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J Stat Phys (2013) 151:458–474
DOI 10.1007/s10955-012-0648-x
Randomness in Competitions
E. Ben-Naim ·N.W. Hengartner ·S. Redner ·F. Vazque z
Received: 20 September 2012 / Accepted: 14 November 2012 / Published online: 27 November 2012
© Springer Science+Business Media New York 2012
Abstract We study the effects of randomness on competitions based on an elementary ran-
dom process in which there is a finite probability that a weaker team upsets a stronger
team. We apply this model to sports leagues and sports tournaments, and compare the the-
oretical results with empirical data. Our model shows that single-elimination tournaments
are efficient but unfair: the number of games is proportional to the number of teams N,
but the probability that the weakest team wins decays only algebraically with N. In con-
trast, leagues, where every team plays every other team, are fair but inefficient: the top √N
of teams remain in contention for the championship, while the probability that the weak-
est team becomes champion is exponentially small. We also propose a gradual elimination
schedule that consists of a preliminary round and a championship round. Initially, teams
play a small number of preliminary games, and subsequently, a few teams qualify for the
championship round. This algorithm is fair and efficient: the best team wins with a high
probability and the number of games scales as N9/5, whereas traditional leagues require N3
games to fairly determine a champion.
Keywords Competitions ·Social dynamics ·Kinetic theory ·Scaling laws ·Algorithms
E. Ben-Naim ()
Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos,
NM 87545, USA
e-mail: ebn@lanl.gov
N.W. Hengartner
Computing and Computer Science Division, Los Alamos National Laboratory, Los Alamos, NM 87545,
USA
S. Redner
Department of Physics, Boston University, Boston, MA 02215, USA
F. Vazquez
Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Str. 38, 01187 Dresden, Germany
Randomness in Competitions 459
1 Introduction
Competitions play an important role in society [1–4], economics [5], and politics. Further-
more, competitions underlie biological evolution and are replete in ecology, where species
compete for food and resources [6]. Sports are an ideal laboratory for studying competitions
[7–10]. In contrast with evolution, where records are incomplete, the results of sports events
are accurate, complete, and widely available [11,12].
Randomness is inherent to competitions. The outcome of a single match is subject to
a multitude of factors including game location, weather, injuries, etc., in addition to the
inherent difference in the strengths of the opponents. Just as the outcome of a single game is
not predictable, the outcome of a long series of games is also not completely certain. In this
paper, we review1a series of our studies that focus on the role of randomness in competitions
[13–17]. Among the questions we ask are: What is the likelihood that the strongest team
wins a championship? What is the likelihood that the weakest team wins? How efficient are
the common competition formats and how “accurate” is their outcome?
We introduce an elementary model where a weaker team wins against a stronger team
with a fixed upset probability q, and use this elementary random process to analyze a se-
ries of competitions [13]. To help calibrate our model, we first determine the favorite and
the underdog from the win-loss record over many years of sports competition from several
major sports. We find that the distribution of win percentage approaches a universal scal-
ing function when the number of games and the number of teams are both large. We then
simulate a realistic number of games and a realistic number of teams, and demonstrate that
our basic competition process successfully captures the empirical distribution of win per-
centage in professional baseball [14,15]. Moreover, we study the empirical upset frequency
and observe that this quantity differentiates professional sports leagues, and furthermore,
illuminates the evolution of competitive balance.
Next, we apply the competition model to single-elimination tournaments where, in each
match, the winner advances to the next round and the loser is eliminated [16]. We use the
very same competition rules where the underdog wins with a fixed probability. Here, we
introduce the notion of innate strength and assume that entering the competition, the teams
are ranked. We find that the typical rank of the winner decays algebraically with the size of
the tournament. Moreover, the rank distribution for the winner has a power-law tail. Hence,
larger tournaments do produce stronger winners, but nevertheless, even the weakest team
may have a realistic chance of winning the entire tournament. Therefore, tournaments are
efficient but unfair.
Further, we study the league format, where every team plays every other team [17]. We
note that the number of wins for each team performs a biased random walk. Using heuristic
scaling arguments, we establish that the top √Nteams have a realistic chance of becoming
champion, while it is highly unlikely that the weakest teams can win the championship. In
addition, the total number of games required to guarantee that the best team wins is cubic in
N. In this sense, leagues are fair but inefficient.
Finally, we propose a gradual elimination algorithm as an efficient way to determine the
champion. This hybrid algorithm utilizes a preliminary round where the teams play a small
number of games and a small fraction of the teams advance to the next round. The number
of games in the preliminary round is large enough to ensure the stronger teams advance. In
1Most of the results reported in this mini-review appear in [13–17]. For clarity, we present a number of
additional plots including Figs. 1,3,5,8. The scaling arguments in Sect. 4are equivalent to those presented
in [17].
460 E. Ben-Naim et al.
the championship round, each team plays every other team ample times to guarantee that the
strongest team always wins. This algorithm yields a significant improvement in efficiency
compared to a standard league schedule.
The rest of this paper is organized as follows. In Sect. 2, the basic competition model
is introduced and its predictions are compared with empirical standings data. The notion
of innate team strength is incorporated in Sect. 3, where the random competition process is
used to model single-elimination tournaments. Scaling laws for the league format are derived
in Sect. 4. Scaling concepts are further used to analyze the gradual elimination algorithm
proposed in Sect. 5. Finally, basic features of our results are summarized in Sect. 6.
2 The Competition Model
In our competition model, Nteams participate in a series of games. Two teams compete
head to head and, at the end of each match, one team is declared the winner and the other as
the loser. There are no ties.
To study the effect of randomness on competitions, we consider the scenario where there
is a fixed upset probability qthat a weaker team upsets a stronger team [2,13]. This prob-
ability has the bounds 0 ≤q≤1/2. The lower bound corresponds to predictable games
where the stronger team always wins, and the upper bound corresponds to random games.
We consider the simplest case where the upset probability qdoes not change with time and
is furthermore independent of the relative strengths of the competitors.
In each game, we determine the stronger and the weaker team from current win-loss
records. Let us consider a game between a team with kwins and a team with jwins. The
competition outcome is stochastic: if k>j,
(k, j ) →(k +1,j) with probability p,
(k, j +1)with probability q, (1)
where p+q=1. If k=j, the winner is chosen randomly. Initially, all teams have zero wins
and zero losses.
We use a kinetic framework to analyze the outcome of this random process [18], taking
advantage of the fact that the number of games is a measure of time. We randomly choose
the two competing teams and update the time by t→t+t, with t =1/(2N), after each
competition. With this normalization, each team participates in one competition per unit
time.
Let fk(t) be the fraction of teams with kwins at time t. This probability distribution must
be normalized, kfk=1. In the limit N→∞, this distribution evolves according to
dfk
dt =p(fk−1Fk−1−fkFk)+q(fk−1Gk−1−fkGk)+1
2f2
k−1−f2
k,(2)
for k≥0. Here we also introduced two cumulative distribution functions: Fk=k−1
j=0fjis
the fraction of teams with less than kwins and Gk=∞
j=k+1fjis the fraction of teams
with more than kwins. Of course, Fk+Gk−1=1. The first two terms on the right-hand-
side of (2) account for games in which the stronger team wins, and the next two terms
correspond to matches where the weaker team wins. The last two terms account for games
between teams of equal strength (the numerical prefactor is combinatorial). Accounting for
the boundary condition f−1≡0 and summing the rate equations (2), we readily verify that
the normalization kfk=1 is preserved. The initial conditions are fk(0)=δk,0.
Randomness in Competitions 461
In contrast to fk, the cumulative distribution functions obey closed evolution equations.
In particular, the quantity Fkevolves according to [13]
dFk
dt =q(Fk−1−Fk)+1
2−qF2
k−1−F2
k,(3)
which may be obtained by summing (2). The boundary conditions are F0=0andF∞=1,
and the initial condition is Fk(0)=1fork>0. We note that the average number of wins,
k=t/2, where k=kkfk, follows from the fact that each team participates in one com-
petition per unit time and that one win is awarded in each game. As k=kk(Fk+1−Fk),
we can verify that dk/dt =1/2 by summing the rate equations (3).
We first discuss the asymptotic behavior when the number of games is very large. In the
limit t→∞, we use the continuum approach and replace the difference equations (3) with
the partial differential equation [19,20]
∂F
∂t +q−(1−2q)F∂F
∂k =0.(4)
According to our model, the weakest team wins at least a fraction qof its games, on average,
and similarly, the strongest team wins no more than a fraction pof its games. Hence, the
number of wins is proportional to time, k∼t. We thus seek the scaling solution
Fk(t) Φk
t.(5)
Here and throughout this paper, the quantity Φ(x) is the scaled cumulative distribution of
win percentage; that is, the fraction of teams that win less than a fraction xof games played.
The boundary conditions are Φ(0)=0andΦ(∞)=1.
We now substitute the scaling form (5)into(4), and find that the scaling function satisfies
Φ[(x −q) −(1−2q)Φ]=0 where prime denotes derivative with respect to x.Thereare
two solutions: Φ=constant and the linear function Φ=(x −q)/(1−2q). Therefore, the
distribution of win percentages is piecewise linear
Φ(x) =⎧
⎪
⎨
⎪
⎩
00≤x≤q,
x−q
p−qq≤x≤p,
1p≤x.
(6)
As expected, there are no teams with win percentage less than the upset probability q,and
there are no teams with win percentage greater than the complementary probability p.Fur-
thermore, one can verify that x=1/2. The linear behavior in (6) indicates that the actual
distribution of win percentage becomes uniform, Φ=1/(p −q) for q<x<p, when the
number of games is very large.
As shown in Fig. 1, direct numerical integration of the rate equation (4) confirms the scal-
ing behavior (5). Moreover, as the number of games increases, the function Φ(x)approaches
the piecewise-linear function given by Eq. (6). However, there is a diffusive boundary layer
near x=qand x=p, whose width decreases as t−1/2in the long-time limit [19].
Generally, the win percentage is a convenient measure of team strength. For example,
Major League Baseball (MLB) in the United States, where teams play ≈160 games dur-
ing the regular season, uses win percentage to rank teams. The fraction of games won is
preferred over the number of wins because throughout the season there are small variations
between the number of games played by various teams in the league.
462 E. Ben-Naim et al.
Fig. 1 The cumulative
distribution Φ(x) versus win
percentage xfor q=1/4 at times
t=100 and t=500. Also shown
for reference is the limiting
behavior (6)
The piecewise-linear scaling function in (6) holds in the asymptotic limits N→∞and
t→∞. To apply the competition model (1), we must use a realistic number of games and
a realistic number of teams. To test whether the competition model faithfully describes the
win percentage of actual sports leagues, we compared the results of Monte Carlo simula-
tions with historical data for a variety of sports leagues [14,15]. In this paper, we give one
representative example: Major League Baseball.
In our simulations, there are Nteams, each participating in exactly tgames through-
out the season. In each match, two teams are selected at random, and the outcome of the
competition follows the stochastic rule (1): with the upset probability q, the team with the
lower win percentage is victorious, but otherwise, the team with the higher win percentage
wins. At the start of the simulated season, all teams have an identical record. We treated the
upset frequency as a free parameter and found that the value qmodel =0.41 best describes
the historical data for MLB (N=26 and t=162). As shown in Fig. 2, the competition
model faithfully captures the empirical distribution of win percentages at the end of the sea-
son. The latter distribution is calculated from all season-end standings over the past century
(1901–2005).
In addition, we directly measured the actual upset frequency qdata from the outcome of
all ≈163,000 games played over the past century. To calculate the upset frequency, we
chronologically ordered all games and recreated the standings at any given day. Then we
counted the number of games in which the winner was lower in the standings at the time of
the current game. Game location and the margin of victory were ignored. For MLB, we find
the value qdata =0.44, only slightly higher than the model estimate qmodel =0.41.
The standard deviation in win percentage, σ,definedbyσ2=x2−x2, is commonly
used to quantify parity of a sports league [21,22]. For example, in baseball, where the
win percentage typically varies between 0.400 and 0.600, the historical standard deviation
is σ=0.084. From the cumulative distribution (6), it straightforwardly follows that the
standard deviation varies linearly with the upset probability,
σ=1/2−q
√3.(7)
There is an obvious relationship between the predictability of individual games and the
competitive balance of a league: the more random the outcome of an individual game, the
higher the degree of parity between teams in the league.
The standard deviation is a convenient quantity because it requires only year-end stand-
ings, which consist of only Ndata points per season. The upset frequency, on the other
Randomness in Competitions 463
Fig. 2 The cumulative
distribution Φ(x) versus win
percentage xfor: (i) Monte Carlo
simulations of the competition
process (1) with qmodel =0.41,
and (ii) Season-end standings for
Major League Baseball (MLB)
over the past century
(1901–2005)
hand, requires the outcome of each game, and therefore involves a much larger number of
data points, Nt/2 per season. Yet, as a measure for competitive balance, the upset frequency
has an advantage [14,15]. As seen in Fig. 3, the quantity σconsists of two contributions:
one due to the intrinsic nature of the game and one due to the finite length of the season. For
example, the large standard deviation σ=0.21 in the National Football League (NFL) is in
large part due to the extremely short season, t=16. Therefore, the upset frequency, which
is decoupled from the length of the season, provides a more accurate measure of competitive
balance [23–27].
The evolution of the upset frequency over time is truly fascinating (Fig. 4). Although
qvaries over a narrow range, this quantity can differentiate the four sports leagues. The
historical data shows that MLB has consistently had the least predictable games, while NBA
and NFL games have been the most predictable. The trends for qfor these sports leagues are
even more interesting. Certain sports leagues (MLB and to a larger extent, NFL) managed to
increase competitiveness by changing competition formats, increasing the number of teams,
having unbalanced schedules where stronger teams play more challenging opponents, or
using a draft where the weakest team can first pick the most promising upcoming talent.
In spite of the fact that NHL and NBA implemented some of these same measures to
increase competitiveness, there are no clear long-term trends in the evolution of the upset
probability in these two leagues. Another plausible interpretation of Fig. 4is that the sports
leagues are striving to achieve an optimal upset frequency of q≈0.4. One may even spec-
ulate that the various sports leagues compete against each other to attract public interest,
and that making the games less predictable, and hence, more interesting to follow is a key
objective in this evolutionary-like process [6,29,30]. In any event, the upset frequency is a
natural and transparent measure for the evolution of competitive balance in sports leagues.
The random process (1) involves only a single parameter, q. The model does not take into
account many aspects of real competitions including the game score, the game location, the
relative team strength, and the fact that in many sports leagues the schedule is unbalanced,
as teams in the same geographical region may face each other more often. Nevertheless,
with appropriate implementation, the competition model specified in Eq. (1) captures basic
characteristics of real sports leagues. In particular, the model can be used to estimate the
distribution of team win percentages as well as the upset frequency.
464 E. Ben-Naim et al.
Fig. 3 The standard deviation σ
as a function of time t.Shown
are results of numerical
integration of the rate equation
(2) with q=1/4. Also shown for
reference is the limiting value
σ∞=1/(4√3)
3 Single Elimination Tournaments
Thus far, our approach did not include the notion of innate team strength. Randomness
alone controlled which team reaches the top of the standings and which teams reaches at
the bottom. Indeed, the probability that a given team has the best record at the end of the
season equals 1/N . Furthermore, we have used the cumulative win-loss record to define
team strength. However, this definition can not be used to describe tournaments where the
number of games is small.
We now focus on single-elimination tournaments, where the winner of a game advances
to the next round of play while the loser is eliminated [16,31]. A single-elimination tour-
nament is the most efficient competition format: a tournament with N=2rteams requires
only N−1 games through rrounds of play to crown a champion. In the first round, there
are Nteams and the N/2 winners advance to the next round. Similarly, the second round
produces N/4 winners. In general, the number of competitors is cut by half at each round
N→N/2→N/4→···→2→1.(8)
In many tournaments, for example, the NCAA college basketball tournament in the
United States or in tennis championships, the competitors are ranked according to some
predetermined measure of their strength. Thus, we introduce the notion of rank into our
modeling framework. Let xibe the rank of the ith team with
x1<x
2<x
3<···<x
N.(9)
In our definition, a team with lower rank is stronger. Rank measures innate strength, and
hence, it does not change with time. Since ranking is strict, we use the uniform ranking
scheme xi=i/N without loss of generality.
Again, we assume that there is a fixed probability qthat the underdog wins the game, so
that the outcome of each match is stochastic. When a team with rank x1faces a team with
rank x2,wehave
(x1,x
2)→x1with probability p,
x2with probability q, (10)
when x1<x
2. The important difference with (1) is that the losing team is now eliminated.
Randomness in Competitions 465
Fig. 4 Evolution of the upset
frequency qwith time. Shown is
data [28] for: (i) Major League
Baseball (MLB), (ii) the National
Hockey League (NFL), (iii) the
National Basketball Association
(NBA), and (iv) the National
Football League (NFL). The
quantity qis the cumulative upset
frequency for all games played in
the league up to the given year. In
football, a tie counts as one half
of a win
Let w1(x) be the distribution of rank for all competitors. This quantity is normalized,
∞
0dxw1(x) =1. In a two-team tournament, the rank distribution of the winner, w2(x),is
given by
w2(x) =2pw1(x)1−W1(x)+2qw1(x)W1(x), (11)
where W1(x) =x
0dy w1(y) is the cumulative distribution of rank. The structure of this
equation resembles that of (2), with the first term corresponding to games where the fa-
vorite advances, and the second term to games where the underdog advances. Mathemati-
cally, there is a basic difference with Eq. (2) in that Eq. (11) does not contain loss terms.
Again, ties are not allowed to occur. By integrating (11), we obtain the closed equation
W2(x) =2pW1(x) +(1−2p)[W1(x)]2.
In general, the cumulative distribution obeys the nonlinear recursion equation
W2N(x) =2pWN(x) +(1−2p)WN(x)2.(12)
Here, WN(x) =x
0dywN(y),andwN(x) is the rank distribution for the winner of an N-
team tournament. The boundary conditions are WN(0)=0andWN(∞)=1. The prefactor 2
arises because there are two ways to choose the winner. The quadratic nature of Eq. (12)re-
flects that two teams compete in each match (competitions with three teams are described
by cubic equations [32–34]). Starting with W1(x) =xthat corresponds to uniform ranking,
w1(x) =1, we can follow how the distribution of rank evolves by iterating the recursion
equation (12). As shown in Fig. 5, the rank of the winner decreases as the size of the tour-
nament increases. Hence, larger tournaments produce stronger winners.
By substituting W1(x) =xinto Eq. (12), we find W2(x ) =(2p)x and in general,
WN(x) =(2p)rx. This behavior suggests the scaling form
WN(x) Ψ(x/x
∗), (13)
where the scaling factor x∗is the typical rank of the winner. This quantity decays alge-
braically with the size of the tournament,
x∗=N−β,β=ln(2p)
ln2 .(14)
When games are perfectly random (upset probability q=1/2), the typical rank of the winner
becomes independent of the number of teams, β(q =1/2)=0. When the games are highly
466 E. Ben-Naim et al.
Fig. 5 The cumulative
distribution of rank. The quantity
WN(x) is calculated by iterating
Eq. (12) with q=1/4
predictable, the top teams tend to win the tournament, β(0)=1. Again, the scaling behavior
(14) shows that larger tournaments tend to produce stronger champions.
By substituting (13)into(12), we see that the scaling function Ψ(z) obeys the nonlocal
and nonlinear equation
Ψ(2pz) =2pΨ (z) +(1−2p)Ψ 2(z). (15)
The boundary conditions are Ψ(0)=0andΨ(∞)=1. From Eq. (15), we deduce the
asymptotic behaviors
Ψ(z)zz→0,
1−Cz
γz→∞,(16)
with the scaling exponent γ=ln(2q)
ln(2p) .Thelarge-zbehavior is obtained by substituting
Ψ(z)=1−U(z) into (15) and noting that since U→0whenz→∞, the correction obeys
the linear equation U(2pz) =2qU(z).
The large-zbehavior of the scaling function Ψ(z) gives the likelihood that a very weak
team manages to win the entire tournament. The scaling behavior (13) is equivalent to
wN(x) (1/x∗)ψ (x/x∗)with ψ(z) =Ψ(z). In the limit z→0, the distribution approaches
a constant ψ(z) →1. However, the tail of the rank distribution is algebraic
ψ(z) ∼z−α,α=1−ln(2q)
ln(2p),(17)
when z→∞. The exponent α>1 increases monotonically with p, and it diverges in the
limit p→1.2
Moreover, the probability that the weakest team wins the tournament, PN=qN, decays
algebraically with the total number of teams, PN=Nln q/ln 2. In the following section, we
discuss sports leagues and find that: (i) the rank distribution of the winner has an expo-
nential tail, and (ii) the probability that the weakest team is crowned league champion is
exponentially small.
The scaling behavior (13) indicates universal statistics when the size of the tournament
is sufficiently large. Once rank is normalized by typical rank, the resulting distribution does
2For deterministic competitions, q=0, the scaling function is exponential ψ(z) =e−z.
Randomness in Competitions 467
Fig. 6 The cumulative
distribution of rank for the
NCAA college basketball
tournament. Shown is the
cumulative distribution W16(x)
versus the rank xfor (i) NCAA
tournament data (1979–2006),
(ii) iteration of Eq. (12)
not depend on tournament size. Further, the scaling law (14) and the power-law tail (17)
reflect that tournaments can produce major upsets. With a relatively small number of upset
wins, a “Cinderella” team can emerge, and for this reason, tournaments can be very excit-
ing. Furthermore, tournaments are maximally efficient as they require a minimal number of
games to decide a champion.
Figure 6shows that our theoretical model nicely describes empirical data [28]forthe
NCAA college basketball tournament in the United States [16]. In the current format, 64
teams participate in four sub-tournaments, each with N=16 teams. The four winners of
each sub-tournament advance to the final four, which ultimately decides the champion. Prior
to the tournament, a committee of experts ranks the teams from 1 to 16. We note that the
game schedule is not random, and is designed such that the top teams advance if there are
no upsets.
Consistent with our theoretical results, the NCAA tournament has been producing major
upsets: the 11th seed team has advanced to the final four twice over the past 30 years.
Moreover, only once did all of the four top-seeded teams advance simultaneously (2008).
Our model estimates the probability of this event at 1/190, a figure that is of the same order
of magnitude as the observed frequency 1/132.
We also mention that in producing the theoretical curve in Fig. 6, we used the upset
frequency qmodel =0.18, whereas the actual game results yield qdata =0.28. This larger dis-
crepancy (compared with the MLB analysis above) is due to a number of factors including
the much smaller dataset (≈7000 games) and the non-random game schedule. Indeed, our
Monte Carlo simulations which incorporate a realistic schedule give better estimates for the
upset frequency [16].
4 Leagues
We now discuss the common competition format in which each team hosts every other team
exactly once during the season. This format, first used in English soccer, has been adopted in
many sports. In a league of size N, each team plays 2(N −1)games and the total number of
games equals N(N −1). Given this large number of games, does the strongest team always
wins the championship?
To answer this question, we assume that each team has an innate strength and rank the
teams according to strength. Without loss of generality, we use the uniform rank distribution
468 E. Ben-Naim et al.
w(x) =1 and its cumulative counterpart W(x)=xwhere 0 ≤x≤1. Moreover, we implic-
itly take the large-Nlimit. Consider a team with rank x. The probability v(x) that this team
wins a game against a randomly-chosen opponent decreases linearly with rank,
v(x) =p−(2p−1)x, (18)
as follows from v(x) =p[1−W1(x)]+qW1(x) [see also Eq. (11)]. Consistent with our
competition rules (1)and(10), the probability v(x) satisfies q≤v≤p.
Since team strength does not change with time, the average number of wins V(x,t) for
a team with rank xgrows linearly with the number of games t,
V(x,t)=v(x)t. (19)
Accordingly, the number of wins of a given team performs a biased random walk: after each
game the number of wins increases by one with probability v, and remains unchanged with
the complementary probability 1 −v. Also, the uncertainty in the number of wins, V ,
grows diffusively with t,
V (x , t) √Dt, (20)
with diffusion coefficient D=v(1−v) [18].
Let us assume that each team plays tgames. If the number of games is sufficiently large,
the best team has the most wins. However, at intermediate times, it is possible that a weaker
team has the most wins. For a team with strength x∗to still be in contention at time t,
the difference between its expected number of wins and that of the top team should be
comparable with the diffusive uncertainty
V(0,t)−V(x
∗,t)∼V (0,t). (21)
We now substitute Eqs. (18)–(20) into this heuristic estimate and obtain the typical rank of
the leader as a function of time,
x∗∼1
√t.(22)
In obtaining this estimate, we tacitly ignored numeric prefactors, including in particular, the
dependence on q.
This crude estimate (22) shows that the best team does not always win the league cham-
pionship. Since t∼N,wehave
x∗∼1
√N.(23)
Since rank is a normalized quantity, the top √Nof the teams have a realistic chance of
emerging with the best record at the end of the season. Thus randomness plays a crucial role
in determining the champion: since the result of an individual game is subject to randomness,
the outcome of a long series of games reflects this randomness.
We can also obtain the total number of games Tneeded for the best team to always
emerge as the champion,
T∼N3.(24)
This scaling behavior follows by replacing x∗in (22) with 1/N which corresponds to the
best team. For the best team to win, each team must play every other team O(N) times!
Randomness in Competitions 469
Fig. 7 The total number of
games Tneeded for the best
team to emerge as champion in a
league of size N. The simulation
results represent an average over
103simulated sports leagues.
Also shown for reference is the
theoretical prediction
Alternatively the number of games played by each team scales quadratically with the size
of the league. Clearly, such a schedule is prohibitively long, and we conclude that the tradi-
tional schedule of playing each opponent with equal frequency is neither efficient nor does
it guarantee the best champion.
We confirmed the scaling law (24) numerically. In our Monte Carlo simulations, the
teams are ranked from 1 to Nat the start of the season. We implemented the traditional
league format where every team plays every other team and kept track of the leader defined
as the team with the best record. We then measured the last-passage time [35], that is, the
time in which the best team takes the lead for good. We define the average of this fluctuating
quantity as T[36,37].AsshowninFig.7, the total number of games required is cubic.
Again, we expect that the probability distribution w(x,t) that a team with rank xhas the
best record after tgames is characterized by the scale x∗given in (22)
w(x, t) (1/x∗)ϕ(x/x∗). (25)
Numerical results confirm this scaling behavior [17]. Since the number of wins performs a
biased random walk, we expect that the distribution of the number of wins becomes normal
in the long-time limit. Moreover, the scaling function in (25) has a Gaussian tail [17]
ϕ(z) ∼exp−const.×z2,(26)
as z→∞.
Using this scaling behavior, we can readily estimate the probability that worst team be-
comes champion (in the standard league format). For the worst team, x∼1, and the corre-
sponding scaling variable in Eq. (25)isz∼√N. Hence, the Gaussian tail (26)showsthat
the probability PNthat the weakest team wins the league is exponentially small,
PN∼exp(−const.×N). (27)
In sharp contrast with tournaments, where this probability is algebraic, leagues do not pro-
duce upset champions. Leagues may not guarantee the absolute top team as champion, but
nevertheless, they do produce worthy champions.
To compare leagues and tournaments, we calculated the probability Pnthat the nth
ranked team is champion for a realistic number of games N=16 and a realistic upset
probability q=0.4(Fig.8). For leagues, we calculated this probability from Monte Carlo
470 E. Ben-Naim et al.
Fig. 8 Leagues versus
tournaments. Shown is Pn,the
probability that the nth-ranked
team has the best record at the
end of the season in the format of
playing all opponents with equal
frequency, and the probability
that the nth-ranked team wins an
N-team single-elimination
tournament. The upset
probability is q=0.4and
N=16
simulations, and for tournaments, we used Eq. (12). Indeed, the top four teams fare better
in a league format while the rest of the teams are better off in a tournament. This behavior
is fully consistent with the above estimate that the top √Nteams have a realistic chance to
win the league.
What is the probability Ptop that the top team ends the season with the best record in a re-
alistic sports league? To answer this question, we investigated the four major sports leagues
in the US: MLB, NHL, NFL, and NBA. We simulated a league with the actual number
of teams Nand the actual number of games t, using the empirical upset frequencies (see
Fig. 3). All of these sports leagues have comparable number of teams, N≈25. Surpris-
ingly, we find almost identical probabilities for three of the sports leagues: (i) MLB with
the longest season and most random games (t=162, q=0.44) has Ptop =0.31, (ii) NFL
with the shortest season but most deterministic games (t=16, q=0.37) has Ptop =0.30,
and (iii) NHL with intermediate season and intermediate randomness (t=80, q=0.41) has
Ptop =0.32. Standing out as an anomaly is the value Ptop =0.45 for the NBA which has a
moderate-length season but less random games (t=80 and q=0.37).
This interesting result reinforces our previous comments about sports leagues competing
against each other for interest and our hypothesis that there are optimal randomness param-
eters. Having a powerhouse win every year does not serve the league well, but having the
strongest team finish with the best record once every three years may be optimal.
5 Gradual Elimination Algorithm
Our analysis demonstrates that single-elimination tournaments have optimal efficiency but
may produce weak champions, whereas leagues which result in strong winners are highly
inefficient. Can we devise a competition “algorithm” that guarantees a strong champion
within a minimal number of games?
As an efficient algorithm, we propose a hybrid schedule consisting of a preliminary round
and a championship round [17]. The preliminary round is designed to weed out a majority
of teams using a minimal number of games, while the championship round includes ample
games to guarantee the best team wins.
In the preliminary round, every team competes in tgames. Whereas the league schedule
has complete graph structure with every team playing every other team, the preliminary
round schedule has regular random graph structure with each team playing against the same
number of randomly-chosen opponents. Out of the Nteams, the Mteams with the largest
Randomness in Competitions 471
number of wins in the preliminary-round advance to the championship round. The number
of games tis chosen such that the strongest team always qualifies. By the same heuristic
argument (21) leading to (22), the top team ranks no lower than 1/√tafter tgames. We
thus require
M
N∼1
√t,(28)
and consequently, each team plays ∼(N/M )2preliminary games. The championship round
uses a league format with each of the Mqualifying teams playing Mgames against every
other team. Therefore, the total number of games, T, has two components
T∼N3
M2+M3.(29)
In writing this estimate, we ignore numeric prefactors, as well as the dependence on the
upset frequency q. The quantity Tis minimal when the two terms in (29) are comparable
[38]. Hence, the size of the championship round M1and the total number of games T1scale
algebraically with N,
M1∼N3/5,and T1∼N9/5.(30)
Consequently, each team plays O(N4/5)games in the preliminary round. Interestingly, the
existence of a preliminary round significantly reduces the number of games from N3to
N9/5. Without sacrificing the quality of the champion, the hybrid schedule yields a huge
improvement in efficiency!
We can further improve the efficiency by using multiple elimination rounds. In this gen-
eralization, there are k−1 consecutive rounds of preliminary play culminating in the cham-
pionship round. The underlying graphical structure of the preliminary rounds is always a
regular random graph, while the championship round remains a complete graph. Each pre-
liminary round is designed to advance the top teams, and the number of games is sufficiently
large so that the top team advances with very high probability. When there are krounds, we
anticipate the scaling laws
Mk∼Nνk,and Tk∼Nμk,(31)
where Mkis the number of teams advancing out of the first round and Tkis the total number
of games. Of course, when there are no preliminary rounds, ν0=1andμ0=3. Following
Eq. (31), the number of teams gradually declines in each round,
N→Nνk→Nνkνk−1→···→Nνkνk−1···ν1→1.(32)
According to the first term in (29), the number of games in the first round scales as
N3/M2
k∼N3−2νk, and therefore, the total number of games obeys the recursion
Tk∼N3−2νk+Tk−1Nνk.(33)
Indeed, if we replace M1with Nν1in Eq. (29) we can recognize the recursion (33). The sec-
ond term scales as Nνkμk−1and becomes comparable to the second when 3 −2νk=νkμk−1.
Hence, the scaling exponents satisfy the recursion relations
νk=3
2+μk−1,and μk=μk−1νk.(34)
472 E. Ben-Naim et al.
Tab le 1 The exponents νkand
μkin Eq. (31)fork≤4k01 2 3 4 ∞
νk03
515
19 57
65 195
211 1
μk39
527
19 81
65 243
211 1
Using ν0=1andμ0=3, we recover ν1=3/5andμ1=9/5 in agreement with (30). The
general solution of (34)is[17]
νk=1−(2/3)k
1−(2/3)k+1,μ
k=1
1−(2/3)k+1.(35)
Hence, the efficiency is optimal, and the number of games becomes linear in the limit
k→∞. For a modest number of teams, a small number of preliminary rounds, say 1–3
rounds, may suffice. As shown in Table 1, with as few as four elimination rounds, the num-
ber of games becomes essentially linear, μ4∼
=1.15.
Interestingly, the result μ∞=1 indicates that championship rounds or “playoffs” have
the optimal size M∗given by
M∗∼N1/3.(36)
Gradual elimination is often used in the arts and sciences to decide winners of design compe-
titions, grant awards, and prizes. Indeed, the selection process for prestigious prizes typically
begins with a quick glance at all nominees to eliminate obviously weak candidates, but con-
cludes with rigorous deliberations to select the winner. Multiple elimination rounds may be
used when the pool of candidates is very large.
To verify numerically the scaling laws (30), we simulated a single preliminary round
followed by a championship round. We chose the size of the preliminary round strictly
according to (31) and used a championship round where all M1teams play against all M1
teams exactly M1times. We confirmed that as the number of teams increases from N=101
to 102,to10
3, etc., the probability that the best team emerges as champion is not only high
but also, independent of N. We also confirmed that the concept of preliminary rounds is
useful for small N.ForN=10 teams, the number of games can be reduced by a factor
>10 by using a single preliminary round.
6 Discussion
We introduced an elementary competition model in which a weaker team can upset a
stronger team with fixed probability. The model includes a single control parameter, the
upset frequency, a quantity that can be measured directly from historical game results. This
idealized competition model can be conveniently applied to a variety of competition formats
including tournaments and leagues. The random competition process is amenable to theo-
retical analysis and is straightforward to implement in numerical simulations. Qualitatively,
this model explains how tournaments, which require a small number of games, can produce
major upsets, and how leagues which require a large number of games always produce qual-
ity champions. Additionally, the random competition process enables us to quantify these
intuitive features: the rank distribution of the champion is algebraic in the former schedule
but Gaussian in the latter.
Randomness in Competitions 473
Using our theoretical framework, we also suggested an efficient algorithm where the
teams are gradually eliminated following a series of preliminary rounds. In each preliminary
round, the number of games is sufficient to guarantee that the best team qualifies to the
next round. The final championship round is held in a league format in which every team
plays many games against every other team to guarantee that the strongest team emerges as
champion. Using gradual elimination, it is possible to choose the champion using a number
of games that is proportional to the total number of teams. Interestingly, the optimal size of
the championship round scales as the one third power of the total number of teams.
The upset frequency plays a major role in our model. Our empirical studies show that
the frequency of upsets, which shows interesting evolutionary trends, is effective in differ-
entiating sports leagues. Moreover, this quantity has the advantage that it is not coupled
to the length of the season, which varies widely from one sport to another. Nevertheless,
our approach makes a very significant assumption: that the upset frequency is fixed and
does not depend on the relative strength of the competitors. Certainly, our approach can
be generalized to account for strength-dependent upset frequencies [39]. We note that our
single-parameter model fares better when the games tend to be close to random, and that
model estimates for the upset frequency have larger discrepancies with the empirical data
when the games become more predictable. Clearly, a more sophisticated set of competition
rules are required when the competitors are very close in strength, as is the case for example,
in chess [40].
Acknowledgements We thank Micha Ben-Naim for help with data collection. We acknowledge support
from DOE (DE-AC52-06NA25396) and NSF (DMR0227670, DMR0535503, & DMR-0906504).
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