By: Thomas O’Farrell
For the first time since the 2010-2011 All-Star game, NBA players will be participating in a televised H.O.R.S.E. competition. H.O.R.S.E. is one of the first basketball games people learn to play, and follows a relatively straightforward set of rules. In a two-player game of HORSE, player A starts. Player A will choose where to shoot from. If they miss the shot, it is now Player B’s turn to choose where to shoot from. If Player A makes the shot, and Player B makes the shot, it is still the Player A’s turn, and Player A will get to choose a new spot to shoot from. But if Player A makes the shot, and Player B misses, Player B gets a letter, and Player A will choose a new spot to shoot from. This cycle continues until one player has five letters, spelling “horse.” However, is worth noting that H.O.R.S.E. has several variations of rules, and even a small tweak in rules could drastically alter the probabilities discussed in the article.
Over the past few years, the NBA has undergone a statistical revolution. The increase in three-pointers and the reduction of mid-range shots has permeated into rec centers across America, as even casual fans have started to play “Moreyball.” However, the game of H.O.R.S.E. has remained largely unchanged. In a traditional game of H.O.R.S.E. most players alternate between mid-range jumpers, 3-pointers, and creative trick shots. This semester I played H.O.R.S.E. against one my friends a few times each week, and followed that exact variety of shots. The result was I lost every game we played. After our semester was turned online due to Coronavirus, I began playing my little sister in H.O.R.S.E. Not only did this allow me to regain my confidence, but it also began my curiosity for the optimal H.O.R.S.E. strategy.
The optimal H.O.R.S.E. strategy should maximize the number of letters your opponent receives each time it is your turn to choose the shot. In the most basic case, a two-player game, consider Player A and Player B are of equal strength. If Player A’s chance of making a shot is p, Player B’s change of missing the shot is 1-p. Because Player A cannot gain any letters when it’s their turn, and Player B can gain multiple letters, Player A is incentivized to have their turn last as long as possible.
During Player A’s turn:
Expected number of letters for B: p(1-p) + p2(1-p) + p3(1-p)…
This geometric series sums to: p(1-p)/(1-p)=p
To maximize the number of letters Player B gets for each time it is player A’s turn, player A should take the highest probability shots possible. This seems very counter intuitive to how a typical HORSE game is played, as people are often taking deep shots or complicated trick shots. While the strategy of taking high-probability “easy” shots would make the game less exciting, and a lot longer, the difference in win percentage is staggering. Assuming Player B will take shots that they will have a 50% chance of making the shots, a Monte Carlo simulation of 100,000 trials is displayed below.