Universal scaling in sports ranking

Ranking is a ubiquitous phenomenon in the human society. By clicking the web pages of Forbes, you may find all kinds of rankings, such as world's most powerful people, world's richest people, top-paid tennis stars, and so on and so forth. Herewith, we study a specific kind, sports ranking systems in which players' scores and prize money are calculated based on their performances in attending various tournaments. A typical example is tennis. It is found that the distributions of both scores and prize money follow universal power laws, with exponents nearly identical for most sports fields. In order to understand the origin of this universal scaling we focus on the tennis ranking systems. By checking the data we find that, for any pair of players, the probability that the higher-ranked player will top the lower-ranked opponent is proportional to the rank difference between the pair. Such a dependence can be well fitted to a sigmoidal function. By using this feature, we propose a simple toy model which can simulate the competition of players in different tournaments. The simulations yield results consistent with the empirical findings. Extensive studies indicate the model is robust with respect to the modifications of the minor parts.


I. INTRODUCTION
Most systems in nature have been perceived as a collection of a huge amount of highly interacting units. Although equipped with different components and interactions, different systems could still possess some commonly shared characteristics. Through analyzing the statistical distributions of varieties of empirical quantities, two main patterns of distributions [1] have been found.
For the first one, there is a typical value, around which most quantities distribute tightly clustered [2], or that is to say, such pattern of distributions are peaked around this typical value. Representative examples include the height of human beings, the speed of cars on the motorway, the intelligence quotient (IQ) test of people, etc. In the traditional IQ test for adults, most people (about 69% of the population) would score in the average range (85-114), a small number (about 26% of the population) would score moderately below average (70-84) and moderately above average (115-129), very high (130 or higher) and very low (below 70) are extremely rare (about 5% of the population).
However, not all quantities could be well characterized by their average values, some would change over an enormous dynamic range, sometimes even many orders of magnitude. Such pattern has absolutely long been familiar with, in the studies regarding the distributions of people's annual incomes [3], word frequencies in text [4], and city sizes [5]. For the city sizes distribution, if cities are ranked by their population from the largest (rank 1) to the smallest (rank N), it is immediately discovered that, only a small number of cities possess the large population, the majority of cities have the small population.
Relationship between rank and city sizes has been found to follow ln rank = α + β ln size, with the slope of the curve β being close to -1, which has been well known as Zipf 's law.
A more general expression of such pattern is the power law distribution, with p(x) = Ax −β , A = e α , and x is the observation of the system. Power law distributions have been considerably widely observed in nature, such as the net worth of the richest individuals in the US [2], the frequencies of occurrence of words in most human languages [2,4,5], the frequencies of family names in most cultures [6], the number of calls received by customs [7,8], the number of bytes of data received from computer users [9], the number of hits on the web sites [10], the number of links to web sites [11], the number of citations received by papers [12], the sizes of computer files (such as the email address books) [13,14], the sizes of earthquakes [15], wars [16], craters on the moon [17] and solar flares [18], the severity of worldwide terrorist attacks [19], the number of species per genus of mammals [20], the number of sightings of birds of different species [21], the sales of books [22], and music recordings [23,24], etc.
As is well known, ranking is a very interesting and ubiquitous phenomenon in the human society as every one tends to seek the best. By clicking the web pages of Forbes you can find all kinds of rankings, from world's most powerful people to world's richest people, from toppaid models to American's top colleges, etc. Our interest here is certainly not the gossip-like topic, but rather whether there are some common patterns in the vastly different ranking systems. Moreover, if yes, can we understand the formalism of such patterns? To facilitate our study we choose a specific kind of ranking systems, sports ranking, in which data are more suitable for analysis. Here players' performance in attending various competitions will be used as the basis of their respective rankings, in terms of scores and/or prize money. Amazingly we find that the distributions of scores and/or prize money follow universal power-laws, with exponents being nearly identical for different sports fields. The universal scalings can be reproduced by our model in which the key mechanism is concerned with win-loss probability distribution for any pair of players. This win-loss probability distribution has been verified by the empirical data. Our model is found to be robust with respect to the small modifications of minor parts.

II. EMPIRICAL RESULTS OF SPORTS RANKING SYSTEMS
To understand how a certain sports ranking system works, let us take tennis as an example. ATP (Association of Tennis Professionals) and WTA (Women's Tennis Association) are world's most successful tennis associations for male and female professionals, respectively. To appear on the ranking systems of ATP or WTA, the number of tournaments a player has to play should reach a minimum, say 10. Tournaments have been divided into several categories, such as grand slams, premier tournaments, international tournaments and year-ending tour championships, mainly based on the prize money. For the most important tournaments such as grand slams, the main draw only consists of 128 players. The entry rule is that if you are top-ranked, then you have more chances to attend the important tournaments. On the other hand, players' good performance will improve their rankings which will in turn entitle them more chances to play tournaments. Since there are so many tournaments each year, for both ATP and WTA, the ranking list of scores and of prize money vary from week to week. Here we are not interested in which specific player is world No.1 in certain sports, but instead the statistical distribution of performance, measured by scores and prize money, of all the member players. What is the form of such a distribution? Is it stable over different time periods? Is it universal? Our data sets cover 12 different sports fields, such as tennis, golf, snooker, and volleyball, etc. All the data are updated up to February 2011. As the sample size of data is small, we adopt the cumulative distribution to reduce the potential statistical errors.

A. Cumulative distribution of scores
A player's score or prize money is a direct measure of his/her performance in various competitions. The higher the score, the better the performance. The statistical distribution of scores or prize money reflects the profile of the performance of all the members belonging to the same association. Every sports field has its own scoring system, hence the orders of scores are not always at the same level. In order to make the distributions of scores or prize money comparable for different sports fields, we rescale the quantities of interest. That is, where S denotes the values of quantities considered, e.g., scores or prize money, and S max is the maximum value of S in the sample, which pertains to the No. 1 player in the ranking list by using S. Cumulative distributions of players' scores or prize money have been shown in Fig. 1 for 12 different sports ranking systems. Amazingly all the distributions share very similar trend, which can be well fitted to the powerlaw with an exponential cutoff as below, where τ and S c are critical exponent and size cutoff, respectively. It should also be noticed that for the same field, all the curves collapse with each other. Values of τ and S c for different sports fields are given in Table 1, where values of τ range from 0.01 to 0.39, and the counterparts of S c , from 0.12 to 0.28. We employ the Kolmogorov-Smirnov (KS) test to quantify how closely the power laws with exponential cutoffs resemble the actual distributions of the observed sets of samples. Based on the observed goodness of fit, the p-value, which is defined to be the probability that the real data are drawn from the hypothesized distribution, is calculated for each set of sample. The p-values given in Table 1 for the statistical significance test are all much larger than 0.1, thereby we could conclude the power laws with exponential cutoffs are reliable fits to the samples of different sports ranking systems.
The evidence of the power-laws in the sports ranking indicates that there is still significant probability to have superman such as Roger Federer in tennis or Tiger Woods in golf. But the prevalent probability is still the players who do not play in the top form. Unlike the human height system, it seems there is no typical player who plays with average level. The power-laws found here are also different from Zipf's law in which the critical exponent is -1, much larger than ours (in absolute value).

B. Pareto principle
The Pareto principle [25], also well known as the 80-20 rule, states that, for many events, roughly 80% of the effects comes from 20% of the causes. Pareto noticed that, 80% of Italy's land was owned by 20% of the population. He carried out such surveys on a variety of other countries further, and to his surprise, the rule was also fulfilled.
The 80-20 rule has also been used to attribute the widening economic inequality, which showed that, the   Table 1.
distribution of global income to be very uneven, with the richest 20% of the world's population controlling 82.7% of the world's income. The 80-20 rule could be applied to many systems, from the science of management to the physical world.
We also check this rule in the sports ranking systems, it is interesting to find that, 20% players indeed possess approximately 80% scores or prize money of the whole system, the ratios we got in different sports ranking sys-tems are shown in Table. 1, values of the ratios are all very close to 0.8.

C. Dependence of win probability on ∆ rank
Here we employ the concept of "win probability" to describe the chances that a player or a team will win when encountering an opponent. For instance, what is What is again her chance against No.2? Theoretically, the chance is much higher in the former case than in the latter one. But the result of a competition is not unknown until it is over, which mainly depends on how the player performs at that specific match. However, the win probability could be solely based on the previous performance of a player against a certain opponent, which then can used to predict her future performance against the same opponent. This might have some applications in betting the result of a match. To simplify the case without loss of generality, we relate the win probability solely to the rank difference of a pair of players. Suppose we now have two players A and B, with A having a higher rank. We will then need to know how likely A can beat B when they meet? This quantity is related to but different from the win percentage we usually refer to. The win percentage depicts the percentage of win of a player over all previous encounters. We assume that the win probability only depends on the rank difference between two players. This means, the probability that No.1 beats No.100 is the same as the one that No.100 beats No.200. Hence, we have the following definition, where ∆r denotes the rank difference (integer), N win (∆r) is the total number of win for the higher-ranked players when the rank difference is ∆r, and N total (∆r) is the total number of matches in which the rank difference between the pair is ∆r. We here emphasize again that the win probability is the probability that the higher-ranked player will win when two players meet. When ∆r is small, say 1, it is difficult to judge which player will win, and in this case P win might approximately equal 0.5. When ∆r is large, for instance 100, P win might approach 1, which means the higher-ranked player is very likely to win. By using the Head to Head records of ATP and WTA, we find that the dependence of P win on ∆r can be well fitted to the sigmoidal function as follows, where a is a parameter dependent on the specific systems. For ATP and WTA, a is 0.021 and 0.032, respectively. The existence of fluctuations is quite natural since even Roger Federer will not win all the matches. The value of a can still tell us some information about how competitive that certain sports is. The smaller a is, the more competitive the sports will be. Let us take WTA and ATP as two examples. When ∆r is 30, the win probability for WTA is nearly 0.7, while the counterpart for ATP is 0.65. This means the game is more unpredictable in ATP than in WTA. It is not strange since men's game is more competitive than women's. We of course wish to test the empirical finding by checking data from other sports fields, but not so many data are available as far as we know. Despite this, this finding will play a key role in our model to follow.

III. A SIMPLE TOY MODEL OF SPORTS RANKING SYSTEMS
What is the origin of the universal scaling in different sports systems? Of course, there have been so many approaches which can explain the origin of power-laws. Some mechanisms or theories are elegant, e.g., random walks [2], self-organized criticality (SOC) [26], etc. It is, however, difficult to try to apply these frameworks to sports ranking systems. We propose a simple toy model, inspired by tennis. Of course, the model may not suit any sports field but does have some general implications. Most importantly, our model can reproduce robust power-laws without having to introduce additional parameters. The rules of the model are defined in the following way, (1) 2 N players are ranked from 1 to 2 N , being assigned random scores drawn from a Gaussian distribution.
(2) For each tournament, all the players have entry permission. Therefore the draw will include 2 N players and in total N rounds. At each round, half of the players will be eliminated when they lose. The rest will enter the next round. The losers at round n will gain score 2 (n−1) . The final champion wins score 2 N .
(3) The key mechanism is to decide which one will lose for a given pair of players. Here our empirical finding will be employed. Namely, when two players meet, the probability that the higher-ranked player will top the lower-ranked opponent is given by 1/(1 + exp(−a * ∆r)), where ∆r is their rank difference, as before.
(4) A new tournament opens up and a new draw is made.
In principle, there is only one parameter in our model, that is a. We can simply call it competition strength. Of course, there are some shortcomings in the model. First, in the actual tournaments not all the players will be accepted. In grand slams there are only 128 players. Second, tournaments can be divided into many categories and may consist of different players. Third, the scoring systems for different tournaments are a little different. For grand slams the scores and prize money are much higher than other tournaments, if the players are eliminated at the same round. We certainly can add these issues into our model in order to test the resilience of the model. At the moment we do not wish to complicate the model by introducing additional parameters. What we need here is a skeleton which may allow us to understand some key features of the specific systems. Namely, if the power-laws with exponential cutoffs can be reproduced through our model, then it is a feasible model. We need not to care about other minor issues.

IV. SIMULATION RESULTS AND DISCUSSIONS
The most important parameter in our model is a, the so-called competition strength. The number of players N p and the number of tournaments N t only have finitesize effects. It is natural to check the dependence of the simulation results on these parameters, which can reflect the resilience of our model.
First of all, we need to test whether the model can reproduce the power-laws of the cumulative distribution of scores. In Fig. 4, N p equals 2048, and N t is 128, while win probability, P win = 1/(1 + exp(−a * ∆r)), with a = 0.021 and 0.032, as given by the empirical data of ATP and WTA, respectively. We find that, the cumulative distributions of scores given by the simulations indeed follow the power-law distributions with exponential cutoff, P (S) ∝ S −τ exp(−S/S c ), with τ = 0.2, 0.22, S c = 0.23, 0.19, respectively for these two samples. Here we notice that the values of the parameters are very close to what are obtained from the experimental data. In the formula of win probability, smaller values of a correspond to more intensive competition. For instance, when a = 0.0001, P win 0.525 for ∆r 1000, which means higher-ranked player only has slightly more chances than the lower-ranked player to win the match between them. While larger values of a suggest that the higher ranked players would win the match with a much larger probability. For example, when a = 2.0, P win 0.88 for ∆rank 1.
Thus here, to analyze the influence of win probability, we simulated our models with different values of a, 0.0001 a 2.0. From Fig. 5, we can find that the cumulative scores distributions change from the power laws with exponential cutoff to exponential. Since when a is very small, such as a = 0.0001, all players nearly win the match randomly, thus the cumulative probabilities of the scores approximately like 1, 1/2, (1/2) 2 , ..., which results the exponential format. For different number of tournaments, N t = 64, 128 and 256, the cumulative distributions of scores are shown in Fig. 6, as seen, one could discover, all the cumulative distributions of scores are power-laws with exponential cutoff, values of the critical exponents τ and size cutoff S c are also very close to those of the empirical results.
In statistical physics, in order to determine the validity of the statistical approach, we often take the thermodynamic limit, in which the number of components N tends to infinity [27]. However, in real world networks, the number of vertices or agents can never be that large, this makes the factor of finite size of paramount importance. For example, even the largest artificial net, the World Wide Web, whose size will soon approach 10 11 Web pages, also shows qualitatively strong finite-size effects [28]. Therefore, here, in order to test the influence of the finite size effect on the final cumulative scores distribution, we considered the transformed score distribution P (S) * S τ versus S/S c , where S c is the characteristic size cutoff. For four different system sizes, such relationships were shown in Fig. 7, which suggest that, the tails of the four curves almost collapsed with each other, thereby, we can conclude the finite size effect is almost negligible.

V. CONCLUSION
In summary, to characterize the intrinsic common features and underline dynamics of ranking systems, we carry out the investigations in an applicable and specific kind of ranking systems, the sports ranking systems, the main results are: (i) The universal scaling law is extensively found in the distributions of scores and/or prize money, in addition, values of the critical exponents are close to each other for 40 samples of 12 sports ranking systems. (ii) Players' scores are discovered to obey the Pareto principle, which means, 20% of players approximately possess 80% of total scores of the whole system. (iii) Win probability is introduced to describe the chance that a player or a team will win when meeting an opponent, we simply relate the win probability solely to the rank difference ∆r, for tennis sport, the win probability has been empirically verified to follow the sigmoid function, P win = 1/(1 + exp(−a * ∆r)). (iv) By employing the empirical features of win probability, we proposed a simple toy model to simulate the real process of the sports systems, the universal scaling could be well reproduced by our model, moreover, this result is robust when we change the values of parameters in the model.
We are expecting to find such similar scaling laws in other ranking systems, and we hope all these results and methods could be well applied to analyzing any type of paired competitions, or solving some practical problems in the ranking systems.