The Price of Anarchy is a theory that has been circulating the nerdy basketball circles that yours truly dabbles in. In short it’s a way to explain how a basketball offense can become more efficient when its best (and most used) players take fewer shots.
This theory might sound familiar if you have ever heard of Bill Simmons’ Ewing Theory. Simmons’ theory attempts to explain how a team can lose its best player and somehow improve. Simmons lays out the theory in his typical satirical and entertaining ways and cites remarkable occurrences of teams improving when superstars get hurt, it also focuses a lot more on psychological issues. Brian Skinner’s recent paper and presentation at the Sloan Sports Analytics Conference begins to explain how something like this could be mathematically possible.
You can read the full article for the technical jargon and mathematical explanations or Skinner’s blog post on the topic. I recommend reading at least one or the other because Skinner explains the concept in great detail (really, read it). The main idea is that basketball is a network problem. Every route to the basket (simplified as every player or shooting option) is a different way to a similar goal: scoring a basket. Some players are more efficient than others but all options are capable of scoring. The catch is that the more an option is used, the less efficient it becomes.
The standard problem used to explain the price of anarchy is rush hour traffic. In the rush hour problem, it becomes clear that the whole community can experience faster commute times if some people choose to take slower routes. The social welfare maximizing outcome prevents other roads from becoming backed up and the average commute for the entire community decreases despite some people experiencing longer commutes. The most extreme example of this is when big cities experience more efficient traffic flow after closing the most traveled roads. (This whole example is explained much better in the original paper and also at Gravity and Levity.)
So on the basketball court the idea is that an offense is most efficient when there is an equal chance that every player on the floor will shoot the ball. Here’s Skinner:
On the basketball court, possessions are like cars. Each one starts at point A (the in-bounds) and attempts to travel to point B (the basket). Different plays are like different roads: each one has a different efficiency that will generally decrease the more it is used. In principle, all of the methodologies and “paradoxes” associated with traffic patterns should be applicable to basketball as well.
This obviously keeps the defense more honest (they can’t just focus on a specific outcome) but it logically holds up without even considering the defense. Skinner’s article explains “skill curves” that attempt to project that the optimum number of shots a star player should take. It’s nearly impossible to accurately graph every player’s skill curve but there is no doubt truth behind this concept.
We see examples of this phenomenon all the time in sports. Some of them could be chalked up as sample size errors but they happen. Just over the last few months in college basketball we saw Michigan State play surprisingly efficient basketball for a stretch without Kalin Lucas. We also saw Notre Dame turn their season around when their best player, Luke Harongody, got hurt. Harongody just happened to take 37% of Notre Dame’s shots when he was on the floor, the highest percentage in the country.
You probably realize where this is going in relation to next year’s Michigan team. Michigan lost two players who accounted for around 60.2% of the team’s shots. A quick glance over KenPom numbers yielded only 3 high major teams that had a pair of players combine for shot% over 60%: Stanford (Fields, Green), Georgia (Thompkins, Leslie), and Notre Dame (Harangody, Abromaitis).
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