By: Eugene Ye
In basketball, points come in three different quantities -- one, two, or three points -- creating a unique situation at the very end of a close game. There exists a shot clock that limits how long a team can take to shoot the ball, so this article will examine the situation in which the shot clock and game clock indicate that there is only enough time to have one more chance to shoot. With one shot left, the attacking team may find itself down two points, leaving them with the choice to either tie the game with a two point shot or win with a three point shot. As in any sport, basketball is a zero-sum game, so this situation can be modelled through game theory. The first way to score would be through free throws. Often, near the end of an NBA game, both teams are in the “bonus” where any defensive foul results in two free throws, potentially tying the game. This would be a rare situation because the defense would refrain from fouling unless the shot was completely uncontested and the chance of it going in was higher than the player making two free throws (as I will discuss in more detail later); as well, it would be unwise for the attacking team to bank their winning chances on a fouling mistake from the defense. This leaves the attacking team with the choice to tie the game with a two point shot/layup or win the game with a three point shot. Similarly, the defending team must decide whether to defend the inside or outside shot. With these different ways of scoring and defending, this situation can be modelled as a simultaneous game to answer the question: is there a dominant strategy during the “clutch” time of an NBA game?
According to NBA.com, these were the league-wide statistics from the 2020/2021 regular season:
To create the simultaneous game, a few assumptions will be made. First, the defense, no matter if they defend the inside or outside, will likely not allow an open or wide open shot so those percentages can be ignored. However, for the three point shot, unless it is from a player like Steph Curry, players will very rarely choose to take a tightly contested three point shot so the two probabilities will be taken from “contested” and “open” shots. The statistics from NBA.com supports this assumption as there are only a little over 150 players who have even attempted such a shot last season and only 72 of those players have a tightly contested 3P% of over 0% (hence why the average and median are so different). If a made two point shot has a payoff of 0.5 and a winning 3 point shot has a payoff of 1, and the probability of each shot were to be assumed to be the average across the NBA, the payoff matrix would look like this:
(Payoff for a tie is 0.5 and payoff for winning is 1.0)
The simultaneous game payoff matrix shows no dominant strategy for either team. A dominant strategy occurs when one strategy in a game is always better than another strategy for one player, no matter how that player's opponents may play. In this case, no strategy is always better and therefore no dominant strategy exists. For the defending team, the best response to a three point shot attempt is to defend against the three, and the best response to a two point shot attempt is to do the opposite. Likewise, the best response for the shooting team is to shoot the three pointer when the defending team is defending against the two pointer, and vice versa. No strategy can be considered dominant, and if a team was to stay consistent with a strategy the defending team could effectively counter it.