The Bradley-Terry model for paired comparisons is a simple and muchstudied means to describe the probabilities of the possible outcomes when individuals are judged against one another in pairs. Among the many studies of the model in the past 75 years, numerous authors have generalized it in several directions, sometimes providing iterative algorithms for obtaining maximum likelihood estimates for the generalizations. Building on a theory of algorithms known by the initials MM, for minorization-maximization, this paper presents a powerful technique for producing iterative maximum likelihood estimation algorithms for a wide class of generalizations of the Bradley-Terry model. While algorithms for problems of this type have tended to be custom-built in the literature, the techniques in this paper enable their mass production. Simple conditions are stated that guarantee that each algorithm described will produce a sequence that converges to the unique maximum likelihood estimator. Several of the algorithms and convergence results herein are new.

https://projecteuclid.org/download/pdf_1/euclid.aos/1079120141

MM algorithms for generalized Bradley-Terry models

Summary. Paired comparison data in which the abilities or merits of the objects being compared may be changing over time can be modelled as a non-linear state space model. When the population of objects being compared is large, likelihood-based analyses can be too computationally cumbersome to carry out regularly. This presents a problem for rating populations of chess players and other large groups which often consist of tens of thousands of competitors. This problem is overcome through a computationally simple non-iterative algorithm for fitting a particular dynamic paired comparison model. The algorithm, which improves over the commonly used algorithm of Elo by incorporating the variability in parameter estimates, can be performed regularly even for large populations of competitors. The method is evaluated on simulated data and is applied to ranking the best chess players of all time, and to ranking the top current tennis-players.

/pdf/parameter-estimation-in-large-dynamic-paired-comparison-2sj4b6b3dw.pdf

Parameter estimation in large dynamic paired comparison experiments

The median procedure in cluster analysis and social choice theory

1. Introduction 2. Binary regression: the logit model 3. Generalized linear models 4. Modeling of binary data 5. Alternative binary regression models 6. Regularization and variable selection for parametric models 7. Regression analysis of count data 8. Multinomial response models 9. Ordinal response models 10. Semi- and nonparametric generalized regression 11. Tree-based methods 12. The analysis of contingency tables: log-linear and graphical models 13. Multivariate response models 14. Random effects models 15. Prediction and classification Appendix A. Distributions Appendix B. Some basic tools Appendix C. Constrained estimation Appendix D. Kullback-Leibler distance and information-based criteria of model fit Appendix E. Numerical integration and tools for random effects modeling.

/pdf/regression-for-categorical-data-5de4fdg1bn.pdf

Regression for Categorical Data

Voting procedures focus on the aggregation of individuals' preferences to produce collective decisions. In practice, a voting procedure is characterized by ballot responses and the way ballots are tallied to determine winners. Voters are assumed to have clear preferences over candidates and attempt to maximize satisfaction with the election outcome by their ballot responses. Such responses can include strategic misrepresentation of preferences. Voting procedures are formalized by social choice functions, which map ballot response profiles into election outcomes. We discuss broad classes of social choice functions as well as special cases such as plurality rule, approval voting, and Borda's point-count method. The simplest class is voting procedures for two-candidate elections. Conditions for social choice functions are presented for simple majority rule, the class of weighted majority rules, and for what are referred to as hierarchical representative systems. The second main class, which predominates in the literature, embraces all procedures for electing one candidate from three or more contenders. The multicandidate elect-one social choice functions in this broad class are divided into nonranked one-stage procedures, nonranked multistage procedures, ranked voting methods, and positional scoring rules. Nonranked methods include plurality check-one voting and approval voting, where each voter casts either no vote or a full vote for each candidate. On ballots for positional scoring methods, voters rank candidates from most preferred to least preferred. Topics for multicandidate methods include axiomatic characterizations, susceptibility to strategic manipulation, and voting paradoxes that expose questionable aspects of particular procedures. Other social choice functions are designed to elect two or more candidates for committee memberships from a slate of contenders. Proportional representation methods, including systems that elect members sequentially from a single ranked ballot with vote transfers in successive counting stages, are primary examples of this class.

Chapter 4 Voting procedures

This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the propos...

Roger R. Davidson

Papers

On Extending the Bradley-Terry Model to Accommodate Ties in Paired Comparison Experiments

Some inconsistencies in judging problems