University of Utah. Mathematical Biology the. Imagine ..... Florida. 5-3. 21. 22. 16. 11. 25. 25. Boise State. 6-1. 39.
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Mathematical Biology University of Utah
The Mathematics of Ranking Schemes or Should Utah Be Ranked in the Top 25? J. P. Keener Mathematics Department University of Utah
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.1/19
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Introduction
Mathematical Biology University of Utah
Problem: How to rank sports teams? The Challenge:
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.2/19
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Introduction
Mathematical Biology University of Utah
Problem: How to rank sports teams? The Challenge: • Uneven paired competition (not all teams play each other)
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.2/19
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Introduction
Mathematical Biology University of Utah
Problem: How to rank sports teams? The Challenge: • Uneven paired competition (not all teams play each other) • The data are sparse. (Each team plays 10 games and there
are over 100 teams)
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.2/19
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Introduction
Mathematical Biology University of Utah
Problem: How to rank sports teams? The Challenge: • Uneven paired competition (not all teams play each other) • The data are sparse. (Each team plays 10 games and there
are over 100 teams) • There is no well-ordering (Team A beats Team B who beats
Team C who beats team A).
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.2/19
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Simple Idea
Mathematical Biology University of Utah
• Suppose team i has Wi wins and Li losses, let
Wi Wi ri = = . Wi + L i Ni Call r = (ri ) the ranking vector.
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.3/19
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Simple Idea
Mathematical Biology University of Utah
• Suppose team i has Wi wins and Li losses, let
Wi Wi ri = = . Wi + L i Ni Call r = (ri ) the ranking vector. • Alternate representation: Set A = (aij ) where aij =
1 Ni
if
team i beat team j, aij = 0 otherwise. Then
r = Ar0 ,
r0 = 1.
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.3/19
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Basic Idea
Mathematical Biology University of Utah
• First modification: To get an indication of strength of
schedule, let
r = A(Ar0 ) = A2 r0 .
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.4/19
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Basic Idea
Mathematical Biology University of Utah
• First modification: To get an indication of strength of
schedule, let
r = A(Ar0 ) = A2 r0 . • Obvious generalization:
An r 0 r= n 0 |A r |
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.4/19
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Basic Idea
Mathematical Biology University of Utah
• First modification: To get an indication of strength of
schedule, let
r = A(Ar0 ) = A2 r0 . • Obvious generalization:
An r 0 r= n 0 |A r | • Question: What if we let n → ∞? • Answer: Define r to be eigenvector of A
Ar = λr The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.4/19
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Mathematical Biology
Method 1:The Eigenvector Method
University of Utah
Suppose that team i has rank ri , and aij is a measure of the result of the game between team i and team j where • 0 ≤ aij ≤ 1, aij = 0 if teams i and j have not played, • aij + aji = 1 if teams i and j have played each other.
Assign a score si to team i
1 X aij rj si = Ni j and then define r so that
λri = si
or
Ar = λr
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.5/19
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Mathematical Biology University of Utah
Method 1:Existence Question Do solutions of this eigenvalue problem exist? The Perron Frobenius Theorem:
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.6/19
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Mathematical Biology University of Utah
Method 1:Existence Question Do solutions of this eigenvalue problem exist? The Perron Frobenius Theorem: • If A has nonnegative entries, then there exists nonnegative
solution of Ar = λr.
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.6/19
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Mathematical Biology University of Utah
Method 1:Existence Question Do solutions of this eigenvalue problem exist? The Perron Frobenius Theorem: • If A has nonnegative entries, then there exists nonnegative
solution of Ar = λr. • If A is irreducible, then r is strictly positive and unique, and λ
is simple and maximal.
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.6/19
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Mathematical Biology University of Utah
Method 1:Existence Question Do solutions of this eigenvalue problem exist? The Perron Frobenius Theorem: • If A has nonnegative entries, then there exists nonnegative
solution of Ar = λr. • If A is irreducible, then r is strictly positive and unique, and λ
is simple and maximal. • r can be calculated using the power method:
An r 0 r = lim , n 0 n→∞ |A r |
r0 = 1
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.6/19
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Choosing aij
Mathematical Biology University of Utah
Ways to specify aij : • aij = 1 if team i beat team j. (W-L record)
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.7/19
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Choosing aij
Mathematical Biology University of Utah
Ways to specify aij : • aij = 1 if team i beat team j. (W-L record) • aij =
Sij Sij +Sji ,
with Sij = points earned by team i vs team j.
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.7/19
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Choosing aij
Mathematical Biology University of Utah
Ways to specify aij : • aij = 1 if team i beat team j. (W-L record) • aij =
Sij Sij +Sji ,
• aij =
Sij +1 Sij +Sji +2 .
with Sij = points earned by team i vs team j.
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.7/19
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Choosing aij
Mathematical Biology University of Utah
Ways to specify aij : • aij = 1 if team i beat team j. (W-L record) • aij =
Sij Sij +Sji ,
• aij =
Sij +1 Sij +Sji +2 . Sij +1 Sij +Sji +2
. 1
0.9
0.8
0.7
0.6 h(x)
• aij = h
with Sij = points earned by team i vs team j.
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5 x
0.6
0.7
0.8
0.9
1
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.7/19
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Method 2:The Nonlinear Fixed Point Method
Mathematical Biology University of Utah
Suppose f (x) is positive, increasing, concave for 0 ≤ x < ∞. Define r as solution of
1 X ri = Fi (r) = f (aij rj ) Ni j where
.05x + x2 f (x) = 2 + .05x + x2 1
0.9
0.8
0.7
f(x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5 x
3
3.5
4
4.5
5
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.8/19
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Mathematical Biology
Method 2:Finding r
University of Utah
Suppose F (x) : Rn → Rn is continuous,
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.9/19
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Method 2:Finding r
Mathematical Biology University of Utah
Suppose F (x) : Rn → Rn is continuous, • positive: F (p) > 0 for all p > 0,
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.9/19
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Method 2:Finding r
Mathematical Biology University of Utah
Suppose F (x) : Rn → Rn is continuous, • positive: F (p) > 0 for all p > 0, • increasing: F (p) > F (q) whenever p > q,
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.9/19
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Method 2:Finding r
Mathematical Biology University of Utah
Suppose F (x) : Rn → Rn is continuous, • positive: F (p) > 0 for all p > 0, • increasing: F (p) > F (q) whenever p > q, • concave: F (tp) > tF (p) for 0 < t < 1, p > 0
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.9/19
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Method 2:Finding r
Mathematical Biology University of Utah
Suppose F (x) : Rn → Rn is continuous, • positive: F (p) > 0 for all p > 0, • increasing: F (p) > F (q) whenever p > q, • concave: F (tp) > tF (p) for 0 < t < 1, p > 0
then the sequence of vectors rk = F (rk−1 ), with r0 = 1 is a monotone decreasing sequence with limn→∞ rn = r where
r = F (r) Remark: This is a generalization of the Perron Frobenius Theorem. The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.9/19
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Method 3:The Probability Method
Mathematical Biology University of Utah
Suppose the probability that team i beats team j is
Sij ri ≈ . πij = ri + rj Sij + Sji Then
Sij rj ≈ Sji ri . Thus, let r minimize
X (Sji ri − Sij rj )2 i,j
subject to
X
ri2 = 1
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.10/19
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Method 3:Finding r
Mathematical Biology University of Utah
Use Lagrange multipliers and minimize
X X (Sji ri − Sij rj )2 − µ( ri2 − 1) i,j
i
equivalently, find r where
Br = µr where
B = (bij ),
bii =
X
2 , Sik
bij = −Sij Sji for i 6= j
k
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.11/19
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Method 3:Finding r
Mathematical Biology University of Utah
Pick λ0 so that B + λ0 I is diagonally dominant. Then, (B + λ0 I)−1 is a positive matrix. From the Perron Frobenius theorem,
(B + λ0 I)−n r0 =r lim n→∞ |(B + λ0 I)−n r 0 |
The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.12/19
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"Linear Models"
Mathematical Biology University of Utah
Remark: This model is one of a class of models called linear models for which
πij = h(ri − rj ) For example, h could be a Gaussian (bell-shaped) or an exponential function. The choice πij =
ri ri +rj
is exponential, since with r = ev
evi −vj evi = . πij = vi v v −v j i j e +e 1+e This is the model (purportedly) used for national tennis, squash, etc. rankings. The Mathematics of Ranking SchemesorShould Utah Be Ranked in the Top 25? – p.13/19
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Method 4:Maximum Likelihood
Mathematical Biology University of Utah
Suppose the outcome of a game is a Bernoulli trail with πij the probability that team i beats team j. If aij represents the outcome of games, then the probability of observing that outcome is
P = Πi