The meaninglessness of backgammon results
Posted By: Timothy Chow
Date: Saturday, 23 June 2012, at 12:53 a.m.
Stick always complains about the meaningless of backgammon results. I decided for fun to run a few numbers.
Suppose there is a World Championship tournament with 128 entrants: 127 donkeys and 1 stick. The donkeys are all equally skillful, and the stick has a 2-to-1 advantage over a donkey in any single five-point match.
Suppose now that the format of the tournament is a seven-round knockout, all five-point matches. What is the probability that the stick will win the tournament?
The answer turns out to be less than 6%.
We can increase the probability that the stick will win the tournament by having players play not just one five-point match, but a best-of-n series of five-point matches where n is some odd number larger than 1. How large does n have to be to give the stick a >50% chance of winning the tournament?
The answer turns out to be n = 15. That is, the stick might need to play as many as 7*15 = 105 five-point matches to claim the World Champion title. If we want to give the stick a 90% chance of winning the tournament then we need to take n = 39.
Part of the reason for these large numbers is that a knockout tournament is a lousy way of determining the most skillful player. Suppose we ran a Swiss tournament instead, where in each round everyone plays a five-point match, and the winner of the match scores 1 point. Then for the stick to have a >50% chance of winning, we'd need about 65 rounds. For a 90% chance of winning we'd need about 150 rounds. Although any given random donkey expects to score only 75 points in a 150-round Swiss, while the stick expects to score 100 points, when you have 127 donkeys, there's still about a 10% chance that the luckiest donkey will outscore the stick.
(Actually the numbers of rounds for the Swiss tournament are slightly inflated because I was too lazy to write an actual simulation; I just assumed that each donkey would have a 50% chance of scoring a point in each round. In a real Swiss the lucky donkeys find themselves playing the stick more often, so it's harder for them to score really high.)
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