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Hearing the recent news about disqualified Badminton players in the ongoing 2012 London Olympics got me wondering about how best to design tournaments to avoid situations where players are incentivized to throw matches. I have no doubt that much has been written about this but I have no idea where to start.

Are there any Arrow-like theorems saying that "ideal tournament design" is impossible, i.e. given some short-ish list of generally agreeable desirable features of a tournament, one proves that they are contradictory?

I'm a novice in this sort of mathematics so feel free to recommend introductory surveys or books as well.

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  • $\begingroup$ Googling "tournament design" yields lots of links. $\endgroup$
    – Jay
    Aug 1, 2012 at 22:49
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    $\begingroup$ I remember something from squash which is an individual sport, but your best five players play their best five, and the results are just added. All you needed to do to crush the opponents, assuming equality of players, is lie about it, have your number 1 play their number 2, your number 2 play their number 3, finally your fifth best gets destroyed by their best but your team wins 4-1. $\endgroup$
    – Will Jagy
    Aug 1, 2012 at 22:59
  • $\begingroup$ @Jay It should go without saying that that was the first thing I tried, but I didn't find any appropriate mathematical articles or anything really accessible to me on the first page or so. Perhaps I didn't look carefully enough???? $\endgroup$
    – j.c.
    Aug 1, 2012 at 23:03
  • $\begingroup$ Googling "tournament design mathematics" yields, among other papers, glicko.net/research/knockout.pdf and emba.uvm.edu/~dinitz/preprints/design_tourney_talk.pdf. $\endgroup$
    – Jay
    Aug 1, 2012 at 23:31
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    $\begingroup$ Asked on MO: mathoverflow.net/questions/40877/… Keep in mind that "tournament" has a precise meaning in combinatorics and may not capture the meaning you want. $\endgroup$
    – Qiaochu Yuan
    Aug 1, 2012 at 23:59

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There are no impossibility results "Arrow like" regarding tournament design but lots of open problems especially in the incomplete information case.

Kay Konrad's book on contests is a good general reference.

As tournaments can be modeled as all-pay auctions, the literature on optimal auction design may also be relevant, see Vijay Krishna's book.

I know the above economics literature well but I am not familiar with the computer science one that may be also relevant: see this thesis https://stacks.stanford.edu/file/druid:qk299yx6689/TV-thesis-final-augmented.pdf

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  • $\begingroup$ Thanks for your answer. If I have a chance I will take a look at the references you suggested. I am not familiar with auctions, so is there an easy way to see that a sports tournament is like an all-pay auction? $\endgroup$
    – j.c.
    Dec 17, 2013 at 18:13
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    $\begingroup$ in an all-pay auction, auction you pay your bid even if you lose. Interpret the bid as cost of effort in the sports game. $\endgroup$
    – Sergio Parreiras
    Dec 17, 2013 at 20:21

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