Optimizing Shot Selection Using Expected Value

The concept of the "efficient shot" in basketball, a shot that maximizes expected points relative to opportunity cost, is grounded in expected value theory. But the naive EV calculation (FG% multiplied by point value) ignores the full opportunity cost structure of possession use. A possession is a scarce resource in basketball: each team typically takes between 95 and 105 possessions per 48-minute game, and each possession ends in a made basket, a missed shot, a turnover, or (if none of the above) a free throw opportunity. A possession consumed by an off-balance mid-range jumper costs the team not just the probability-weighted points from that particular shot, but the entire offensive possession, including the alternative of a better shot, a penetration that draws a foul, or a pass to an open shooter.

This means the correct question is not "which shot type has the highest EV?" but rather "given our personnel and the defensive coverage we're facing, what is the probability of creating each type of shot opportunity, and how does that translate into expected points per possession?" The distinction matters because teams can't simply choose to shoot only corner threes: there are geometric constraints on how many corner-three opportunities can be created in a given offensive system, and pursuing corner threes more aggressively reduces the element of surprise that makes them efficient in the first place.

This investigation constructs the full expected-value landscape of the NBA shot chart using publicly available shot log data, then examines the optimization problem of how teams should allocate shot attempts across zones to maximize expected scoring, accounting for the diminishing returns that arise from defensive adjustment.

We use NBA shot data from the 2022-23 regular season, grouping shots into zones defined by the official NBA shot chart geography: restricted area, paint (non-RA), mid-range (left, center, right by range band), corner three (left and right), and above-the-break three (left, center, right). For each zone, we calculate (1) league-average FG%, (2) expected points per attempt (FG% × point value), and (3) league-wide attempt frequency (shots from this zone as a fraction of all shots).

We then solve the shot selection optimization problem in two forms. The unconstrained version finds the shot mix maximizing expected points per attempt, which has the trivial solution of allocating everything to the highest-EV zone. The more meaningful constrained version models defensive response: as teams shift more attempts toward high-EV zones, defenses over-rotate toward those zones, reducing FG% as a function of attempt share. We model this as a linear penalty: each percentage point increase in zone attempt share reduces FG% by an empirically estimated coefficient derived from cross-sectional regressions of zone efficiency on attempt rate across teams.

Player-specific analysis uses the Basketball Reference advanced stats table to segment players into quartiles by three-point shooting percentage, then recomputes the optimal shot mix for each quartile under the same constrained optimization framework.