# Comparison of 11-Point and 21-Point Match FormatsAugust 6, 2001

The change to 11-point games raises the question of which 11-point match formats should replace the best-of-3 and best-of-5 21-point match formats that we are all familiar with. To answer this, we will consider various different ways of comparing match formats.

### Point Counts

One way to compare different formats is to count the number of points you need to score to win the match. A better way is to note that a game to 21 is really a best-of-40-points game. Since 40 is an even number, games can end in a tie, which necessitates the deuce rules to break the tie. Since deuce can theoretically go on forever, let’s ignore this complication for the moment by assuming that none of the games go to deuce. Table 1 gives the number of points needed to win the match and the total number of points in the match (ignoring deuce) for various 11-point and 21-point match formats.

Table 1. Point Counts for Various Match Formats
Points
to
Win Game
Games
in
Match
Points
Needed to
Win Match
Total
Points
in Match
21 121 40
21 342120
21 563200
11 111 20
11 322 60
11 533100
11 744140
11 955180
111166220

### Expected Number of Points

Rather than just counting points, we can consider the expected (or average) number of points in a match. If one player is so much better than the other that he or she wins every point, then this is just the number of points needed to win the match (the values for which are in Table 1). However, as the two players get closer in level, the expected length of a match increases, with the longest matches being expected for two players that are equally matched. Table 2 gives the expected number of games and expected total points for various match formats for two equally matched players. These values include deuce games (once the game reaches deuce, the expected number of additional points in the match turns out to be four).

Table 2. Expected Number of Points for Various Match Formats
Points
to
Win Game
Games
in
Match
Expected
Games
Expected
Total Points
21 11.00 37.24
21 32.50 93.09
21 54.13153.60
11 11.00 18.83
11 32.50 47.07
11 54.13 77.67
11 75.81109.44
11 97.54141.95
11119.29174.97

### Probability of Upset

The final method that we will use to compare different formats is to consider the probability of an upset (i.e. the probability that the better player will lose the match). The longer the match, the smaller the probability of an upset. Note that if the two players are equally matched, then neither player is better and each player is equally likely to win the match, regardless of the match format. Thus we must consider matches between unequal players. Table 3 gives the probability of an upset for various match formats and various values for the probability of the better player winning a point.

Table 3. Probability of an Upset for Various Match Formats
Points
to
Win Game
Games
in
Match
Probability of
Winning a Point
0.520.540.560.580.60
21 10.400.300.210.140.09
21 30.350.210.120.060.02
21 50.310.160.070.020.01
11 10.420.350.280.220.16
11 30.390.280.190.120.07
11 50.360.230.140.070.03
11 70.340.200.100.040.02
11 90.320.170.080.030.01
11110.300.150.060.020.00

### Conclusions

The methods of comparing formats give similar results, although the count of the number of points needed to win a match differs somewhat from the others (and seems to be the crudest of the methods). Overall, the 21-point best-of-3 format is halfway between the 11-point formats of best-of-5 and best-of-7. The 21-point best-of-5 format is between the 11-point formats of best-of-9 and best-of-11, but is somewhat closer to the best-of-9 format.

Thus either the 11-point best-of-5 format or the 11-point best-of-7 format is a reasonable replacement for the 21-point best-of-3 format. The 11-point best-of-9 format is the best replacement for the 21-point best-of-5 format.

### Technicalities

All the probabilities were calculated using standard (and reasonable for this purpose) modeling assumptions, i.e. the probability of winning a point is constant (thus independent of the score) and is the same for both players. We could allow the probability of winning a point to depend on which player is serving, but doing so does not affect the conclusions. 