How many Games are Expected to be Played?
In a previous post I discussed the expected number of rallies in a game and a match. In this post I will discuss the number of games per match in dependence of the probabilities to win a rally or a game. This analysis is still under the assumption of independent and identically distributed outcomes of each rally or game.
How to Calculate the Expected Number of Games
The calculation is rather simple as there are only six possible combinations of won and lost games in a match:
- Won, Won
- Lost, Won, Won
- Won, Lost, Won
- Lost, Won, Lost
- Won, Lost, Lost
- Lost, Lost
Given the probability to win a game \(P_{Game}(p)\) the expected number of games can be calculated using these six combinations:
\[\begin{eqnarray} \bar{N}_{games per match}(p) &=& 2 \cdot P_{Game}(p) \cdot P_{Game}(p) \\ &+& 3 \cdot (1-P_{Game}(p)) \cdot P_{Game}(p) \cdot P_{Game}(p) \\ &+& 3 \cdot P_{Game}(p) \cdot (1-P_{Game}(p)) \cdot P_{Game}(p) \\ &+& 3 \cdot (1-P_{Game}(p)) \cdot P_{Game}(p) \cdot (1-P_{Game}(p)) \\ &+& 3 \cdot P_{Game}(p) \cdot (1-P_{Game}(p)) \cdot (1-P_{Game}(p)) \\ &+& 2 \cdot (1-P_{Game}(p)) \cdot (1-P_{Game}(p)) \\ \end{eqnarray}\]Correspondingly the average difference of won games can be calculated by substituting the numbers of games with the difference of won games:
\[\begin{eqnarray} \bar{N}_{games per match}(p) &=& (+2) \cdot P_{Game}(p) \cdot P_{Game}(p) \\ &+& (+1) \cdot (1-P_{Game}(p)) \cdot P_{Game}(p) \cdot P_{Game}(p) \\ &+& (+1) \cdot P_{Game}(p) \cdot (1-P_{Game}(p)) \cdot P_{Game}(p) \\ &+& (-1) \cdot (1-P_{Game}(p)) \cdot P_{Game}(p) \cdot (1-P_{Game}(p)) \\ &+& (-1) \cdot P_{Game}(p) \cdot (1-P_{Game}(p)) \cdot (1-P_{Game}(p)) \\ &+& (-2) \cdot (1-P_{Game}(p)) \cdot (1-P_{Game}(p)) \\ \end{eqnarray}\]Plots for Game Probabilities
The following plots show the results for different probabilities to win a single game.
Average
We see that for two equally strong players there is an average number of 2.5 games per match, which corresponds to a 50% chance of playing three games. For more one-sided matches the average decreases towards a value of 2, which would correspond to a match that always consists of only two games and never of three games.
Difference
The average difference of games shows an almost straight curve, only a small bend is visible.
Results
This plot shows the different game results. A result of 2-0 means the match was won in straight games, 2-1 means it was won in three games, and so on.
For a game probability of 50% all curves meet as then all game results have the same probability. The shares of three game matches have the highest value at game probabilities of 1/3 and 2/3 respectively. Going to very large or very small game probabilities the share of the 2-0 and 0-2 results becomes dominant.
Plots for Rally Probabilities
Using the results from the post Probability to Win a Game we can transform these plots into plots depending on the probability to win single rallies. We do this transformation due to the fact that probabilities for single rallies are usually more familiar than probabilities for games, as there are many more rallies per match. Due to the magnification effect we expect the plots to be compressed horizontally.
Average
As expected we see a much narrower peak. For a rally probability of 40% the match lasts on average a bit less than 2.2 games. This is consistent with the game probability of close to 10% for the outsider as found when calculating his or her probabilty to win a game. The outsider will have two chances to win a game, so there is an approximate chance of close to 20% of winning one of the first two games, and hence extending the match by one more game.
Difference
Also the difference shows a much compressed curvature. The S-shape is clearly visible in this plot.
Results
Again the plot is compressed horizontally. The peaks for the three-game shares now peak at about 47% and 53&.
Conclusion
The more similar the strengths of two players or pairs are, the more games a match will have. This post showed the dependence under the i.i.d-assumption.
The results of this post could be of importance to tournament organizers as the time a match is expected to last is determined to a great extent by the number of games played, at least in the current scoring system. A tournament organizer could aim to distribute matches to different courts in a way such that matches expected to last more games will be on courts with fewer matches. Thus the variances of expected numbers of games played per court could be minimized.