Alex Holowczak wrote: ↑
Wed Apr 11, 2018 3:50 pm
Chris Goodall wrote: ↑
Wed Apr 11, 2018 3:40 pm
Because you're not doing the maths properly. You're adding the numbers together as if they're time-invariant. But the round in which you're expected to face seed p
has a significant bearing on your probability of actually facing seed p
in that round. If you're seeded 128, you've got a 100% chance of facing the top seed in round 1. There's no possible way of getting out of it, even if the top seed only has 1 pair of shorts. If you're one of the 64 players in the opposite half of the draw to the top seed, your chance of facing the top seed in the final is significantly less than 100%, because of the significant possibility that someone in the preceding 6 rounds will have knocked off the top seed for you, as indeed Xiangzhi did to Carlsen in round 3.
This is true, but if you were to have percentages as the probability of which seed gets to the Final, then 1 would be at the top of the list; even if that number would be < 50%.
I think it would be possible to tweak my spreadsheet to reflect the odds based on the Elo ratings of the players who entered. If you then ran 1000 simulations using those odds, you might get another useful piece of information. But that's beyond both my Excel skills and time available...
I've done some Excelling, and the upshot is I still think I'm right, but for different reasons.
That being pencilled in to face p
in a later round decreases your chance of facing p
at all is a factor, but not necessarily the decisive factor. You can create scenarios where this is the decisive factor (a single "monster" and 127 players of equal strength), but you can also create scenarios where your wibbly wobbly line shows up (Zipf).
There's another factor though, which is: at what point are you comparing each player's projected route difficulty? If you're comparing it before round 1, then the relationship is linear: everyone plays the seed that is 129 minus their own seed.
If you're comparing it after
round 1, then you have to make sure you're comparing like with like. If you're the 128th seed and you've just beaten the top seed, your future route through the tournament may appear to be easier than the 127th seed's route was at the start of the tournament. But that doesn't take into account that the 127th seed's route changed when you beat the top seed. Instead of going through the top seed, the 127th seed's route now goes through you. Or more likely, it goes through the 4th seed. You can't compare your own route in a world where you've beaten the top seed, to everyone else's route in a world where you haven't.
EDIT - this paper
seems to confirm that, if you define envy-freeness
as the absence of any player who would prefer to swap seedings with a lower-seeded player before the tournament starts, it's always possible to contrive some set of win probabilities that violates the envy-freeness condition, such as my "monster" competition that I mentioned above. Intuitively, if there's a monster in your competition, facing the monster as late as possible is to your advantage.
Research question, then: what can we say about the envy-freeness of a tournament where the distribution of playing strengths is log-normal, like it is for elite chess players (until you get to the top 10 in the world, who are all a bit undergraded relative to the rest because they don't have as many higher-graded players they can draw with)?