In the comments to my match-fixing post earlier this week, Elihu Feustel commented:

There are almost no situations where a best of 3 match is a favorite to go to three sets. If the market priced a player as greater than 50% to win in exactly 3 sets, that alone is compelling evidence of match fixing.

In Monday’s questionable Challenger match, not only did the betting markets believe that the match was likely to go three sets, it picked a specific winner in three sets.

It takes only a bit of arithmetic to see why Elihu’s point is correct. Let’s say two players, A and B, are exactly evenly matched. Each one has a 50% chance of winning the match and a 50% chance of winning each set. Thus, the odds that A wins the match in straight sets are 25% (50% for the first set multiplied by 50% for the second). The odds that B wins in straights are the same. The probability that the match finishes in straight sets, then, is 50% (25% for an A win + 25% for B), meaning that the odds of a three-set match are also 50%.

As soon as one player has an edge, the probability of a three-set match goes down. Consider the scenario in which A has a 70% chance of winning each set. The odds that player A wins in straight sets are 49% (70% times 70%) and the odds that B wins in straight sets are 9% (30% times 30%). Thus, there’s a 58% chance of a straight-set victory, leaving a 42% chance of a three-setter.

This simple approach makes one major assumption: each player’s chances don’t change from one set to another. That probably isn’t true. It seems most likely to me that the player who wins the first set gets stronger relative to his opponent, perhaps because he gains confidence, or because his opponent loses confidence, or because he figures he doesn’t have much chance of winning. (I’m sure this isn’t true in *all* matches, but I suspect it applies often enough.)

If it’s true that the probability of the second set is dependent–even slightly–on the outcome of the first set, the likelihood of a three-setter decreases even further.

**Probability in practice**

As expected, far fewer than half of tour-level matches go three sets. (I’m considering only best-of-three matches.) So far this year, 36% of ATP best-of-threes have gone the distance, while only 32% of Challenger-level matches have done so.

In fact, men’s tennis has even fewer three-set matches than expected. For every match, I used a simple rankings-based model to estimate each player’s chances of winning a set and, as shown above, the odds that the match would go three sets. For 2014 tour-level matches, the model–which assumes that set probabilities are independent–predicts that 44% of matches would go three sets. That’s over 20% more third sets that we see in practice.

There are two factors that could account for the difference between theory and practice. I think both play a part:

- Sets aren’t independent. If winning the first set makes a player more likely to win the second, there would be fewer three-setters than predicted.
- There’s usually a bigger gap between players than aggregate numbers suggest. On paper, one player might have a 60% chance of winning the match, but on the day, one player might be tired, under the weather, unhappy with his racquets, uncomfortable with the court … or playing his best tennis, in a honeymoon period with a new coach, enjoying friendly calls from home line judges. The list of possible factors is endless. The point is that for any matchup, there are plenty of effectively random, impossible to predict variables that affect each player’s performance. I suspect that those variables are more likely to expand the gap between players–and thus lower the likelihood of a three-setter–than shrink it.

**A note on outliers**

Despite the odds against three-setters, some players are more likely go three than others. Among the 227 players who have contested 100 or more ATP best-of-threes since 1998, 20 have gone the distance in 40% of more of their matches. John Isner, tennis’s most reliable outlier, tops the list at 47.4%.

Big servers don’t dominate the list, but Isner’s presence at the top isn’t entirely by chance. After John, Richard Fromberg is a close second at 46.7%, while Goran Ivanisevic is not far behind at 43.0%. Mark Philippoussis and Sam Querrey also show up in the top ten.

It’s no surprise to see these names come up. One-dimensional servers are more likely to play tiebreaks, and tiebreaks are as close to random as a set of tennis can get. Someone who plays tiebreaks as often as Isner does will find himself losing first sets to inferior opponents and winning first sets against players who should beat him.

That randomness not only makes it more likely the match will go three sets, it’s also something the players are aware of. If Isner drops a first-set tiebreak, he realizes that he still has a solid chance to win the match–losing the breaker doesn’t mean he’s getting outplayed. If there is a mental component that partially explains the likelihood of the first-set winner taking the second set, it doesn’t apply to players like him.

Still, even Big John finishes sets in straights more than half the time. Every other tour regular does so as well, so it would take a very unusual set of circumstances for a betting market–or common sense–to favor a three-set outcome.