parity and predictability of competitions: nonlinear ...ebn/talks/sports.pdf · federico vazquez,...
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Parity and Predictability of Competitions: Nonlinear Dynamics of Sports
Eli Ben-Naim
Complex Systems Group & Center for Nonlinear StudiesLos Alamos National Laboratory
Federico Vazquez, Sidney Redner (Los Alamos & Boston University)
2006 Dynamics Days Asia Pacific, Pohang, Korea, July 12, 2006
Talk, papers available from: http://cnls.lanl.gov/~ebn
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Plan
• Parity of sports leagues
• Theory: competition model
• Predictability of competitions
• Competition and social dynamics
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What is the most competitive sport?
Football
Baseball
Hockey
Basketball
American football
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What is the most competitive sport?
Football
Baseball
Hockey
Basketball
American football
Can competitiveness be quantified?How can competitiveness be quantified?
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• Teams ranked by win-loss record
• Win percentage
• Standard deviation in win-percentage
• Cumulative distribution = Fraction of teams with winning percentage < x
Parity of a sports leagueMajor League Baseball
American League2005 Season-end Standings
! =!!x2" # !x"2
F (x)0.400 < x < 0.600
! = 0.08
In baseball
x =Number of wins
Number of games
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Data
• 300,000 Regular season games (all games)
• 5 Major sports leagues in US, England
sport league full name country years games
soccer FA Football Association England 1888-2005 43,350
baseball MLB Major League Baseball US 1901-2005 163,720
hockey NHL National Hockey League US 1917-2005 39,563
basketball NBA National Basketball Association US 1946-2005 43,254
american football NFL National Football League US 1922-2004 11,770
source: http://www.shrpsports.com/ http://www.the-english-football-archive.com/
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0 0.2 0.4 0.6 0.8 1x
0
0.2
0.4
0.6
0.8
1
F(x)
NFLNBANHLMLB
Standard deviation in winning percentage
data theory
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0 0.2 0.4 0.6 0.8 1x
0
0.2
0.4
0.6
0.8
1
F(x)
NFLNBANHLMLB
Distribution of winning percentage clearly distinguishes sports
!
•Baseball most competitive?•American football least competitive?
data theory
Standard deviation in winning percentage
00.050.100.150.200.25
MLB FA NHL NBA NFL
0.210
0.1500.1200.1020.084
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• Two, randomly selected, teams play
• Outcome of game depends on team record
- Better team wins with probability 1-q
- Worst team wins with probability q
- When two equal teams play, winner picked randomly
• Initially, all teams are equal (0 wins, 0 losses)
• Teams play once per unit time
Theory: competition model
(i, j)!!
(i + 1, j) probability 1" q
(i, j + 1) probability qi > j
!x" =12
q =
!1/2 random1 deterministic
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• Probability distribution functions
• Evolution of the probability distribution
• Closed equations for the cumulative distribution
Boundary Conditions Initial Conditions
Rate equation approach
dgk
dt= (1! q)(gk!1Gk!1 ! gkGk) + q(gk!1Hk!1 ! gkHk) +
12
!g2
k!1 ! g2k
"
gk = fraction of teams with k wins
Gk =k!1!
j=0
gj = fraction of teams with less than k wins Hk = 1!Gk+1 =!!
j=k+1
gj
G0 = 0 G! = 1 Gk(t = 0) = 1
better team wins worse team wins equal teams play
Nonlinear Difference-Differential Equations
dGk
dt= q(Gk!1 !Gk) + (1/2! q)
!G2
k!1 !G2k
"
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An exact solution• Winner always wins (q=0)
• Transformation into a ratio
• Nonlinear equations reduce to linear recursion
• Exact solution
dGk
dt= Gk(Gk !Gk!1)
dPk
dt= Pk!1
Gk =Pk
Pk+1
Gk =1 + t + 1
2! t2 + · · · + 1
k! tk
1 + t + 12! t
2 + · · · + 1(k+1)! t
k+1
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0 0.5 1 1.5 2x
0
0.2
0.4
0.6
0.8
1
F(x
)
k=10k=20k=100scaling theory
Long-time asymptotics
• Long-time limit
• Scaling form
• Scaling function
Gk !k + 1
t
F (x) = x
Seek similarity solutionsUse winning percentage as scaling variable
Gk ! F
!k
t
"
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Scaling analysis• Rate equation
• Treat number of wins as continuous
• Stationary distribution of winning percentage
• Scaling equation
dGk
dt= q(Gk!1 !Gk) + (1/2! q)
!G2
k!1 !G2k
"
Gk+1 !Gk "!G
!k
Gk(t)! F (x) x =k
t
[(x! q)! (1! 2q)F (x)]dF
dx= 0
!G
!t+ [q + (1! 2q)G]
!G
!k= 0
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Scaling analysis• Rate equation
• Treat number of wins as continuous
• Stationary distribution of winning percentage
• Scaling equation
dGk
dt= q(Gk!1 !Gk) + (1/2! q)
!G2
k!1 !G2k
"
Gk+1 !Gk "!G
!k
Gk(t)! F (x) x =k
t
[(x! q)! (1! 2q)F (x)]dF
dx= 0
!G
!t+ [q + (1! 2q)G]
!G
!k= 0
Inviscid Burgers equation!v
!t+ v
!v
!x= 0
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Scaling solution• Stationary distribution of winning percentage
• Distribution of winning percentage is uniform
• Variance in winning percentage
F (x) =
!""#
""$
0 0 < x < qx! q
1! 2qq < x < 1! q
1 1! q < x.
f(x) = F !(x) =
!""#
""$
0 0 < x < q1
1! 2qq < x < 1! q
0 1! q < x.
! =1/2! q"
3
q 1! q
1
F (x)
x
q 1! qx
f(x)
12q ! 1
!"!
q = 1/2 perfect parityq = 1 maximum disparity
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0 0.2 0.4 0.6 0.8 1x
0
0.2
0.4
0.6
0.8
1
F(x)
Theoryt=100t=500
0 200 400 600 800 1000
t
0.1
0.2
0.3
0.4
0.5
!
14!
3
NFL
MLB
League games
MLB 160
FA 40
NHL 80
NBA 80
NFL 16
Approach to scaling
•Winning percentage distribution approaches scaling solution•Correction to scaling is very large for realistic number of games•Large variance may be due to small number of games
Numerical integration of the rate equations, q=1/4
Variance inadequate to characterize competitiveness!!(t) =
1/2! q"3
+ f(t) Large!
t!1/2
t!1/2
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0 0.2 0.4 0.6 0.8 1x
0
0.2
0.4
0.6
0.8
1F(x)
NFLNBANHLMLB
The distribution of win percentage
•Treat q as a fitting parameter, time=number of games•Allows to estimate qmodel for different leagues
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• Upset frequency as a measure of predictability
• Addresses the variability in the number of games
• Measure directly from game-by-game results
- Ties: count as 1/2 of an upset (small effect)
- Ignore games by teams with equal records
- Ignore games by teams with no record
The upset frequency
q =Number of upsetsNumber of games
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1900 1920 1940 1960 1980 2000year
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
q
FAMLBNHLNBANFL
The upset frequency
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1900 1920 1940 1960 1980 2000year
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
q
FAMLBNHLNBANFL
The upset frequencyLeague q qmodel
FA 0.452 0.459
MLB 0.441 0.413
NHL 0.414 0.383
NBA 0.365 0.316
NFL 0.364 0.309
Football, baseball most competitiveBasketball, American football least competitive
q differentiatesthe different
sport leagues!
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1900 1920 1940 1960 1980 2000year
0.08
0.10
0.12
0.14
0.16
0.18
0.20
0.22
0.24
0.26
0.28
!
NFLNBANHLMLBFA
1900 1920 1940 1960 1980 2000year
0.30
0.32
0.34
0.36
0.38
0.40
0.42
0.44
0.46
0.48
q
FAMLBNHLNBANFL
Evolution with time
•Parity, predictability mirror each other•American football, baseball increasing competitiveness•Football decreasing competitiveness (past 60 years)
! =1/2! q"
3
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Century versus Decade
0.25
0.30
0.35
0.40
0.45
0.50
FA MLB NHL NBA NFL
0.3640.365
0.414
0.4410.452
Century (1900-2005)
0.25
0.30
0.35
0.40
0.45
0.50
FA MLB NHL NBA NFL
0.39
0.34
0.430.45
0.41
Decade (1995-2005)
Football-American Football gap narrows from 9% to 2%!
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0.44 0.46 0.48 0.5 0.52 0.54 0.56x
0
0.2
0.4
0.6
0.8
1F
(x)
MLB
All-time team records
•Provides the longest possible record (t~13000)•Close to a linear function
NY Yankees (0.567)
! = 0.024q = 0.458
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Discussion
• Model limitation: it does not incorporate
- Game location: home field advantage
- Game score
- Upset frequency dependent on relative team strength
- Unbalanced schedule
• Model advantages:
- Simple, involves only 1 parameter
- Enables quantitative analysis
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Conclusions• Parity characterized by variance in winning percentage
- Parity measure requires standings data
- Parity measure depends on season length
• Predictability characterized by upset frequency
- Predictability measure requires game results data
- Predictability measure independent of season length
• Two-team competition model allows quantitative modeling of sports competitions
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Competition and Social Dynamics
• Teams are agents
• Number of wins represents fitness or wealth
• Agents advance by competing against age
• Competition is a mechanism for social differentiation
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• Agents advance by competition
• Agent decline due to inactivity
• Rate equations
• Scaling equations
The social diversity model
k ! k " 1 with rate r
dGk
dt= r(Gk+1 !Gk) + pGk!1(Gk!1 !Gk) + (1! p)(1!Gk)(Gk!1 !Gk)! 1
2(Gk !Gk!1)2
(i, j)!!
(i + 1, j) rate p
(i, j + 1) rate 1" pi > j
[(p + r ! 1 + x)! (2p! 1)F (x)]dF
dx= 0
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Social structures1. Middle class
Agents advance at different rates
2. Middle+lower classSome agents advance at different rates
Some agents do not advance
3. Lower classAgents do not advance
4. Egaliterian class All agents advance at equal rates
Bonabeau 96
Sports
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Publications
• Parity and Predictability of CompetitionsE. Ben-Naim, F. Vazquez, S. RednerJ. Quant. Anal. in Sports, submitted (2006)
• What is the Most Competitive Sport?E. Ben-Naim, F. Vazquez, S. Rednerphysics/0512143
• Dynamics of Multi-Player GamesE. Ben-Naim, B. Kahng, and J.S. KimJ. Stat. Mech. P07001 (2006)
• On the Structure of Competitive SocietiesE. Ben-Naim, F. Vazquez, S. RednerEur. Phys. Jour. B 26 531 (2006)
• Dynamics of Social DiversityE. Ben-Naim and S. RednerJ. Stat. Mech. L11002 (2005)
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“I do not make predictions,
especially not about the future.”
Yogi Bera