Who's #1? The Science of Rating and Ranking

Who's #1? The Science of Rating and Ranking

Who's #1? The Science of Rating and Ranking

Who's #1? The Science of Rating and Ranking


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.


A natural approach in the science of ratings is to first rate individual attributes of each team or participant, and then combine these to generate a single number that reflects overall strength. in particular, success in most contests requires a strong offense as well as a strong defense, so it makes sense to try to rate each separately before drawing conclusions about who is #1.

But this is easier said than done, especially with regard to rating offensive strength and defensive strength. the problem is, as everyone knows, playing against a team with an impotent defense can make even the most anemic offense to appear stronger than it really is. and similarly, the apparent defensive prowess of a team is a direct consequence on how strong (or weak) the opposition’s offense is. in other words, there is a circular relationship between perceived offensive strength and defensive strength, so each must take the other into account when trying to separately rate offensive power and defensive power. How this is accomplished is the point of this chapter.

Od Objective

The objective is to assign each team (or participant) both an offensive rating (denoted by O > 0 for team i) and a defensive rating (denoted by d > 0 for team i) that are derived from statistics accumulated from past play. Given any particular sport or contest, there are generally many statistics that might be considered, but scores are the most basic statistics, and actually they do a pretty good job. Moreover, once you understand how our od method uses scores to produce offensive and defensive ratings, you will then easily see how to use statistics other than scores to build your own systems. For these reasons, we will describe our od rating technique in terms of scores.

Od Premise

Suppose that m teams are to be rated, and use scoring as the primary measure of performance. Team j is considered to be a good offensive team and deserves a high offensive rating when it scores many points against its opponents, particularly opponents with high defensive ratings. Similarly, team i is a good defensive team and deserves a high defensive rating when it holds its opponents, particularly strong offensive teams, to low scores. Consequently, offensive and defensive ratings are intertwined in the sense that each offensive rating is some function f of all defensive ratings, and each defensive rating is some function g of all offensive ratings.

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