Pythagorean Wins — AUDL
Applying the Pythagorean Formula for Wins used in other major sports to the AUDL.

“Okay. People who run ball clubs, they think in terms of buying players. Your goal shouldn’t be to buy players, your goal should be to buy wins. And in order to buy wins, you need to buy runs.” -Peter Brand, Moneyball
Intro
The Pythagorean formula for expected wins is a common statistic used across sports. It has nothing to do with geometry but the name was applied due to the similarity with the Pythagorean theorem. I apply this formula to the AUDL and talk about how it can be used as a foundation for other stats and analysis in the future.
Background
The quote above is from the movie Moneyball. Peter Brand is describing to Billy Beane the logic derived from Bill James’ Pythagorean formula for predicting team’s winning percentage in baseball. This formula can be written as follows.

The idea behind the formula is that the number of games a baseball team should win in a season is dependent on the number of runs scored and runs allowed. The “1.83” coefficient has been determined empirically through historical baseball data. And this “1.83” coefficient is key to thinking about how one would buy wins in the Moneyball quote above. In the MLB, adding ~10 runs to a team’s total runs scored for a season corresponds with one additional win when using the Pythagorean formula. This is the basis for how WAR — wins above replacement (player)— is calculated in baseball.
Daryl Morey was the first to adapt the formula for the NBA. Recently, Football Outsiders also starting publishing a stat called Pythagorean wins for the NFL that uses the same formula.

The coefficients in the formula for each sport are different due to the distributions of scores and winning percentages for teams. Consider that doubling an opponent’s score is common in the NFL and MLB. Not so much in the NBA. At the same time, we commonly see teams win 75% or more of their NBA games in a season while the 116 win 2001 Seattle Mariners team only won 71.6% of their games.
Calculating Pythagorean Formula — AUDL
So I wanted to look at AUDL season by season data to calculate the Pythagorean formula for ultimate. Keep in mind that this formula relates specifically to the AUDL and the AUDL rule set.
To calculate the AUDL coefficient in the formula (the exponent in the above formulas), I collected every team’s regular season W-L record, points scored, and points allowed. From this data, we can reconstruct the Pythagorean formula, run a regression, and determine the AUDL coefficient.
We need to first reconstruct the Pythagorean formula as follows.

- PSt = average number of points scored per game for team t
- PAt = average number of points allowed per game for team t
- Pt = PSt / PAt
- c = Pythagorean coefficient
- t = index for an individual season for each team
Then take the natural log of both sides.

We can use this formula to fit a regression and determine the AUDL coefficient “c” for the Pythagorean formula.
The following chart shows the results fitting this data for the AUDL. There are 103 team seasons included in the data set from the 2012–2017 AUDL regular seasons. Note that 9 team seasons were removed because they had either 0 wins or 0 losses. I compiled the data from a combination of ultianalytics.com, Leaguevine, and the AUDL website. The data aren’t 100% accurate due to the multiple sources, but they are good enough to give us a strong estimate of the AUDL coefficient for the formula.


Takeaways
So how should we think about this 5.90 coefficient for the AUDL? Well I think it should be used as a foundation for additional stats. For instance let’s consider wins above replacement (WAR) in the AUDL. This takes us back to the Moneyball quote from the start of the article. Maybe we can change it to the following,
“Okay. People who run AUDL clubs, they think in terms of flying in elite players. Your goal shouldn’t be to fly in elite players, your goal should be to buy wins. And in order to buy wins, you need to buy breaks or buy more offensive holds.”
How many more breaks or holds are needed to achieve one more win in the AUDL? The average AUDL team scores 22.4 points per game (average over all regular seasons 2012–2017). Assuming 14 games in a regular season, we can calculate how many more “points” a team would need to score to achieve one more regular season win. I use “points” in quotes here because I think it’s better to analyze in terms of offensive holds or defensive breaks.
To achieve one more win, an AUDL team needs to find ~15 more points than their opponents for the season. This corresponds with finding ~7.5 more breaks in a season or finding ~7.5 more offensive holds in a season (both contribute+/- 2 to point differential).
Applying to Individual Players — AUDL
So the next logical step here is to consider individual statistics in the AUDL, apply a reasonable formula, and determine a WAR statistic or Win-Shares statistic for players. While I think this would be awesome, I worry that it is really hard to make the leap from goals/assists/takeaways to how many more breaks or holds a player produces. This is much easier in the MLB and even the NBA where plays are more discreet.
One statistic that could help us isolate individual performance in the AUDL is the Real Plus/Minus (RPM) statistic used in basketball. I think there are some challenges to applying this technique to ultimate, but it could be something interesting to investigate.
Regardless, I think understanding the Pythagorean formula for wins in the AUDL is a great starting place for further analysis of the league. As always, let me know what you think.

