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Chess players are typically rated by a single number, their Elo rating, which suggests that their skills are totally ordered and thus transitive. But I am wondering if there have ever been three (well-known) chess players A, B, C such that, according to general opinion or some statistically significant calculation, A is more likely to win against B, B is more likely to win against C, and C is more likely to win against A.

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    I've seen some literature on this in past about these "triads of players," usually with the narrative that style was the driving factor. I'll look for those references if I can. Aug 25, 2022 at 23:47
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    Kasparov, Kramnik and Shirov. Kramnik overall played well against Kasparov and has a plus score against him in classical chess, but Kramnik overall struggled against Shirov (I am very positive that Shirov has a plus score, at least in his prime against Kramnik), and Kasparov completely dominated Shirov, I don't think Shirov ever won a classical game against Kasparov. So we have Kramnik > Kasparov > Shirov > Kramnik and so on...
    – Akavall
    Aug 26, 2022 at 1:29
  • @SecretAgentMan: So do I. I think Tal was one of the three. No idea where to look, though. Aug 26, 2022 at 7:57
  • @Akavall Did Kramnik really struggle against Shirov, though? His FIDE statistics suggest otherwise: ratings.fide.com/profile/4101588/…
    – Hauptideal
    Aug 26, 2022 at 13:26
  • @Hauptideal, I am not sure that I understand what FIDE statistics refer too, according to these statistics Kramnik never lost a game with white, but he certainly did in the 1998 candidates match chessgames.com/perl/chessgame?gid=1266208. According to chessgames.com/perl/…, Kramnik still has a plus score vs Shirov, but it mostly in later years when Shirov was past his prime.
    – Akavall
    Aug 26, 2022 at 16:49

2 Answers 2

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Found it! "Canonical" answer from "Dynamic Decision Making in Chess", p.9. Author Boris Gelfand quotes Beliavsky: Tal, Korchnoi and Stein were blitzing "Loser stands up" style, and all the time Tal lost to Korchnoi, Korchnoi to Stein and Stein to Tal.

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  • Amusingly, already in the next chess book I read the next day, there was the next circle suggested... Oct 1, 2022 at 19:17
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To draw statistical conclusions, you would need a large enough sample. You will definitely find a lot of examples of "rock-paper-scissors"-configurations (RPSC) of players if you dig the head-to-head data, but this will very likely be due to pure chance and other factors.

For example, Nepo had a positive score against Carlsen prior to their world championship match (and they had played a decent number of games up to that point). However, in this count, there were also games that both players played in their youth, before reaching their current strength and development.

Nepo, despite having a huge talent for the game, slacked off, while Carlsen studied chess intensely. It would have been easy to find an RPSC at that point. Only find an opponent to who Nepo lost at some point in his life and who also played Magnus (and probably lost). One such players is Mateusz Bartel. He won a quite spectacular game against Nepo in 2015. Bartel lost against Carlsen in the Bundesliga in 2008. There you have your RPSC with well-known players (there are much more and probably better examples - I just took the first fitting player in alphabetical sorting).

Is this statistically "significant"? No. And with more games played, Nepo does not have a positive score against the stronger (w.r.t. Elo) player Carlsen. This should happen to about every such configuration you find in the data: have more games played, and the stronger players will eventually win the majority of the games. This is what Elo measures.

Now you could ask: what about style, or "most-feared-opponents"? If you only look at the top level (2600+), you'll find that those players are generally strong, without major weaknesses, displaying a universal style. Besides, the concept of "style" usually gets oversimplified. GM Karsten Müller very recently published an interesting book about player types and explains, that positional/tactical isn't a good/sufficient classification.

There, he analyzes the playing styles of world champions and that most of them have a similar style, with very few exceptions. The most notable is Tal, but according to Müller, players with his style could not succeed in winning the world championship matches in today's times with defensive techniques being better developed and chess engine preparation. Therefore, you will most likely have to move to lesser-known players (probably IM level) or go back in time and compare historical players.

I would reject any such configuration between 2600+ players in today's time as a statistical fluke.

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