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Match Equity Formula Reviewed and Revised -- Pre-Crawford Score Adjustments
Posted By: Rick Janowski In Response To: Match Equity Formula Reviewed and Revised (Rick Janowski)
Date: Tuesday, 18 June 2013, at 6:10 p.m.
This thread covers the normal scores in a match. Crawford and Post-Crawford Scores will be included in a separate thread. The original Janowski Rule for these scores is given below:
Match equity for the leader, 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 modification listed in order of importance as follows:
A. Adjustment for premature attainment of 100% probability.
B. Modification to the form of the equation.
C. 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, 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/Summary: 3 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%)
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