Pythagoras theorem is easily one of the most popularly known mathematical and geometric equations that we have also read about during a growing up years. It’s related to the length of three different sides of any right-angled triangle.
Understanding Pythagorean expectation
Almost 2000 years after it was originally introduced, Pythagoras theorem or Pythagoras equation was reworked by Bill James, the renowned baseball analyst, as a foundation for his Pythagorean expectation, through which he tried explaining a team’s true chances of winning based on the runs of points scored by it, instead of solely based on its actual winning percentage. Understanding this expectation can give you a fair amount of edge in your sports betting endeavours, regardless of whether you bet offline or in some online casino.
Its formula is:
Win percentage = (runs or points scored ^ X) / (runs or points scored ^ X + runs or points allowed ^ X)
Looking at it closely we’ll observe that scoring events are actually comparatively more in number than the wins. It appropriately explains the actual ability of a side, as scores/goals may not arrive always at the most beneficial point of time in a game.
It’s possible for a team to score when it’s already in lead in a match, or it may concede a goal when it’s leading by a narrow margin. This distribution of the timing and placement of goals conceded or scored may often unsustainably depress or inflate the team’s winning record, and thus its league position, more so if we are looking at comparatively smaller samples.
Simply put, a team which outperforms its Pythagorean expectation can be considered as being lucky, and one which underperforms unlucky. However, neither of these conditions is guaranteed to continue for very long.
The concept of Pythagorean expectation has rapidly spread from the sport of baseball, and has gone on to include American football, soccer and basketball. You’ll see its active implementation in several online betting environments such as Bet365, William Hill etc.
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Usage of Pythagorean expectation in the field of soccer betting
Unlike as in case of the sports events where ties or draws rarely occur, football or soccer throw up all sorts of unique challenges when it comes to application of the Pythagorean approach.
Looking at it closely, you’ll observe that draws or ties are actually the biggest challenge when you use Pythagorean expectation in the field of football betting. It is closely followed by the importance of the goal scoring environment wherein the matches are played. Considering the huge variety of leagues in the field of soccer or football, you are always going to witness high-scoring matches in some leagues compared to the others.
In addition, football teams may sometimes see themselves playing with less than 11 players owing to red cards, which obviously impacts the scoring events and hence odds at portals like Bet365. Many such issues have been successfully addressed by a paper written by Howard Hamilton.
The initial exponent used by James was 2, very much in line with his equation’s origin, but changing the exponent X’s value, helped in reducing the root mean square error between the forecasted number of wins by James and the actual number of wins. 1.83 is the value which is normally used in the field of baseball, rather than 2.
A similar approach is adopted in football, with the discrepancy between the actual outcomes and Pythagorean expectation approaching a minimum at the exponent of 1.35, instead of 2.
The win percentage is fairly straightforward in events wherein there are lesser chances of drawn games. However, it’d include a significant number of draws when it comes to football matches. Hence, win percentage is usually dependent on the percentage of possible points that may possibly be won, in order to account for a side’s ability to pick up points in drawn games despite not scoring in those matches.
If we look at a season comprising of 38 games, having 3 points per win, there are around 114 possible points up for grabs. So, a side with a scoring record which implies an actual winning percentage of 50%, may expect to end the season with a total of 57 league points, under this adaptation.
Some more refinements may include changing the exponent for the numerator and denominator components in this equation, and incorporating a term which consists of the total number of goals scored for and against in the exponent to permit variations in the goalscoring environment.
Football teams that score as well as allow fewer number of goals are more likely to have more number of drawn games, compared to the ones that score and concede more number of goals. You can ascertain all such facts by looking at the statistics of any offline or online casino or bookmaker.
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The extent by which the RMS error between reality and expectation of the actual number of points, is reduced by the selection of the exponent, is revealed by the corresponding decrease in the RMSE, with the models becoming more refined.
Therefore, exponent 2 results in RMSE of around 10 points for every team, in case of the 2014-15 edition of the English Premier League. It goes down to 6 points for every team, if 1.35 was instead used, and further down to 4.4 points, if there was an inclusion of goal environment in the exponent.
Gauging the total number of points over a season
Pythagorean expectation or Pythagoras theorem find its most widespread usage in the field of soccer, in figuring out if the number of points scored by a team in a particular season is actually justified by its scoring records. Hence, it may normally serve as an excellent performance indicator in any subsequent campaign.
The goal difference of a team serves the same purpose when it comes to separating the possibly unsustainable luck element from something like a repeatable skill.
The 65 points scored by Newcastle in the 2011 – 12 season was almost 10 points in excess of what was figured out with the help of a regular Pythagorean expectation, for a team that had scored 56 goals in total and had conceded 51.
It was highly unlikely for any side that had scored 5 goals more than it had conceded, to repeat the number of single goal victories scored by Newcastle, accompanied by its few very heavy defeats. Hence, it wasn’t too surprising when they didn’t get many points in the 2012 – 13 season.
We carried out a study of various English Premier League campaigns wherein top rated teams fell short or exceeded their Pythagorean expectation. It was found that the majority of these teams showed equal splits, which is expected if luck was considered a major factor in the under or over performance of a team.
Manchester United were found to be one of the noteworthy exceptions, as the total number of points scored by them at the end of the league exceeded their conceding and scoring record in precisely 17 of the 23 English Premier League seasons it was a part of.
A constant factor in this Manchester United record was obviously the presence of Sir Alex Ferguson. An additional research paper also supports the case for Manchester United’s amazing ability of scoring winning goals, especially in the last few minutes of football matches.
Hence, there is some amount of tentative evidence available which suggest that Manchester United, under Alex Ferguson, owed its apparent over achievement partly to the presence of its highly effective manager. One of major reasons why MU was always the starting favourites at many sports betting portals like Bet365 etc.
If we look at the Liverpool’s performance on the other hand, we’d notice that in the team has overall been an underachiever in 18 of its 23 English Premier league seasons, but the evidence of a recurring trend seems less compelling.
8 of the 18 years they had underperformed, were with a difference of 3 points or less. Their average underperformance was found to be 1.7 points per season, which is almost half of the average over performance of Manchester United.
Although it may be enticing to narratively validate a side’s ability of consistently performing below or over its Pythagorean expectation, in several cases, a team’s subsequent performance in the league is more or less in line with its previous Pythagorean expectation, compared to its previous actual point total.
Profitable trends in Pythagorean expectation
If we look over the entire Premier League era, the actual number of points scored in a particular season have a slightly better correlation to the Pythagorean expectation derived from the previous year, rather than the actual number of points scored in the previous season.
Refer to the stats of any online casino (with regard to EPL), and you’ll figure that the most notable EPL over performance came in the 1992 – 93 season when Norwich managed to finish third on the table, scoring 72 points, in that Premier League (comprising of 42 games at that time). This was despite having scored 61 goals in total, and conceding 65. It had won 16 of its games by a margin of one goal, but its Pythagorean expectation was merely 55 points. The team dropped to the twelfth position on the table, with 53 points, in the 1993 – 94 Premier League season.
If you look closely, you’ll observe that out of the 10 teams that exceeded their expectations the most in the history of English Premier League, 8 of them witnessed a major loss in total number of points in the subsequent season. On the other hand, 9 out of 10 most underperforming football teams (compared to their Pythagorean expectation) scored more number of points in the next season.
Liverpool, Tottenham and Chelsea got lucky in the 2014 – 15 English Premier League season, exceeding their Pythagorean expectation by 7, 9 and 9 points respectively! Hence, with more than normal level of luck, these teams were expected to gain fewer number of points in the 2015 – 16 season.
Everton, Southampton and Leicester on the other hand were quite unlucky and were expected to exceed their total number of points from the 2014 – 15 season to a more respectable number in the 2015 – 16 edition.