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: We have assumed that competitiveness is each individual's inherent characteristic, which changes in a much longer time scale compared to outcomes of competition, and we relate the latter to ranks. The idea is that although a contestant's rank fluctuates over tournaments, it will correctly reflect her true competitiveness in the long run. Even if the competitiveness may interact with actual tournament results, it will usually be related to a cumulative measure of performance that mainly reflects low-frequency, i.e., long-term behaviour. For example, we have calculated the Kendall tau rank correlation coefficient15, denoted by τ, to see how the accumulated amounts of prize money change their relative positions between two successive tournaments (Fig. 6). If a certain pair of contestants keep their relative positions, they are said to be concordant, and discordant otherwise. The coefficient τ is defined as the number of concordant pairs minus that of discordant pairs, divided by the total number of possible pairs. Beginning with the same initial amount of money for every contestant, which is set to zero, we run fifty tournaments in a row, accumulating the prize money for each individual. A contestant's accumulated money from a series of tournaments determines her performance in the next tournament in such a way that r = (N − i)/(N − 1) is assigned to the contestant when she has the ith largest accumulated amount. The relative positions of two equal amounts are random. In spite of this variability, the ranks of the accumulated money get stabilised after 20 or 30 tournaments in all the cases considered (Fig. 6), and the resulting P(k) is almost identical to the static-r case for each Γ. Still, one may ask what happens if their time scales approach each other so that a current rank directly affects performance at the next tournament, provided that the tournaments are regular events. Even if an individual's rank fluctuates over time, it might still be possible for this correlation between successive tournaments to reproduce the power-law tail part of P(k). In fact, this question is not really well-posed because a knockout tournament leaves many contestants' ranks undetermined except a few prize winners, and this is the fundamental advantage of a knockout tournament. We nevertheless suppose that a player's competitiveness at the next time step is a nondecreasing function of the current performance, say, rt+1 = R(nt), where nt is the number of wins in the tournament at time t, and R is a nondecreasing function between zero and one. Since r determines how many rounds the contestant can go through, the distribution of nt+1 is essentially a function of nt. The situation is actually boring because the same contestant wins the first place all the time, but we may exclude this exceptional contestant from our consideration. We begin with noting that any tournament results in a distribution of nt as , which is the initial distribution of the next tournament at time t + 1. The corresponding cumulative portion of contestants with results below nt is thus . As above, if f(x) is the Heaviside step function with f(0) = 1/2, the chance to win the first round for a contestant that passed nt rounds at the previous tournament is . The first term represents the probability to meet an opponent with the same nt, and the factor of one half originates from f(0). The distribution of nt at the next round is p1(nt) = 2w0(nt)p0(nt). We can repeat this procedure to obtain a general expression as with g(x) ≡ 2x. By definition, we have If k is not very small, pk(nt) converges to a certain function of y ≡ k − nt with a maximum around y ≈ 0 [Fig. 7(a)]. The conditional probability to reach k and stop there for given nt is found as with [Fig. 7(b)]. We observe that qk(nt) can also be described as a certain function V(y) when . Moreover, we find that for any nt. In other words, the time series {nt ≥ 0} can be roughly described as a biased random walk towards the origin. Since this holds true for anyone, each contestant's average result will be rapidly equalised by the bias so we predict that the probability distribution P(k) will be narrow. This prediction is well substantiated by numerical results shown in Fig. 8, where P>(k) is drawn in a semi-log plot. Therefore, in terms of the time scale of competitiveness, the power-law shape of P(k) is observable when competitiveness changes much more slowly compared to the frequency of tournaments. |
View Article: PubMed Central - PubMed
Affiliation: Department of Physics, Pukyong National University, 608-737 Busan, Korea.