The Underhanded Free Throw
Source article: Like Father, Like Son, Like Granny? A Case for Underhand Free Throws
The world of basketball is filled with fiercely competitive athletes who will do almost anything to give themselves the upper hand on the court. Whether it’s waking up at 5 in the morning to run extra miles, staying late to take extra shots, or grinding out practice drills until they want to collapse, great NBA players will do whatever it takes to become better at their craft. Now, imagine if there was a radically different technique that would immediately improve their game. Imagine that this technique was incredibly consistent, easy to learn, and thoroughly tested. The introduction of a technique like this seems to be a no-brainer for both the players to practice and for the coaches to teach. So, why, when Rick Barry came out in the 60’s and 70’s with his incredibly successful underhanded free throw shot (that netted him a >90% free throw percentage) did it not immediately get picked up by the entire league? Despite thorough research done more recently by Larry Silverberg, a professor of mechanical and aerospace engineering at North Carolina State that showed the consistency, and lower margin for error of the underhanded throw did it not take off? Silverberg concluded that a player who has a free throw percentage of less than 50% would likely be better off making the switch to an underhanded free throw, and yet, despite many powerhouse players with bad free throw shot percentages, none of them have ever attempted an underhand shot.
In this class, we have been introduced to game theory, and the idea of payoffs resulting from the different actions of these players. Let’s say in our game, there are 2 players, the player and the coach. Consider this game a kind of sudden death where the result of the entire game comes down to a final free throw. The coach loves to win and knows about the efficacy of the underhanded free throw and has been teaching his players to use it in games. If the player shoots overhanded and wins, the coach receives a payoff of 10, and if the player shoots underhanded and they win, the coach receives a payoff of 11. Losing gives the coach a payoff of 0 if the player shoots overhanded and 1 if the player shoots underhanded. The player receives a payoff of 10 for a win and 0 for a loss. However, the player incurs a cost of c (backlash for shooting a shot that looks “silly”) and a benefit of d (coach giving him a slap on the back for following his advice) whenever he shoots an underhanded shot. In this game, c represents the media backlash and embarrassment that an NBA player anticipates from using the unorthodox under-handed shot. This game is represented by the following 2×2 matrix.
Throwing free throws is a scary business. Even with an underhanded shot, the chance of hitting the free throw is still decently below 100%. So, when the player comes to the line to take the shot, he knows that he is either going to win or he is going to lose. If the player wins and shoots underhanded, his payoff is 10-c+d. If the player wins and shoots overhanded, his payoff is 10. Given that the player can’t be sure whether or not he will make the free throw at all, and knowing that the backlash for looking silly is likely much greater than the praise from his coach (c>d), the player realizes that his payoff for shooting overhanded is strictly higher than his payoff for shooting underhanded and thus will always choose to shoot overhanded. In other words, choosing to shoot an overhanded shot strictly dominates the choice of an underhanded shot. When the player shoots overhand, he is either failing to account for the increased odds of his victory, or his cost c is greater than his coaches praise d (it is easy to imagine a situation where the player more highly values his public image than his coaches praise). We can also see that if the player’s c was sufficiently high (c>10) he would rather lose and look normal than win and look silly. We can also see that no matter how highly the coach values winning, he can’t force a player to deviate from their overhanded shot if their c is sufficiently high. In our game, perhaps Rick Barry has a c £ 0. Since Rick Barry doesn’t care what people think about his shot, his payoff is 10+d which is higher than the alternative overhand shot payoff of 10, and thus his choice of using an underhanded shot strictly dominates the choice of shooting an overhanded shot.
Game theory is a fascinating subject to study. When one looks close enough, almost anything can be represented as a game and as a result can be written down in a way that is open to mathematical interpretation.