Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

Everyone naturally wants to know “Who’s #1?” We are grateful that Jim Keenerhas written a lively featured review telling us about ranking and the follow-up toLangville and Meyer’s 2006 best-seller Google’s PageRank and Beyond. I was a littleconcerned about the issue, however, when the next review turned out to be aboutgambling and was by a poker player, once well known in Seattle. As usual, however,we’ve again come up with a wonderful variety of expert opinions on lots of serious newapplied math monographs and textbooks deserving your attention. We thank them fortheir advice!

Bob O’MalleySection Editor

bkreview@amath.washington.edu

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SIAM REVIEW c© 2012 Society for Industrial and Applied MathematicsVol. 54, No. 4, pp. 827–847

Book Reviews

Edited by Robert E. O’Malley, Jr.

Featured Review: Who’s # 1? The Science of Rating and Ranking. By Amy N. Lang-ville and Carl D. Meyer. Princeton University Press, Princeton, NJ, 2012. $29.95. xvi+247 pp.,hardcover. ISBN 978-0-691-15422-0.

“What do you do for a living, Mr. Keener?” It is a question that I often get at partiesand am reluctant to answer, knowing that “I am a mathematician” will be an immedi-ate conversation stopper. Sometimes, however, I get to tell a story about how mathe-matics underlies many of the important changes we have had in our technology-drivenculture. Given the chance, I can regale you with accounts of how mathematics led todiscoveries of new drugs, automatic cardiac defibrillators (AEDs), and secure wirelessencryption. However, nothing holds the attention of my nontechnical audience betterthan the Google story (at least when I mention that Sergey Brin and Larry Page,cofounders of Google, are, according to Forbes Magazine, worth $16.7 billion each).

In my telling, the Google story is a story of ranking, in this case the ranking ofwebsites by Google’s ranking program PageRank. As you probably know, what Brinand Page were able to do (among other things) was to devise a method by whichthey could rank the results of a web search so that the most important “hits” couldbe listed first. The result is that when you do a Google search, it is usually the casethat what you are looking for (or what advertisers want you to see) is listed withinthe first few items. But what you may not know is that at its core the Google searchengine is the world’s largest and most computationally demanding application of thePerron–Frobenius theorem.

Ranking has become a feature of our daily lives. Not only do we regularlyrely on Google to rank the results of a search, we rely on Amazon to recommendproducts, Netflix to recommend movies, and Facebook to tell us what other peoplelike most. If you are a college football fan, then you know about the BCS (BowlChampionship Series) and its dependence on a combination of computer rankingsand polls to determine the participants in the major college bowl games. There arerankings for all competitive sports, including basketball, gymnastics, tennis, baseball,racquetball, golf, chess, and video gaming. As academicians, we pay attention todepartmental rankings by U.S. News and impact factors of the journals in which wepublish our latest and greatest results.

In their first book on the subject of ranking systems [3], Langville and Meyer pro-vided a history of the development of web search engines and the mathematics behindthem. In this most recent book, they describe the general problem of ranking, with aprimary emphasis on the ranking of sports teams, primarily football and basketball.

The problem of ranking sports teams (or competitors) is easy to state: Find anordering of competitors that is reflective of how good they are, based on informationabout the outcome of head-to-head competitions. Typically, a well-ordering does not

Publishers are invited to send books for review to Book Reviews Editor, SIAM, 3600 Market St.,6th Floor, Philadelphia, PA 19104-2688.

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828 BOOK REVIEWS

exist since there are almost always situations in which competitor A beat competitorB who beat competitor C who beat competitor A. Furthermore, many competitionsare unbalanced, since not all teams compete against all other teams an equal numberof times and there are often teams that do not compete directly against each other.

Other important ranking problems are those faced by Amazon and Netflix: Finda ranking of products based on reviews by consumers that accurately predicts howwell you will like a product and that can make reliable recommendations.

The building of a ranking system is like any other exercise in applied mathematics.First, one must identify a problem that is quantifiable, in this case, find a linearordering of competitors that is reflective of how good they are. Next, develop amodel, i.e., a system of equations that incorporates the data with the ranking asthe unknown variables, then develop numerical algorithms to solve the system ofequations accurately and efficiently. After a solution is found, there will be weaknessesor inconsistencies in the results, so one iterates on this process until an acceptablesolution is found.

Langville and Meyer walk their readers through this process several times, de-scribing models that are currently used by different people or organizations. Theystart with a description of the basic idea of the model to determine a ranking. Forfootball or basketball, one begins with win-loss records or the scores from head-to-headcontests. Then the authors describe the mathematics needed to solve the problem.After evaluating the results, they suggest improvements or modifications to the modelto account for features that were not incorporated in the original formulation. Someof these modifications for football and basketball include accounting for home fieldadvantage, accounting for changes over the course of a competitive season, accountingfor differences between offensive and defensive strengths, etc. They explore the conse-quences of some of these modifications but also suggest other modifications that inter-ested readers can pursue on their own, thus making this exploration nicely open ended.

A quick overview of the main methods described by Langville and Meyer is asfollows: Massey’s method finds the least squares solution of the overdetermined systemof equations dij ≡ ri− rj = yij , where ri is a rating of team i and yij is the differencein the scores from a contest between the two teams. Colley’s method defines the ratingvector r as the solution of a linear system of equations Ar = y, where the entries ofA are related to the number of times teams play, and the vector y is related to thedifference between wins and losses for each team. Keener’s method (not surprisingly,the one I like best) defines the rating vector to be the positive eigenvector of a matrixA whose entries are related to the scores of contests between teams. This is also themethod that underlies Google’s PageRank. Two other closely related methods, theMarkov method and User Preference Ratings, look for a rating that is the positiveeigenvector of a matrix A (actually a fixed point of A, since the column sums of Aare all 1), where the entries of A are reflective of the outcome of a contest betweencompetitors or, in the case of product preferences, the result of surveys of consumers.

Finally, Elo’s method (used for chess) updates ratings iteratively taking

rnewi = roldi +K(Sij + f(doldij )),

where Sij is a measure of the outcome of the most recent contest between teams iand j, Sij + Sji = 1, and f(d) is a logistic function 0 < f < 1.

This book can be used effectively in several ways. The mathematical hobbyist(with an undergraduate background in linear algebra and MATLAB) can easily startdeveloping her own ranking system for the competition of most interest. Langville and

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BOOK REVIEWS 829

Meyer devote a significant amount of attention to their own experiences with studentsand colleagues trying to predict the outcome of the NCAA basketball tournament(March Madness). They also include a chapter on where and how to get data, allof which is extremely helpful in getting started yourself. The book is, however, tootechnical to be appreciated by a true lay reader.

The book could be used to supplement a course on linear algebra and/or numeri-cal linear algebra. Mathematical topics include least squares (regression) methods andnormal equations (they don’t mention, but easily could, the Fredholm Alternative),eigenvector methods, and, associated with this, the Perron–Frobenius theorem, irre-ducibility and connectivity, Markov methods and stochastic matrices, and reorderingoptimization problems. In the process, the reader is exposed to a variety of numericalissues and solution techniques, including Gaussian elimination, Cholesky decompo-sitions, iterative methods, including the power method and GMRES, and a varietyof other optimization techniques. Of course, along with these important numericalmethods come an appreciation for and skill in MATLAB programming.

The book could also be used as the basis for a short topics course or undergradu-ate research project on ranking, or it could be used in a modeling class as an exampleof how mathematical modeling is done.

In addition to describing the mathematics of ranking, the book is full of inter-esting tidbits that add to the pleasure of its reading. Langville and Meyer devote asignificant amount of attention to the practical issues surrounding ranking systemsand the many questions (sometimes controversial) we all have relating to rankingmethods. Different ranking algorithms give different answers, and where the stakesare high (such as in deciding which team goes to a bowl game), these differences canbe important. Their comparisons of different methods are quite illuminating. Theyalso describe the algorithm used by U.S. News to rank universities and departmentaldegree programs and state that this is one situation in which a more sophisticatedranking algorithm would be preferable. They also point out that some ranking systemscan be manipulated to give skewed results [1]. For example, they describe methodsthat web designers have used to increase the Google ranking of their websites, andways in which the ranking system designers have eliminated some of these issues.

But the one question that appears throughout the book, and that is also thenumber one question that people ask when talking about ranking systems, is “canranking systems make you rich?” Of course, what is meant by this question is “canthe results of a ranking system be used in a predictive fashion to improve the odds ofwinning a bet on the outcome of the competition?” This is, for some people, the holygrail of ranking systems.

Langville and Meyer devote a significant amount of attention to this question,including a section that is perhaps my favorite, entitled, “Can Keener Make youRich?” Of course, I could have answered that question quite easily, having heardmany times the question, “if you are so smart, how come you are not rich?” Langvilleand Meyer do a nice job of explaining why one should not expect to do particularlywell using a ranking system as an algorithm to place bets.

First and foremost, the reason is that with a ranking system one is using datafrom relatively few competitions to predict the probability that team A will beat teamB on a particular occasion. At best, one might hope for a long-term answer (i.e., anestimate of the percentage of wins and losses over a long time), but to expect highaccuracy for individual contests is not reasonable.

Furthermore, most betting these days is not a simple matter of determining thewinner (except perhaps in horse racing), but involves point spreads, or plus/minus

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830 BOOK REVIEWS

margins. Langville and Meyer give an illuminating discussion of point spreads andwhy not to expect ranking systems to do particularly well with point spreads. In anentire chapter on point spreads, they are careful to distinguish between a ranking asa measure of past performance and as a predictor of future performance.

But there is, as they say, more than one way to skin a cat, and, in this case, morethan one way to try to use ranking systems to get rich. Without making an explicitlist, Langville and Meyer mention a number of these. So, to help those of you whohave such aspirations, I have come up with my own ranking of how to make moneywith ranking systems. In reverse order:

10. Use the results of a ranking system to predict the outcome of sporting events.Place bets on the basis of these predictions. Lose less money than otherpeople who do not use ranking systems.

9. Use the results of a ranking system to predict the outcome of NCAA MarchMadness. Win the ESPN March Madness Competition. (So far, this has nothappened.)

8. Write scholarly articles on ranking systems, publish in prestigious journals[2], get tenure and a (small) pay raise.

7. Write excellent books about ranking systems (like this one), publish with aprestigious publisher, wait for books to become bestsellers, collect royalties.

6. Sell the results of your ranking system to a major news organization, write asyndicated column.

5. Develop a ranking system that is interesting to the U.S. Intelligence andmilitary effort, but unfortunately is classified. Be sure not to tell anyone thedetails of what you are doing.

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BOOK REVIEWS 831

4. Demonstrate how smart you are by developing good ranking systems. Gethired by a startup company that wants clever algorithm and software devel-opers.

3. Use the results of ranking systems to rate and promote products that yousell.

2. Use the results of the ranking system on an Internet website and sell adver-tising and/or bias the outcome of the ranking based on advertising sales.

1. Use the data from an Internet site to rank preferences of site users, start acompany like Google.

REFERENCES

[1] D. N. Arnold and K. K. Fowler, Nefarious numbers, Notices Amer. Math. Soc., 58 (2011),pp. 434–437.

[2] J. P. Keener, The Perron–Frobenius theorem and the ranking of football teams, SIAM Rev., 35(1993), pp. 80–93.

[3] A. N. Langville and C. D. Meyer, Google’s PageRank and Beyond: The Science of SearchEngine Rankings, Princeton University Press, Princeton, NJ, 2006.

JAMES KEENER

University of Utah

The Doctrine of Chances: ProbabilisticAspects of Gambling. By Stewart N. Ethier.Springer, New York, 2010. $99.00. xiv+816 pp.,hardcover. ISBN 978-3-540-78782-2.

I have been associated with the casino in-dustry for many years. I have served as aconsultant for casinos, I have written morethan 200 magazine articles on mathemat-ics and poker, I designed and taught acourse on mathematics for a casino man-agement program at a university numeroustimes, I played blackjack as a card counterfor six years, and I have played poker formany years in card rooms throughout NorthAmerica.

The aforementioned experience has ledme to identify three essentially separategroups of people with an interest in themathematics of gambling and with the po-tential to be interested in purchasing a bookon the subject. One group, of course, is com-posed of people with strong backgrounds inmathematics. Most of these people are, infact, mathematicians, but there are peoplefrom other disciplines—primarily computerscience—scattered among them.

Another group, albeit small, is com-posed of people in a program dealing with

casino management. The entire industry islargely based on applied probability, so itcertainly makes sense that people whosegoals include decision-making positions inthe casino world should have some ideaabout the mathematics forming the foun-dation of that world. One could provide astrong argument in support of this belief,but I shall not attempt to do so here.

A serious problem for this group arisesfrom the fact that such programs arenormally offered through business schools.Most of the students in these programs haveweak mathematical backgrounds and not alot of mathematical talent. This creates achallenge for designing and teaching an ap-propriate course.

I have slowly become aware of a thirdgroup over many years of meeting peoplein casinos and card rooms. These are peo-ple who are intelligent and articulate andwho spend some time thinking about whatthey are up against when playing gamesin a casino. Most of the people to whomI am referring are either blackjack players,poker players, or dealers. However, a feware interested in other casino games.

Their mathematical backgrounds varywildly. Some have studied mathematics at

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