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).
Sims and Harris also took more shots than any other duo in the Big Ten and Michigan is also the only team to lose both of their top two shot takers in the Big Ten. Here’s a breakdown of how many shots each Big Ten team’s “top two duos” took:
| Team | Shot% |
| Michigan | 60.2 |
| Wisconsin | 58.1 |
| Ohio St. | 57.1 |
| Purdue | 57.0 |
| Indiana | 55.2 |
| Illinois | 51.4 |
| Michigan St. | 50.6 |
| Northwestern | 50.5 |
| Penn St. | 50.4 |
| Minnesota | 49.8 |
| Iowa | 46.4 |
Not surprisingly, the most common thought that fans have about Michigan’s season next year is doom. When Michigan won, Sims and Harris typically had monster games. When Michigan lost, they disappointed. In this era of superstars, they are all we remember. Not to mention the other players on the team either regressed from their freshman year or had inconsistent freshman seasons.
Sims and Harris took an average of 28 shots per game last season. Obviously, one or even two people won’t step up to take all of these shots. Instead they will be more optimally distributed among the team.
All returning players that are expected to be key cogs in next year’s offense took less than 16% of Michigan’s shots when they were on the floor, well below even an equal distribution of 20%. The fact that they didn’t get take their shots in the last two years is a bit troubling but there’s no doubt that the dynamic of next year’s team will be different.
Ideally next year’s team will look something like John Beilein’s West Virginia teams. His 2005 team had six players take over 19% of the team’s shots when they were on the floor. Similarly his 2007 team had 4 freshmen and sophomores in a group of 6 players that took over 18% of the team’s shots when they were on the floor. Beilein’s offenses over his last three years at West Virginia ranked 18th, 12th, and 13th in Pomeroy’s rankings and were in many ways the epitome of balanced offense.
There are also some flaws with this analysis… None of Novak, Douglass, Morris, or Lucas-Perry was able to hit shots at a particularly efficient rate last season. They all had an eFG% between 43 and 48 percent, making the likelihood that they will become more effective with more shots unlikely. The greatest hope here is that last year was some kind of sophomore slump but we won’t find that out until next year.
Expecting an elite offense next year would be ludicrous however the Price of Anarchy gives us at least some reason for hope. It’s unlikely that Michigan will have a player that takes over 25% of their shots next year. The question is what happens to the multitude of players who will add a couple more shots per game.
This is an over simplistic model because there are also other factors effecting a game beyond shooting. Sims and Harris accounted for around half of Michigan’s rebounding and rebounding might be Michigan’s biggest concern next year. It also doesn’t consider other abilities like Harris’ 27.7% assist rate.
There are a lot of shots available next year and someone is going to have to take them. Under John Beilein, Michigan has maximized their possessions by limiting turnovers which means that the shot attempts will be there. Novak, Douglass, Lucas-Perry, Morris, Vogrich, McLimans, Morgan, Hardaway, Smotrycz, Horford, and any other freshmen that enter the picture will get their looks. The scary part is that we haven’t seen half of next year’s play college basketball and not all balanced teams are good teams, you still have to make shots.
