“The truth will set you free. But not until it is finished with you.”
― David Foster Wallace, Infinite Jest
I've been reading a lot of David Foster Wallace, starting with some of his articles and essays and leading up to the infinitely-lengthy Infinite Jest. Wallace is a remarkable writer, and his article on tennis star Roger no middle initial Federer that gives name to this posting is a wonderful description of how, even among the supremely gifted players at the top of the international tennis circuit, Federer is every so slightly more gifted, and how the accumulation of these small gifts lead to wonderful "Federer moments" of exquisite play beyond the highest level. It's also led to 17 Grand Slam (Australian Open, French Open, Wimbledon, US Open) championships and a reasonable argument that Federer is the greatest tennis player of all time.
Both Flesh and Not describes the small advantages in speed, sense and angle that add up to enough to win specific points, but gives no sense that such points are ubiquitous - Federer has had remarkable winning streaks, but not the kind my son calls Triple Bagels (6-0, 6-0, 6-0), and in racking up his wins, it's not like he's winning all his games by shutout (or "at love", as the tennis aficionados say). So, how has he done it? His advantages are subtle. Is the accumulation of such small advantages really enough to add up to "the greatest of all time?" They are - and in this posting we'll go into a bit of why.
"A chessgame is won with the gradual accumulation of small advantages."
― Wilhelm Steinitz, World Chess Champion (1886 - 1894)
Roger Federer may not be winning his games, sets and matches at love, but with 17 Grand Slam titles he has spent a lot of time at the top of the pyramid -- far more than might seem obvious, even for a magnificent player who at his best might have won maybe 60% of the points in his matches. Surely 60% of the points = 60% of the games = 60% of the matches, doesn't it?
Not so fast. Tennis has a curious scoring system - basically "first to 4-points, but you have to win by 2." We'll assume that each point is an individual event (with no causal link to any other point), thus we can model tennis games with discrete Markov processes.
A quick word about Markov and our analysis. Markov processes (named for Russian mathematician Andrey Markov and often referred to as "Markov chains") are terrific modeling tools for systems that transition from state to state with a finite number of countable states. For our tennis example here the "states" will be points, games, sets, and matches. Markov processes are said to be "memoryless" - the next state depends only on the current state and not on any sequence of preceding events or states. If the system that we're modeling (here Roger Federer playing tennis points/games/sets/matches) conforms to these rules, then the Markov approach can offer lots of interesting quantitative information: probability of winning, expected number of points/games/sets played and more. But first let's see what the flow of a game looks like in modeling:
So here we start at the top of the graph with a score of love-love (0-0), and you can follow the graph through the progression of points to the outcome -- either a win for Player A, or a win by Player B. This is great -- we can number the vertices of the graph and put this into a Markov Process model. First let's number the vertices, here:
And now we can apply a discrete Markov process model (courtesy of Mathematica 9). In the model we'll set Federer's point-winning percentage to 60% (.60), and thus giving his opponents 40% of the points. A single-line Matematica equation, and we have our basic model here:
Super -- as far as it goes. But what does it tell us? If Federer wins 60% of the points, doesn't he (obviously) win 60% of the games, sets and matches? Let's see what our model tells us:
Now we're on to something! So, by our model, a winning probability of 60% for each point in a tennis game gives us a game-win 73.6% of the time. And if a winning probability of 60% for each point leads to a win in 74% of the games, what does a 74% game-win-probability give us? As it turns out we can apply a discrete Markov process to games and sets, too. The process graph for a full tennis set (shown below) is a bit more complicated than for a game, but the Markov process works similarly, and here's what we get turning 60% points-wins into 74% game-wins:
Again, as before you can start at 0-0 and work your way through the different game-results to a completed set. The Markov process for a tennis set is a bit more cumbersome, but works similarly to the game Markov process and is shown below:
The key thing to note in the process is that our input "p" is no longer the 60% point-win-percentage, but rather the 73.6% game-win-percentage. If the Federer of our model wins 73.6% of his games, what percentage of his sets will he win? Let's see...
Now we're getting somewhere - winning 60% of points may not sound all that great, but a set-win percentage of almost 95% sounds more like the stuff of the greatest of all time. Moreover we're still we're not quite done -- we have "game, set" calculated, so let's extend the analysis to see what we get for "Game, Set, MATCH."
Match play is pretty simple -- and here we'll use Wimbledon-style matches -- best of 5 sets, first to three wins, wins. The Markov process is also similar to the processes we've seen for games and sets, and is shown below:
Serve it up and you can see how a Roger Federer, here modeled as winning 60% of the points in his matches can be (as he did that over most of a decade) the greatest of all time:
Here 60% of the points give us (or gives Roger, in our model) wins in 99.8% of his matches. Now, of course there are some caveats here:
- The 60% number is a very round figure, taken from the guesstimate that Federer in his prime won 70+% of points on serve, and may have won almost 50% even on return-of-service
- Federer at 30 years of age is not the Federer of 25, but even last year he won the Wimbledon final with 151/288, or 52.43% of points won.
- If we run 52.43% through our Markov models, we can estimate that Federer would win 56% of his games, 62% of his sets, and 72% of his matches - and even those statistics are for an older Federer, taken from a single Finals match against one of the top-4 players in the world.
David Foster Wallace writes wonderfully about "Federer Moments" in Roger Federer as Religious Experience, and the wonder shown here in our Markov models is not only Federer's magnificent play, but the incomparable consistency at that exalted level for the decade that saw him win 4 Australian Opens, 1 French Open, 7 Wimbledon Championships and 5 US Opens. The models here show, not "Federer Moments" but the "Federer Edge" -- just a little bit better than anyone in the world, point-after-point, game-over-game, match-over-match for a decade of play. We can't know what lies ahead, but we can plan for it -- the Wimbledon Men's Singles Final this year is on July 7.
"In an era of specialists, you're either a clay court specialist, a grass court specialist, or a hard court specialist...or you're Roger Federer."
― Jimmy Connors