Wimbledon

The Championships (better known as “Wimbledon”), one of the world’s premier bracketed tournaments, is due to start in a few days.

I’ll forgo a full FEPS analysis of Wimbledon’s bracket format. Presumably the All England Lawn Tennis & Croquet Club has thought things through very carefully, and settled upon a format that meets its needs. At a guess, I imagine that the two main factors were fairness (A), honoring the traditions of the event, and spectacle, maximizing the economic value of the event, not necessarily in that order.

One of the distinctive things about the Wimbledon bracket is its use of partial seeding. Only the top 32 players in the 128-player main draw are seeded, with the other 96 being allocated their initial bracket positions by blind draw. This means that even the top-seeded player could draw the 33rd-best player in the draw in the first round, and that the lowliest qualifier could draw another lowly qualifier in the first round. And the 32 players fortunate enough to be seeded are still slotted into the bracket in a somewhat random way.

Seeding of any kind tends to improve the fairness (C) coefficient of a tourney. This post reports the results of some simulations that show how big this effect is, and compares it to other possible ways of seeding (or failing to seed).

The parameters for the simulation are these: Luck is set at unity, and somewhat reduced by having each of the matches in the 128 bracket played best of five (as in the mens draw in the real tournament). Wimbledon is decidedly an elite tournament, so I’ve applied an elite threshold of zero – there will be no conspicuously weak players in the draw, only conspicuously strong ones.

For the payout schedule, I’ve used the actual payouts for the men’s and women’s draws for the 2017 event. There are, in order from first-round losers to the champions:

£35K, £57K, £90K, £147K, £275K, £550K, £1.1M, and £2.2M 

These numbers are a little misleading, as the financial rewards for the top players come much more from endorsements than from prize money.

I’ve simulated the tourney with four different seeding patterns. The first three are:

1. An unseeded, blind draw tourney;
2. A fully-seeded tourney; and
3. A partly seeded tourney, with 32 players seeded in the conventional pattern, and the other 96 drawn at random.

All three of these are for purposes of comparison – none of them is the pattern actually in use for Wimbledon.

The seeding for Wimbledon (and other elite tennis tourneys) is similar to pattern 3, with 32 seeded players spaced out in the bracket with 96 unseeded players, in such a way that no seeded player can meet another of the seeds until the third round. But it differs from the conventional seeding pattern (shown, for example, in 32upper), in that there’s a random element to the way the seeds are ordered.

In a conventional seeding pattern, assuming that all of the seeded teams won in the first two rounds, the third round pairings would be 1 v. 32, 2 v. 31, 3 v. 30, … , 16 v. 17. In Wimbledon, there are layers to the protection afforded the seeds.

Wimbledon, in contrast, places the seeds randomly in several layers. The 1 and 2 seeds are in their conventional positions at the top and the bottom of the bracket. The 3 and 4 seeds are on the same lines as they would be for the conventional bracket, but they can appear in either order. Thus, though according to the standard pattern 1 and 4 would always be in the upper half of the draw, and 2 and 3 in the lower half, at Wimbledon it’s equally likely that the upper half will have 1 and 3 and the lower half 2 and 4.

Additional seeds are placed in three more layers: 5 – 8, 9 – 16, and 17 – 32. Thus, for example, the 9th through 16th seeds collectively occupy the same lines as they would in the standard pattern, but they may be entered in any order. In the conventional pattern (assuming that all matches are won by the lower-numbered seed), the number one seed should encounter, in order, seeds 32, 16, 8, and 4 on the way to a final showdown with number 2. But with the random element within each of the layers, in 2016 Wimbledon the number one seed would encounter seeds 28, 13, 6, and 3. In each round, as this draw had it, the top seed would encounter a slightly better opponent. Thus, this random element somewhat mitigates the advantage enjoyed by the top seeds, to the advantage of lower seeds.

I was unaware of this odd layered seeding approach when I drew up the specifications for my new simulator, and so it’s not something I can test directly. So, to represent the layered-seeding approach, I’ve chosen to simulate the actual seeding for the 2016 men’s draw.

In the table below, I’ve reported the fairness (C) statistic for the tourney as a whole, along with the average money winnings that can be expected for various skill levels. I show the top 10 players, then skip to skill ranks 31-34 to show the effect on players near the cutoff for the 32 seeding patterns, and finally ranks 125-128 to show what happens at the bottom of the distribution.

	Blind draw	Fully seeded	32 seeds	Wimbledon 2016
fairness (C)	7.29	23.38	16.84	16.06
1	£1,466,919	£1,628,877	£1,629,446	£1,549,930
2	£815,619	£1,017,709	£1,016,220	£1,084,794
3	£576,160	£704,718	£702,264	£615,596
4	£443,462	£500,624	£500,355	£617,256
5	£363,554	£402,369	£398,943	£350,255
6	£307,666	£346,478	£346,047	£280,361
7	£267,365	£285,043	£282,976	£298,646
8	£235,898	£229,948	£226,644	£296,629
9	£212,376	£207,139	£204,799	£195,886
10	£195,667	£200,617	£197,488	£224,370
31	£80,155	£74,410	£77,704	£84,051
32	£78,933	£72,787	£76,250	£78,805
33	£77,315	£71,491	£58,499	£58,561
34	£75,919	£70,643	£57,676	£57,807
125	£39,308	£35,002	£38,398	£38,398
126	£39,188	£35,000	£38,247	£38,304
127	£39,029	£35,000	£38,216	£38,197
128	£38,938	£35,000	£38,153	£38,125

These numbers need to be taken with a pinch of salt. Each represents 100,000 trails, which is ordinarily enough to wash out most of the variability of the results, but where some of the mean payouts themselves are in the millions, one should assume that last couple of digits are not significant. (I would have divided everything by 1000, but I couldn’t bear to get rid of the extra 2 pound expectancy for the 125th ranked player in the fully seeded tourney). But some of the apparent anomalies in the table are really in the data. For example, the superior result for the 10th ranked player with the Wimbledon seeding pattern reflects that the 2016 draw was kind to the number 10, who would ordinarily encounter 23, 7, 2, and 3 on his way to the finals but, as it happened, encountered 24, 8, 4, and 2. This is not a benefit that will reliably accrue to number 10s in the future.

As shown in the table, any kind of seeding enhances fairness (C) enormously. Seeding all 128 lines rather than just 32 matters little to the best players, but it makes an enormous difference around the 32nd place cutoff for seeding, and it adds substantially to the expectation of the players at the very bottom of the table, who have a chance to win a round against a similarly low-ranked opponent. These lowest ranked players do much better, though not quite as well as they would in a complete blind draw, where they occasionally will draw another low-ranked opponent in the second round.

So, does this experiment vindicate partial seeding?

That’s not for me to say, of course, but I think that there is indeed a case for it. It makes very little difference at the very top of the table, but it does push an appreciable amount of money down to the lower ranks. This will compromise fairness (C), of course, but I think the recompense of bringing some interestingly close matches into the early rounds is substantial.

From the point of view of the organizers (and the broadcasters), this means that there will be more interesting tennis to watch in the first week, but only a negligible chance that the method will precipitate the early departure of Roger Federer. I’d make that a win-win.

The benefits, if any, of the eccentric tiered allocation of the seeds are less clear.