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### Synopses & Reviews

#### Publisher Comments

*Who's #1?*is an excellent survey of the fundamental ideas behind mathematical rating systems. Once a realm of sports enthusiasts, ranking things is becoming a vital tool in many information-age applications. Langville and Meyer compare and contrast a variety of models, explaining the mathematical foundations and motivation. Readers of this book will be inspired to further explore this exciting field."

**--Kenneth Massey, Massey Ratings**

"Langville and Meyer provide a rigorous yet lighthearted tour through the landscape of ratings methodologies. This is an enjoyable read that looks at ratings through the lens of sports, but also touches on how ratings affect our everyday lives through movies, Web search, online shopping, and other applications."**--Chris Volinsky, member of the winning Netflix Prize team**

"*Who's #1?* provides a much-needed synthesis of the methods used for ranking and rating things like sports teams, movies, politicians, and more. There is a ton of interest in this topic, and readers now have one place to look for a comprehensive treatment of the different approaches."**--Wayne L. Winston, author of Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football**

"This highly accessible book gives readers a comprehensive account of the different mathematical ranking techniques across many different disciplines, and will appeal to everyone from researchers to sports statistics junkies."**--Sep Kamvar, author of Numerical Algorithms for Personalized Search in Self-organizing Information Networks**

#### Synopsis

*Who's #1?*offers an engaging and accessible account of how scientific rating and ranking methods are created and applied to a variety of uses.

Amy Langville and Carl Meyer provide the first comprehensive overview of the mathematical algorithms and methods used to rate and rank sports teams, political candidates, products, Web pages, and more. In a series of interesting asides, Langville and Meyer provide fascinating insights into the ingenious contributions of many of the field's pioneers. They survey and compare the different methods employed today, showing why their strengths and weaknesses depend on the underlying goal, and explaining why and when a given method should be considered. Langville and Meyer also describe what can and can't be expected from the most widely used systems.

The science of rating and ranking touches virtually every facet of our lives, and now you don't need to be an expert to understand how it really works. *Who's #1?* is the definitive introduction to the subject. It features easy-to-understand examples and interesting trivia and historical facts, and much of the required mathematics is included.

#### Synopsis

A website's ranking on Google can spell the difference between success and failure for a new business. NCAA football ratings determine which schools get to play for the big money in postseason bowl games. Product ratings influence everything from the clothes we wear to the movies we select on Netflix. Ratings and rankings are everywhere, but how exactly do they work? *Who's #1?* offers an engaging and accessible account of how scientific rating and ranking methods are created and applied to a variety of uses.

Amy Langville and Carl Meyer provide the first comprehensive overview of the mathematical algorithms and methods used to rate and rank sports teams, political candidates, products, Web pages, and more. In a series of interesting asides, Langville and Meyer provide fascinating insights into the ingenious contributions of many of the field's pioneers. They survey and compare the different methods employed today, showing why their strengths and weaknesses depend on the underlying goal, and explaining why and when a given method should be considered. Langville and Meyer also describe what can and can't be expected from the most widely used systems.

The science of rating and ranking touches virtually every facet of our lives, and now you don't need to be an expert to understand how it really works. *Who's #1?* is the definitive introduction to the subject. It features easy-to-understand examples and interesting trivia and historical facts, and much of the required mathematics is included.

#### Synopsis

"*Who's #1?* is an excellent survey of the fundamental ideas behind mathematical rating systems. Once a realm of sports enthusiasts, ranking things is becoming a vital tool in many information-age applications. Langville and Meyer compare and contrast a variety of models, explaining the mathematical foundations and motivation. Readers of this book will be inspired to further explore this exciting field."--Kenneth Massey, Massey Ratings

"Langville and Meyer provide a rigorous yet lighthearted tour through the landscape of ratings methodologies. This is an enjoyable read that looks at ratings through the lens of sports, but also touches on how ratings affect our everyday lives through movies, Web search, online shopping, and other applications."--Chris Volinsky, member of the winning Netflix Prize team

"*Who's #1?* provides a much-needed synthesis of the methods used for ranking and rating things like sports teams, movies, politicians, and more. There is a ton of interest in this topic, and readers now have one place to look for a comprehensive treatment of the different approaches."--Wayne L. Winston, author of *Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football *

"This highly accessible book gives readers a comprehensive account of the different mathematical ranking techniques across many different disciplines, and will appeal to everyone from researchers to sports statistics junkies."--Sep Kamvar, author of *Numerical Algorithms for Personalized Search in Self-organizing Information Networks*

#### Synopsis

*Who's #1?*offers an engaging and accessible account of how scientific rating and ranking methods are created and applied to a variety of uses.

Amy Langville and Carl Meyer provide the first comprehensive overview of the mathematical algorithms and methods used to rate and rank sports teams, political candidates, products, Web pages, and more. In a series of interesting asides, Langville and Meyer provide fascinating insights into the ingenious contributions of many of the field's pioneers. They survey and compare the different methods employed today, showing why their strengths and weaknesses depend on the underlying goal, and explaining why and when a given method should be considered. Langville and Meyer also describe what can and can't be expected from the most widely used systems.

The science of rating and ranking touches virtually every facet of our lives, and now you don't need to be an expert to understand how it really works. *Who's #1?* is the definitive introduction to the subject. It features easy-to-understand examples and interesting trivia and historical facts, and much of the required mathematics is included.

#### Synopsis

*Who's #1?*is an excellent survey of the fundamental ideas behind mathematical rating systems. Once a realm of sports enthusiasts, ranking things is becoming a vital tool in many information-age applications. Langville and Meyer compare and contrast a variety of models, explaining the mathematical foundations and motivation. Readers of this book will be inspired to further explore this exciting field."--Kenneth Massey, Massey Ratings

"Langville and Meyer provide a rigorous yet lighthearted tour through the landscape of ratings methodologies. This is an enjoyable read that looks at ratings through the lens of sports, but also touches on how ratings affect our everyday lives through movies, Web search, online shopping, and other applications."--Chris Volinsky, member of the winning Netflix Prize team

"*Who's #1?* provides a much-needed synthesis of the methods used for ranking and rating things like sports teams, movies, politicians, and more. There is a ton of interest in this topic, and readers now have one place to look for a comprehensive treatment of the different approaches."--Wayne L. Winston, author of *Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football *

"This highly accessible book gives readers a comprehensive account of the different mathematical ranking techniques across many different disciplines, and will appeal to everyone from researchers to sports statistics junkies."--Sep Kamvar, author of *Numerical Algorithms for Personalized Search in Self-organizing Information Networks*

### About the Author

**Amy N. Langville**is associate professor of mathematics at the College of Charleston.

**Carl D. Meyer**is professor of mathematics at North Carolina State University. They are the authors of

*Google's PageRank and Beyond: The Science of Search Engine Rankings*(Princeton).

### Table of Contents

Purpose xiii

Audience xiii

Prerequisites xiii

Teaching from This Book xiv

Acknowledgments xiv

Chapter 1. Introduction to Ranking 1

Social Choice and Arrow's Impossibility Theorem 3

Arrow's Impossibility Theorem 4

Small Running Example 4

Chapter 2. Massey's Method 9

Initial Massey Rating Method 9

Massey's Main Idea 9

The Running Example Using the Massey Rating Method 11

Advanced Features of the Massey Rating Method 11

The Running Example: Advanced Massey Rating Method 12

Summary of the Massey Rating Method 13

Chapter 3. Colley's Method 21

The Running Example 23

Summary of the Colley Rating Method 24

Connection between Massey and Colley Methods 24

Chapter 4. Keener's Method 29

Strength and Rating Stipulations 29

Selecting Strength Attributes 29

Laplace's Rule of Succession 30

To Skew or Not to Skew? 31

Normalization 32

Chicken or Egg? 33

Ratings 33

Strength 33

The Keystone Equation 34

Constraints 35

Perron-Frobenius 36

Important Properties 37

Computing the Ratings Vector 37

Forcing Irreducibility and Primitivity 39

Summary 40

The 2009-2010 NFL Season 42

Jim Keener vs. Bill James 45

Back to the Future 48

Can Keener Make You Rich? 49

Conclusion 50

Chapter 5. Elo's System 53

Elegant Wisdom 55

The K-Factor 55

The Logistic Parameter ? 56

Constant Sums 56

Elo in the NFL 57

Hindsight Accuracy 58

Foresight Accuracy 59

Incorporating Game Scores 59

Hindsight and Foresight with ? = 1000, K = 32, H = 15 60

Using Variable K-Factors with NFL Scores 60

Hindsight and Foresight Using Scores and Variable K-Factors 62

Game-by-Game Analysis 62

Conclusion 64

Chapter 6. The Markov Method 67

The Markov Method 67

Voting with Losses 68

Losers Vote with Point Differentials 69

Winners and Losers Vote with Points 70

Beyond Game Scores 71

Handling Undefeated Teams 73

Summary of the Markov Rating Method 75

Connection between the Markov and Massey Methods 76

Chapter 7. The Offense-Defense Rating Method 79

OD Objective 79

OD Premise 79

But Which Comes First? 80

Alternating Refinement Process 81

The Divorce 81

Combining the OD Ratings 82

Our Recurring Example 82

Scoring vs. Yardage 83

The 2009-2010 NFL OD Ratings 84

Mathematical Analysis of the OD Method 87

Diagonals 88

Sinkhorn-Knopp 89

OD Matrices 89

The OD Ratings and Sinkhorn-Knopp 90

Cheating a Bit 91

Chapter 8. Ranking by Reordering Methods 97

Rank Differentials 98

The Running Example 99

Solving the Optimization Problem 101

The Relaxed Problem 103

An Evolutionary Approach 103

Advanced Rank-Differential Models 105

Summary of the Rank-Differential Method 106

Properties of the Rank-Differential Method 106

Rating Differentials 107

The Running Example 109

Solving the Reordering Problem 110

Summary of the Rating-Differential Method 111

Chapter 9. Point Spreads 113

What It Is (and Isn't) 113

The Vig (or Juice) 114

Why Not Just Offer Odds? 114

How Spread Betting Works 114

Beating the Spread 115

Over/Under Betting 115

Why Is It Difficult for Ratings to Predict Spreads? 116

Using Spreads to Build Ratings (to Predict Spreads?) 117

NFL 2009-2010 Spread Ratings 120

Some Shootouts 121

Other Pair-wise Comparisons 124

Conclusion 125

Chapter 10. User Preference Ratings 127

Direct Comparisons 129

Direct Comparisons, Preference Graphs, and Markov Chains 130

Centroids vs. Markov Chains 132

Conclusion 133

Chapter 11. Handling Ties 135

Input Ties vs. Output Ties 136

Incorporating Ties 136

The Colley Method 136

The Massey Method 137

The Markov Method 137

The OD, Keener, and Elo Methods 138

Theoretical Results from Perturbation Analysis 139

Results from Real Datasets 140

Ranking Movies 140

Ranking NHL Hockey Teams 141

Induced Ties 142

Summary 144

Chapter 12. Incorporating Weights 147

Four Basic Weighting Schemes 147

Weighted Massey 149

Weighted Colley 150

Weighted Keener 150

Weighted Elo 150

Weighted Markov 150

Weighted OD 151

Weighted Differential Methods 151

Chapter 13. "What If . . ." Scenarios and Sensitivity 155

The Impact of a Rank-One Update 155

Sensitivity 156

Chapter 14. Rank Aggregation-Part 1 159

Arrow's Criteria Revisited 160

Rank-Aggregation Methods 163

Borda Count 165

Average Rank 166

Simulated Game Data 167

Graph Theory Method of Rank Aggregation 172

A Refinement Step after Rank Aggregation 175

Rating Aggregation 176

Producing Rating Vectors from Rating Aggregation-Matrices 178

Summary of Aggregation Methods 181

Chapter 15. Rank Aggregation-Part 2 183

The Running Example 185

Solving the BILP 186

Multiple Optimal Solutions for the BILP 187

The LP Relaxation of the BILP 188

Constraint Relaxation 190

Sensitivity Analysis 191

Bounding 191

Summary of the Rank-Aggregation (by Optimization) Method 193

Revisiting the Rating-Differential Method 194

Rating Differential vs. Rank Aggregation 194

The Running Example 196

Chapter 16. Methods of Comparison 201

Qualitative Deviation between Two Ranked Lists 201

Kendall's Tau 203

Kendall's Tau on Full Lists 204

Kendall's Tau on Partial Lists 205

Spearman's Weighted Footrule on Full Lists 206

Spearman's Weighted Footrule on Partial Lists 207

Partial Lists of Varying Length 210

Yardsticks: Comparing to a Known Standard 211

Yardsticks: Comparing to an Aggregated List 211

Retroactive Scoring 212

Future Predictions 212

Learning Curve 214

Distance to Hillside Form 214

Chapter 17. Data 217

Massey's Sports Data Server 217

Pomeroy's College Basketball Data 218

Scraping Your Own Data 218

Creating Pair-wise Comparison Matrices 220

Chapter 18. Epilogue 223

Analytic Hierarchy Process (AHP) 223

The Redmond Method 223

The Park-Newman Method 224

Logistic Regression/Markov Chain Method (LRMC) 224

Hochbaum Methods 224

Monte Carlo Simulations 224

Hard Core Statistical Analysis 225

And So Many Others 225

Glossary 231

Bibliography 235

Index 241

### Product Details

- ISBN:
- 9780691154220
- Binding:
- Hardcover
- Publication date:
- 02/26/2012
- Publisher:
- Princeton University Press
- Language:
- English
- Pages:
- 272
- Height:
- 10 in
- Width:
- 7 in
- Thickness:
- .75
- Illustration:
- Yes
- Author:
- Author:
- Author:
- Author:
- Author:
- Author:
- Media Run Time:
- B
- Subject:
- Applied Science and Engineering
- Subject:
- Popular science
- Subject:
- Mathematics
- Subject:
- Mathematics | Probability and Statistics