## Wednesday, August 28, 2013

### Victoria Duval Pulls off the Upset with a Key Six Point Streak

The first big upset of the 2013 US Open occurred Tuesday night, with 296th ranked Victoria Duval defeating 11th ranked (and 2011 US Open champion) Samantha Stosur. Match highlights, plus Victoria's giddy post-game interview, here.

I had been playing around with an in-match win probability model for tennis (similar to my NBA model), and this seemed like a good test case. Tennis probability is actually easier to model than fixed timed sports like basketball and football. For tennis, all you really need to specify is the probability of winning a point (both when serving and returning). The rest is just a calculation of the branching probabilities (somewhat tedious, but straightforward).

As with the NBA, I am not the first to attempt this. Jeff Sackmann (of HeavyTopspin.com) compiled win probability graphs for a large sample of 2011 matches. For a more math-centric view of things (with infinite series!), take a look at Ian Stewart's collection of mathematical recreation essays, Game, Set and Math (my original introduction to the application of math to sports). Or check here for a closed formula solution for winning a single game.

### Point Probabilities

Pre-game odds for tennis matches tend to vary, but as best I could tell, Victoria Duval was about an 11-1 underdog (8.1% probability) to win the match. From that number I can back into an implied point probability using my win probability model. An 8.1% match probability would imply that Victoria Duval had a 44.3% probability of winning each point. Using standard splits between serve and return results in an implied point probability of 56.6% when serving and 32.0% when returning (these numbers are kind of rough, so I'm open to suggestions).

Armed with these probabilities and the point by point detail, I can then chart out the progress of the match below. The format is nearly identical to my NBA win probability graphs, see here for background. Use the sliders at the bottom of the graph to zoom in. Mouseover the graph for score details.

 Victoria Duval 5 6 6 Excitement 5.8 Samantha Stosur 7 4 4 Comeback 207

Some observations:

### The Low Point

Although Duval started off with an 8.1% win probability, her in-game probability actually dropped as low as 0.5% in the second set.  In other words, her odds of winning the match at that point were 207 to 1.  Duval's low point occurred with Stosur at game point, threatening to take a 5-2 lead in the second set, having already won the first. But Duval was able to battle back and break Stosur's serve, bringing the set score to 3 games to 4. Duval went on to win the set 6 games to 4.

### The Turning Point

The turning point of the entire match is readily apparent on the graph.  With the third set tied at 3 games, Stosur was serving and had won the first point. At that point, Duval's win probability was 22.2%. Duval then proceeded to win six straight points, breaking Stosur's serve to take a 4-3 lead and a 30-0 lead in the 8th game of the set. Her win probability at the end of that streak was 74.4%, a 52.2% increase. Duval went from a 3 to 1 underdog to a 1 to 3 favorite in just six points.

### The Endgame

While Duval clearly had the upper hand at that point, there was still drama to come. Duval nearly broke serve while up 5-4, falling behind 15-40 to Stosur. Her win probability dropped below 50%. Duval then won the next two points, forcing deuce. The win probability then zig-zagged several times as the match swung from deuce to advantage-Duval to advantage-Stosur to advantage-Duval to advantage-Stosur and then finally back to advantage-Duval and match point.

If there's interest, I can probably generate these graphs for other interesting US Open matches.