Between my posts on marble sports and beauty pageants, luck versus skill in sports seems to be something of a theme of this blog. Today I want to talk about tennis — but instead of writing descriptively, I want to be prescriptive: I’ll advocate for a change to the rules of tennis that would make it more of a game of skill, and probably more fun to watch, too.

But first, a little philosophy on the role of luck in sports. Imagine two soccer teams — call them the Excellent Etruscans and the Mediocre Minoans. The Etruscans play the Minoans at the European Championships, and the Etruscans dominate the match: they are in possession of the ball 60% of the time and complete almost twice as many passes. They play at a level where they have a 1.5% chance of scoring a goal per minute of play; meanwhile, the Minoans only have a 1% chance per minute. If they played forever, you’d expect the Etruscans to score 50% more goals.

But a soccer match isn’t infinite: it lasts 90 minutes. And as it happens, the Minoans win the match 1-0. This isn’t all that surprising: given their level of play, there was a 25% chance that the Minoans would beat the Etruscans (and another 28% chance of a tie).

To me, this is disappointing: the Etruscans played much better, and yet the Minoans got lucky and won. To a Minoan fan, the outcome is a hollow victory: the team played worse but happened to win anyway; to an Etruscan fan, it is an undeserved disappointment. Ideally, the winner of a match should be the team that played better that day. To whatever extent the format of soccer doesn’t live up to this ideal, I view that as an inadequacy of the format.

One pretty simple way to address this inadequacy is to extend the length of soccer matches. Of course this has its limits: athletes can only play for so long before exhausting themselves. On the other hand, suppose you could reduce the role of luck without making the game any more strenuous; why wouldn’t you?

This brings me to tennis. Unlike soccer, I actually played and followed tennis for quite a while and feel qualified to have an opinion on its rules. And my opinion is that it should be harder to serve.

For those unfamiliar, I’ll briefly describe how tennis works. (If you think you’ll find this boring, read this short footnote^{1} and skip to the section below with the charts.)

A **point** in tennis begins with one person “**serving**” the ball. The other person returns the serve, and then the point is played out.

Points are organized into “**games**“; the winner of a game is the first player to four points, win by two. **The same player serves the ball throughout the game**, and the server alternates from game to game. Games are organized into “**sets**“; to winner of a set is the first player to six games, win by two; but if a set reaches 6-6 then the players play a “**tiebreak**“, which is a special long game where the players alternate who serves. Finally (typically), whoever wins two sets wins the **match**.

In tennis, the player serving the ball has a large advantage. I thought that this was the case in every sport with a server: badminton, squash, volleyball, etc. — for the same reason that going first is often an advantage in games such as chess. But this turns out to be false: in volleyball the returner has an advantage, and in badminton it’s basically even. In tennis, however, the server’s probability of winning a point ranges from 56% (in women’s doubles) to 65% (in men’s doubles).

This translates to a much higher probability of the server winning a *game*, since a game has several points. That probability is particularly high for players who build their strategy around serving really well. Sometimes this leads to crazy matches like the 2010 Wimbledon match between John Isner and Nicolas Mahut, where *168 consecutive games* were won by the server, and which finished with Isner winning 70 games to 68 in the final set.

(Wimbledon’s rules differ somewhat from the usual rules of tennis, which is why that set didn’t have a tiebreak. The rules have since changed as a direct result of this match.)

The fact that the server has a large advantage makes tennis somewhat “soccer-like”. You can think of a *break of serve* — that is, winning a game when your opponent is serving — as analogous to a goal in soccer. It is really common for a set to have one break of serve total, and whoever gets that break wins the set — sort of how 1-0 is a common score in soccer matches. This “low sample size” means that tennis has a higher-than-ideal amount of luck. Worse, many sets end in tiebreaks. A tiebreak is sort of like a penalty shootout in soccer: while skill is involved, it really is just a way to break a tie, and the outcome is fairly random.

What would happen if we changed the rules of tennis in a way that took away the server’s advantage? I wrote a simulation to figure this out. Here’s how the simulation worked:

- Call the two players “Positive” and “Negative”. I started the simulation by giving Positive some
**skill advantage**(i.e. Positive is the better player). Positive represents the player who is playing better*that day*(i.e. who “deserves” to win), not the player who is usually better. A reasonable skill advantage number might be 0.1. - I also pre-determined a
**server advantage**; the appropriate number here is 0.18 for women and 0.33 for men (this is based on the data in the chart above). - Sequentially, for every point in the match, to determine its winner:
- I took Positive’s skill advantage. I then added the server advantage if Positive was serving, and subtracted it if Negative was serving. The resulting number represents who had an advantage that point.

- Then I added a normally distributed random number with standard deviation 1. This new number represents
**luck**during that point. Any given point has a lot of luck — whether the ball lands just slightly in or just slightly out, etc.

- If the resulting number was positive, Positive won the point; if it was negative, Negative won the point.

I ran this simulation 100,000 times each for different levels of server advantage to see how the probability of an upset (i.e. Negative winning despite being worse) changed.

This chart shows the probability of winning when one player was a little better than the other — enough to give them 2:1 odds. Getting rid of the server advantage would decrease the probability of an upset by anywhere from 1% (in women’s singles) to 2.5% (in men’s doubles). This isn’t nothing, but it’s not too exciting.

What if one player is substantially better than the other?

At this skill differential, getting rid of the server advantage would decrease the probability of an upset in men’s singles from 17% to 14% — that’s almost one-fifth! What if one player is much better than the other?

This is the sort of skill differential you see when a top player (like Novak Djokovic) plays a #100 or #200 player. In this case the absolute difference in probability of an upset is smaller (only around 2%) for men’s singles — but the relative decrease (7.5% to 5.5%) is very large. Essentially this means that top players would lose early a lot less.

What sort of change could be made to reduce the server advantage? You could try a number of things, like making the server stand farther back, but I think the most natural remedy is to make the service box (the box in which the serve has to land) smaller. It probably wouldn’t be by much — maybe a few inches — but the result would be tennis matches with a pretty different dynamic. In addition to reducing the role of luck, reducing the server advantage would make the game more exciting. Right now, when watching tennis you can often be pretty confident about who will win any given game (the server); with this change, every game would be competitive. Put another way, tennis would feel a little more like a high-scoring game like basketball and a little less like soccer.

^{1. In each point, the person serving the ball has a substantial advantage. To gain the upper hand, a player must win several points in close proximity while returning the opponent’s serve. This happens rarely, effectively making tennis a “low-scoring” sport like soccer, increasing the role of luck.↩}

Based on your description, it seems like tennis has a lot of ‘intermediate rounding’. Whoever wins the most points wins the game, whoever wins the most games wins the set, whoever wins the most sets wins the match. It seems like if you want to cut down noise, you should just have whoever wins the most points win, regardless of how they are spread out over time.

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Originally this sounded entirely right to me. I wrote a simulation to verify and see how much noise gets cut as a result, and discovered that it’s a little more complicated. Under the current rules, the player who wins the match typically wins around 80 points — but it’s typically more if the match is close and less if it’s not close. Having sets is a way to end on-average less competitive matches more quickly. So really, if you make the winner whoever reaches 80 points first, you’re decreasing the sample size of close matches (thus introducing noise) while increasing the sample size of non-close matches (which rarely affects the outcome). Maybe you could fix this by having the rules be something like “First to 100 points, but if you’re ahead by 15 at any point in time then you win”. I might write a follow-up post about this, so thanks for the idea!

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Worth noting that you could plausibly implement this change without modifying any tennis court, just by specifying that balls which touch the line are out on serve, rather than in (as for any other point).

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