Ali, I., W. Cook, and M. Kress "On the minimum violations ranking of a tournament" Management Science 32:6 1986 660-672 Soren P. Sorensen Nov 12 '99 Amoako-Adu, B., H. Marmer, and J. Yagil "The Effeciency of Certain Speculative Markets and Gambler Behavior" Journal of Economics and Business 37 1985 Soren P. Sorensen Nov 12 '99 Anderson, L.B., "Paired Comparisons," Chapter 17 in Operations Research and the Public Sector, Volume 6 of Handbooks in Operations Research and Management Science, S. M. Pollock, M. H. Rothkopf, and A. Barnett, eds., Amsterdam, London, New York, Tokyo: North-Holland, 1994, 585-620 Lowell Anderson Sep 4 '01 Annis, David H. and Bruce A. Craig (2005) "Hybrid Paired Comparison Analysis, with Applications to the Ranking of College Football Teams," Journal of Quantitative Analysis in Sports: Vol. 1 : Iss. 1, Article 3. Christopher M. Rump Jan 9 '07 Annis, David H. "Dimension Reduction for Hybrid Paired Comparison Models" Journal of Quantitative Analysis in Sports 3:2, April 2007 John Trono May 1, 2007 Annis, David H. and Samuel S. Wu "A Comparison of Potential Playoff Systems for NCAA I-A Football" Jim Ashburn May 25, 2007 Ashburn, James and Paul Colvert "A Bayesian Mean-Value Approach with a Self-Consistently Determined Prior Distribution for the Ranking of College Football Teams" Feb. 2006 Jim Ashburn Mar 24 '06 Bassett, Gilbert W. "Robust Sports Ratings Based on Least Absolute Errors." The American Statistician. May 1997:1-7 Kenneth Massey Sep 10 '99 Bethel, Roy "A Solution to the Unequal Strength of Schedule Problem." 2005. NFL Results. Boand, William F. "Azzi Ratem." 1930. Booklet. and "Azzi Ratem - Boand Football Ranking System." 1950. Booklet. An explanation of a mathematical rating system. Ratings of leading teams. Anton Grobani 1975 Boand, William F. "National Football Champions." 1941. Booklet. Facts, ratings since 1924. Anton Grobani 1975 Boginski, Vladimir, Sergiy Butenko and Panos M. Pardalos "Matrix-based Methods for College Football Rankings" 2003 Jim Ashburn Nov 23 '05 Bradley, R. A. and Terry, M. E. "Rank analysis of incomplete block designs. I. The method of paired comparisons", Biometrika (volume 39, 1952) pages 324-45. The idea was first proposed (I've been told) in Zermelo, E. (1929). Die Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung. Mathematische Zeitschrift 29, 436-60. David Rothman Dec 2 '98 Bühlmann, Hans and Peter J. Huber, "Pairwise Comparison and Ranking in Tournaments," The Annals of Mathematical Statistics, Received Oct. 1962, 501-510. Jim Ashborn Apr 24 '09 Callaghan, Thomas, Mason A. Porter, and Peter J. Mucha "Random Walker Ranking for NCAA Division I-A Football", Georgia Institute of Technology 2003 Thomas Callaghan (an undergrad), Mason Porter (a postdoc), and I have recently studied a relatively simple-to-explain ranking scheme, in terms of individual idealized voters changing their minds according to the win/loss outcomes of games. It may not be mathematically completely novel (in particular, it is a subclass of the "direct methods" described in Keener's SIAM Review article), but we believe the interpretation in terms of these model voters is new. Peter J. Mucha Nov 1 '03 Callaghan, Thomas, Peter J. Mucha, and Mason A. Porter "The Bowl Championship Series: A Mathematical Review" Sept. 2004 Notices of the AMS 887-893 Jim Ashburn Nov 23 '05 Cassady CR, Maillart LM, Salman S "Ranking sports teams: A customizable quadratic assignment approach" INTERFACES 35 (6): 497-510 NOV-DEC 2005 (from Science Citation) Christopher M. Rump Jan 9 '07 Chebotarev PY, Shamis E "Preference fusion when the number of alternatives exceeds two: indirect scoring procedures" JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS 336 (2): 205-226 MAR 1999 (from Science Citation) Christopher M. Rump Jan 9 '07 Conner, G. R. and C. P. Grant "An Extention of Zemelo's Model for Ranking by Paired Comparison" Euro. Jnl of Applied Mathematics, 11:3, June 2000 225-247 Jim Ashburn Jan 14 '02 Cook, W., I. Golan and M. Kress "Heuristics for ranking players in round robin tournaments" Computers and Operations Research 15:2 1988 135-144 Soren P. Sorensen Nov 12 '99 Dickinson, Frank G. "Dickinson's Football Ratings." 1941. Booklet. An explanation of his rating system, with team ratings from 1924-40. Anton Grobani 1975 Eaton, Morris L., "Some Optimum Properties of Ranking Procedures," The Annals of Mathematical Statistics, Received July 1966, 124-137. Jim Ashburn Apr 24 '09 Elo, Arpad E., The Rating of Chess Players Past and Present 2nd Ed. , New York, Arco Pub., 1986 Kenneth Massey Feb 9 '00 Fahrmeir, L. and Tutz, G. "Dynamic Stochastic Models for Time-Dependent Ordered Pair-Comparison Systems," Journal of the American Statistical Association, 89, 1438-49 (1994) Peter R. Wolfe Dec 4 '00 Fainmesser, I., Fershtman, C. and Gandal, N. "A Consistent Weighted Ranking Scheme with an Application to NCAA College Football Rankings," 2009. Neil Gandal May 2 '09 Farmer, Christopher J. "Probabilistic Modelling in Multi-Competitor Games" Univ. of Edinburgh 2003 Jim Ashburn Nov 23 '05 Fluek, John A. and James F. Korsh, "A Generalized Approach to Maximum Likelihood Paird Comparison Ranking," The Annals of Statistics, 3(4), 1975 846-861. Jim Ashburn Apr 24 '09 Ford, LR Jr. "Solution of a Ranking Problem from Binary Comparisons." American Mathematical Monthly, 64(8) part 2, 1957, p. 28-33. Peter R. Wolfe Sep 16 '01 Gimpel, Kevin "Beating the NFL Football Point Spread" Jim Ashburn Dec 19 '07 Glickman, Mark E. "Dynamic Paired Comparison Models with Stochastic Variances" Jim Ashburn Jan 14 '02 Glickman, Mark E. "Parameter Estimation in Large Dynamic Paired Comparison Experiments" Jim Ashburn Jan 14 '02 Glickman, M. E., and H. Stern "A State-Space Model for National Football League Scores" Journal of the American Statistical Association 93 1998 Soren P. Sorensen Nov 12 '99 Goddard, S. "Ranking in tournaments and group decission making" Management Science 29:12 1983 1384-1392 Soren P. Sorensen Nov 12 '99 Grafton, Clive. "Junior College Football Rating Systems." Statistics Bureau of the National Junior College Athletic Association. 1955. Booklet. The development of the junior college football rating systems and a comparison between the Williamson and Gridiron Index Systems. Anton Grobani 1975 Grobani, Anton. "Guide to Football Literature", Gale Research Company, Detroit, 1975. The entries credited here to Anton Grobani are from pages 177-178 of this book. Eric M. Kemp Oct 10 '99 Glickman, M.E. and Stern, H.S. "A State-Space Model for National Football League Scores," Journal of the American Statistical Association, 93, 25-35 (1998) Peter R. Wolfe Dec 4 '00 Gupta, Shanti S. and Milton Sobel. "On a Statistic which arises in Selection and Ranking Problems." The Annals of Mathematical Statistics, pp. 957-967 (1957) Jim Ashburn May 19 '10 Harville, David. "The Use of Linear-Model Methodology to Rate High School or College Football Teams." Journal of the American Statistical Association. June 1977: 278-289. Kenneth Massey Sep 10 '99 Harville, David. "Predictions for NFL Games Via Linear-Model Methodology." Journal of the American Statistical Association. Sep. 1980: 516-524. Kenneth Massey Sep 10 '99 Harville DA and Smith MH, "The Home-Court Advantage: How Large and Does it Vary from Team to Team?". The American Statistician Vol. 48 (1), p.22-28 (1994). Peter R. Wolfe Sep 14 '99 Harville, David. "College Football: A Modified Least-Squares Approach to Rating and Prediction." American Statistical Proceedings of the Section on Statistics in Sports, 2001. [Available Spring 2002.] Peter R. Wolfe Sep 16 '01 Harville, David. "The Selection and/or Seeding of College Basketball or Football Teams for Postseason Competition: a Statistician's Perspective." American Statistical Proceedings of the Section on Statistics in Sports, 2000, p. 1-18. Peter R. Wolfe Sep 16 '01 Harville DA "The selection or seeding of college basketball or football teams for" postseason competition JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 98 (461): 17-27 MAR 2003 (from Science Citation) Christopher M. Rump Jan 9 '07 Helms Athletic Foundation. "National Football Champions" 1942. Booklet. The record of each team picked by Helms as the national champion from 1883-1941. Sketches of great coaches. Anton Grobani 1975 Hochbaum, Dorit S. "Ranking Sports Teams and the Inverse Equal Paths Problem." Department of Industrial Engineering and Operations Research and Walter A. Haas School of Business, University of California, Berkeley Jim Ashburn May 19, 2010 Houlgate, Deke. "Ten Thousand and One Scores." 1938. Paperbound. Teams ratings based on scores of major games from 1928-37. Anton Grobani 1975 Huang, Tzu-Kuo and Ruby C. Weng and Chih-Jen Lin "Generalized Bradley-Terry Models and Multi-Class Probability Estimates" Journal of Machine Learning Research 7 (2006) 85-115 Jim Ashburn May 25, 2007 Huber, Peter J., "Pairwise Comparison and Ranking: Optimum Properties of the Row Sum Procedure," The Annals of Mathematical Statistics, Received Nov. 1962, p. 511-520. Jim Asburn April 24 '09 Jech, Thomas. "The Ranking of Incomplete Tournaments: A Mathematician's Guide to Popular Sports." American Mathematical Monthly, 90(4), 1983, p. 246-66. Peter R. Wolfe Sep 16 '01 Karlin, Samuel, Mathematical Methods & Theory in Games, Programming, & Economics, Dover Publications, March 1992 Kenneth Massey Feb 9 '00 Karlis, D. and I. Nizoufras "Statistical Modelling for Soccer Games: the Greek League" 1998 Jim Ashburn Jan 14 '02 Keener, James P., "The Perron-Frobenius Theorem and the Rating of Football Teams", SIAM Review (volume 35, issue 1, March 1993) pages 80-93. This reviews the author's pet rating systems, one of which includes a system formulated on a least likelihood principle using the same formula as Rothman does, but with a slightly different grading function. The use of the least likelihood principle with the form A/(A+B), along with iteration methods for calculating the numbers is actually due to Zermelo, from 1926. The author also mentioned the 'retrodiction criterion' (fitting the past whereas the 'prediction criterion' fits the future) and assessed his rating systems according to it. All of this was done only for Division I games for 1989. It wouldn't be too hard to write a paper updating the situation, particularly assessing the different rating methods out there according to how well they do on all 680-plus teams and 3500 games, instead of just Division I-A. The ability to separate out the Division I-A, I-AA, II and III teams is a critical test for any rating method. I suspect that 2 or 3 of the 4 methods the author describes will fail the test. One of them can't be used at all since it breaks down when there's more than one weakly connected component. This is a similar problem that Rothman's rating method would experience if it were converted to a win's-only rating method (which is why he uses point margins to a limited extent). The author does, by the way, present enough detail for anyone interested (with enough computing power) to carry out the ratings computation. Mark Hopkins Dec 19 '97 Kuonen, Diego "Statistical Models for Knock-out Soccer Tournaments" Winter 1996/97 Jim Ashburn Jan 14 '02 Knorr-Held, L. "Dynamic Rating of Sports Teams," The Statistician, 49, 261-76 (2000) Peter R. Wolfe Dec 4 '00 Leake, RJ. A Method for Ranking Teams: With an Application to College Football. In Management Science in Sports, ed. Machol RE et al. Amsterdam: North-Holland Publishing Co., 1976. p. 27-46. Peter R. Wolfe Jun 12 '99 Lebovic JH, Sigelman L "The forecasting accuracy and determinants of football rankings" INTERNATIONAL JOURNAL OF FORECASTING 17 (1): 105-120 JAN-MAR 2001 (from Science Citation) Christopher M. Rump Jan 9 '07 Martinich, Joseph "College Football Rankings: Do Computers Know Best?" Interfaces 32(5) pp. 85-94 SEP-OCT 2002 An analysis of the systems used in the 2001 BCS. Frank Alder Aug 7 '03 McFarland, William N. "An Examination of Football Scores." Waverly Press. 1932. A rating system devised for game scores. Ratings of leading teams. Anton Grobani 1975 Mease, David "A Penalized Maximum Likelihood Approach for the Ranking of College Football Teams Independent of Victory Margins" Department of Statistics, Univerisity of Michigan, Ann Arbor, MI The American Statistician, Nov. 2003. Menke, Joshua and Tony Martinez "A Bradley-Terry Artificial Neural Network Model for Individual Ratings in Group Competitions" Computer Science Departments, Brigham Young University 2006 Mosteller, Frederick. "Collegiate Football Scores, U.S.A." Journal of the American Statistical Association. 65, pp.35-48 (1970). This paper is not completely on topic, but it is a very interesting analysis of the data contained in college football scores, written by the grand old man of the Department of Statistics at Harvard College. Peter R. Wolfe Sep 19 '99 Mundfrom DJ, Heiny RL, Hoff S "Power ratings for NCAA division II football" COMMUNICATIONS IN STATISTICS-SIMULATION AND COMPUTATION 34 (3): 811-826 2005 (from Science Citation) Christopher M. Rump Jan 9 '07 NCAA. NCAA Football Statistics including links to the Football Record Books. Nutting, Andrew "The Marginal Accuracy of Weekly College Football Rankings: Are There Diminishing Returns?" September 2006 Andrew Nutting Apr 25 '07 Palomino, F., L. Rigotti and A. Rustichini "Skill, Strategy and Passion: an Empirical Analysis of Soccer" 1998 Jim Ashburn Jan 14 '02 Park, Juyong and M. E. J. Newman "Network-based ranking system for U.S. college football" Journal of Statistical Physics, P10014 (2005) and the talk at NetSci 06 Juyong Park Aug 24 '06 Qin, Dennis "Predictions and Rankings in College Football." 2009 Jim Ashburn May 19 '10 Reid, Michael B. "Least Squares Model for Predicting College Football Scores" Univ. of Utah 2003 Jim Ashburn Nov 23 '05 Rue, H. "A Fast and Exact Simulation Algorithm for General Gaussian Markov Random Fields" 1999 Jim Ashburn Jan 14 '02 Rue, H. and 0. Salversen "Prediction and Retrospective Analysis of Soccer Matches in a League" 1999 Jim Ashburn Jan 14 '02 Singh, Jagbir and Thompson WA Jr. "A Treatment of Ties in Paired Comparisons." The Annals of Mathematical Statistics, 39(6), 1988, p. 2002-15. Peter R. Wolfe Sep 16 '01 Sinuany-Stern Z "Ranking of Sports Teams via the AHP" JOURNAL OF THE OPERATIONAL RESEARCH SOCIETY 39 (7): 661-667 JUL 1988 (from Science Citation) Christopher M. Rump Jan 9 '07 Smith, Warren D. "Rating Systems for GamePlayers, and Learning" NEC July 1994. Jim Ashburn Jan 14 '02 Srivastava, M.S. and J. Ogilvie. "The Performance of some Sequential Procedures for a Ranking Problem." The Annals of Mathematical Statistics 39:3 pp. 1040-1047 (1968) Jim Ashburn May 19 '10 Stefani, Raymond T. "Football and Basketball Predictions Using Least Squares" IEEE Transactions on Systems, Man, and Cybernetics 7 1977 Soren P. Sorensen Nov 12 '99 Stefani, Raymond T. "Improved Least Squares Football, Basketball, and Soccer Predictions." IEEE Transactions on Systems, Man, an Cybernetics. Feb 1980: 116-123 Kenneth Massey Sep 10 '99 Stern, Hal. "A Continuum of paired comparisons models." Biometrika. 77(2), pp.265-73 (1990). Peter R. Wolfe Sep 19 '99 Stern, Hal. "On the Probability of Winning a Football Game." The American Statistician. August 1991:179-183. Kenneth Massey Sep 10 '99 Stern, Hal. "Who's Number one? - Rating Football Teams" in Proceedings of the Section on Statistics in Sports, 1992, p. 1-6 Peter R. Wolfe Sep 17 '99 Stern, Hal. "Who's Number 1 in College Football?.. And How Might We Decide?" Chance, Summer, 1995:7-14. Kenneth Massey Sep 10 '99 Stern, Hal. "Statistics in Sports" 1997 Chance Lecture Series Go to http://www.dartmouth.edu/~chance/ChanceLecture/contents.msri.html and click on "Statistics in Sports" (requires RealPlayer). I've found a neat audio/video recording of a 1997 lecture by Hal Stern on "Probability and Statistics in Sports," complete with slides. The ~50 min lecture covers several sports but includes a discussion on ranking college football teams using least-squares. Stern includes a brief discussion of the 1997 Sagarin rankings, which he shows to be highly correlated with least-squares. At the conclusion of the talk, Stern presents several references for useful publications on sports statistics. Ladany and Machol, 1977: Optimal Strategies in Sports. Machol, Ladany, and Morrison, 1976: Management Science in Sports. Bennett: Statistics in Sport. Chance Magazine (column beginning in Vol. 10) Eric M. Kemp May 18 '02 Stern, Hal S. "American Football" Iowa State University, Dec. 24, 1997 Jim Ashburn Dec 19, '07 Stern, H.S., and Mock, B. "College Basketball Upsets: Will a 16-seed Ever Beat a 1-seed?" Chance, 11, 26-31 (1998) Peter R. Wolfe Dec 4 '00 Stern HS "Statistics and the college football championship" AMERICAN STATISTICIAN 58 (3): 179-185 AUG 2004 (with several discussion papers and a followup reply) (from Science Citation) Christopher M. Rump Jan 9 '07 Stern, H.S. (2006) "In Favor of A Quantitative Boycott of the Bowl Championship Series," Journal of Quantitative Analysis in Sports: Vol. 2 : Iss. 1, Article 4. Christopher M. Rump Jan 9 '07 Stone, Daniel F. "Testing Bayesian Updating with the AP Top 25" John Hopkins University, Oct. 2007 Jim Ashburn Dec 19, '07 Thomas, I.B. "Method of Ranking College Football Teams." Allen, Lane and Scott. 1922. Booklet. A point system, using won and loss records. Results for 1921. Anton Grobani 1975 Thompson, Mark. "On Any Given Sunday: Fair Competitor Orderings with Maximum Likelihood Methods." Journal of the American Statistical Association. 70, pp. 536-41 (1975). Peter R. Wolfe Sep 19 '99 Tong, Y. L. "An Adaptive Solution to Ranking and Selection Problems." The Annals of Statistics 6:3 pp.658-672 (1978) Jim Ashburn May 19, 2010 Trono, John A. "Applying the Overtake and Feedback Algorithm." Dr. Dobb's Journal 357, pp. 36-41 (Feb. 2004). The Overtake and Feedback (OAF) algorithm "home page" John A. Trono Oct 27, '04 Trono, John A. "An Effective Nonlinear Rewards-Based Ranking System" Journal of Quantitative Analysis in Sports 3:2, April 2007 John A. Trono May 1, '07 West, Brady T. and Madhur Lamsal. "A New Application of Linear Modeling in the Prediction of College Football Bowl Outcomes and the Development of Team Ratings." Journal of Quantitative Analysis in Sports Jim Ashburn May 19, 2010 Wilkins R "Electrical Networks and Sports Competition" ELECTRONICS AND POWER 29 (5): 414-418 1983 (from Science Citation) Christopher M. Rump Jan 9 '07 Wilson, R. L. Ranking College Football Teams: A Neural Network Approach. Interfaces 25:4, 1995. p.44-59. Peter R. Wolfe Jun 12 '99 Wilson, R. L. "The 'Real' Mythical College Football Champion" Operation Research / Management Science Today 1995 24-29 Soren P. Sorensen Nov 12 '99 Zuber, R. A., J. M. Gander, and B. D. Bowers. "Beating the Spread: Testing the Efficiency of the Gambling Market for National Football League Games" Journal of Political Economy 93 1985 Soren P. Sorensen Nov 12 '99