Returns to inequality in sports

Now that the last of my posts on returns to income inequality is up seems like the time for a quick reflection on the concept overall and how well it explained the success of teams.

The returns to inequality

The NBA is where the inequality of a team appears to make a difference in the expected success. This fits with the narrative that teams need to have a star (or several) rather than a surplus of role players. In all of the other leagues analyzed it does not make a significant difference. The NFL and MLB show a negative coefficient. Inequality harms a team in those two leagues. The NHL – most similar to the NBA in salary structure and individual player leverage – is the only other league to show a positive correlation between inequality and team performance.

This whole analysis is necessarily limited. The cumulative build-up of a team’s salaries can only tell us so much (R-squared values MLB=0.13, NBA=0.32, NFL=0.07, NHL=0.13) about the way they perform on the field/ice/court. It is a prediction, sometimes made years before, and made either under duress as part of a bidding process for a free agent or dictated by the terms of the collective bargaining agreement to a draft pick.

Still, it is interesting that one of the coefficients was significant while two others were close (p-value 0.2) after controlling for overall team spending. Even if it just confirmed what people already “knew” it was interesting enough for me.

Looking at a metric more-strictly focused on performance like WAR for baseball or Win Shares for basketball is problematic because end-of-season numbers incorporate the ups and downs of actual performance, so the team’s sum total matches to the performance. For 2012 (or 2012-13 for basketball) the WAR correlation with run differential is 0.89 while the Win Shares correlation with point differential is 0.997. These metrics are very good at allocating out the runs (points) to match their actual totals after the fact.

Unfortunately for us, the effects of a transcendent star making others better – or of a well-balanced team attacking weak links in opposing defenses – are already baked into these backward-looking metrics. To be useful we would need to look at the pre-season expected totals. Perhaps in a future post.

Returns to inequality in the NHL

Take a look over here if you want to get the background for this series, otherwise read on. 

Sports + Numbers Prediction: "I am guessing that returns to inequality are strong here too, with relatively high leverage of the individual players resembling the NBA more than the NFL or MLB." 

The data 

To see the impact of inequality we will look at each team’s Gini coefficient against their winning percentage, controlling for team spending. The resulting equation gives us an r-squared value of 0.13 with only salary spending being significant (P-value of 0.00008) while the Gini coefficient comes in at a P-value of 0.21.

Payroll vs Points % (total points / potential points) - NHL 2009-10 to 2012-13

For every million dollars in team spending the expected increase in winning percentage is 0.00397. For a team that spends $10 million more than a comparable team – all else equal – we would expect them to win 3 additional games (or win 2 more with two additional overtime losses (or win 1 more with four additional overtime losses (or win the same number but have six additional overtime losses))).

Gini vs Points % (total points / potential points) - NHL 2009-10 to 2012-13

On inequality the - insignificant - coefficient is 0.19. Within the range of Gini coefficients in baseball (0.22 to 0.47) this would mean a difference of 8 points (4 wins but I’ll spare the rest) from the most equal to the least equal (more wins to the least equal). Not nothing but not exactly a huge impact. The gap in payroll ($30 million to $71 million) projects to a gap of nearly 27 points.

Payroll vs Gini (color-coded by points %) - NHL 2009-10 to 2012-13

Returns to inequality in MLB

Take a look over here if you want to get the background for this series, otherwise read on.

Sports + Numbers Prediction: "This is anyone’s guess. The returns to inequality – after controlling for the wide distribution in overall team salary – might be strong or they might not. I don’t have a good feel for it so this will be more of a fact finding mission."

The data

To see the impact of inequality we will look at each team’s Gini coefficient against their winning percentage, controlling for team spending. The resulting equation gives us an r-squared value of 0.13 with only salary spending being significant (P-value of 0.00001) while the Gini coefficient comes in at a P-value of 0.18.

Payroll vs Winning % - MLB 2008-2012

For every million dollars in team spending the expected increase in winning percentage is 0.000654. For a team that spends $10 million more than a comparable team – all else equal – we would expect them to win an additional game.

Gini vs Winning % - MLB 2008-2012

On inequality the - insignificant - coefficient is -0.14. Within the range of Gini coefficients in baseball (0.35 to 0.66) this would mean a difference of 7 wins from the most equal to the least equal (more wins to the most equal). Not nothing but not exactly a huge impact. The gap in payroll ($19 million to $206 million) projects to a gap of nearly 20 wins.

Payroll vs Gini (color-coded by winning %) - MLB 2008-2012

Returns to inequality in the NFL

Take a look over here if you want to get the background for this series, otherwise read on.

Sports + Numbers Prediction: "I would think the returns to inequality are high here, but not exactly as a proxy for having stars. The NFL’s cap structure essentially forces teams to play rookies and younger, pre-contract extension players heavily and supplement them with selected veterans. The catch is that nearly all teams have a big young player population so the difference between one that works out and one that doesn’t might not be visible in the salary distribution."

The data

To see the impact of inequality we will look at each team’s Gini coefficient against their winning percentage, controlling for team spending. The resulting equation gives us an r-squared value of 0.07 with only salary spending being significant (P-value of 0.0007) while the Gini coefficient comes in at a P-value of 0.57.

Payroll and Winning %

For every million dollars in team spending the expected increase in winning percentage is 0.00417. For a team that spends $15 million more than a comparable team – all else equal – we would expect them to win an additional game.

Gini and Winning %

On inequality the coefficient is -0.17. Even though the P-value tells us this that Gini is insignificant, the fact that the variance from perfect equality to perfect inequality is slightly less than three wins should tell us all we need to know. 

Gini and Payroll, color coded by winning percentage

Returns to inequality in the NBA

Take a look over here if you want to get the background for this series, otherwise read on.

Sports + Numbers Prediction: "My guess is the NBA will have the biggest returns to inequality as a proxy for teams having stars. With those stars they are not able to afford middling salaries for role players and drop quickly down to minimum salary or exception-level players."

The data

To see the impact of inequality we will look at each team’s Gini coefficient against their winning percentage, controlling for team spending. The resulting equation gives us an r-squared value of 0.32 with both terms being significant (P-values of 0.000008 for spending and 0.00004 for Gini coefficient).

Payroll and Winning %

For every million dollars in team spending the expected increase in winning percentage is 0.005. For a team that spends $20 million more than a comparable team – all else equal – we would expect them to win an additional eight games.

Gini and Winning %

On inequality the coefficient is 0.67. In theory this would tell us that a team with a single superstar making all of the money (stay with me, I understand this is not possible under the NBA CBA) would have 55 more wins than a team with all players making exactly the same salary. In practice we have variation within a much smaller band ranging from 0.09 to 0.56 (see table below for details) so the projected difference if those two teams had the same salary would be 25 games. 

Gini and Payroll, color coded by winning percentage