Universal statistics of the knockout tournament.
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We assign a real number called competitiveness to each contestant and find that the resulting distribution of prize money follows a power law with an exponent close to unity if the competitiveness is a stable quantity and a decisive factor to win a match.Otherwise, the distribution is found narrow.The existing observation of power law distributions in various kinds of real sports tournaments therefore suggests that the rules of those games are constructed in such a way that it is possible to understand the games in terms of the contestants' inherent characteristics of competitiveness.
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Affiliation: Department of Physics, Pukyong National University, 608-737 Busan, Korea.
ABSTRACT
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We study statistics of the knockout tournament, where only the winner of a fixture progresses to the next. We assign a real number called competitiveness to each contestant and find that the resulting distribution of prize money follows a power law with an exponent close to unity if the competitiveness is a stable quantity and a decisive factor to win a match. Otherwise, the distribution is found narrow. The existing observation of power law distributions in various kinds of real sports tournaments therefore suggests that the rules of those games are constructed in such a way that it is possible to understand the games in terms of the contestants' inherent characteristics of competitiveness. Related in: MedlinePlus |
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Mentions: This implies a tendency that P(k) usually exhibits a power law with an exponent close to unity but that randomness makes the tail shorter. Suppose that f(x) has a finite resolving power, quantified by a characteristic width Γ over which f(x) rapidly increases. The Heaviside step function corresponds to a limiting case of Γ → 0. We can predict the followings when Γ is finite but sufficiently small: At the beginning of the competition, the width σ of pn(r) is much greater than Γ, so f(x) effectively serves as a step function. The above analysis shows that σ decreases as 2−n so it becomes comparable with Γ after ν ~ log2(1/Γ) rounds. Thereafter, the decrease of σ slows down. Finally, when after many rounds, the survivors' competitiveness is irrelevant and the outcomes are mostly determined by pure luck. Therefore, a natural guess for P(k) would be with kΓ ~ O(zν) and γ in equation (1). This functional form is confirmed in our numerical simulations (Fig. 4). This distribution can also be derived from the maximum entropy principle as in equation (10) but with an additional constraint on Σk ln k1314, which corresponds to the total number of fixtures in this context. The above argument can be pursued further by employing the following f(x): where the exponential functions make it possible to explicitly evaluate the integral. Then, the winning chance is given as which approaches c0(r) = r as Γ → 0 and c0(r) = 1/2 as Γ → ∞, as expected. As above, this yields which is normalised to unity as . This result is quite suggestive, because equation (16) modifies equation (4) at n = 1 by adding O(Γ) when and subtracting the same amount when [Fig. 5(a)]. In short, p0(r) becomes flatter when r is close to 0 or 1. If we take one step further, the low-r correction becomes less important and we find where we have left only the dominant correction of O(Γ) [Fig. 5(b)]. For general n, the result up to the correction of O(Γ) is inductively found as This implies that the finite resolution is most noticeable among highly competitive players with , whereas the story looks similar to the case of perfect resolution when (1 − r) is small but still much larger than Γ. |
View Article: PubMed Central - PubMed
Affiliation: Department of Physics, Pukyong National University, 608-737 Busan, Korea.