Djokovic as GOAT? / Follow-up: Tennis, numbers, and reasoning

In light of Djokovic now being set to overtake Federer in weeks-at-number-one and having just taken his 18th major, while Nadal has caught up with Federer at 20, it is time to briefly revisit a text on how to determine the tennis GOAT (Part II in a series)—or rather on why doing so is next to impossible.

As this blog is closed-ish, I will not dig deep into details or re-analyze what is said in the old text, but I do note that:

1. I still consider weeks-at-number-one the best of the “easy” proxies. If we apply this proxy, Djokovic would (in just a few weeks) be the GOAT (of at least the Open Era). This especially as he is a fair bit younger than Federer (and a-year-or-so* younger than Nadal).

   *Here and elsewhere, note that I will not do any fact checking either. There might be minor errors here and there, but nothing that changes the “big picture”.

2. I would still rate Federer’s career as the better overall, but not by that much and, again, Djokovic is the younger. Certainly, while Federer’s longevity is (was?) extreme, it appears that both Nadal and Djokovic are similar—possibly, even better.
3. Federer’s dominance at his height was almost unsurpassable, and that might in the end be the strongest argument pro-Federer in a GOAT discussion and/or in a discussion of who was the best among the “Big Three”.
4. Nadal’s fatal flaw remains that he has achieved too little (relatively speaking!) outside of clay and that he has mostly been second to either Federer or Djokovic at any given time. I can still see no true case for Nadal being more than the “Clay GOAT”. My old estimate of “Federer > Djokovic > Nadal” might now be “Federer = Djokovic > Nadal”, or Federer marginally ahead of Djokovic or Djokovic marginally ahead of Federer.

   However, Nadal has improved in the comparison of feats that formed Part III of the aforementioned series. The comparison made there was based on 12 French-Open titles, while he now stands at 13. (On the other hand, Djokovic reaching 9 Australian Opens, at a lesser age and on a more competitive surface, weakens the accomplishment in comparison.)

5. The already tricky comparisons are made trickier by the effects of COVID, which include several weakened playing fields, including for Nadal’s 13th French Open and, maybe, the current/2021 Australian Open for Djokovic; a canceled Wimbledon (Djokovic reigning champion; Federer a strong victory candidate, had he played*); and a long period where the ATP ranking** was frozen or otherwise used exceptional rules.

   *Independent of the COVID issue, Federer appears to have taken portions of 2020 off for an injury break or operation or similar. I have not followed tennis in enough detail after 2019 to say for certain.

   **But I suspect that Djokovic would have remained at number one even with the regular rules, and would still be set to take over in weeks-at-number-one.

Skimming through the articles of the series, I note at least one faulty math statement (others might very well be present):

In Part I, I say that “For instance, the probability that the sum of two fair and six-sided dice exceeds* seven is 5/12 a priori but 5/6 given that we already know that one of the dice came up six.”, which is correct in the first half but not in the second: I had my mind on a scenario where one die (dice?) is thrown, it comes up six, and then the other die is thrown. As the order is not specified, another view is necessary. To this, there are 11 (independent) outcomes with at least one six, viz 1–6, 2–6, …, 5–6, 6–6, 6–5, .., 6–2, 6–1. Of these, all but two (6–1, 1–6) exceed seven and the true probability, barring other errors on my behalf, should be 9/11. Looking at the difference, 9/11 – 5/6 = (54 – 55) / 66 = – 1 / 66, making the new result slightly smaller. (The difference is an implicit, faulty, double-counting of 6–6, which unlike e.g. the 5–6/6–5 pair only appears once.)

*Used in the “strictly greater” sense. Another weakness is that this formulation could be interpreted as “greater or equal”. In the latter case, both the old and the new “given that” probability is 1, as the event is unavoidable. (The probability for the first half of the statement would rise to 7/12.)

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