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Universal statistics of the knockout tournament.

Baek SK, Yi IG, Park HJ, Kim BJ - Sci Rep (2013)

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, Pukyong National University, 608-737 Busan, Korea.

ABSTRACT
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.

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(a) Probability distribution of r at the 5th round when f(x) is the Heaviside step function, equation (2). The data points are obtained numerically by simulating 104 tournaments with N = 212 and the line shows our analytic prediction in equation (4). (b) Average value of r at the nth round, where the data points are obtained numerically and the line represents equation (5).
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f3: (a) Probability distribution of r at the 5th round when f(x) is the Heaviside step function, equation (2). The data points are obtained numerically by simulating 104 tournaments with N = 212 and the line shows our analytic prediction in equation (4). (b) Average value of r at the nth round, where the data points are obtained numerically and the line represents equation (5).

Mentions: We can extract various useful information from this probability density function. For example, the average competitiveness after the nth round is and therefore the width of pn(r) decreases as σ ~ 2−n. A contestant with r passes the nth round but not the next one with probability where we have used wk = ck and the sum over n is normalised to unity for any r between zero and one. The average prize money for this person with r can thus be calculated as As shown in Fig. 2, qn has a peak at and the summations above can be approximated as If kn = zn, it means that in the vicinity of r = 1. Note that we have approximated r as unity at the denominator of equation (8). Therefore, Zipf's plot shows a power law with slope −log2z, leading to P(k) ~ k−γ with γ = (log2z)−1 + 1 due to the relationship between Zipf's plot and P(k)12. This exactly coincides with equation (1) derived for a single tournament. We have numerically performed tournaments and the results confirm validity of our analysis as shown in Fig. 3, where the numerical calculations of c5(r) and 〈r〉n agree perfectly with the analytic results. The detailed procedure of our simulation is explained in Methods.


Universal statistics of the knockout tournament.

Baek SK, Yi IG, Park HJ, Kim BJ - Sci Rep (2013)

(a) Probability distribution of r at the 5th round when f(x) is the Heaviside step function, equation (2). The data points are obtained numerically by simulating 104 tournaments with N = 212 and the line shows our analytic prediction in equation (4). (b) Average value of r at the nth round, where the data points are obtained numerically and the line represents equation (5).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3824171&req=5

f3: (a) Probability distribution of r at the 5th round when f(x) is the Heaviside step function, equation (2). The data points are obtained numerically by simulating 104 tournaments with N = 212 and the line shows our analytic prediction in equation (4). (b) Average value of r at the nth round, where the data points are obtained numerically and the line represents equation (5).
Mentions: We can extract various useful information from this probability density function. For example, the average competitiveness after the nth round is and therefore the width of pn(r) decreases as σ ~ 2−n. A contestant with r passes the nth round but not the next one with probability where we have used wk = ck and the sum over n is normalised to unity for any r between zero and one. The average prize money for this person with r can thus be calculated as As shown in Fig. 2, qn has a peak at and the summations above can be approximated as If kn = zn, it means that in the vicinity of r = 1. Note that we have approximated r as unity at the denominator of equation (8). Therefore, Zipf's plot shows a power law with slope −log2z, leading to P(k) ~ k−γ with γ = (log2z)−1 + 1 due to the relationship between Zipf's plot and P(k)12. This exactly coincides with equation (1) derived for a single tournament. We have numerically performed tournaments and the results confirm validity of our analysis as shown in Fig. 3, where the numerical calculations of c5(r) and 〈r〉n agree perfectly with the analytic results. The detailed procedure of our simulation is explained in Methods.

Bottom Line: 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.

View Article: PubMed Central - PubMed

Affiliation: Department of Physics, Pukyong National University, 608-737 Busan, Korea.

ABSTRACT
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.

Show MeSH
Related in: MedlinePlus