NBA and Game Theory: Optimal Late-Game Strategy and Three-Point Dependency
The inception of the three-point shot was originally thought to be insignificant in the grand scheme of basketball strategy. Teams relied heavily upon a tall, strong center who could dominate close to the basket for easy shots at earning two points. Nevertheless, it’s introduction to the professional game brought about several changes in how many situations could be approached. Consider, for example, a situation in which a team is down by two points and theoretically has the last possession of the game (barring any turnover or misusage of the shot clock timer). Without the three-point shot, the losing team really only has one strategy, with two opposing outcomes based on whether the strategy pans out: send the game into overtime, or miss and take the loss. However, with the introduction of the three-point shot, the losing team now has a third possible outcome: take the three-point shot, and have a chance at winning the game, or miss and take the loss.
Given these two possible strategies, it is worthwhile to consider what the losing team might do to grant themselves the best possible outcome. One might say that if they go for the tie and overtime (i.e. take the two-point shot), they grant themselves a higher probability of winning the game by earning the extra time to take a lead. However, Ruminski clarifies this misconception by constructing a game with two players out of the end-game situation. The two players in this game are the offensive team (who is supposedly losing by 2), and the defensive team. Using statistics detailing the probabilities of making open and contested two-point and three-point shots, and giving the defensive team the strategy of either defending the two-point shot and the three-point shot closely, we see that the game turns into this two-by-two scenario depiction. Clearly, there is no dominant strategy here, so Ruminski goes on to find the mixed equilibrium. Overall, he found that if the defending team chooses to defend the three-pointer less than 80% of the time, the losing team should choose to shoot the three-pointer, and likewise, the defending team should fear the three-pointer more than sending the game to overtime. As a result, we see that three-pointers introduction to the professional game causes a heavy weighting on the extra point in close situations.
This dependency on the three-pointer is not just limited to end-game situations, however. In the past two years, we’ve seen a persistent turn towards incorporating the three-point shot more frequently into teams’ offenses, and the payoffs being tangible. Therefore, similar to how Ruminski constructed his end of game situation, we could construct a game out of each possession to show how putting more significance on the three-point shot could boost a team’s probability of winning each possession and accumulating enough of a lead to secure the game. However, this is heavily reliant on the team’s three-point shooting percentages and assist-to-turnover ratio, but we do have precedence for this sort of strategy working, especially in the past year (Golden State Warriors).
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