Contents
Chapter 1. Introduction to Ranking
Social Choice and Arrow’s Impossibility Theorem
The Running Example Using the Massey Rating Method
Advanced Features of the Massey Rating Method
The Running Example: Advanced Massey Rating Method
Summary of the Massey Rating Method
Summary of the Colley Rating Method
Connection between Massey and Colley Methods
Strength and Rating Stipulations
Forcing Irreducibility and Primitivity
Hindsight and Foresight with ξ = 1000, K = 32, H = 15
Using Variable K-Factors with NFL Scores
Hindsight and Foresight Using Scores and Variable K-Factors
Losers Vote with Point Differentials
Winners and Losers Vote with Points
Summary of the Markov Rating Method
Connection between the Markov and Massey Methods
Chapter 7. The Offense–Defense Rating Method
Alternating Refinement Process
Mathematical Analysis of the OD Method
The OD Ratings and Sinkhorn–Knopp
Chapter 8. Ranking by Reordering Methods
Solving the Optimization Problem
Advanced Rank-Differential Models
Summary of the Rank-Differential Method
Properties of the Rank-Differential Method
Solving the Reordering Problem
Summary of the Rating-Differential Method
Why Is It Difficult for Ratings to Predict Spreads?
Using Spreads to Build Ratings (to Predict Spreads?)
Chapter 10. User Preference Ratings
Direct Comparisons, Preference Graphs, and Markov Chains
The OD, Keener, and Elo Methods
Theoretical Results from Perturbation Analysis
Chapter 12. Incorporating Weights
Chapter 13. “What If . . .” Scenarios and Sensitivity
The Impact of a Rank-One Update
Chapter 14. Rank Aggregation–Part 1
Graph Theory Method of Rank Aggregation
A Refinement Step after Rank Aggregation
Producing Rating Vectors from Rating Aggregation-Matrices
Summary of Aggregation Methods
Chapter 15. Rank Aggregation–Part 2
Multiple Optimal Solutions for the BILP
Summary of the Rank-Aggregation (by Optimization) Method
Revisiting the Rating-Differential Method
Rating Differential vs. Rank Aggregation
Chapter 16. Methods of Comparison
Qualitative Deviation between Two Ranked Lists
Kendall’s Tau on Partial Lists
Spearman’s Weighted Footrule on Full Lists
Spearman’s Weighted Footrule on Partial Lists
Partial Lists of Varying Length
Yardsticks: Comparing to a Known Standard
Yardsticks: Comparing to an Aggregated List
Pomeroy’s College Basketball Data
Creating Pair-wise Comparison Matrices
Analytic Hierarchy Process (AHP)
Logistic Regression/Markov Chain Method (LRMC)