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Who's #1?

Book Description

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.

Table of Contents

  1. Cover
  2. Half title
  3. Title
  4. Copyright
  5. Contents
  6. Preface
    1. Purpose
    2. Audience
    3. Prerequisites
    4. Teaching from This Book
    5. Acknowledgments
  7. Chapter 1. Introduction to Ranking
    1. Social Choice and Arrow’s Impossibility Theorem
    2. Arrow’s Impossibility Theorem
    3. Small Running Example
  8. Chapter 2. Massey’s Method
    1. Initial Massey Rating Method
    2. Massey’s Main Idea
    3. The Running Example Using the Massey Rating Method
    4. Advanced Features of the Massey Rating Method
    5. The Running Example: Advanced Massey Rating Method
    6. Summary of the Massey Rating Method
  9. Chapter 3. Colley’s Method
    1. The Running Example
    2. Summary of the Colley Rating Method
    3. Connection between Massey and Colley Methods
  10. Chapter 4. Keener’s Method
    1. Strength and Rating Stipulations
    2. Selecting Strength Attributes
    3. Laplace’s Rule of Succession
    4. To Skew or Not to Skew?
    5. Normalization
    6. Chicken or Egg?
    7. Ratings
    8. Strength
    9. The Keystone Equation
    10. Constraints
    11. Perron–Frobenius
    12. Important Properties
    13. Computing the Ratings Vector
    14. Forcing Irreducibility and Primitivity
    15. Summary
    16. The 2009–2010 NFL Season
    17. Jim Keener vs. Bill James
    18. Back to the Future
    19. Can Keener Make You Rich?
    20. Conclusion
  11. Chapter 5. Elo’s System
    1. Elegant Wisdom
    2. The K-Factor
    3. The Logistic Parameter ξ
    4. Constant Sums
    5. Elo in the NFL
    6. Hindsight Accuracy
    7. Foresight Accuracy
    8. Incorporating Game Scores
    9. Hindsight and Foresight with ξ = 1000, K = 32, H = 15
    10. Using Variable K-Factors with NFL Scores
    11. Hindsight and Foresight Using Scores and Variable K-Factors
    12. Game-by-Game Analysis
    13. Conclusion
  12. Chapter 6. The Markov Method
    1. The Markov Method
    2. Voting with Losses
    3. Losers Vote with Point Differentials
    4. Winners and Losers Vote with Points
    5. Beyond Game Scores
    6. Handling Undefeated Teams
    7. Summary of the Markov Rating Method
    8. Connection between the Markov and Massey Methods
  13. Chapter 7. The Offense–Defense Rating Method
    1. OD Objective
    2. OD Premise
    3. But Which Comes First?
    4. Alternating Refinement Process
    5. The Divorce
    6. Combining the OD Ratings
    7. Our Recurring Example
    8. Scoring vs. Yardage
    9. The 2009–2010 NFL OD Ratings
    10. Mathematical Analysis of the OD Method
    11. Diagonals
    12. Sinkhorn–Knopp
    13. OD Matrices
    14. The OD Ratings and Sinkhorn–Knopp
    15. Cheating a Bit
  14. Chapter 8. Ranking by Reordering Methods
    1. Rank Differentials
    2. The Running Example
    3. Solving the Optimization Problem
    4. The Relaxed Problem
    5. An Evolutionary Approach
    6. Advanced Rank-Differential Models
    7. Summary of the Rank-Differential Method
    8. Properties of the Rank-Differential Method
    9. Rating Differentials
    10. The Running Example
    11. Solving the Reordering Problem
    12. Summary of the Rating-Differential Method
  15. Chapter 9. Point Spreads
    1. What It Is (and Isn’t)
    2. The Vig (or Juice)
    3. Why Not Just Offer Odds?
    4. How Spread Betting Works
    5. Beating the Spread
    6. Over/Under Betting
    7. Why Is It Difficult for Ratings to Predict Spreads?
    8. Using Spreads to Build Ratings (to Predict Spreads?)
    9. NFL 2009–2010 Spread Ratings
    10. Some Shootouts
    11. Other Pair-wise Comparisons
    12. Conclusion
  16. Chapter 10. User Preference Ratings
    1. Direct Comparisons
    2. Direct Comparisons, Preference Graphs, and Markov Chains
    3. Centroids vs. Markov Chains
    4. Conclusion
  17. Chapter 11. Handling Ties
    1. Input Ties vs. Output Ties
    2. Incorporating Ties
    3. The Colley Method
    4. The Massey Method
    5. The Markov Method
    6. The OD, Keener, and Elo Methods
    7. Theoretical Results from Perturbation Analysis
    8. Results from Real Datasets
    9. Ranking Movies
    10. Ranking NHL Hockey Teams
    11. Induced Ties
    12. Summary
  18. Chapter 12. Incorporating Weights
    1. Four Basic Weighting Schemes
    2. Weighted Massey
    3. Weighted Colley
    4. Weighted Keener
    5. Weighted Elo
    6. Weighted Markov
    7. Weighted OD
    8. Weighted Differential Methods
  19. Chapter 13. “What If . . .” Scenarios and Sensitivity
    1. The Impact of a Rank-One Update
    2. Sensitivity
  20. Chapter 14. Rank Aggregation–Part 1
    1. Arrow’s Criteria Revisited
    2. Rank-Aggregation Methods
    3. Borda Count
    4. Average Rank
    5. Simulated Game Data
    6. Graph Theory Method of Rank Aggregation
    7. A Refinement Step after Rank Aggregation
    8. Rating Aggregation
    9. Producing Rating Vectors from Rating Aggregation-Matrices
    10. Summary of Aggregation Methods
  21. Chapter 15. Rank Aggregation–Part 2
    1. The Running Example
    2. Solving the BILP
    3. Multiple Optimal Solutions for the BILP
    4. The LP Relaxation of the BILP
    5. Constraint Relaxation
    6. Sensitivity Analysis
    7. Bounding
    8. Summary of the Rank-Aggregation (by Optimization) Method
    9. Revisiting the Rating-Differential Method
    10. Rating Differential vs. Rank Aggregation
    11. The Running Example
  22. Chapter 16. Methods of Comparison
    1. Qualitative Deviation between Two Ranked Lists
    2. Kendall’s Tau
    3. Kendall’s Tau on Full Lists
    4. Kendall’s Tau on Partial Lists
    5. Spearman’s Weighted Footrule on Full Lists
    6. Spearman’s Weighted Footrule on Partial Lists
    7. Partial Lists of Varying Length
    8. Yardsticks: Comparing to a Known Standard
    9. Yardsticks: Comparing to an Aggregated List
    10. Retroactive Scoring
    11. Future Predictions
    12. Learning Curve
    13. Distance to Hillside Form
  23. Chapter 17. Data
    1. Massey’s Sports Data Server
    2. Pomeroy’s College Basketball Data
    3. Scraping Your Own Data
    4. Creating Pair-wise Comparison Matrices
  24. Chapter 18. Epilogue
    1. Analytic Hierarchy Process (AHP)
    2. The Redmond Method
    3. The Park-Newman Method
    4. Logistic Regression/Markov Chain Method (LRMC)
    5. Hochbaum Methods
    6. Monte Carlo Simulations
    7. Hard Core Statistical Analysis
    8. And So Many Others
  25. Glossary
  26. Bibliography
  27. Index