As an award-winning 2007 advertising campaign by a German auto manufacturer emphasized, we love making lists. We make “best of” lists of places to visit, favorite movies, athletes, academic institutions, and great companies to work for, not to mention black lists of things to avoid. All of these lists have something in common: they start with the item in first place, followed by the rest.

But how is that first element determined? For most lists, some form of rating and ranking methods come into play. For example, many different criteria are employed to rank the results of a search engine query or to establish priorities among tasks to be done. In rare cases, formal criteria can be established to optimize the ranked list with respect to the desired criteria. Most often, however, programmatic ranking involves a lot of tinkering and playing with model parameters to obtain a reasonable result. And, as the authors note, this is what makes rating and ranking such an attractive area for research.

Actually, rating and ranking are two related problems. Rating techniques assign numerical scores to each item in the list, which creates a ranked list once items are sorted according to their numerical score. Other ranking methods produce order in a list based on different criteria.

In this clearly written monograph, Langville and Meyer survey some of the most popular methods employed to create ranked lists, with a particular emphasis on those employed in sports, including examples from NCAA college football (the Bowl Championship Series ratings), NFL playoffs, and NCAA college basketball (the March Madness brackets). Even though most examples are cast in terms of sports, for which statistics are widely available and easy to interpret, the ranking techniques described here can be extrapolated to many other situations.

Roughly the first half of the book is devoted to analyzing the mathematics behind different ranking methods and their key properties. Sports team ranking methods by Massey, Colley, and Keener are discussed, as well as the Elo system (famous for ranking chess players) and the Markov method (the foundation of Google’s PageRank). The authors also discuss a rank aggregation method known as the offense-defense rating method, which combines offensive and defensive ratings into a single ranked list. Another chapter is devoted to “ranking by reordering” methods, nonscalable techniques that pose the ranking problem as a formal optimization problem.

The second half of the book focuses on ranking problems. Short chapters provide interesting comments on subjects such as predicting point spreads using relative differences in the rating (an application in the sports realm, where “beating the spread is the Holy Grail for those in the betting world”), dealing with user preferences (a key feature in recommender systems, which must rank recommendations according to users’ profiles), handling ties (a minor problem in some circumstances), incorporating weights (for example, giving more weight to the recent past when tuning results), and analyzing the sensitivity of various methods to small changes. The book ends with two chapters on rank aggregation methods (aggregating several ranked lists into one, as meta-search engines do for the Web) and statistical techniques to compare alternative rankings (basically, Kendall’s tau and Spearman’s weighted footrule). The authors provide pointers to different sports data servers on the Web, which provide statistics that can be downloaded for testing and developing your own rating and ranking methods.

While the chapters on particular methods are thorough and provide all of the details necessary to understand their foundations and what to expect from them, the chapters on related topics are more superficial, simply highlighting some of the issues that should be addressed when fine-tuning one’s own ranking methods. Nevertheless, these are always informative and provide an accessible introduction to many interesting topics.

Ranking is an inherent problem whenever we compile a list. This readable book distills many different ideas and methods for ranking lists. If this is something you do or would like to do, you will find this monograph a valuable addition to your personal library.