NBA Small Ball and the Nash Equilibrium
It is abundantly clear from the Golden State Warriors’ championship season last year and historic 73 win season this year that small ball is here to stay. Under the direction of Coach of the Year Steve Kerr, the Warriors have terrorized the league the past two years with their most potent small ball lineup, the famous “Death Lineup”. But what exactly is small ball, anyway?
According to sportingcharts.com, small ball is a style of play in which “size is sacrificed in favor of speed and three-point shooting. The team typically puts out a three-guard lineup or a lineup without a center.” By sacrificing size and physical strength in the paint for speed and marksman shooting, a team can outmaneuver their more bulky adversaries, creating spacing and easy baskets.
As one can clearly see from the above shot chart of the Warriors’ death lineup, the shooting average of the small ball lineup is well above league average in the majority of areas on the court (with above average shooting indicated by green). Looks great doesn’t it? Can we use game theory to find the best response to this lineup?
In his article discussing game theory’s applications to basketball, Ari Caroline uses Nash Equilibria to explain how two competing teams can reach a steady-state equilibrium.
In Caroline’s example above, we have a standard payoff table, where red indicates that Team A has an advantage while negative scores, in blue and in parentheses, indicate that Team B has an advantage. In the middle, we find the sole equilibrium point in the table—these two line ups match up so well that any unilateral change in lineup by other team results in a worse result for that team. Therefore, as in any Nash Equilibrium, the best strategy for either team is to keep playing their respective lineups.
Caroline goes on to elaborate and refine the payoff matrix until arriving at equilibrium lineups that closely resemble the prototypical basketball lineup—2 guards, two forwards, and a center.
Now the question remains—in such a table, with one team being the Warriors, is there a equilibrium point that the death lineup forms part of? I don’t have any data to work with, but it would be interesting to what points, if any, occur. Would such an equilibrium point consist of two small ball lineups, so that the best way to counter small ball would be with another small ball lineup?
Basketball is not easy to model with game theory. Lineup effectiveness in real life is clearly inversely proportional with time played. In Caroline’s example, while it makes logical sense that lineup fives of both teams should play each other the whole game, as there is no incentive to unilaterally change lineups, players would tire and effectiveness would decrease at varying rates, altering the table. Additionally, there are so many possible lineups and a relatively small sample space. Without knowing the payoffs of all the different lineup combinations, it is difficult to ascertain anything for sure.
http://wagesofwins.com/2012/11/06/an-introduction-to-nash-equilibrium-lineups/
http://www.sportingcharts.com/dictionary/nba/small-ball.aspx