Braess’ Paradox and ‘The Ewing Theory’
In game 2 of the 1999 NBA Eastern Conference Championships, Knicks star Patrick Ewing tore his Achilles tendon, resulting in him missing the remainder of the playoff series. Despite the public perception that the Knicks would quickly falter and lose the series, they rallied to win three of their next four games, qualifying for the NBA Finals. Bill Simmons, then a sports writer at ESPN, dubbed the phenomenon of a team losing a star player and actually improving ‘The Ewing Theory,’ citing multiple examples throughout all of sports where this, seemingly, has been the case. Brian Skinner, then a Physicist at the University of Minnesota, hypothesized that ‘The Ewing Theory’ could have a networks explanation, and he applied networks concepts to a basketball offense to prove his point. Specifically, Skinner considers a dribble penetration play.
In Skinner’s example, either Player 1 or Player 2 starts with the ball and dribbles towards the hoop. Player 1 is successful at dribbling towards the hoop with probability 1-0.5x, where x is the fraction of the time that Player 1 is asked to dribble towards the hoop, and Player 2 is successful at dribbling towards the hoop with probability 0.5. Then, the player who dribbled towards the hoop can either shoot or pass to Player 5. Player 1 shoots successfully with probability 0.5, and Player 2 shoots successfully with probability 1-0.5y, where y is the fraction of the time that Player 2 is asked to shoot. Player 5 can then either shoot with probability of success 0.75-0.25z, where z is the fraction of the time Player 5 is asked to shoot, or pass to either of Player 1 or Player 2 for the shot. Passes are always completed successfully. The following is a diagram of the team’s offensive options, where black edges indicate passes, blue edges indicate dribbles to the hoop, and red edges indicate shots:
In Nash Equilibrium, Player 2 should never dribble to the hoop, as that strategy is dominated by Player 1 dribbling to the hoop, so x = 1. Then, Player 1 should never shoot, as that strategy is dominated by Player 3 shooting, so Player 1 passes to Player 3. Then, Player 3 and Player 2 employ a mixed strategy to determine who should shoot, resulting in y = 2/3 and z = 1/3. Multiplying through, Player 1 succeeds in dribbling to the hoop with probability 1/2, and Player 2 and Player 3 each shoot successfully with probability 2/3, so the team scores successfully with probability 1/3.
Suppose, instead, Player 3 is no longer part of the offense.
In this simpler, symmetric Nash equilibrium, x = y = 0.5, and the team scores successfully with probability 3/8, which is paradoxically greater than the probability when Player 3 was part of the offense. Framed in this way, this basketball play is an example of Braess’ Paradox, in which adding a road to a road network can actually increase, rather than decrease, traffic times. Despite the fact that Player 3 was a star player, shooting successfully with probability 2/3, his presence in the network decreases overall offensive performance for the team at Nash Equilibrium.
Using a networks framework and allowing for the possibility of Braess’ Paradox-adjacent phenomena, it could be possible to explain lots of sports phenomena in which a star player leaves but the team improves. I suggest further studies of the matter involving a wide range of sports in addition to more statistical evidence to go along with the theory.
Sources:
Simmons, Bill. “Ewing Theory 101.” ESPN, ESPN Internet Ventures, 9 May 2001, https://www.espn.com/espn/page2/storypage=simmons%2F010509a.
Skinner, Brian. “The Price of Anarchy in Basketball.” ArXiv.org, School of Physics and Astronomy, University of Minnesota, 7 Nov. 2011, https://arxiv.org/abs/0908.1801.