Abstract: This paper examines the optimality of the shooting decisions of National Basketball Association (NBA) players using a rich data set of shot outcomes. The decision to shoot is a complex problem that involves weighing the continuation value of the possession and the outside option of a teammate shooting. We model this as a dynamic mixed-strategy equilibrium. At each second of the shot clock, dynamic efficiency requires that marginal shot value exceeds the continuation value of the possession. Allocative efficiency is the additional requirement that at that “moment”, each player in the line-up has equal marginal efficiency. To apply our abstract model to the data we make assumptions about the distribution of potential shots. We first assume nothing about the opportunity distribution and establish a strict necessary condition for optimality. Adding distributional assumptions, we establish sufficient conditions for optimality. Our results show that the “cut threshold” declines monotonically with time remaining on the shot clock and is roughly in line with dynamic efficiency. Over-shooting is found to be rare, undershooting is frequently observed by elite players. We relate our work to the usage curve literature, showing that interior players face a generally steeper efficiency trade off when creating shots.
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