by Ben Blatt

Perhaps the NBA was hoping for something out of a Larry Bird versus Michael Jordan McDonalds commercial when it debuted the NBA H-O-R-S-E competition two years ago. Instead, they got an excruciatingly slow paced game in which the players had a combined 38.6% shooting percentage. However, there was another factor that caused the game to struggle; from a mathematical standpoint, the rules of the NBA H-O-R-S-E encouraged a competitively uninteresting game.

The problem with the NBA rules is the lack of a punishment if both of your opponents make the shot. In the variant of H-O-R-S-E I analyzed earlier, if both of your opponents make the called shot then the next player gets to call a shot. These rules lead to an ideal strategy of taking a shot you’ll make about half the time. In the NBA competition, if both of your opponents make the called shot you simply get to call another shot. This leads to no motivation to try a difficult shot. Since you are playing two opponents, if you call an easy shot your opponents are more likely to miss it than you are. Even if they both make it, you can simply keep calling an easy shot. If all players wanted to play strategically and maximize their chances of success, they would always take easy jump shots and layups which would lead to an extremely boring competition.

Another factor that pushes players to take easy shots is that a round in which you make your shot but your first opponent misses is not ideal. This is because the third player would then get to call a shot, which would make you vulnerable to receive a letter. Instead, it is much better to make your called shot, have the first opponent make the shot, and the second opponent miss.

This intuition can be explained mathematically as well. Each shot can be viewed as an independent event with probability ‘p’ of success. Since the shot caller is choosing the difficulty of the shots, he is also choosing the value of ‘p’. This assumes that the probability of a shot is based on the shot type and not the shooter. While this is not realistic, among similarly skilled players the results should only vary slightly. With this in mind, the expected value (in terms of number of letters received by all opponents) when calling a shot with probability p can be represented as a sum.

The probability that a letter is given out in any round, given that the round occurs, is p((1-p)-p(1-p)). This is simply the sum of two different probabilities: the probability that player one makes the shot but player two misses, p(1-p), plus the probability player one and two make the shot but player three misses, (p(p(1-p)). Under NBA rules, only one player can get a letter before a new shot is called. In order for round k to occur, the first player must make the first k shots and player two must make k-1 shots. This is because as long as player two makes his shot, player one will get to call an additional shot. If player one misses his shot, then he loses possession.

This can be simplified and then maximized. It turns out under NBA rules the percentage shot that maximizes opponents’ letters is one made 99.99% of the time (the equation is undefined at 100%). The smartest thing for a player to do under these rules is to take a shot that they’ll make virtually every time.

So if this is the ideal strategy, surely the NBA players figured this out and played accordingly? In order to find out if this was true, I undertook some of the most arduous and painstakingly boring research of my life: watching both H-O-R-S-E competitions. I organized all called shots into three categories. ‘Easy’ shots were defined as any type of free-throw, layup, or close jump shot. ‘Medium’ shots were defined as any type of shot in the difficulty of a three-pointer. ‘Hard’ shots were defined as any type of half-court or equally outrageous shot.

The distribution of called shots in the NBA H-O-R-S-E was 26.5% ‘easy’ shots, 51.8% ‘medium’ shots, and 21.7% ‘hard’ shots. The average number of letters given out in one round when the shot caller called each shot was 0.32,0.33, and 0.17 respectively. Clearly the ‘hard’ shots were unwise, although the ‘medium’ shots had a higher expected value than the ‘easy’ shots. However, this was only the average value after one round. Over 27% of the time a player called an ‘easy’ shot, he got to be the next person to call the shot. Looking at it as a geometric series, this brings the actual expected value for an ‘easy’ shot to .43 letters. When a player called a ‘medium’ shot, he got to be the next person to call the shot just 13% of the time bringing its expected value to .38 letters. A player never got to call another shot after calling a ‘hard shot’. While not extreme, the difference between giving out .43 letters to your opponent compared to .38 or .17 letters vindicates the strategy that more difficult shots should be avoided.

The players would have been better off taking easy shots. It was probably unwise for Rajon Rondo to take a shot from atop the scorer’s table. It was probably even more unwise for Omri Casspi and Kevin Durant to attempt the same shot after watching Rondo fail. Although the players did not realize (or were uninterested in) the most strategic way to play, if the NBA ever decides to bring the H-O-R-S-E competition back it would be best off changing its rules.

*Ben Blatt can be contacted at bbblatt@gmail.com*.

Very nice analysis!

If you were to “fix” the game of HORSE (so perhaps a 20% shot were optimal) how would you do it?

Interesting question. As shown in the original HORSE analysis, a 53% shot is ideal under the rules that the ball get turned over if everyone makes it. However, I see how this might still not be terribly interesting as it would not encourage outrageous trick shots. If we wanted to decrease the favorable shot chance even lower, towards 20%, one possibility is to give each opponent two tries to make a called shot. I worked this out quickly for a three player match and got the sum{k=1 to infinity} p^k(1-(p^2+2p(1-p))^2)^k-1(n-1)(1-p). This has a maximum value at p=35%. While not quite at 20%, giving opponents additional chances to attempt the same shot would certainly encourage more difficult shot calling.

And this is combined with the “if all opponents make it, the next player gets to call”?

Also, how might the probabilities change with larger groups playing? I’ve got 4 brothers and we sometimes play with as many as 8. In that case, it seems that you’d want to stay away from risk even more, just to make sure that at least SOMEBODY gets a letter. (The odds that all 7 make it being quite low).

You are correct in your assumption of ““if all opponents make it, the next player gets to call” with the note that making one of two shots counts as “making it” in the rules outlined in the comments above.

You’re also right in your assumption that playing with many people makes risk even less worthwhile. Under standard rules, in an eight person game a 66% shot is ideal. The modification of “must make one out of two” would help lower that value. Any other modification that would limit the opportunity of the shot caller and increase the opportunities of the other players would lower this percentage as well.