Efficient Frontier Optimization of NBA Shot Selection Using Markov Reward Processes

Abstract

This paper asks whether modern NBA shot selection can be evaluated as a dynamic portfolio problem rather than as a collection of isolated shot attempts. We combine a possession-level Markov reward process with both a mean-variance shot allocation objective and a defense response cost. The model separates rim attempts, midrange attempts, corner threes, and noncorner threes, then values each candidate shot diet by immediate scoring, continuation after offensive rebounds, empirical game-level variance, and a timeout-response stability cost. Because fastbreak value depends on geometry that is not fully observed in standard play-by-play, we optimize the no-fastbreak sample first and then add back observed fastbreak shot selection as a fixed transition component. Our results indicate that the league aggregate operates near the efficient frontier, with marginal improvements to be made by slightly decreasing perimeter volume and mid range attempts while increasing rim attempts. Team-level analyses reflect this general trend. Ultimately, these findings suggest the three-point revolution has pushed volume slightly past its optimal boundary. By incorporating outcome variance and continuation effects, we show that optimizing solely for expected value mathematically overstates the benefit of perimeter shots.