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Basketball as a Network Problem

In basketball, a logical conclusion is to take the highest-percentage shot each time down the court in order to play an efficient game. But when using network theory to analyze basketball offense, the analysis suggests that taking the highest-percentage shot each time may not lead to the most efficient game possible. In Brian Skinner’s paper “The price of anarchy in basketball”, he represents basketball as a network problem to show that the short term highest efficiency decision does not necessarily lead to the optimal outcome.

The basketball network problem is analogous to the classic traffic network problem that we studied in class. Each offensive possession has a starting point (the in-bounds pass) and an ending point (the basket). The individual offensive possessions are analogous to individual cars in the traffic network. Each possession takes place through a pathway, which represents the sequence of passes and movements leading to a shot attempt. There are numerous plays, or pathways, through this network, each with a different efficiency that is a function of the number of possessions where that specific play is run. The efficiency of the play as a function of number of possessions in the basketball network is equivalent to the travel time of a road as a function of the number of cars in the traffic network. When more cars use a particular road, the travel time increases. If a team runs the same play too many times, the defense will adjust and be able to anticipate the play, lowering the efficiency of that play.

 exnetwork

The efficiency of a shot is defined by a basketball statistic known as “true shooting percentage” (TS%), which is a field goal percentage adjusted for free throws and three point shots. As expected, the TS% of a particular player decreases as the player’s usage rate (fraction of team’s shots taken) increases. This happens because a high volume shooter is more likely to take bad shots, whereas a player who picks his shots more carefully is more likely to take higher percentage shots. On a team with five players, we assume that the team’s best player is the one with the highest TS%. The logical conclusion would be to give the ball to the team’s best player more often to improve the odds of scoring. However, if we simplify the network to five paths, with each of the paths representing one of the five players shooting, the Nash equilibrium will occur when each of the five pathways have the same TS%. The team’s best player is effectively just as good as the rest of his teammates. From this we can conclude that the optimal solution is to limit the team’s best player’s usage rate such that his TS% is higher than his teammates, leading to a greater overall TS% as a team. We can then extend this logic to saying that a team should not run its best play too often, or else the efficiency of that play will become the same as other less efficient plays.

tspercent

In actuality, most coaches and athletes do not simply use their most efficient strategy every single time like Nash equilibrium would suggest. While they probably do not think of their strategies in terms of network theory, the ideas of saving a team’s best play for a key situation, a pitcher saving his best pitch to keep batters off guard, or football teams running the ball to keep the defense guessing, are all examples of the same type of reasoning that the above network problem uses.

 One interesting scenario that arises from this network problem is the “Ewing Theory”. In 2001, ESPN.com writer Bill Simmons wrote about the “Ewing Theory”. One of Simmons’s friends was convinced that when Patrick Ewing, an NBA great, was out due to injury or foul trouble, his teams played better. While Simmons gave psychological and team chemistry reasons for why the “Ewing Theory” occurs, it is possible that there is also a networks explanation for this. Similar to Braess’s paradox where the removal of an efficient pathway can improve traffic flow, the removal of a good player from a team can actually improve the overall efficiency of a team. Of course, in sports there are much more factors that cannot be accurately represented in such a network representation, but it is interesting to see how network theory can provide explanations for scenarios in sports.

References

“The price of anarchy in basketball” – Brian Skinner

http://arxiv.org/pdf/0908.1801v4.pdf

“Ewing Theory Revisited” – Bill Simmons

http://grantland.com/features/ewing-theory-revisited/

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