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Match Equity Formula Reviewed and Revised  Crawford & PostCrawford Score Adjustments
Posted By: Rick Janowski
Date: Tuesday, 18 June 2013, at 7:17 p.m.
In Response To: Match Equity Formula Reviewed and Revised (Rick Janowski)
This thread covers the Crawford and PostCrawford scores. The normal preCrawford scores were covered in the previous thread.
Crawford Scores: The original Janowski Rule for these scores is given below :
Match equity for the leader, M = 0.55 +0.55*D/(T + 2) = M = 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 RockwellKazaross 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, M = 0.525 + 0.57*D/(T + 2) = M = 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.
PostCrawford Scores: These scores are rarely if ever needed in making match play predictions 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 2away, M = 0.512 = 0.500 plus maximum free drop allowance of 0.012 (derived from RockwellKazaross MET). All other scores may be derived from trailer even number of points away scores as follows:
PostCrawford Odd Away Scores (T >= 3): The match equity is approximately equal to the Crawford score where T is reduced by 1, e.g., PostCrawford score for 1 away 5 away = Crawford score for 1 away 4 away. Note that the in most cases postCrawford score at odd away scores are slightly less 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 postCrawford scores where the doubling cube kills or at least partially eradicates them.
PostCrawford Even Away Scores (T >= 4): The match equity may be derived from the Crawford score where T is reduced by 2, e.g., PostCrawford score for 1 away 6 away = Crawford score for 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 effectively there is effectively no free drop allowance at 1 away 14 away (in reality there will be a negligible difference of course).
Conclusion/Summary: Use of the proposed modifications for predicting Crawford and PostCrawford scores, maximum errors should be reduced from 1.9% to 0.9%, in comparison with the older formulae.

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