Who's No.1?

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