The Summer of Jeff

Single Set Win Expectancy Tables

Posted in tennis by Jeff on December 2, 2010

Yesterday I gave a (flawed) teaser of a Markov analysis of tennis.  I believe I’ve got the bugs cleared out.  What I’ve coded is a generalized solution to determine the probability of a player winning a set, given (a) the current score, (b) who serves next, and (c) the probability of each player’s holding serve.

For the purposes of a generic table, (c) is problematic.  To be truly generic, we would assume that the server holds no advantage.  That’s clearly wrong.  Even though the server’s advantage varies considerably depending on the skill of the server (and the returner), it seems like a more useful table would reflect the average likelihood that a player holds his serve.

It seems fundamental, but tennis stats just aren’t that mainstream, so I really don’t know what the ‘average’ likelihood of holding serve is.  The assumption I used yesterday was that the server wins 63% of points, which translates to 80% of service games.

I’m going to try something a bit different.  Below are two tables, one of which assumes that the server wins 65% of points (83% of service games), and the other assumes that the server wins 60% of points (73.5% of service games).  Very approximately, I think we can call these “hard court” and “clay court” tables, respectively.

The Single Set WinEx Table

The first two columns show a score.  The next column is either ‘s’ for ‘serving’ or ‘r’ for ‘returning.’  For example, ’1, 0, s’ gives the probability that a player wins the set if he is serving with a 1-0 lead, while ’1, 0, r’ gives the likelihood that the player wins the set if he is returning with a 1-0 lead.

When an even number of games have been played, ‘s/r’ is in the third column, because the probabilities are the same.

The 4th and 5th columns show the probability of winning or losing for my ‘hard court’ estimate, and the 6th and 7th give the probabilities for my ‘clay court’ estimate.

GmSc        Hard (0.65)   Clay (0.60)
            p(W)   p(L)   p(W)   p(L)  
0  0  s/r  50.0%  50.0%  50.0%  50.0%  
0  1  s    44.7%  55.3%  42.7%  57.3%  
0  1  r    24.1%  75.9%  29.8%  70.2%  
0  2  s/r  18.8%  81.2%  22.6%  77.4%  
0  3  s    13.6%  86.4%  15.6%  84.4%  
0  3  r     4.7%  95.3%   8.3%  91.7%  
0  4  s/r   2.4%  97.6%   4.2%  95.8%  
0  5  s     0.8%  99.2%   1.4%  98.6%  
0  5  r     0.2%  99.8%   0.5%  99.5%  

1  0  r    55.3%  44.7%  57.3%  42.7%  
1  0  s    75.9%  24.1%  70.2%  29.8%  
1  1  s/r  50.0%  50.0%  50.0%  50.0%  
1  2  s    44.2%  55.8%  42.0%  58.0%  
1  2  r    21.7%  78.3%  27.7%  72.3%  
1  3  s/r  15.9%  84.1%  19.7%  80.3%  
1  4  s    10.2%  89.8%  12.0%  88.0%  
1  4  r     2.9%  97.1%   5.5%  94.5%  
1  5  s/r   1.0%  99.0%   1.9%  98.1%  

2  0  s/r  81.2%  18.8%  77.4%  22.6%  
2  1  r    55.8%  44.2%  58.0%  42.0%  
2  1  s    78.3%  21.7%  72.3%  27.7%  
2  2  s/r  50.0%  50.0%  50.0%  50.0%  
2  3  s    43.5%  56.5%  40.9%  59.1%  
2  3  r    18.6%  81.4%  24.7%  75.3%  
2  4  s/r  12.1%  87.9%  15.7%  84.3%  
2  5  s     5.9%  94.1%   7.2%  92.8%  
2  5  r     1.2%  98.8%   2.6%  97.4%  

3  0  r    86.4%  13.6%  84.4%  15.6%  
3  0  s    95.3%   4.7%  91.7%   8.3%  
3  1  s/r  84.1%  15.9%  80.3%  19.7%  
3  2  r    56.5%  43.5%  59.1%  40.9%  
3  2  s    81.4%  18.6%  75.3%  24.7%  
3  3  s/r  50.0%  50.0%  50.0%  50.0%  
3  4  s    42.7%  57.3%  39.4%  60.6%  
3  4  r    14.4%  85.6%  20.4%  79.6%  
3  5  s/r   7.1%  92.9%   9.7%  90.3%  

4  0  s/r  97.6%   2.4%  95.8%   4.2%  
4  1  r    89.8%  10.2%  88.0%  12.0%  
4  1  s    97.1%   2.9%  94.5%   5.5%  
4  2  s/r  87.9%  12.1%  84.3%  15.7%  
4  3  r    57.3%  42.7%  60.6%  39.4%  
4  3  s    85.6%  14.4%  79.6%  20.4%  
4  4  s/r  50.0%  50.0%  50.0%  50.0%  
4  5  s    41.5%  58.5%  36.8%  63.2%  
4  5  r     8.5%  91.5%  13.2%  86.8%  

5  0  r    99.2%   0.8%  98.6%   1.4%  
5  0  s    99.8%   0.2%  99.5%   0.5%  
5  1  s/r  99.0%   1.0%  98.1%   1.9%  
5  2  r    94.1%   5.9%  92.8%   7.2%  
5  2  s    98.8%   1.2%  97.4%   2.6%  
5  3  s/r  92.9%   7.1%  90.3%   9.7%  
5  4  r    58.5%  41.5%  63.2%  36.8%  
5  4  s    91.5%   8.5%  86.8%  13.2%  
5  5  s/r  50.0%  50.0%  50.0%  50.0%  
5  6  s    41.5%  58.5%  36.8%  63.2%  
5  6  r     8.5%  91.5%  13.2%  86.8%  

6  5  r    58.5%  41.5%  63.2%  36.8%  
6  5  s    91.5%   8.5%  86.8%  13.2%  
6  6  s/r  50.0%  50.0%  50.0%  50.0%
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  1. [...] to working out the single set tables, this was a [...]

  2. [...] Jeff on December 4, 2010 If you’re joining in progress, here’s an intro, along with single set tables and single game [...]

  3. [...] last few days I’ve presented a lot of theory. My win expectancy tables are based on what you might call a “video game” model of [...]


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