Novak Djokovic vs Roger Federer 2011 US open Semifinal Match Point
In an epic five-set semifinal match at the 2011 US open, Novak Djokovic, the world No. 1 player, defeated Roger Federer, the world No.3 player 6-7 (7), 4-6, 6-3, 6-2, 7-5. There was much attention to this matchup as it truly was a clash of titans; Djokovic, the best player this year was facing a second grand slam semifinal match since the French Open a few months earlier this year against Federer, the most dominant male player ever in tennis history who reigned the courts without competition only about a few years ago.
The match had been greatly anticipated by many tennis fans over the world, and the two players did not disappoint their crowd. In the end, the astonishing display of the five setter match came down to just one point in the fifth set when Djokovic was facing two match points, down 15-40 at game score 3-5 in the fifth set. It wouldn’t be an overstatement to say that this first match point determined the outcome of the entire match. Federer served out wide for the match, and Djokovic took a rare full swing at his return cross court, and it barely nicked the line. The following video shows this point: http://www.youtube.com/watch?v=1EGz8AeqAPY
It was once in a lifetime shot for most tennis players, and it just so happened to Djokovic at a semifinal of a grand slam, and he made it. No one can deny that it was more of a “gamble” shot than a deliberate, intentional one, but Djokovic took a risk, and it certainly paid off. One could never know whether or not Djokovic considered any game theory behind his decision before this shot, but it would prove to be the strict best response at that particular situation for him. And after this point was given to Djokovic, the tide has completely turned, and he would win the next four games to seal the fifth set victory 7-5.
Now let’s analyze the point at 15-40 match point. Consider it as a two player game. Federer is serving his first serve, and he has two options. He can either serve wide or down the middle (he often hits an ace either way in a match point). Djokovic, in returning position, must choose either to anticipate a wide serve or down the middle serve before Federer serves, especially because Federer is known as a very strong server, and this was his first serve that reaches up to around 120~130mph (if djokovic stands in the middle of the two directions, he will likely lose the point). Also, Djokovic has two options for each direction he chooses; he can either gamble and take a full swing at the serve, which will reduce the probability of returning it in the court but will increase his chance of winning the point if it does go in, or play safe and hit a usual return, which will increase the probability of returning it in the court but will decrease his chance of winning the point even if it does go in, since Federer will likely take advantage of a soft return. Now we can produce a table with the two players and their possible choices, and the outcome as the probability that each player will win the point after the ball is returned by Djokovic.(So if Djokovic chooses a wrong side to return, his probability will be 0, and Federer’s probability will be 1, presented as (0,1)). The probability for each player to win the point in a “safe return” situation will be determined based on match statistics.
Figure 1 below shows the first serve statistics of Federer during the match. The left portion of the picture shows the number of points won, and on the right the number of serves in. As you can see, he hits a lot of first serve out wide (32 out of 56) on deuce side because it opens up the court for his strong forehand winner or volley. You can also deduce the probability Federer wins the point on first serve when he serves out wide (23 out of 32) or (0.72), and down the middle (9 out of 14) or (0.643) on deuce side. It is safe to not acknowledge the between the wide and down-the-middle serves because Federer rarely ever hits anything but extreme wide or down-the-middle first serve in a match point.
Figure 1
The probability that any “gamble” shot will go in, even for a top trained athlete such as Novak Djokovic, can be assumed to be about once every two shots (half and half). The resulting outcome with any gamble shot can be shown as (0.5,0.5) because if it does go in, Djokovic will win the point, and if it doesn’t, Federer automatically wins the point. Combining all the above statistics, the following table can be produced. The outcome is always (P(Djokovic wins), P(Federer wins)) and the sum of each outcome’s pair must be 1.
The best outcome for Djokovic is (0.5, 0.5) at most, and it occurs at both (wide, gamble) and (middle, gamble). Just looking at the table above, one might conclude that returning middle either safe of gamble way is the strict best response because middle, safe has a better probability than wide, safe (0.357>0.28). But this table did not account for the probability of the direction that Federer will serve to. From just a shear intuition, Federer would want to serve out wide because he felt the most comfortable serving that side all night during the match (32 out of 56) and has had the highest percentage winning from serving that side (72%). Djokovic must have known this and without a doubt, was anticipating a wide serve. So really, the options he had was to either return wide with a “gamble” shot, or play safe. He of course chose the option that was a dominant strategy (0.5, 0.5), and it was arguably the best decision of his career.
Regarding this incredible shot, all that the 2011 US open champion (Djokovic went on to defeat Nadal in the final to win the tournament) could say was fairly nominal: “The forehand return, I cannot explain to you because I don’t know how it happened,” Djokovic said. “I read his serve and I was on the ball and I had to hit it hard, and it got in, luckily for me.”
But his two sentences more or less summarize the detailed game theory behind it.
