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. 1. Introduction. In

Editor: The Bradley-Terry model for obtaining individual skill from paired comparisons has been popular in many areas. In machine learning, this model is related to multi-class probability estimates by coupling all pairwise classification results. Error correcting output codes (ECOC) are a general framework to decompose a multi-class problem to several binary problems. To obtain probability estimates under this framework, this paper introduces a generalized Bradley-Terry model in which paired individual comparisons are extended to paired team comparisons. We propose a simple algorithm with convergence proofs to solve the model and obtain individual skill. Experiments on synthetic and real data demonstrate that the algorithm is useful for obtaining multi-class probability estimates. Moreover, we discuss four extensions of the proposed model: 1) weighted individual skill, 2) home-field advantage, 3) ties, and 4) comparisons with more than two teams. Keywords: Bradley-Terry model, Probability estimates, Error correcting output codes, Support Vector Machines

In image quality assessment, preference for various image processing algorithms or treatments is often determined using paired comparisons. In this experimental design, pairs of images processed by different algorithms or "treatments" are presented to a judge. The preferred treatment is selected and a tally is kept of the number of times each treatment is preferred to another. It is possible to estimate an interval scale for treatments in a hypothetical psychological space using this method. There are two...

This report studies rating systems: systems that produce quantitative measures, called "ratings," of the ability of players in a league, based on game results. By "quantitative", it is meant that win odds for a game between two players in the league may be estimated from their ratings. We consider both `static' and `dynamic' systems. The latter update the ratings after each game. Attention is given to noise in rating systems and to the distribution of ratings in the player population. Some real-world data is also included. This subject may be of interest to gamblers, gaming leagues, psychologists, consumer groups, and industry.

this paper, we analyze a natural extension of Zermelo's model resulting from a singular perturbation. We show that this extension produces a ranking for arbitrary (nonnegative) outcome matrices and retains several of the desirable properties of the original model. In addition, we discuss computational techniques and provide examples of their use. 1 Introduction

he tournaments, and 4 players only competed in 1 tournament. The World Cup participants who competed in 3 or more tournaments were 5.2 A Model for Chess Game Outcomes 78 Tournament Dates Number of Competitors Brussels, Belgium April 1, 1988 -- April 22, 1988 18 Belfort, France June 14, 1988 -- July 3, 1988 16 Reykjavik, Iceland October 3, 1988 -- October 24, 1988 18 Barcelona, Spain March 30, 1989 -- April 20, 1989 17 Rotterdam, Netherlands June 3, 1989 -- June 24, 1989 16 Skelleftea, Sweden August 12, 1989 -- September 3, 1989 16 Table 5.1: World Cup Chess Tournaments, 1988--1989 contenders for monetary prizes. Table 5.2 lists the players and indicates the tournaments in which each player competed. For each game in the World Cup, the data consists of the players involved in the game, the outcome of the game (win, loss or draw), an indication of which player played the white pieces (the player with the white pieces moves first), and the tournament in which the game oc

We study maximum likelihood estimation for the statistical model for both directed and undirected random graph models in which the degree sequences are minimal sufficient statistics. In the undirected case, the model is known as the beta model. We derive necessary and sufficient conditions for the existence of the MLE that are based on the polytope of degree sequences, and wecharacterize in a combinatorial fashion sample points leading to a nonexistent MLE, and non-estimability of the probability parameters under a nonexistent MLE. We formulate conditions that guarantee that the MLE exists with probability tending to one as the number nodes increases. By reparametrizing the beta model as a log-linear model under product multinomial sampling scheme, we are able to provide usable algorithms for detecting nonexistence of the MLE and for identifying non-estimable parameters. We illustrate our approach on other random graph models for networks, such as the Rasch model, the Bradley-Terry model and the more general p1 model of Holland and Leinhardt (1981).

The existence of maximum likelihood estimates in the Bradley-Terry model and its extensions

A method is given for quantitatively rating the social acceptance of different options which are the matter of a preferential vote. The proposed method is proved to satisfy certain desirable conditions, among which there is a majority principle, a property of clone consistency, and the continuity of the rates with respect to the data. One can view this method as a quantitative complement for a qualitative method introduced in 1997 by Markus Schulze. It is also related to certain methods of one-dimensional scaling or cluster analysis.

Random utility theory models an agent’s preferences on alternatives by drawing a real-valued score on each alternative (typically independently) from a parameterized distribution, and then ranking the alternatives according to scores. A special case that has received significant attention is the Plackett-Luce model, for which fast inference methods for maximum likelihood estimators are available. This paper develops conditions on general random utility models that enable fast inference within a Bayesian framework through MC-EM, providing concave loglikelihood functions and bounded sets of global maxima solutions. Results on both real-world and simulated data provide support for the scalability of the approach and capability for model selection among general random utility models including Plackett-Luce. 1