# Fairness at the Western and Southern

So how does the seeding system in use at the Western and Southern (and most important professional tennis tournaments) affect the fairness of their brackets? First, we need to consider how the basic seeding structure affects the outcome. In a subsequent post, I’ll finally extend the analysis to this year’s actual Western and Southern.

Let’s first put down some benchmark numbers. Here are two reference points – how a tournament of this size performs as a blind draw, and how it performs with complete (and perfectly accurate) seeding. In every case reported here, the tournament is run with 56 players on a 64 bracket with conventionally-spread byes, using the parameters established here for professional tennis.

Blind Draw: f(C) = 0.276; f(B) = 19.4
f(b:A-F) = 19.4, 5.5, 3.2, 0.2, 0.1, 0.002

As a blind draw, there is no seeding of any kind. Fairness of both kinds suffers, a bit, from the presence of the eight byes. Without the byes, f(B) would show only noise, with a value at 1.00 or less. The round-by-round f(b) analysis shows that the unfairness is severe in the first round, where the byes are, but diminishes in later rounds, and is essentially gone by round D.

Full, Perfect Seeding: f(C) = 0.196; f(B) = 116.9
f(b:A-F) = 116.0, 71.3, 47.0, 26.3, 13.2, 5.3

This is the basic tradeoff you get from seeding. f(C) is considerably better, and f(B) is considerably worse. The type-B unfairness is also more lingering, though it does diminish a little.

The round-by-round analysis hints at one of the more problematical features of full seeding – uninteresting matches in the early rounds. To create more interest in the early-round contests, while still preserving most of the fairness (C) enhancing feature of the seeding, one can seed only some of the teams, and use blind draw to fill out the rest of the bracket.

16 Seeded Teams: f(C) = 0.198; f(B) = 117.4
f(b:A-F) = 117.4, 74.4, 48.2, 26.4, 13.3, 5.3

Fairness (C) is diminished only a little. Surprisingly, though fairness (B) is also a little worse, and the round-by-round shows essentially the same pattern as the fully-seeded format.

But the similarity in the numbers for fairness (B) does not mean that there is no effect to reducing the number of seeded teams that get protected. There is a substantial effect, but it lands differently on the various parts of the skill distribution. The chart shows the change in expectation between full seeding and 16-team seeding, but the seeded position.

The expectation of the top eight seeds goes up – the value of the bye is greater when the potential opposition in the round that’s avoided is better. But the next eight seeds see their expectations diminish because their early opponents are, on average better. The next few players – those from 17 to 23 in the skill ranking – suffer the greatest loss in expectation. They were at least partially protected in the first round with full seeding, but have no protection at all with only 16 seeds. Finally, almost the entire lower half of the skill distribution sees their expectations rise. With full seeding, they will always be playing a better player in the first round, but with only 16 seeds there’s a chance that they’ll draw a weak first-round opponent. And the gain is greatest for the 32 and 33 seeds, who are otherwise fated to meet the 1 or 2 seeds in an early round.

These, then, are the baseline fairness expectations for 56-player professional tennis tournaments, be they blind draw, fully seeded, or partially seeded. In the next post, I’ll use these baselines to assess the fairness of one particular tournament: the 2017 Western and Southern men’s draw, showing how features peculiar to that one tournament – the tiered seeding, the mis-seeding of several players, and the dislocation caused by late scratches.