Can money buy wins? Team correlation alone can't tell you
In the previous post, I linked to the Wages of Wins study that found a correlation of less than 0.4 between single-season NBA team payroll and wins. The authors have argued that because the correlation is low, we can conclude that money can't buy wins.
That got me thinking ... is that really true? It occurred to me that if the team payrolls vary in a narrow range, that should reduce the correlation, because there's less room for the relationship to make itself evident.
To check that, I ran an experiment. I set up a situation where payroll was 100% correlated with talent. Then, I simulated 30 independent seasons of 82 games, first, where the salary distribution was wide, and, then, where the salary distribution was narrow.
First, the wide distribution. Teams vary between .300 and .700 in a distribution shaped like the roof of a house. (Technically, talent was taken as .300 + (rnd/5) + (rnd/5), where "rnd" is uniform in (0,1).). Here are five correlation coefficients from successive runs. (These are r's; square them to get r-squareds.)
.78 .76 .86 .90 .84
Now here's the narrow distribution. Teams vary from .450 to .550:
.36 .32 .35 .07 .11
Clearly, the variability of payroll makes a huge difference in the correlation.
We can make the correlation come out as low as we want, just by reducing the teams' spending variance. If a salary cap and floor forced every team to spend within, say, 1% of each other, the correlation would probably be very close to zero.
And, therefore, the conclusion that low correlation implies inability to price talent is just not true. Here, payroll buys talent with 100% correlation, and there is no doubt that a team that chooses to spend more will win more games. And that's true whether the payroll/wins correlation is .7, or .3, or .1, or even zero.
It sounds illogical, but it's true: the correlation between team spending and wins, taken alone, is not enough to tell you anything about whether team spending can buy wins.