# A New Way of Looking at Lottery Matches

When Rafael Nadal was eliminated from the US Open last week, a bit of bad luck was involved. He won only two fewer points than his opponent, Fabio Fognini, claiming 49.7% of the total points played. In his career up to that point, Rafa had won 8 of 18 matches in which he won between 49% and 50% of total points. It doesn’t take much to flip the result of such a match.

Matches in which neither player wins more than 51% of points represent nearly one in ten contests on the ATP tour. As Michael Beuoy demonstrated last year, those matches are very much up for grabs: the player with the most points wins less than 65% of the time.

In writing about the small subset of matches in which the loser wins a higher percentage of return points than the winner, Carl Bialik has coined the useful term “lottery matches.” However, Bialik has limited the term to those matches that have an unexpected result. I’d like to expand the definition a bit to all those tight matches that could go either way, even if the player who wins the most points ends up winning as expected.

(A quick side note: Bialik prefers comparing return points, the building blocks of his Dominance Ratio metric. Matches are won a bit more frequently when the winner’s DR is below 1.0 than when he wins fewer than 50% of total points played. These metrics often overlap, of course. To make this arcane subject a bit more accessible, I’m going to stick with the traditional total-points-won stat.)

As Beuoy showed, matches aren’t guaranteed to go to the player who wins the most points unless that guy wins at least 53% of points. (Even then, there’s a slight possibility of an upset, but it’s sufficiently rare that, for today’s purposes, I’m going to ignore it.) 52.5% is much better than 50.5%, but at 52.5%, you’re still going to lose about one of every 25 matches.

By extending the “lottery match” umbrella to all those matches in which neither player wins 51%, 52%, or even 53% of total points, we acknowledge that none of these matches are sure things, and we can look at a broader range of matches to determine whether players are winning as many tight matches as they should. Further, by considering such a category of tight matches, we’ll be able to identify those men who play a lot of them–and by doing so, leave themselves vulnerable to lucky upsets.

Winning the lottery (matches)

Let’s start with the broadest category: all matches in which neither player won more than 53% of total points. These represent everything from true toss-ups at 50% to near-guarantees at 52.9%. Using Beuoy’s model, we can take the total points won from each of these matches and calculate the likelihood that the player with the greater number of points won the match.

Nadal, for instance, is one of the more effective players in these tight matches. Going into the US Open, he had played 168 of them, winning 115. By taking the total points won from each of these matches, we find that he “should have” won only 102.5 of them, meaning that by some combination of clutch play and luck, he’s outperformed expectations by 12%.

Among active players with at least 100 of these matches, Nadal ranks an impressive fourth overall, behind John Isner, Fognini, and Jurgen Melzer. Novak Djokovic and Andy Murray are just inside the top 20, exceeding expectations by 6% and 5%, respectively, while Roger Federer is much further down the list, winning 7% fewer of these tight matches than he should.

Finding Fed on the negative end of this list is a surprise, since Federer, Nadal, and Isner are among the very, very few players who consistently beat expectations in tiebreaks. Tiebreak skill should be closely related to outperforming expectations in tight matches. In any event, my collaborator on a related project, Ryan Rodenberg, has written at length about Federer’s lack of success in some lottery matches.

When we narrow the focus to matches in which neither player won more than 51% of points–true toss-up matches–Nadal is still among the best. In fact, the top four of Rafa, Fognini, Melzer, and Isner remains the same, as each of those players has won between 36% and 38% more often than they should in contests with these extremely slim margins.  Once again, Djokovic and Murray are positive, at +16% and +6%, respectively, while Federer trails far behind, at -9%.

Careening downward

A big advantage of using the broader, 53-percent-of-points definition of lottery matches is that it gives us a larger sample to work with. Nadal has only played 27 matches in his career when the loser won more points than the winner did, and only 40 when neither player topped 51% of total points won.

In the 53% category, though, Nadal has amassed several matches each year of his career, allowing us to look at more meaningful trends. Each season from 2005-11, he averaged about 15 tight matches per year, and won at least one more than we would’ve expected of him, often two or three. Since the beginning of last year, though, he’s played 25, winning only 13 when he should have won 16.

Even with the bigger sample, these are small margins. If Nadal comes roaring back next year and beyond, again winning more close matches than expected, we’ll ultimately see these two seasons as outliers. Yet most of Nadal’s peers post surprisingly consistent records in tight matches. In the last decade, Djokovic and Murray have each had only one season each below -10%, and Federer has reliably underperformed, never reaching +7% for a full season. Not every player is as good in these matches as Nadal, but the ones who do excel post roughly similar numbers from one year to the next.

The bigger picture

Winning tight matches is useful, but as Federer’s experience demonstrates, it’s hardly necessary. And in the case of Fognini, exceeding expectations in lottery matches is hardly sufficient for more general success.

Even better than winning tight matches is winning easy matches, and a useful side effect of studying lottery matches is generating measurements of who plays them the most–and, of course, the least.

Lottery matches–again, those in which neither player wins more than 53% of points–represent fewer than 20% of Rafa’s career matches. His 19.7% rate of close contests is lower than any other player since 2000 (minimum 100 matches). In this category, the big four are bunched together as expected. Among active players, Federer is second lowest, Djokovic is third, and Murray is eighth. Kei Nishikori and David Ferrer are also among the top ten.

At the other end of the spectrum, we find the usual big-serving suspects. Vasek Pospisil tops the list at 49.5%, with Ivo Karlovic (44.5%), Isner (41.9%), and Jerzy Janowicz (40.5%) filling out the top four.

Analyzing the results of very close matches–whichever definition you prefer–is a useful way of identifying players on lucky or unlucky streaks, or even those who appear to play particularly well on big points. However, the more meaningful metric–certainly the one that more closely correlates with elite-level success–is the one that tells us who is avoiding tight matches. The only thing better than luck is not needing it.

## One thought on “A New Way of Looking at Lottery Matches”

1. Elihu Feustel says:

I’d be curious to see what percent of return points Isner wins in tiebreakers versus other return points in that set. If Isner rests during his opponents’ service games, but not opponents’ service in tiebreakers, you might see a meaningful difference.