Glossary

Arrow’s Impossibility Theorem Ken Arrow’s famous theorem about the four conditions that every voting system (and hence rating system) should satisfy. The impossibility label comes from his proof that no voting system can satisfy all four conditions simultaneously.

average rank a rank-aggregation technique where the integers representing the rank in several rank-ordered lists are averaged to create a rank-aggregated list.

authority rating a rating given to a node in a directed graph based on the quality and quantity of its inlinks.

bias-free this description refers to a method’s avoidance of the potential rating problem created when strong teams run up the score against weak teams.

Borda count a common rank-aggregation technique introduced by Jean-Charles de Borda in 1770 in which each candidate receives a score that is equal to the number of candidates he or she outranks for each ranked list. The scores from each ranked list are summed for each candidate and a single number is created. The candidates are then ranked by their total Borda Count.

BCS (Bowl Championship Series) the governing body for NCAA college football that assigns a BCS rating to each team and selects bowl participants based on these ratings.

bracketology fans of a sport complete a bracket predicting outcomes in a tournament before play officially begins.

centroid method a very inexpensive rating system that uses the centroid of a score difference matrix as the rating vector.

Colley method a rating system developed by Dr. Wesley Colley, which modifies the simplest and oldest rating system of the winning percentage as the rating for each team. His slight modification to this system revolves around the equation r_i = (1 + w_i)/(2 + t_i) and remedies some of the flaws of the winning percentage method.

Colleyized Massey method a rating system which recognizes the connection between the Massey and the Colley methods. It incorporates properties of both methods by recognizing the relationship between the two that is given by the formula C = 2I + M.

combiner method a rank-aggregation technique that minimizes the effect of outliers or anomalies that seem inconsistent with the rankings in other lists.

concordant pair given two ranked lists, a pair of items appearing in both lists is called concordant if the relative ranking of the two items is the same in both lists; i.e., if item i is ranked above (or below) item j in both lists, the (i, j) pair is concordant. Concordant pairs are used in the Kendall tau measure of rank correlation.

Condorcet winner the winner if a third candidate were to drop out of the race. For example, Al Gore was the Condorcet winner of the 2000 election in Florida. Gore would have been the overall winner had Ralph Nader not taken votes away from him and allowed George W. Bush to win.

dangling node a node that has no outlinks. In the sports context this is an undefeated team.

defensive rating vector a vector created by the OD method that contains the defensive ratings of all teams being rated.

discordant pair given two ranked lists, a pair of items appearing in both lists is called discordant if the relative ranking of the two items in the two lists does not agree; i.e., if item i is ranked above item j in one list, yet below in the other list, then the (i, j) pair is discordant. Discordant pairs are part of the Kendall tau measure of rank correlation.

Elo method a rating system based on a linear update rule that was developed by Arpad Elo to rate and rank chess players worldwide. The system has since been adapted and adopted for other applications including soccer.

evolutionary optimization a technique that uses the principles of Darwinian evolution (survival of the fittest, mating, and mutating) to solve a difficult optimization problem.

full ranked list a list that contains an arrangement of all elements in the set being ranked.

HITS an acronym for Hypertext Induced Topic Search, which was created by Jon Kleinberg to rank linked documents. This algorithm is one of the most popular and successful methods for ranking webpages and is used in the present-day search engine Ask.com.

hub rating a rating given to a node in a directed graph based on the quality and quantity of its outlinks.

independence of irrelevant alternatives Arrow’s second requirement for a voting system that states that changes in the order of candidates outside the subset should not affect the ranking of two candidates relative to each other.

Keener method a rating system developed by James Keener, which uses the dominant eigenvector of a (possibly massaged) point score matrix as the rating vector.

Kemenization an inexpensive refinement step that can be applied to the results of a rank-aggregation procedure to improve the agreement of the aggregated ranking with the input rankings.

Kendall tau measure a measure τ of rank correlation that was invented by Maurice Kendall to determine the deviation between two ranked lists; −1 ≤ τ ≤ 1. If two lists are in perfect agreement, then τ = 1 and if one list is the reverse of the other, then τ = −1.

MAD mean absolute deviation between the percentage of games a team wins during the season and the ratio of its cumulative points allowed over its cumulative points scored.

Markov method a rating system that uses the stationary vector of a Markov chain to rate, and hence, rank items.

Massey method a rating system developed by Ken Massey that uses the method of least squares to rate, and hence rank, teams. The Massey method revolves around the equation r_i − r_j = y_k, which expresses the idea that the ratings of the two teams ideally predicts the margin of victory in a contest between them.

OD a modified HITS algorithm that creates a directed graph in which each node represents a team and the links represent the interaction between the two teams. Each directed link has two interpretations: one related to the team’s offensive ability and the other related to the team’s defensive ability.

non-dictatorship Arrow’s fourth requirement for a voting system that states that no single voter should have disproportionate control over an election.

offensive rating vector a vector created by the OD method that contains the offensive ratings of all of the teams being rated.

PageRank a rating vector created by Google and their Markov method for ranking webpages.

Pareto principle Arrow’s third requirement for a voting system, which states that if all voters choose A over B, then a proper voting system should always rank A ahead of B.

partial list contains a subset of the items under consideration. This is as opposed to a full list, which contains all items.

permutation a vector of length n containing the integers 1 through n. In the ranking context, the order of these integers corresponds to an item’s rank position.

plurality voting a voting method in which each voter submits just a lone vote for their top choice.

point spread the point differential between two teams in a given matchup.

preference list a voting method in which each voter submits a list of candidates in a ranked order.

ranked list a list of n items that contains a permutation of the integers 1 through n where each integer gives the rank position of the corresponding item.

rank-aggregation the process of using mathematical techniques to create one robust aggregated list from several individually ranked lists.

rating list a list of scalars representing the rating of the corresponding items in the list. A rating list, when sorted, produces a ranked list.

retrodictive scoring a method for comparing various ranking methods. Retrodictive scoring uses a method’s ranked list to predict the winners of past matchups.

smart ranking method a ranking method that has a short warm-up period before it starts making accurate predictions; i.e., the method does not require a great deal of data before predicting well.

top-k list a partial list of length k that contains a ranking of only the top k items, i.e., 1^st through k^th place.

unrestricted domain Arrow’s first requirement for a voting system that states that every voter should be able to rank the candidates in any arrangement he or she desires.

vig short for vigorish, is the mechanism by which bookies make their money. It is simply the fee charged for the bookmaker’s service.

winning percentage a very simplistic method for ranking items by the percentage of wins out of the total number of pair-wise relationships.