## Is ‘luck’ persistent? May 25, 2011

Posted by tomflesher in Baseball, Economics.
Tags: , , ,

I’ve been listening to Scott Patterson’s The Quants in my spare time recently. One of the recurring jokes is Wall Street traders’ use of the word ‘Alpha’ (which usually represents abnormal returns in finance) to refer to a general quality of being skillful or having talent. That led me to think about an old concept I haven’t played with in a while – wins above expectation.

As a quick review, wins above expectation relate a team’s actual wins to its Pythagorean expectation. If the team wins more than expected, it has a positive WAE number, and if it loses more than expected, it has wins below expectation, or, equivalently, a negative WAE. It’s tempting to think of WAE as representing a sort of ‘alpha’ in the traders’ sense – since the Pythagorean Expectation involves groups of runs scored and runs allowed, it generates a probability that a team with a history represented by its runs scored/runs allowed stats will win a given game. If a team has a lot more wins than expected, it seems like that represents efficiency – scoring runs at crucial times, not wasting them on blowing out opponents – or especially skillful management. Alternatively, it could just be luck. Is there any way to test which it is?

It’s difficult. However, let’s break down what the efficiency factor would imply. In general, it would represent some combination of individual player skill (such as the alleged clutch hitting ability) and team chemistry, whether that boils down to on- or off-field factors. Assuming rosters don’t change much over the course of the year, then, efficiency also shouldn’t change much over the course of the year. Similarly, if a manager’s skill was the primary determinant of wins above expectation, then for teams that don’t change managers midyear, we wouldn’t expect much of a change throughout the course of the season. Most managers work up through the minors, so there probably isn’t a major on-the-job training effect to consider.

On the other hand, if wins above expectation are just luck, then we wouldn’t need to place any restrictions on them. Maybe they’d change. Maybe they wouldn’t. Who knows?

In order to test that idea, I pulled some data for the American League off Baseball Reference from last season. I split the season into pre- and post-All-Star Break sets and calculated the Pythagorean expectation (using the 1.81 exponent  referred to in Wikipedia) for each team. I found WAE for each team in each session, then found each team’s ‘Alpha’ for that session by dividing WAE by the number of games played. Basically, I assumed that WAE represented extra win probability in some fashion and assumed it existed in every game at about the same level. The results:

$\begin{tabular}{ | l | c | c | c| r | } \hline Team & WAE1 & Alpha1 & WAE2 & Alpha2 \\ \hline NYY & 0.823 & 0.009 & -2.474 & -0.033 \\ \hline TBR & -0.5 & -0.003 & 0.207 & 0.003 \\ \hline BOS & 0.494 & 0.006 & 0.900 & 0.012 \\ \hline TEX & -1.041 & -0.012 & 0.291 & 0.004 \\ \hline CHW & 2.379 & 0.027 & -0.244 & -0.003 \\ \hline DET & 3.918 & 0.046 & -4.706 & -0.062 \\ \hline MIN & -1.67 & -0.019 &.3.693 & 0.05 \\ \hline LAA & 3.83 & 0.042 & -2.860 & -0.040 \\ \hline TOR & -0.202 & -0.002 & 1.555 & 0.021 \\ \hline OAK & -1.939 & -0.022 & -2.418 & -0.033 \\ \hline KCR & 0.023 & 0.000 & 1.976 & 0.027 \\ \hline SEA & 0.225 & 0.003 & 2.188 & 0.03 \\ \hline CLE & -2.096 & -0.023 & 0.907 & 0.012 \\ \hline BAL & -1.028 & -0.012 & 8.900 & 0.120 \\ \hline \end{tabular}$

As is evident from the table, a whopping 10 out of the 14 teams see a change in the sign of Alpha from before the All-Star Game to after the All-Star Game. The correlation coefficient of Alpha from pre- to post-All-Star is -.549, which is a pretty noisy correlation. (Note also that this very closely describes regression to the mean.) It’s not 0, but it’s also negative, implying one of two things: Either teams become less efficient and/or more badly managed, on average, after the break, or Alpha represents very little more than a realization of a random process, which might just as well be described as luck.