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Winning probability doesn't depend on match length?!

Posted By: Timothy Chow
Date: Monday, 5 June 2023, at 1:30 p.m.

In the current issue of the USBGF PrimeTime magazine, Art Benjamin, Richard Stubbe, and Susan Martonosi have an article in which they examine the effect of PR difference and match length on the winning probability.

They begin by noting the "conventional wisdom": Larry Kaufmann, in a 1991 Inside Backgammon article, predicted that if your Elo rating is X and your opponent's Elo rating is Y, then your probability of winning an N-point match is

1/(1 + 10(–(X – Y)√N)/2000))

In particular, your probability of winning the match is predicted to increase as the match length increases. Another piece of "conventional wisdom" is that a PR difference of 3 corresponds to an Elo difference of 100. Putting these two pieces of conventional wisdom together, we obtain predictions that, for example, with a PR difference of 5, the stronger player's win probability is about 60% for a 5-point match and about 66% for a 13-point match.

The surprise in the article is that the authors analyzed 19,000 actual matches from Backgammon Studio (courtesy of Terje Pedersen) and found that, in the range 5 ≤ N ≤ 17, winning probability was essentially independent of match length!

This is a very surprising result, and when I corresponded with Art Benjamin, I found that he didn't really believe it himself, but wasn't sure what the explanation was.

It doesn't seem possible that winning probability can be totally independent of match length, but I'm willing to believe that it increases more slowly as a function of N than Kaufmann predicted. I have two questions for you all:

1. Does anyone know how Kaufmann derived his formula? Back issues of Inside Backgammon seem to be hard to come by.

2. What other studies are there of the effect of match length on winning probability? I found some relevant articles on Backgammon Galore, such as The Backgammon rating system by Matti Rinta-Nikkola and Ratings: A Mathematical Study by Douglas Zare and Adam Stocks, but would be interested in more references, as well as any anecdotal experiences people have.

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