## Win Probability and Volatility in Tennis

Last week, I presented three win expectancy graphs of the same up-and-down tennis match. Tennis players and fans have an intuitive sense of the most “important” points in a match, and we can quantify that as well.

Again we can mimic win probability graphs from baseball–this time, I’ve added a measure of “volatility” for each point in the match. Volatility is defined the difference in win probability between two hypotheticals: that the server wins the next point, and that the returner wins the next point. In other words, if the score is 30-15, volatility is the difference between the WP at 40-15 and at 30-30.

For today, we’re going to stick with the Roddick-Pavel Davis Cup five-setter from 2006. Here’s a recap of that match, if you want a better idea of what you’re looking at.

The green line is Roddick’s win probability, the purple line is volatility, and the spaces demarcate each set. These numbers follow from the assumptions that the players are equal and that the server wins 65% of points.

In many ways, tennis is simpler than other sports for which you can show win probability and volatility. Note the even ups-and-downs toward the end of the first set–that’s a deuce game where Pavel won each deuce-court point, but took a few tries to nail down the game. The win probability at deuce doesn’t change within the game, so a long deuce game, such as the last game of this match, involves a lot of predictable zig-zagging.

Also predictable are the high volatility peaks in the final set. A break of serve doesn’t count for much in the second set of a five-set match, but it makes a huge difference in the decider.

The peak volatility (26.5%) came in the third game of the final set, with Roddick serving 1-1, 30-40. Had he won the point and reached deuce, his win probability would’ve gotten near 50%, as he would be likely to hold serve to 2-1. But if Pavel won, the break knocked Roddick’s chances down to 21.7%, down a break in the fifth. A similar moment came at 30-40 in the previous game when Pavel faced the same problem and a 24.2% volatility.

The volatility plummeted as Pavel kept winning to 5-1, but once Roddick crept back, it crossed 20% three more times, all at deuce in Pavel’s 5-4 service game.

I’ll be interested to see how the volatility graph looks in a variety of other situations–particularly deciding set tiebreaks in which the ultimate winner fights off multiple match points.

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