[jeffsco]

Imagine a tournament to choose the best tennis player. If you imagine these dice as a simple model of a tennis player, there can be no best tennis player. So tennis tournaments make no sense. This is paradoxical if you believe in tennis rankings.

[CamperBob2]

It makes sense to me that the best athlete cannot be determined in some scenarios, or that the term "best" is simply meaningless. Due to training and/or genetics, player A might be strong in areas where player B is weak. Likewise, player B might be strong in areas where player C is weak.

Neither of these statements gives us any insight into what will happen if player A faces player C, but a traditional tournament structure might well keep A and C from ever facing each other.

[mziel]

It's not. Let's assume that ">" means probability of winning over 0.5

Federer > Djokovic (well not lately)

Djokovic > Nadal

Nadal > Federer

Everything depends on the draw. Same with the initial choice of dice.

[wolfgke]

> This is paradoxical if you believe in tennis rankings.

So the solution is to go the rational way and stop believing in ranking if you claim that tennis ranking is based on a mathematical model that has the non-transitivity property. Where exactly is the problem?

[quietplatypus]

Uhh, what if I told you that like most people with common sense and who realize that the initial conditions of a tournament are fixed in advance and that the ranking at the end only applies for that tournament and there are many other factors that influence who wins and loses in each match, I take tennis rankings with a grain of salt?

Hell I can even accept that player A > B > C > A if each has different skills and specialities that they bring to the table, without trapping myself and others into declaring it's some "paradoxical nonsense". Do we say rock < paper < scissors < rock and declare it's the end of mathematics?

[rumbo]

Does anyone anywhere say that if something may appear to some people paradoxical (have you ever thought about what paradoxical means?) it is the end of mathematics?

[Veedrac]

A more revealing model might be a race, since one's performance is not obviously linked to another's and there's only one variable being measured (time to completion).

[quietplatypus]

Not even true in that case, because races often come down to luck with pit stops and how aggressive other drivers are in protecting their space. If you can't pass the dude without rear ending him, you're not going to pass him, doesn't matter how better your "single-racer" driving skills or faster your car is on paper.

Even in footraces, the competitors still have influence on each other with drafting and late starting and whatnot.