Introducing pairwise Pythagorean expectations

Jay Heumann, “An improvement to the baseball statistic ‘Pythagorean Wins'”, Journal of Sports Analytics, 2(1): 49-59, 2016. [PDF]

Abstract: This paper introduces a new version to the commonly used “Pythagorean wins” statistic, which improves on the traditional method by means of mathematical adjustment.  The new statistic is called the “pairwise” Pythagorean wins, and it is demonstrated, over a 30-year data set, to have a smaller overall root mean square error than the original Pythagorean formula.

Just when I thought I was out…they pull me back in!

Michael Corleone, The Godfather (Part III)

The Pythagorean expectation and its extensions into soccer and other sports have been a part of my life since late 2009, and sometimes it seems in some precincts of the soccer analytics community that it’s all I’m known for.  Just when I thought that I had pursued that line of research for as long as I could, along comes a paper that promises to open up new veins of work.  That paper, which I had the pleasure of reviewing for the Journal of Sports Analytics, is what I will write about in this post.

The author of the article is Dr. Jay Heumann, who is, as best as I could determine, a professional mathematician living in the Washington, D.C., area.  Dr. Heumann received his undergraduate, master’s, and doctorate degrees in mathematics from Columbia, Wisconsin, and University of British Columbia, respectively, with an emphasis on number theory.  That’s a very solid academic pedigree in mathematics.

The central point of the article is that Heumann takes the “expectation” part of Pythagorean expectation seriously, identifies why it is not an expectation in the mathematical sense, and creates a remedy to make it one.

Let’s make an aside and define what a statistical expectation is.  An expectation — or expected value — is the sum of the possible values of an event multiplied by their respective probabilities.  For a discrete random variable \(X\), the expectation is

E[X] = \sum_{i=0}^{\infty} x_i p_i

There are a number of properties associated with expected values, but the most relevant ones for this paper are the scalar and linearity properties.  The scalar property means that the expected value of a scalar is itself:

E[C] = C

and the linearity property means that the expectation of the sum of two or more random variables is the sum of their expectations taken separately:

E[X + Y] = E[X] + E[Y]

Now, the Pythagorean is the expected number of wins (or points), so we can express it as \(E[X_i]\) for team \(i\).  If we consider wins and losses as the only possible outcomes, the sum of the expected wins for all of the teams in the league is the total number of games played in the league — a constant.  In mathematical language,

E[X_1] + E[X_2] + \ldots + E[X_N] = E[X_1 + X_2 + \ldots + X_N] = E[C] = C

It is this expression that Heumann shows to be false when applied to all of the teams in the league.  In fact, we have no way of ensuring that the total number of Pythagorean wins or points adds up to the total number of games played or points available.  I’ve seen this in my own work on Pythagoreans — the sum of the residual between expected and actual points, over all teams, rarely adds up to zero — but chalked it up to the artifacts of approximation.

Heumann goes on to propose a remedy: decompose the Pythagorean into a sum of head-to-head matchups between all of the teams in the league.  Such a formulation satisfies the scalar and linear properties of a statistical expectation as the total number of Pythagorean wins equals the number of games played between the two teams.  This is the pairwise Pythagorean.

Heumann demonstrates the pairwise Pythagorean on thirty years of Major League Baseball data (1960-1990).  In a comparison of the classic and pairwise Pythagorean with default exponents of 2.0 and 1.83 (the observed “optimal” Pythagorean exponent), the pairwise Pythagorean showed a 3-6% improvement in RMSE, with the difference more apparent in the divisional era of MLB (post-1968).

The pairwise Pythagorean is a fine addition to our knowledge about expected performance in league competitions, but its implementation requires more data than previously.  The original expectation only required the number of games played by each team and the total number of runs (or goals) scored and allowed.  With the pairwise Pythagorean, all of the match results in a competition are necessary.  Maybe some will feel that the relative simplicity of implementing the classic Pythagorean is worth the trade-off of slightly reduced accuracy.  But we also use the Pythagorean to evaluate over- and under-performance of teams, and relative performances that add to zero over all teams is more coherent.

When I think about my own work on applying the Pythagorean for soccer, several questions come to mind:

  • How can the pairwise Pythagorean be applied to competitions with three possible results?
  • In modern soccer competitions, points are not shared in case of a draw.  How can the expectation be altered to account for that?
  • Does the optimal Pythagorean exponent change in this formulation?  And in what direction?
  • How does the formulation change under pairwise conditions?  The classic formulation had to be re-derived from the basic assumptions to make applicable to soccer.  Does that have to be repeated?

I hope these issues that I’m raising show that Dr. Heumann has opened up some interesting paths for further work through his paper.  Go read it; I think you will like it and, hopefully, some new ideas will come to your mind as well.


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