The Hot Hand in Reverse at 30-40

Italian translation at settesei.it

30-40 is the most common break point score in professional men’s tennis. It occurs about 15% more often than 40-AD, 30% more often than 15-40, and more than three times as often as 0-40.

It seems that all 30-40s are not created equal. Within the microcosm of a single game, the momentum can swing either way: 30-40 could be the result of a fight to 30-30 followed by a lapse by the server; it could emerge when the server fights back from 0-40.

Regardless of an individual game’s history, the outcome of all points at 30-40 should be created equal. At that score, the server has proven himself skilled enought to win two points against his opponent’s three. In theory, the sequence doesn’t matter any more than it would in a series of coin flips.

Yet anecdotally, it seems that the sequence does matter. Coming from 30-30, the server may feel that he just lost focus for a moment. From 0-40, the returner may feel that he’s due after missing his first two opportunities. (Or to support the opposite hypothesis, the server may have gained confidence by fighting off the first two breakers.)

Regardless of the conventional wisdom, this is now something we can test. If tennis players are completely consistent from one point to the next, the route to 30-40 shouldn’t matter. If they are susceptible to mental ebbs and flows (in predictable ways, anyway), the route to 30-40 should affect how often these break point chances are converted.

15-40 or 30-30?

Let’s start with the simplest possible question. Whenever a game reaches 30-40, the previous point was either 15-40 or 30-30. From 15-40, the server has regained the momentum, though the returner may feel he has a golden opportunity. From 30-30, the returner has the momentum, but the server may feel he can regain control with a single swing of the racquet.

It turns out that there isn’t much difference between the two. From 2011 grand slam men’s singles matches, we have 2136 games in which the score reached 30-40. (Not 40-AD, as 40-AD points must follow deuce.) 890 of those games went through 15-40, while the other 1246 went through 30-30.

In the 15-40 games, the break point at 30-40 was converted 41.2% of the time. In 30-30 games, the break point was converted 40.2% of the time. This gives a slight edge to the “returner sees a golden opportunity” hypothesis, but it is hardly overwhelming evidence.

Love-40

If we look further into each game’s history, two points back, we can compare 0-40 games to the alternatives. Of the 2136 games that reached 30-40, not even 10% passed through 0-40. In those 206 games that passed through 0-40 en route to 30-40, the third break point was converted a whopping 45.1% of the time.

There’s also a noticeable difference between the two other three-point scores. More than half of 30-40 games pass through 15-30; in those 1310 games, the 30-40 break point was converted 41% of the time. But when the game passed through 30-15 before the server lost two consecutive points, the break point was converted only 38.3% of the time.

While the evidence isn’t conclusive, it suggests a sort of reverse hot-hand effect: The player who won the most of the first three points has the best chance of winning at 30-40; the player who won the last two does not.

The same argument even extends to the first two points: If the server reached 30-0, then loses the next three points, the break point is converted only 34.9% of the time. In other words, if a game passes through 30-0 en route to 30-40, you’re better off betting on the guy who just lost the last three points.

If there is a qualitative explanation for this, it might be that fighting off break points requires more mental energy; after coming back from 0-40 (or even 15-40, maybe even 15-30) to 30-40, the server may not have much left. Alternatively, it may require more physical energy; perhaps a rush to 0-40 serves as a wake-up call to the server that he must fight harder to stay in the game. If he does (and if he succeeds in the staying in the game), he’s still competing against the superman who won the first three points of the game. I’m automatically skeptical of explanations of this sort, largely because it would be just as easy to generate stories to support the opposite conclusion. But in this case, at least they explain a quantitative finding.

Another possible explanation may not be as likely, but it is a bit more amusing. Economists and statisticians like to poke fun at the general populace and its innumeracy. Most people think that if you’ve flipped a coin ten times and it has come up heads every time, the odds are better than 50% that it will come up tails on the next flip. After all, it’s “due.”

Perhaps tennis players feel the same way. If a server falls to 0-40, then saves two break points, maybe the returner feels that he’s due. It’s true that the returner is very likely to break at 0-40, but by the time the server saves two breakers, both players start from a clean slate: it’s just as if a coin were flipped five times, with three consecutive heads followed by two tails. But if the coin thinks it’s due … all bets are off.