Those medalling kids: knockout tournaments and who wins what?

(This piece is based on a paper I read recently... but I can't find a reference for it. If you know which paper I mean, please let me know and I'll update.)

There's a reason tennis knockout draws are seeded. I'll get to why in a moment. But first, let's do a thought experiment.

Imagine you have four people competing in a knockout tournament -- let's call them Alice, Bob, Clive and Doris. Alice is clearly the best player, followed by Bob, who is much better than Clive, who will always beat Doris.

Clearly, Alice will win every game she plays and walk off with the gold, assuming she passes the doping control. (I have my suspicions, personally.) The most interesting question, then, is how likely is Bob to get the silver medal he deserves?

Well, you'd think it would be certain in that kind of set-up, since he always beats Clive and Doris. However, in an unseeded draw, it's not -- 1/3 of the time, he'll have to settle for a bronze. That's because sometimes he's unlucky enough to meet Alice before the final -- and ends up thrashing Doris in the bronze medal match without ever getting the chance to beat Clive to a pulp.

'Sometimes' isn't very precise, is it? Why is it a 1/3 chance? The simplest way to think of it is to consider Alice's possible opponents in the first round: she's equally likely to play against any of the other three, so the probability of meeting Bob in particular is 1/3.

A bigger knockout draw

When there are more people, things get even worse for poor Bob. The Olympic tennis tournament has 64 players in. If it were unseeded, he'd have only a 32/63 chance -- only a smidge more than 50-50 -- of winning a silver medal. His chances of a bronze are also small -- only 16/63. The other 15/63 times, he meets Alice before the semifinal and goes off empty-handed, muttering under his breath and making comments along the lines of 'I'm not saying Alice is a drugs cheat, but...'.

How about Clive? He has a decent chance of silver, in fact, but it's not easy to work out: Bob has to be in Alice's half of the draw (31/63) and Clive in the other half (32/62), making a probability of 996/3906, or 25.5%. His bronze chances are pretty good, too -- as long as he avoids both Alice and Bob until the semis, he'll be on the podium. The chances of that are pretty good -- I reckon a little more than 25% again.

(Turning to a football knockout tournament: when a team in the Champions League wins the FA Cup -- an unseeded draw -- the runner-up qualifies for the UEFA cup. That's dicey: even without upsets, there's barely a 50-50 chance of them being the second-best team. But I digress.)

This is why tennis knockout tournaments are seeded: it guarantees that the best players are kept apart until the later rounds -- which means that barring upsets, you'll always have an Alice-Bob final and Clive will always get the bronze. The best players[1] will end up winning -- which is the whole point of a tournament, isn't it?


[1] as long as the seedings are correct, which is a whole other argument.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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