Analytics expert, Mark Taylor, pinpoints the advantages that the Pythagorean expectation may have if used in soccer betting and its potential profits when used on long-term markets.
Mathematics can be widely used in soccer betting. Among other things, the Pythagoras’ theorem, which relates to the length of the three sides of a right-angled triangle, is a method which may bring huge profits in soccer betting.
How does the Pythagorean expectation work?
Primarily used for baseball betting, as an attempt to explain how likely is a team to win in terms of the points or runs they score and allow, the Pythagoras’ theorem served as a basis for the development of this new Pythagorean expectation. Thus, it can be used to predict more than one team’s actual winning percentage.
The equation is as follows:
Win % = (points or runs scored ^X) / (points or runs scored ^X + points or runs allowed ^X).
This method, as previously mentioned, was used primarily in baseball, and later on, it included basketball and American football. Now, it is used with soccer as well.
It takes into account scoring events, rather than wins, since scorings are more numerous and they best depict the team’s true strength and abilities. Scoring may come when the team is already very ahead as opposed to its opponent, or ‘surrender’ when they lead by a narrow margin. All of these events may boost or decrease their winning record changing their position in the league.
Therefore, we can look at these events as something that is relative and changeable and it can be concluded that when a team performs better than their Pythagorean expectation, it may be considered just ‘lucky’ and vice versa.
How to use Pythagorean expectation in soccer betting?
The biggest challenge with the Pythagorean expectation is the possibility of a draw in a match unlike certain other sports where this is rarely the case. To these challenge, we can also add the goal scoring environment, as well the fact that sometime, the matches are to be played with less than 11 players due to red cards and all of these circumstances may, to high extent, distort the overall performance of a team.
The deviation between Pythagorean expectation and actual outcomes in this sport approaches a minimum at an exponent of 1.35. Since in soccer there will always be a huge amount of draws, the percentage of win often equals the percentage of possible points that a team may win in order to account for a side to be able to pick up a point in a drawn game, although not having scored a point in that match.
Following the adaptation, a team may expect to end a season with 57 league points if they had a true win percentage of 50 %, in a 38-game season with three points for a win. In all the season there is a possibility of 114 points at stake.
Other adaptations include, varying the exponent for the components of the denominator and numerator in the equation, as well as including a term that comprises the number of goals scored and allowed in the exponent to allow for variations in the goal-scoring environment.
Consequently, draws usually happen for teams, both of which score and allow few goals. The opposite is true for teams which score and concede in greater numbers.
The extent to which the choice of exponent reduces the RMS error between the expectation and reality of actual points is shown by the gradual decrease in RMSE.
An exponent of two gives a RMSE of nearly 10 points per team for the 2014/15 Premier League, this is then reduced to six points if 1.35 is used and additionally to just over 4.4 points if goal environment is included in the exponent.
Measuring points over a season
The most significant use of the Pythagorean expectation in soccer is to separate the unsustainable element of luck, i.e. if the scoring records of one team corresponds to the number of points that the team scored in one season. Therefore, these statistics will be used to see how one team will perform in the subsequent period.
Newcastle won 65 points in 2011/12 and this was nearly 10 more than it was expected by a typical Pythagorean expectation for a team that scored 56 goals and conceded 51.
As a team which could only score five more goals than they concede, it was highly unlikely that the high number of goal victories and the few defeats will be repeated. Therefore, as expected, they achieved fewer points in 2012/13.
Below is a chart which shows the equal splits if luck played the major role in the over or under performance of a team:
| Team | Number of overperforming seasons | Number of underperforming seasons | Average points above or below expectation |
| Arsenal | 11 | 12 | 0.0 |
| Aston Villa | 10 | 13 | 0.3 |
| Crystal Palace | 3 | 3 | 0.0 |
| Chelsea | 13 | 10 | 0.4 |
| Everton | 7 | 16 | -0.6 |
| Leicester City | 4 | 5 | -1.8 |
| Liverpool | 5 | 18 | -1.7 |
| Manchester City | 8 | 10 | -1.1 |
| Manchester United | 17 | 6 | 3.1 |
| Newcastle United | 11 | 10 | 0.8 |
| Norwich City | 5 | 2 | 3.3 |
| Southampton | 4 | 12 | -1.4 |
| Stoke City | 5 | 2 | 1.7 |
| Sunderland | 7 | 7 | -0.5 |
| Swansea City | 1 | 3 | -1.7 |
| Tottenham Hotspur | 14 | 9 | 1.2 |
| Watford | 0 | 2 | -2.4 |
| W.B.A | 3 | 6 | -0.6 |
| West Ham United | 12 | 7 | 1.1 |
Due to the constant factor in Manchester United, Sir Alex Ferguson, Manchester United may be noted as an exception to this because their final league points are over their scoring and conceding record in 17 of their 23 Premier League seasons. And statistically, it appears however that it is indeed their ability to score winning goals in the last minutes of a match.
On the contrary, Liverpool was the underachiever in 18 of 23 seasons.
In general, how one team subsequently performs in a league has to do more with keeping with their previous Pythagorean expectation than their previous actual point total.
Valuable Pythagorean expectation trends
In 1992/93 Norwich finished third with 72 points in the then 42-game Premier League, despite scoring 61 goals and conceding 65. They won 16 games by a single goal margin, but their Pythagorean expectation was just 55 points, and in 1993/94 they dropped to 12th, winning 53 points.
Chelsea, Tottenham and Liverpool had some amount of luck in 2014/15, when they overperformed their Pythagorean expectation by 9.9 and 7 points respectively. So, with more usual levels of luck, they could expect to gain fewer points in the upcoming seasons.
On the other hand, ‘unlucky’ Leicester upcoming seasons of the Premier League.