Live by the Three, Die by the Three? The Price of Risk in the NBA

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Matthew Goldman

Justin M. Rao


Abstract: An important problem facing a basketball team is determining the right proportion of 2 and 3 point shots to take. With many possessions remaining, a team should maximize points—a 3-pointer is simply worth 1.5 2-pointers. 3-point attempts have roughly double the per-shot variance as 2-point attempts, but a team should be “risk neutral.” As time remaining decreases, the trailing team should place an increasingly positive value on risk; the opposite holds for the leading team. Our game theoretic analysis yields a testable optimality condition: 3-point success rate must fall relative to 2-point success rate when a team’s preference for risk increases. Using four years of play-by-play data, we find strong evidence this condition holds for the trailing team only. As a lead decreases, the leading team should become more risk-neutral, but teams in this circumstance actually tighten up and become more risk averse, contrary to what their risk preferences ought to be to maximize the chance of winning the game. We also show that if the offense shoots more 3’s as it becomes risk-loving this implies the attack can be varied more readily than the defensive adjustment. 3-point usage does increase with the trail team’s preference for risk, but actually falls for the leading team. Teams get it right when losing and wrong when winning. We also find a strong motivating effect of losing—the trailing teams displays an overall boost in efficiency for both shot types.