3.2.2 What'S The Probability Of Winning?
Description
This article is from the Table Tennis (Ping Pong) FAQ, by ttennis@bu.edu with numerous contributions by others.
3.2.2 What'S The Probability Of Winning?
We want to extract from the data the probability of winning a match as a
function of the difference in ratings of the two players. Let's look at the
distribution of the matches by rating.
-------------------------------------------------------------
Rating | Pre | Adjusted | Post
difference |-------------------------------------------------
| Matches Upsets | Matches Upsets | Matches Upsets
-------------------------------------------------------------
0- 299 | 973 272 | 1126 260 | 1123 212
300- 599 | 229 15 | 275 4 | 283 1
600- 899 | 69 1 | 86 0 | 80 0
900-1199 | 11 0 | 17 0 | 18 0
1200-3000 | 3 0 | 6 0 | 6 0
-------------------------------------------------------------
The reason there are fewer total matches in the "Pre" column is that we
have excluded those matches that involve an unrated player. For our
purposes, the main thing to notice is how few matches there are with large
rating differences and how few of them are upsets. Hence any estimate we
calculate for the probability of winning when there are large rating
differences will be of questionable accuracy. Of course we are using only 8
tournaments; there are over 200 tournaments per year.
TECHNICAL STUFF
To proceed we need a model for the probability of winning a nonhandicap
match as a function of the rating difference. This gets technical for
awhile. We will use a logistic model. Let D be the rating difference, P be
the probability of winning a nonhandicap 2 out of 3 match, and b be the
model parameter. The form of the logistic model is
P( D ) = exp( bD ) / ( 1 + exp( bD ) )
We fit the model to each of the three sets of data by maximum likelihood.
Here is the result.
------------------ Ratings | b ---------|-------- Pre | 0.00795 Adjusted | 0.01115 Post | 0.01517 ------------------
Each model lets us calculate the probability of winning a nonhandicap 2 out
of 3 match for any difference in rating. Given standard assumptions
(probability of winning a point is independent of the score and of who is
serving) a probability of winning a nonhandicap 2 out of 3 match
corresponds to a probability of winning a point.
This suggests how to calculate a handicap chart. Pick one of the three
models. Pick a rating difference. Convert this to the probability of
winning a nonhandicap 2 out of 3 match using the model. Convert this to the
probability of winning a point. Now find the handicap such that the
probability of winning a handicap match is 0.5 (i.e., the handicap match is
fair to both players). By the way, my 386 computer (no coprocessor) needed
about an hour to compute the charts.
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