The Summer of Jeff

7-point Tiebreak Win Expectancy Tables

Posted in tennis by Jeff on December 4, 2010

If you’re joining in progress, here’s an intro, along with single set tables and single game tables.

A Markov model of a tiebreak gets a little tricky–depending on where you are in the breaker, the remaining points could be evenly split between the two servers, off by one, or even off by two.  But that’s my problem…below you’ll find the win expectancy at every possible score in a tiebreak for each of three different types of players/surfaces.

The three sets of percentages are for three types of evenly matched players.  “0.6” refers to a pair of players who win 60% of service points, which could model counterpunchers or stronger servers on a slower surface.  0.65 is probably about average for a hard court match, and 0.7 is a battle between two dominant servers.  The table quantifies what we already know: The more effective the servers, the more valuable each mini-break.

To read the table, start with the first three columns of any row. The second row, for instance, shows percentages for a player who is serving (“s”) at 0-1. In the 0.60 model, he has a 40.8% chance of winning the tiebreak.

              0.60            0.65            0.70         
              p(W)   p(L)     p(W)   p(L)     p(W)   p(L)  
0  0  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
0  1    s    40.8%  59.2%    41.8%  58.2%    42.7%  57.3%  
0  1    r    36.2%  63.8%    34.7%  65.3%    32.9%  67.1%  
0  2  s/r    27.0%  73.0%    26.5%  73.5%    25.6%  74.4%  
0  3    s    18.4%  81.6%    18.6%  81.4%    18.6%  81.4%  
0  3    r    15.1%  84.9%    13.7%  86.3%    12.1%  87.9%  
0  4  s/r     8.5%  91.5%     8.0%  92.0%     7.2%  92.8%  
0  5    s     3.8%  96.2%     3.7%  96.3%     3.5%  96.5%  
0  5    r     2.8%  97.2%     2.3%  97.7%     1.8%  98.2%  
0  6  s/r     0.7%  99.3%     0.6%  99.4%     0.5%  99.5%  

1  0    r    59.2%  40.8%    58.2%  41.8%    57.3%  42.7%  
1  0    s    63.8%  36.2%    65.3%  34.7%    67.1%  32.9%  
1  1  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
1  2    s    40.0%  60.0%    41.0%  59.0%    42.0%  58.0%  
1  2    r    35.0%  65.0%    33.3%  66.7%    31.4%  68.6%  
1  3  s/r    24.9%  75.1%    24.3%  75.7%    23.4%  76.6%  
1  4    s    15.7%  84.3%    15.9%  84.1%    15.8%  84.2%  
1  4    r    12.4%  87.6%    11.1%  88.9%     9.5%  90.5%  
1  5  s/r     5.9%  94.1%     5.4%  94.6%     4.8%  95.2%  
1  6    s     1.7%  98.3%     1.7%  98.3%     1.5%  98.5%  
1  6    r     1.2%  98.8%     0.9%  99.1%     0.7%  99.3%  

2  0  s/r    73.0%  27.0%    73.5%  26.5%    74.4%  25.6%  
2  1    r    60.0%  40.0%    59.0%  41.0%    58.0%  42.0%  
2  1    s    65.0%  35.0%    66.7%  33.3%    68.6%  31.4%  
2  2  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
2  3    s    38.9%  61.1%    40.0%  60.0%    41.2%  58.8%  
2  3    r    33.3%  66.7%    31.5%  68.5%    29.4%  70.6%  
2  4  s/r    22.2%  77.8%    21.5%  78.5%    20.6%  79.4%  
2  5    s    12.1%  87.9%    12.3%  87.7%    12.3%  87.7%  
2  5    r     9.0%  91.0%     7.8%  92.2%     6.5%  93.5%  
2  6  s/r     2.9%  97.1%     2.6%  97.4%     2.2%  97.8%  

3  0    r    81.6%  18.4%    81.4%  18.6%    81.4%  18.6%  
3  0    s    84.9%  15.1%    86.3%  13.7%    87.9%  12.1%  
3  1  s/r    75.1%  24.9%    75.7%  24.3%    76.6%  23.4%  
3  2    r    61.1%  38.9%    60.0%  40.0%    58.8%  41.2%  
3  2    s    66.7%  33.3%    68.5%  31.5%    70.6%  29.4%  
3  3  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
3  4    s    37.3%  62.7%    38.7%  61.3%    40.0%  60.0%  
3  4    r    30.9%  69.1%    28.9%  71.1%    26.6%  73.4%  
3  5  s/r    18.2%  81.8%    17.6%  82.4%    16.6%  83.4%  
3  6    s     7.2%  92.8%     7.4%  92.6%     7.4%  92.7%  
3  6    r     4.8%  95.2%     4.0%  96.0%     3.2%  96.9%  

4  0  s/r    91.5%   8.5%    92.0%   8.0%    92.8%   7.2%  
4  1    r    84.3%  15.7%    84.1%  15.9%    84.2%  15.8%  
4  1    s    87.6%  12.4%    88.9%  11.1%    90.5%   9.5%  
4  2  s/r    77.8%  22.2%    78.5%  21.5%    79.4%  20.6%  
4  3    r    62.7%  37.3%    61.3%  38.7%    60.0%  40.0%  
4  3    s    69.1%  30.9%    71.1%  28.9%    73.4%  26.6%  
4  4  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
4  5    s    34.8%  65.2%    36.5%  63.5%    38.2%  61.9%  
4  5    r    27.2%  72.8%    24.9%  75.1%    22.4%  77.7%  
4  6  s/r    12.0%  88.0%    11.4%  88.6%    10.5%  89.5%  

5  0    r    96.2%   3.8%    96.3%   3.7%    96.5%   3.5%  
5  0    s    97.2%   2.8%    97.7%   2.3%    98.2%   1.8%  
5  1  s/r    94.1%   5.9%    94.6%   5.4%    95.2%   4.8%  
5  2    r    87.9%  12.1%    87.7%  12.3%    87.7%  12.3%  
5  2    s    91.0%   9.0%    92.2%   7.8%    93.5%   6.5%  
5  3  s/r    81.8%  18.2%    82.4%  17.6%    83.4%  16.6%  
5  4    r    65.2%  34.8%    63.5%  36.5%    61.9%  38.2%  
5  4    s    72.8%  27.2%    75.1%  24.9%    77.7%  22.4%  
5  5  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%  
5  6    s    30.0%  70.0%    32.5%  67.5%    35.0%  65.0%  
5  6    r    20.0%  80.0%    17.5%  82.5%    15.0%  85.0%  

6  0  s/r    99.3%   0.7%    99.4%   0.6%    99.5%   0.5%  
6  1    r    98.3%   1.7%    98.3%   1.7%    98.5%   1.5%  
6  1    s    98.8%   1.2%    99.1%   0.9%    99.3%   0.7%  
6  2  s/r    97.1%   2.9%    97.4%   2.6%    97.8%   2.2%  
6  3    r    92.8%   7.2%    92.6%   7.4%    92.7%   7.4%  
6  3    s    95.2%   4.8%    96.0%   4.0%    96.9%   3.2%  
6  4  s/r    88.0%  12.0%    88.6%  11.4%    89.5%  10.5%  
6  5    r    70.0%  30.0%    67.5%  32.5%    65.0%  35.0%  
6  5    s    80.0%  20.0%    82.5%  17.5%    85.0%  15.0%  
6  6  s/r    50.0%  50.0%    50.0%  50.0%    50.0%  50.0%

Of course, a tiebreak can continue past 6-6. Any tie (e.g. 9-9) means that each player has a 50% chance of winning. Any other score (e.g. 8-9, 11-10) has the same probabilities as 6-5 or 5-6.

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