E. Ben-Naim, F. Vazquez, S. Redner
Theoretical Division and Center for Nonlinear Studies,
Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Department of Physics, Boston University, Boston, Massachusetts 02215
What is the most competitive team sport? We answer this question via a statistical survey of game results [1,2,3,4]. We relate parity with predictability, and propose the likelihood of upsets as a measure of competitiveness.
We studied the results of all regular season competitions in 5 major professional sports leagues in England and the United States (table I): the premier soccer league of the English Football Association (FA), Major League Baseball (MLB), the National Hockey League (NHL), the National Basketball Association (NBA), and the National Football League (NFL). NFL data includes the short-lived AFL. We considered only complete seasons, with more than 300,000 games in over a century [5].
The winning fraction, the ratio of wins to total games, quantifies
team strength. Thus the distribution of winning fraction measures the
parity between teams in a league. We computed , the fraction of
teams with a winning fraction of
or lower at the end of the
season, as well as
, the standard deviation in winning fraction. Here
denotes the average over all teams and all years
using season-end standings. For example, in baseball where the winning
fraction
typically falls between
and
, the variance
is
. As shown in figures 1 and 2a, the winning fraction
distribution clearly distinguishes the five leagues. It is narrowest
for baseball and widest for football.
Do these results imply that MLB games are the most competitive and NFL
games the least? Not necessarily! The length of the season is a
significant factor in the variability in the winning fraction. In a
scenario where the outcome of a game is completely random, the total
number of wins performs a simple random walk, and the standard
deviation is inversely proportional to the square root of the
number of games played. Generally, the shorter the season, the larger
. Thus, the small number of games is partially responsible for
the large variability observed in the NFL.
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To account for the varying season length and reveal the true nature of
the sport, we set up mock sports leagues where teams, paired at
random, play a fixed number of games. In this simulation model, the
team with the better record is considered as the favorite and the team
with the worse record is considered as the underdog. The outcome of a
game depends on the relative team strengths: with the ``upset
probability'' , the underdog wins, but otherwise, the favorite
wins. Our analysis of the nonlinear master equations that describe
the evolution of the distribution of team win/loss records shows that
decreases both as the season length increases and as games
become more competitive, i.e., as
increases [6]. In a
hypothetical season with an infinite number of games, the winning
fraction distribution is uniform in the range
and as a
result,
.
We run Monte Carlo simulations of these artificial sports leagues,
with sport-specific number of games and a range of values. We
then determine the value of
that gives the best match between the
distribution
from the simulations to the actual sports
statistics (figure 1). Generally, we find good agreement between the
simulations results and the data for reasonable
values.
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To characterize the predictability of games, we followed the
chronologically-ordered results of all games and reconstructed the
league standings at any given day. We then measured the upset
frequency by counting the fraction of times that the team with the
worse record on the game date actually won (table I). Games between
teams with no record (start of a season) or teams with equal records
were disregarded. Game location was ignored and so was the margin of
victory. In soccer, hockey, and football, ties were counted as 1/2 of
a victory for both teams. We verified that handling ties this way did
not significantly affect the results: the upset probability changes by
at most
(and typically, much less) if ties are ignored.
We find that soccer and baseball are the most competitive sports with
and
, respectively, while basketball and football,
with nearly identical
and
, are the least. There
is also good agreement between the upset probability
,
obtained by fitting the winning fraction distribution from numerical
simulations of our model to the data as in figure 1, and the measured
upset frequency (table I). Consistent with our theory, the variance
mirrors the bias,
(figures 2a and 2b). Tracking the
evolution of either
or
leads to the same conclusion: NFL
and MLB games are becoming more competitive, while over the past 60
years, FA displays an opposite trend.
In summary, we propose a single quantity, , the frequency of
upsets, as an index for quantifying the predictability, and hence the
competitiveness of sports games. We demonstrated the utility of this
measure via a comparative analysis that shows that soccer and baseball
are the most competitive sports. Trends in this measure may reflect
the gradual evolution of the teams in response to competitive pressure
[7], as well as changes in game strategy or rules [8].
Our model, in which the stronger team is favored to win a game
[6], enables us to take into account the varying season length
and this model directly relates parity, as measured by the variance
with predictability, as measured by the upset likelihood
. This connection has practical utility as it allows one to
conveniently estimate the likelihood of upsets from the more
easily-accessible standings data. In our theory, all teams are equal
at the start of the season, but by chance, some end up strong and some
weak. Our idealized model does not include the notion of innate team
strength; nevertheless, the spontaneous emergence of
disparate-strength teams provides the crucial mechanism needed for
quantitative modeling of the complex dynamics of sports competitions.
We thank Micha Ben-Naim for assistance in data collection and acknowledge support from DOE (W-7405-ENG-36) and NSF (DMR0227670 & DMR0535503).