A statistical approach to the aggregation of votes

This thesis analyses the problem of the aggregation of individual orderings on a finite set of alternatives into a collective ranking. We assume that a true ranking exists but it is not known. The individual orderings represent the opinions of experts on the true ranking. They may err, but the probability that they compare correctly two alternatives is higher than one half. Their votes are treated as observations of a process that may contain errors. We apply the maximum likelihood principle to these votes, i.e. we find the ranking that is most likely to be the true one.; The first contribution of this thesis consists in relaxing the hypothesis that the probability of comparing correctly two alternatives is the same for all pairs of alternatives. We adopt the more realistic point of view that this probability increases with the distance, in terms of position, between the alternatives in the true ranking. We then obtain results quite different from those obtained under constant probabilities.; With the maximum likelihood approach, one simply minimizes the probability of an error. The second contribution of the thesis consists in the introduction of the notion of a loss function to weigh the errors by order of gravity. The optimal ranking minimizes expected loss. This is a new approach to the aggregation of votes. A numerical analysis of the case of three alternatives and of an example with four alternatives indicates, however, that this other approach yields results that are not too different from those of the maximum likelihood approach.; The expected loss approach is then applied to the data from an Olympic figure skating competition. Having obtained estimates of the parameters of an increasing probability function, we compute the optimal ranking for this competition, under different loss functions that depend on a single parameter. Once again, the optimal ranking is often the most likely ranking. The final ranking actually obtained by the skaters in this competition minimizes the expected loss for specifications of the loss function that can hardly be justified. Other rules fare generally better.; A ranking method must often be chosen in advance of a competition and is to be used over a long period of time. A question of interest is thus how different ranking methods would fare in repeated competitions with respect to different loss functions. In other words, what is the ex ante expected loss of a ranking method? This is the last question addressed in this thesis. After the computation of the expected loss for some known rules, we reach the conclusion that no single rule outperforms the others for every number of alternatives, every loss function and every probability function.