This is Regressing, a numbers-minded column by our clever friends at the Harvard Sports Analysis Collective. Today: The NBA's H-O-R-S-E competition was doomed to fail.
Perhaps the NBA was hoping for something along the lines of that Larry Bird-Michael Jordan McDonald's commercial when it debuted the H-O-R-S-E competition during All-Star weekend two years ago. What the league got instead was an excruciatingly slow affair in which the players shot a combined 38.6 percent — a sort of mid-'90s Cavs-Knicks game, except with trick shots, which is probably why H-O-R-S-E got dropped from All-Star weekend. In a way, it was the NBA's own fault. The rules of NBA H-O-R-S-E encouraged a competitively uninteresting game.
The problem with the rules is that a shot caller isn't punished if both of his opponents make the shot, too. In the variant of H-O-R-S-E I analyzed earlier, if both your opponents make the called shot, 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. An easy shot is a smart shot. Because you have two opponents, the likelihood that at least one will miss is pretty good. And 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, and everyone watching at home would start flipping channels.
That's not the only incentive to take easy shots. Consider: 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 getting a letter. It's better to make your called shot, have the first opponent make the shot, too, 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." 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.
This can be simplified and then maximized. It turns out that under NBA rules the shot that maximizes opponents' letters is a shot that's made 99.99 percent of the time (the equation is undefined at 100 percent). 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 represents the ideal strategy, surely the NBA players figured this out and played accordingly, right? In order to determine 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 with 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 percent easy shots, 51.8 percent medium shots, and 21.7 percent hard shots. The average number of letters given out in one round when the shot caller called each shot was .32, .33, and .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. When a player called an easy shot, he got to call the next shot 27 percent of the time. 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 called the next shot just 13 percent 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 with .38 or .17 letters is enough to encourage a strategy of avoiding the difficult — but more entertaining — shots.
It was probably unwise for Rajon Rondo to take a shot from atop the scorer's table. It was probably even worse for Omri Casspi and Kevin Durant to attempt the same shot after watching Rondo fail. So maybe the players weren't completely attuned to the rules' incentive to be lame. But these guys compete for a living. They adapt to whatever the game encourages them to do. If the NBA ever decides to bring back H-O-R-S-E, it'd better fix the rules first.
The Harvard Sports Analysis Collective is a student club dedicated to quantitative analysis of sports strategy and business. Follow them on Twitter, @Harvard_Sports. If you have any comments or ideas for future columns, email them to email@example.com.