Theory of Point Differences
As I recently discussed the advantages and disadvantages of point-handicap betting I will now discuss theoretical issues with point differences.
Point differences are elemental to point-handicap betting. I published my thoughts on the aspects related to betting in the post Advantages and Disadvantages of Point-Handicap Betting. In this post I will just discuss the stochastic side.
How to Calculate the Point Difference for a Match
The point difference is obtained by summing the differences for all games in a match. For example a match ending 21-10 17-21 22-20 will have differences of +11, -4, +2 for the games, which will then add up to +9. Alternatively one could sum all points a player or pair won in all games and then calculate the difference. In the example above, the first player or pair won 21+17+22 = 60 points, while its opponent won 10+21+20 = 51 points, giving a difference of 60 - 51 = +9.
How to Estimate the Expected Point Difference
Averages
Under the assumption that each rally is won with the constant probability \(p\) by one player, the expected average for the point difference for a game can be calculated as follows:
\[\begin{eqnarray} \bar{\Delta}_{game} (p) &=& \sum\limits_{n_2 = 0}^{19} \, (21-n_2) \cdot \dbinom{20+n_2}{20} p^{20} (1-p)^{n_2}\cdot p \\ &+& \sum\limits_{i = 0}^8 (2) \cdot \dbinom{20+20}{20} p^{20} (1-p)^{20} \cdot \left[ 2p(1-p) \right]^i \cdot p^2 \\ &+& (1) \cdot \left. \left. \dbinom{20+20}{20} p^{20} (1-p)^{20} \cdot \right[2 p (1-p) \right]^9 \cdot p \\ &+& (-1) \cdot \left. \left. \dbinom{20+20}{20} p^{20} (1-p)^{20} \cdot \right[2 p (1-p) \right]^9 \cdot (1-p) \\ &+& \sum\limits_{i = 0}^8 (-2) \dbinom{20+20}{20} p^{20} (1-p)^{20} \cdot \left[ 2p(1-p) \right]^i \cdot (1-p)^2 \\ &+& \sum\limits_{n_1 = 0}^{19} \, (n_1-21) \dbinom{n_1+20}{n_1} p^{n_1} (1-p)^{20}\cdot (1-p) \end{eqnarray}\]The first line corresponds to games won with a score of twenty-one to something, the second line corresponds to games won with a difference of two points after extra points while the third line corresponds to games won with a score of 30-29. The following lines correspond to the same games in reverse order but lost. All probabilities are then weighted by the difference of points, given by the term in the first parentheses.
The average for a match can then by calculated using the formula
\[\begin{eqnarray} \bar{\Delta}_{match} (p) &=& p_{game}^2 \cdot 2 \cdot \left[ \bar{\Delta}_{won\,game} \right] \\ &+& 2 p_{game}^2 (1-p_{game}) \cdot \left[ 2 \cdot \bar{\Delta}_{won\,game} + \bar{\Delta}_{lost\,game} \right] \\ &+& 2 p_{game} (1-p_{game})^2 \cdot \left[ \bar{\Delta}_{won\,game} + 2 \cdot \bar{\Delta}_{lost\,game} \right] \\ &+& (1-p_{game})^2 \cdot \left[ 2 \cdot \bar{\Delta}_{lost\,game} \right], \end{eqnarray}\]where \(p_{game}\) is the probability to win a game given the probability \(p\) to win each rally. \(\bar{\Delta}_{won\,game}\) and \(\bar{\Delta}_{lost\,game}\) are the average point differences from the formula above but only taking won or lost games into account respectively. The first line then corresponds to a game won in two straight games, the second line to the two ways the match can be won in three games. The third and fourth line correspond to the same sequences of games but for matches that are ultimately lost.
Medians
The median can be calculated using similar formulas. The match or game outcomes can be ordered by the point difference and then, starting with the lowest differences, summed until a combined probability of 50% is reached.
Plots
The following plots show the average point difference and the median for different probabilities \(p\) to win each rally.
In both plots we can see a S-shape, starting at the lowest possible difference for a rally probability of zero, giving point differences of zero for rally probabilities of 50% and going to the maximal value for rally probabilities approaching 100%. The overall S-shape is however rather weak. There is also for the most parts no big difference between the average and the median.
Games
The step at 50% is quite distinct. Even increasing the rally percentage from 50% to 50.003% is sufficient to increase the median to +2, because the probability of won games increases above 50% by a fraction. Thus the median is now the smallest point difference in the won matches. The influence of matches ending 30-29, i.e. with a difference of +1, is present, but not visible in the graph.
Matches
The step at 50% is almost invisible for the point difference for a match. This is understandable because a game is either won or lost, but in a match multiple games can soften this distinction. In general the curve for the median is much smoother than in the plot for games. For probabilites between 40 and 60% the median tends to deviate more from zero as the average.
Tables
The following tables show the average and median point difference for various probabilities of winning a game and winning a match.
Games
A player winning games with a probability of 95% will have a 37.68% probability to win an individual rally. His average point difference will be +8.25, the median of the point difference will be +9 in his favour.
| Game Probability |
Rally Probability |
Average Difference |
Median Difference |
|---|---|---|---|
| 95% | 62.32% | +8.25 | +9.0 |
| 90% | 59.64% | +6.69 | +7.0 |
| 85% | 57.82% | +5.55 | +6.0 |
| 80% | 56.37% | +4.59 | +5.0 |
| 75% | 55.09% | +3.71 | +4.0 |
| 70% | 53.96% | +2.91 | +4.0 |
| 65% | 52.91% | +2.15 | +3.0 |
| 60% | 51.91% | +1.42 | +2.0 |
| 55% | 50.95% | +0.70 | +2.0 |
| 50% | 50.00% | +0.00 | +0.0 |
Matches
The table gives the same data as the table above, but using probabilites to win entire matches.
| Match Probability |
Rally Probability |
Average Difference |
Median Difference |
|---|---|---|---|
| 98% | 60.35% | +15.34 | +16.0 |
| 96% | 58.83% | +13.71 | +14.0 |
| 94% | 57.85% | +12.56 | +13.0 |
| 92% | 57.10% | +11.62 | +12.0 |
| 90% | 56.47% | +10.79 | +12.0 |
| 88% | 55.93% | +10.05 | +11.0 |
| 86% | 55.45% | +9.35 | +10.0 |
| 84% | 55.01% | +8.69 | +10.0 |
| 82% | 54.62% | +8.09 | +9.0 |
| 80% | 54.25% | +7.51 | +9.0 |
| 78% | 53.90% | +6.94 | +8.0 |
| 76% | 53.56% | +6.39 | +8.0 |
| 74% | 53.25% | +5.86 | +7.0 |
| 72% | 52.94% | +5.33 | +7.0 |
| 70% | 52.65% | +4.82 | +7.0 |
| 68% | 52.36% | +4.32 | +6.0 |
| 66% | 52.08% | +3.81 | +6.0 |
| 64% | 51.80% | +3.32 | +5.0 |
| 62% | 51.53% | +2.84 | +4.0 |
| 60% | 51.27% | +2.36 | +4.0 |
| 58% | 51.01% | +1.88 | +3.0 |
| 56% | 50.76% | +1.41 | +3.0 |
| 54% | 50.50% | +0.93 | +2.0 |
| 52% | 50.25% | +0.47 | +1.0 |
| 50% | 50.00% | +0.00 | +0.0 |
Distribution
The following plots show the distribution of the point difference for rally probabilities \(p\) of 50%, 55% and 60%, first for single games, then for whole matches. The vertical dashed line indicates the position of the median.
Games
The distributions look like Gaussian or Poisson distributions with a part ripped out and stacked on top of the bins bordering the ripped out part.
The large bin for point differences of -2 and +2 might surprise. For a rally percentage of 50%, about 25% of games are decided with a point difference of exactly two points. The bin with a point difference of plus or minus one is too small to be visible in the plots.
Matches
First we can observe the gap for point differences around zero. For two-game matches differences of zero or one point are impossible, differences of two or three points would require at least one game to end 30-29, which makes these differences very unlikely. For three game matches these point differences are pussible though.
For a rally probability of 50% we observe a distribution symmetric around zero. With increasing rally probabilities the curve moves to higher differences, becoming higher at the peak and narrower. Not that the ranges of the y-axes changes between the plots.
In the plot for a probability of 50%, we can also see that differences of four are more likely than outcomes differing by five points. This can be explained by the large number of games ending with a point difference of two, as explained above.
The second plot can explain, why the median tends to have a larger absolute value than the average. From the table above, we see that a rally probability of 55% corresponds to a match probability of 84%, an average point difference of +8.9 and a median point difference of +10. So there is a difference of about 1.3 points between the average and the median. Looking at the plot we can see the asymmetry in the distribution. Many bins to the left of the median are for more distant from the median than the bins on the tight side. Thus tehe average will be drawn to the left, i.e. to smaller values.
Odds
Finally as a small cheat sheet we will give the median point differences and the implied match probability for different combinations of odds often observed. First Player Odds and Second Player Odds denote the odds for the two players or pairs in the match. The Match Probability shows the implied winning probability for the first player or pair. The Median Point Difference is given in the last column.
| Match Probability | First Player Odds |
Second Player Odds |
Median Point Difference |
|---|---|---|---|
| 98% | 1.01 | 11.00 | +16.0 |
| 90% | 1.08 | 7.00 | +12.0 |
| 80% | 1.20 | 4.33 | +9.0 |
| 70% | 1.36 | 3.00 | +7.0 |
| 60% | 1.53 | 2.37 | +4.0 |
| 55% | 1.72 | 2.00 | +2.0 |
| 50% | 1.85 | 1.85 | +0.0 |
Conclusion
This theoretical foundation is basic knowledge to every bookmaker and professional gambler. However note that all this relies on the assumption of independent and identically distributed outcomes of the individual rallies. Deviations from this assumption could lead to opportunities for sports bettors.
It can also have practical applications for active players. The tables for the average and median point differences can serve as a start point when a better player plays a weaker player. Suppose the better player wins 90% of the games, a handicap of 7 points, i.e. letting the weaker player start with a score of 7, would give each player an approximately equal chance to win the game.