Match Equity Formula Reviewed and Revised
Posted By: Rick Janowski
Date: Tuesday, 18 June 2013, at 6:01 p.m.
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 MET 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 Magazin, 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 This surprised me somewhat as I had assumed my old formulae were pretty much out-of-date.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 will be provided in two separate threads.
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