Foreword by Scott E Page

In this brilliant, wide ranging book, Péter Érdi, an award winning teacher and scholar, takes up the phenomena of rankings and ratings (I’ll get to the difference in a moment). A computational scientist, Érdi proves equally skilled as a social observer revealing deep implications of the ubiquitous rankings and ratings created by social and mainstream media. We assign far to much credibly to numerical rankings that may, at their core, be subjective impressions. Even more troubling, as we change our behavior to move up those lists, we allow ourselves to be manipulated by the rankings.

Rankings and these behavioral responses to them occur across the socio-technical landscape. To grasp the breadth of his inquiry, take a moment and leaf through the index. Here’s one sampling taken back to front: the Wong Baker pain scale, US News and World Report University Rankings, Scar (yes, from the {\em Lion King}), recommendation letters, the illusion of objectivity, the Hungarian National Soccer Team, Erdős numbers, ELO Chess Ratings, and Campbell’s Law. The A’s alone include Jane Austen, Aristotle, and Arrow’s Impossibility Theorem.

That sampling of topics only hints at the fun in store for you. Throughout, Péter ‘s fertile, lively mind is on full display. You are in for a treat. Though the book takes up technical topics, Péter’s writing manages to be bright, funny, and clear. By book’s end, many readers may contemplate catching the train to Kalamazoo or Budapest with the hopes of meeting up with Péter to learn more about preferential attachment mechanisms, bounded rationality, social neuroscience, the psychology of list making, or the applications of network statistics. For those of you who know Péter, particularly his former students, reading the book will remind you of his boundless, generous curiosity. The book like Péter, is informative, deep, thought provoking, and joyful.
The best rankings rely on objective criteria. Rankings of the tallest buildings, largest Northern Pike, and fastest motorcycles can be accepted on face value. However, even objective criteria may, when viewed under a microscope, contain elements of subjectivity. The official height of a building includes the building’s towers if they are integral to the building. The spire on the Freedom Tower in New York counts while the two antennae atop the Willis Tower in Chicago do not. The integral portion of a building lies in the eye of the beholder. And that’s where the problems start, with the inclusion of subjectivity.

Subjectivity allows us to rank as we see fit. In the movie version of Thomas Wolfe’s “The Right Stuff”, a reporter asks the astronaut Gordon Cooper, played by a young Dennis Quaid, to name the best pilot he ever saw. Cooper first rambles about pictures on a wall in a place that no longer exists and hurtling steel, all as a prelude to naming the Chuck Yeager. When Cooper realizes that the press wants a story and has little interest in the truth of the matter, he breaks into a toothy smile and says, “ who’s the best pilot, I ever saw? well, uh, your lookin’ at him.”

{\em Air and Space Magazine} would beg to differ. They do not rank Gordon Cooper in the top ten, though Yeager does rank. \’Erdi would be quick to point out that both Cooper’s ranking and the magazine’s, just like the ubiquitous rankings we find on the World Wide Web—the top ten beaches, the top eight Belgian ales, and the top seven dog breeds—-are subjective. Some person, or group of people, made up an ordering and then justified it using criteria derived after the fact. Nevertheless, these rankings confer a degree of authority—ah the power of numbers.

Yet, as Érdi shows, in most of the important cases, objective rankings are not possible. Permit me a brief foray into formalism. Formally, a ranking is a {\em complete, asymmetric, and transitive relation}. Complete means that it compares any two things. Asymmetric means that it ranks each item either above or below every other: either you like beets more than carrots or carrots more than beets. Transitive means that if A is preferred to B, and B to C, then A must also be preferred to C.
As logical as transitivity may seem, it can be violated by collections of rankings. In a Condorcet Triple, a majority rule vote of three people each of whom has transitive preferences, results in A defeating B, B defeating C, and C defeating A. Majority rule voting becomes an instantiation of the Rock–Paper–Scissors game. In other words, even if each person has a consistent ranking, that in no way implies that a collective ranking exists.

A similar problem arises if the items we wish to rank possess multiple dimensions. Magazines rank restaurants by evaluating the quality of their food, their ambiance, and the professionalism of their staff. They then assign numbers to each restaurant on each dimension and sum them to produce a rating. Out of a total score of 30, one restaurant may score 28, while another scores 27. These numbers, as \’Erdi points out, are subjective. One person’s five out of five may be another person’s four out of five. What appears scientific is largely made up.

In fairness, often the scores are a mix of objective and subjective. Such is the case in the US News and World Report’s college rankings which take into account the number of classes a college offers with fewer than nineteen students and the overall faculty to student ratio (both objective) as well as ranking by deans (subjective). To create ratings (which can then be simplified to a ranking), US News then attaches a weight to each of these criteria. How do they come up with the weights? Again, those are just made up based on common sense. So once again, what appears scientific is in fact subjective.

An immediate consequence of this method is that a college can improve its ranking by limiting enrollments in some classes to only nineteen students. Doing so improves the school’s ranking, Keep in mind that no empirical evidence supports a significant loss in learning from adding a twentieth student. US News just choose the number nineteen. To see the pernicious effects of just this one ranking criterion go to almost any college web page and you will see that they advertise the number of classes with nineteen or fewer students. Colleges prevent students from taking classes (sorry, you’re number twenty) so as not to hurt their rankings.

It follows, paradoxically, that even well intentioned attempts to identify the best of us may bring out the worst in some of us, as we distort ourselves to improve our rankings. Thus, the greater importance we attach to these mostly subjective rankings, the more we produce behavioral distortions. With this book, Professor \’Erdi has done us a service. He has taught us to think more deeply, yet done so with a captivating examples and a light hand.
Scott E Page
Leonid Hurwicz Collegiate Professor of Complex Systems, Political Science, and Economics, University of Michigan-Ann Arbor. External Faculty, The Santa Fe Institute