Wednesday, January 16, 2013

How Much Is a Win Worth to an NBA Team?

Last month, I used J.C. Bradbury's free agent valuation method to determine how many wins the Red Sox expected Mike Napoli and Shane Victorino to contribute to the team in 2013. That worked fine, but suppose we want to build a similar model for the NBA. Again, we'll use the basic system Bradbury outlines in "The Baseball Economist" (ch. 13). Here, Bradbury found a relationship between revenue, wins, and the size of the city a franchise plays in.

All three of those variables are readily available. For city size, we'll use the population of the metropolitan statistical area (MSA) each team plays its home games in, as reported in the 2010 U.S. Census*. Revenue is available through Forbes' Business of Basketball listings. This data is almost exactly one year old -- suggesting that it covers the 2010-2011 season, and not the recent lockout-shortened 2011-2012 season. This is better for our purposes; I don't want the compressed schedules and reduced number of games to interfere with my results.

* - And the Canadian equivalent for Toronto, with the hope that the two have very similar methodologies.

The last element -- wins -- are also easy to find. However, we're not interested in wins per se but rather in what each member of a team contributes to those wins. Bradbury uses run differential for baseball, and we could use point differential for the NBA, but the dynamics of a basketball game make it difficult to determine exactly how many points an individual player contributes on offense (think of Ray Allen, taking more open shots this year thanks to opponents doubling James and Dwyane Wade) and prevents on defense (think of Serge Ibaka, intimidating opponents into changing their shot even when he can't actually block it).

Instead, we'll use win shares, as originally defined in Dean Oliver's Basketball on Paper and calculated by Basketball-Reference. Win shares are calculated such that the total win shares earned by a team should approximate the number of wins a team has. In other words, when the contributions of all the players on the 1995-96 Bulls team that won 72 games are included, you should end up with a total of 72 win shares. By this measure, Michael Jordan was worth just over 20 wins* to this team, and Scottie Pippen was worth over 12.

The record for win shares in a season belongs to Kareem Abdul-Jabbar's 1971-72 campaign, when he earned 25.4.

Naturally, this statistic correlates very well with wins: the sum of each team member's win shares (which I'll refer to as "team win shares") has a correlation coefficient of 0.97 with wins, where a correlation coefficient of 1 would represent a perfectly straight line.

Now that we have a substitute for wins, we can relate revenue to team win shares and city size. When we do this, we find that this relation is not as strong as it is in baseball. We can find a weak linear relationship:

Revenue = 1.287 * Team Win Shares + 3.45 * MSA + 59.77

Each of these cofficients is statistically significant (p < 0.05), but the adjusted R^2 value is only 0.38. This means that less than 40% of the variance in revenue is explained by win shares and city size. By contrast, the adjusted R^2 value of the model we built for MLB revenue is around 0.67 -- two-thirds of the variance in MLB revenue is explained by market size and winning alone. Of course, given the aggressive revenue-sharing model and salary cap employed by the NBA, it's not surprising that these two factors don't play as large a role in the NBA as they do in MLB.

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