabling all scores in a 15-point match (including Crawford and Post-Crawford) to be predicted within 0.9% of the Rockwell-Kazaross MET values (and to within 1.5% in a 19-point match). Details of the proposed adjustments are provided below. This section covers the normal scores in a match. Crawford and Post-Crawford Scores will be covered in a separate section. The original Janowski rule for these scores is given below:

Match equity for the leader is

M = 0.5 +
 0.85 × D T + 6

where D is the difference in score and T is the number of points needed by the trailer.

There are three proposed modifications, listed in order of importance as follows:

1. Adjustment for premature attainment of 100% probability.
2. Modification to the form of the equation.
3. Leader 2-away and 3-away considerations.

These will now be discussed.

### A. Adjustment for premature attainment of 100% probability

This is the main source of error associated with the limitations of linear formula at high probabilities approaching 100%. Typically, these errors may become significant for probabilities above 0.9. The specific proposal is to change the expression where the basic formula predicts M > 0.88 (optimised from the data) from various match scores. The proposed adjustment is:

When M > 0.88, use instead M= 0.66 × M + 0.33

Or perhaps expressed more simply, where M predicted is greater than 0.88, reduce by an amount equal to 0.34 × (M − 0.88).

With this modification alone the maximum error in a 15-point match should reduce from 3.1% to 1.6%. Interestingly, this methodology was adopted in the “Das Backgammon Magazin” article, but was unnecessary for modelling the Woolsey MET values.

### B. Modification to the form of the equation

I tested a number of different forms for both the numerator and denominator in the original expression D/(T+6) but found that still appears to be the optimal choice to minimise overall error, assuming the equation should be of a linear form. However, a small improvement occurs when the constant is changed from 0.85 to 0.87. Consequently, the proposed revised form of the equation is:

Match equity for the leader is

M = 0.5 +
 0.87 × D T + 6

### C. Leader 2-away and 3-away considerations

The 2-away scores and 3-away scores appear to be those most difficult to model by linear approximation formula presumably because their close proximity to match end or Crawford scores leaves some reflected discontinuities and odd/even effects. Investigations indicate that 2-away scores act more like theoretical 1.9-away scores and 3-away scores act more like 3.1-away scores in a linear model. Investigations however show that only one adjustment is desirable to improve overall accuracy — the 3-away score. A small improvement may be made by assuming 3-away scores are 3.1-away in calculating the difference in score D.

### Conclusion and summary

Three proposed modifications are made to improve the overall accuracy of predicted match equities. Proposal A, the high probability adjustment, is by far the most important accounting for about 70% of the overall improvement. Proposals B and C in comparison are somewhat optional — the individual should decide whether the greater complexity is worth a maximum possible gain of 0.7%. (i.e., from 1.6% to 0.9%). ## Crawford and Post-Crawford Score Adjustments

### Crawford scores

The original Janowski rule for these scores is given below:

Match equity for the leader is

M = 0.55 +
 0.55 × D T + 2
= 0.55 + 0.55 ×
 T − 1 T + 2

where D is the difference in score and T is the number of points needed by the trailer.

Considering the Crawford score match equities from the Rockwell-Kazaross MET, a significant reduction in maximum error (from 1.9% to 0.9%) may be effected if the following expression is used instead:

Match equity for the leader is

M = 0.525 +
 0.57 × D T + 2
= 0.525 + 0.57 ×
 T − 1 T + 2

Unlike normal match scores, there is no practical need to make an adjustment for premature attainment of 100% probability.

### Post-Crawford scores

These scores are rarely if ever needed in making match-play predictions, but it may be of some academic interest to the reader (perhaps calculation of gammon or backgammon value for crucial checker play decisions but it seems remote). These are discussed below:

At 1-away 1-away of course M = 0.500 at DMP. At 1-away 2-away, M = 0.512 = 0.500 plus maximum free drop allowance of 0.012 (derived from Rockwell-Kazaross MET). All other scores may be derived from trailer even number of points away scores as follows:

### Post-Crawford odd-away scores (T ≥ 3)

The match equity is approximately equal to the Crawford score where T is reduced by 1. E.g., the post-Crawford equity at 1-away 5-away equals the Crawford equity at 1-away 4-away. Note that in most cases the post-Crawford equities at odd-away scores are slightly less than their Crawford equivalents, but the effect is very marginal and may be ignored.

The maximum difference of about 0.4% occurs at 1-away 4-away (curiously no reduction occurs at 1-away 2-away scores. At each subsequent even-away score the difference reduces by about 0.1%. This effect relates to extra equity at Crawford scores from undoubled backgammons not realised at post-Crawford scores where the doubling cube kills or at least partially eradicates them.

### Post-Crawford even-away scores (T ≥ 4)

The match equity may be derived from the Crawford score where T is reduced by 2. E.g., the post-Crawford equity at 1-away 6-away equals the Crawford equity at 1-away 4-away. However, at these scores a free drop allowance should also be considered.

The maximum free-drop allowance (= 0.012) should be added to the probability at 1-away 2-away. At each subsequent even-away score the allowance may be considered to be reduced by 20% of the maximum value, i.e., 0.096 at 1-away 6-away) such that there is effectively no free-drop allowance at 1-away 14-away (in reality there will be a negligible difference of course).

### Conclusion and summary

By using the proposed modifications for predicting Crawford and post-Crawford equities, maximum errors should be reduced from 1.9% to 0.9% in comparison with the older formulae.

## References 