Sponsored by the Institute for Mathematical Behavioral Sciences and
the Center for Decision Analysis
December 4, 2004 - Social Science Plaza A, room 2112



Some Variations on the Elo System for Rating Chess Ability

William H. Batchelder
Professor of Cognitive Sciences and Institute for Mathematical Behavioral Sciences
University of California at Irvine

Arpad Elo invented, albeit somewhat informally, a system to rate chess ability in the early 1960s. The system has developed into current practice internationally, and it is used to rate over a hundred thousand chess players on the same scale. The system is used to pair players during tournaments as well as to determine qualification levels and invitations for future tournaments. The system takes as input the results of chess games- win, loss, or draw- and, as a function of the prior ratings of the players and the game results; it computes revised ratings for the players. Further, given the ratings of the players, it can calculate predictions of match and round-robin competition. One way to think of the system is that it is a statistically based, tracking system for monotone paired-comparison systems that undergo dynamic, but not explicitly modeled, changes over time. Chess players value their ratings in much the same way that folks value money, and especially the elderly players would resist the incorporation of an explicit, age-based model of the change. This observation suggests some axioms for formalizing a tracking system that our group has studied, and I will present some results and open problems.

Rating Professional Golfers

Scott M. Berry
Berry Consultants

The rating of professional golfers has become a very important exercise within the golf community.  These ratings are used for tournament qualifying, which is crucial for the success of golfers.  A "computerized" ranking is used to rank golfers worldwide.  In this talk I will discuss these rankings--specifically the use of money won, combining results from multiple tours, and the time horizon used.  I develop my own golf rankings which are optimal in predicting future golf tournament success.

Sequencing Effects in Jai-Alai Games and Seeding Effects in Tennis Tournaments

Bernard Grofman
Department of Political Science and Institute for Mathematical Behavioral Sciences
University of California at Irvine

The application of decisision theoretic and statistical modeling to sports has covered many topics, including the impact of alternative sports scoring and ranking rules (e.g., what happens in ice skating, how best to decide which teams should make the playoffs or be invited to Bowl games), perceptual biases among bettors (e.g., underbetting favorites), evaluating optimal sports strategies (e.g., when to bunt), and looking at the nature of competition functions as a function of sampling frames (e.g., the impact of point, game, set, match in tennis on magnifying the impact on outcomes of small differences between players); projecting trends in sports performance (e.g., time to run the mile, gender differences), etc. Here we provide some �toy� models of the impact of sequencing effects on the outcomes of sports contests, by looking at order effects in jai-alai matches, on the one hand, and seeding of favorites effects in tennis (and other sports) tournaments, on the other hand. The ideas we present are basically in the nature of �throw-aways,� suggesting topics where more sophisticated analyses and clear theorematic results would be desirable. But even more importantly, they also are fun topics for teaching intro statistics.

Ranking College Football or Basketball Teams: Some Objectives and a Modified Least-Squares Approach

David A. Harville
Research Staff Member Emeritus
Mathematical Sciences Department, IBM

Rankings play a role in the selection of college football teams for inclusion in BCS bowl games and in the selection of college basketball teams for inclusion in postseason tournaments. Such rankings affect the teams that are being ranked, and should be based on a ranking system having certain attributes, including accuracy, appropriateness, impartiality, unobtrusiveness, nondisruptiveness, verifiability, and comprehensibility. A system possessing all of these attributes, except for unobtrusiveness, can be achieved by applying least squares to a statistical model in which the expected difference in score in each game is modeled as a difference in team effects plus or minus a home field/court advantage. The potential obtrusiveness of this approach can be largely eliminated by introducing modifications to reward winning per se and to eliminate any incentive for �running up the score�.

General Blotto and Competitive Games

Scott Page
Professor, Senior Research Scientist, Ctr. for Political Studies
Assoc. Director, Ctr. for the Study of Complex Systems, Northwestern

In this paper, we discuss a generalization of the Colonel Blotto game that we call General Blotto.  We use this game to show the prevalence of cycles in sporting competition.

From Voting to Judging Sports Events

Donald G. Saari
Distinguished Professor of Mathematics and Econonomics and Director, Insitute for Mathematical Behavioral Sciences
University of California, Irvine

The judging in several sports is based on a subjective evaluations.  In some sports, such as figure skating, the schemes reflect the history of abuse, strategic behavior, and just plain cheating.  While "reforms" in these sports have been directed toward countering abuse, we must wonder whether, even when all judges act sincerely,  the final outcome has anything to do with how they collectively rank the contestants.  By appealing to recent results in the voting literature, I will raise doubts about the accuracy of the outcomes, and then I will make a "reform" proposal.

Modeling Contests

Stergios Skaperdas
Professor of Economics and Institute for Mathematical Behavioral Sciences
Univerity of California at Irvine

Contests are games in which players compete for a prize by exerting effort so as to increase their probability of winning. We will first review axiomatic and stochastic approaches to justifying contest success function. We will then discuss some results that emphasize applications to athletic contests.

What is the point of the Bowl Championship Series?

Hal Stern
Professof Statistics and Institute for Mathematical Behavioral Sciences
University of California at Irvine

The US college football championship is determined each winter by the Bowl Championship Series (BCS), a set of four college football games and an associated ranking system that helps to determine the participants in the games.  One of the games is pre-designated as the national championship game and the top two teams play in that game.  The BCS has been controversial since its implementation prior to the 1998 season with each controversy provoking a change in the rating formula.  Here we talk about statistical ideas that might help.