## Braess’s Paradox in Basketball

Source: The price of anarchy in basketball, Brian Skinner, https://arxiv.org/pdf/0908.1801.pdf

I really enjoyed learning about Network Traffic and, in particular, found Braess’s Paradox incredibly interesting. I looked into it a bit more and found that it not only applied to vehicular traffic, but it also applies to electric grids, food webs, and sports. Specifically, Braess’s Paradox can be found in basketball. To illustrate, consider a network that travels from the beginning of a possession to a possession ending in a basket. Since there are five players on the court for a team at a given time, there are five possible routes to score a basket during a possession. This could be modeled as shown below.

From there, it’s possible to think about the efficiency of a route as a player’s field goal percentage. Thus, the team’s field goal percentage represents the efficiency of the team or network. When a team has a star player, one who is significantly better than their teammates, two interesting developments arise.

First, since the player is a star, they are better and thus more efficient, so thus their route is more efficient than their teammates. However, we can imagine that if a player is asked to shoot an exorbitant amount, they will become less effective. Thus, we would want a variable, say x, to keep track of how many shots the star shoots, and the efficiency of the player would be some initial field goal percentage – (x / some fatigue factor). This relationship between volume and efficiency is similar to what we investigated in class when finding Nash Equilibria. Interestingly, the Nash Equilibria of a basketball team suggests that a star player should shoot the same number of shots as their teammates, which feels counter-intuitive since they are (by definition) better and more efficient than their teammates.

Another interesting relationship between Network Traffic and basketball appears in the form of Braess’s Paradox. In lecture, we discussed Braess’s Paradox in the context of adding a road to a pre-existing travel network. We found that adding a road with 0 travel time increased the total travel time. In basketball, this is applied in reverse to demonstrate the effect that a star player has on the team’s efficiency. It was found that by removing a star player from a team, the team actually becomes more efficient, and thus better. This is analogous to a shortcut in a traffic network that becomes overused, and is especially interesting in this era of basketball where teams are trying to collect as many star players as possible.