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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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Schematic illustration of a tournament with four contestants A, B, C, and D.Contestant B has competitiveness rB and gets prize money kB = z2 because she has defeated A and C. Likewise, C gets kC = z1 because she has won only a single match against D.
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f1: Schematic illustration of a tournament with four contestants A, B, C, and D.Contestant B has competitiveness rB and gets prize money kB = z2 because she has defeated A and C. Likewise, C gets kC = z1 because she has won only a single match against D.

Mentions: In this work, we instead focus on statistical analysis of a specific system of competition, i.e., the knockout tournament among inhomogeneous participants. Our main point is that a large part of statistics is universal in the sense that it is independent of most details of the game but already determined by the tournament structure. Let us consider a player's number of wins denoted by n, for example. When the tournament has been finished, the distribution of n denoted by P(n) is always an exponentially decreasing function of n. It is a purely geometric property of the tournament tree independent of any details of the game, loosely mapped to the critical percolation on a binary tree10. If the prize money is highly skewed towards the best players, similarly to real sports tournaments, one can assume that the prize money kn after winning n rounds is also an exponential function of n, that is, kn ~ zn (Fig. 1). Combining these two, one finds that the distribution P(k) ~ k−γ with and this mechanism belongs to combination of exponentials according to Newman11. If z gets very large, γ converges to unity, yielding P(k) ~ k−1. As z → 1, on the other hand, γ diverges because P(k) approaches the distribution function of n, which is an exponential function. In fact, if z < 2, the total amount of prize money gets unbounded as the number of contestants grows, which means that the organiser of this tournament has a risk of bankruptcy. This explains why kn has to be such a rapidly increasing function of n, and we see that the feasible range of γ is between one and two. Moreover, if there is a typical number of prize winners, z is effectively very large, driving γ to unity. This is a simple prediction for a single tournament. In other words, this analysis corresponds to gathering data of prize money distributed over many tournaments without identifying who was who. The actual statistics collected in this way, however, will not be very interesting to us, and it is usually more meaningful to consider individual-based statistics: Even for a team sport, each team may be regarded as an individual. It is notable that Deng et al. resolve this problem by introducing the notion of ranks, belonging to individuals, and also by assuming that a player's winning probability against another is a function of their rank difference. Following this approach, we will see how our simple prediction in equation (1) can be reproduced on average in the individual-based statistics.


Universal statistics of the knockout tournament.

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

Schematic illustration of a tournament with four contestants A, B, C, and D.Contestant B has competitiveness rB and gets prize money kB = z2 because she has defeated A and C. Likewise, C gets kC = z1 because she has won only a single match against D.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f1: Schematic illustration of a tournament with four contestants A, B, C, and D.Contestant B has competitiveness rB and gets prize money kB = z2 because she has defeated A and C. Likewise, C gets kC = z1 because she has won only a single match against D.
Mentions: In this work, we instead focus on statistical analysis of a specific system of competition, i.e., the knockout tournament among inhomogeneous participants. Our main point is that a large part of statistics is universal in the sense that it is independent of most details of the game but already determined by the tournament structure. Let us consider a player's number of wins denoted by n, for example. When the tournament has been finished, the distribution of n denoted by P(n) is always an exponentially decreasing function of n. It is a purely geometric property of the tournament tree independent of any details of the game, loosely mapped to the critical percolation on a binary tree10. If the prize money is highly skewed towards the best players, similarly to real sports tournaments, one can assume that the prize money kn after winning n rounds is also an exponential function of n, that is, kn ~ zn (Fig. 1). Combining these two, one finds that the distribution P(k) ~ k−γ with and this mechanism belongs to combination of exponentials according to Newman11. If z gets very large, γ converges to unity, yielding P(k) ~ k−1. As z → 1, on the other hand, γ diverges because P(k) approaches the distribution function of n, which is an exponential function. In fact, if z < 2, the total amount of prize money gets unbounded as the number of contestants grows, which means that the organiser of this tournament has a risk of bankruptcy. This explains why kn has to be such a rapidly increasing function of n, and we see that the feasible range of γ is between one and two. Moreover, if there is a typical number of prize winners, z is effectively very large, driving γ to unity. This is a simple prediction for a single tournament. In other words, this analysis corresponds to gathering data of prize money distributed over many tournaments without identifying who was who. The actual statistics collected in this way, however, will not be very interesting to us, and it is usually more meaningful to consider individual-based statistics: Even for a team sport, each team may be regarded as an individual. It is notable that Deng et al. resolve this problem by introducing the notion of ranks, belonging to individuals, and also by assuming that a player's winning probability against another is a function of their rank difference. Following this approach, we will see how our simple prediction in equation (1) can be reproduced on average in the individual-based statistics.

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