Maximum Likelihood Approach to Vote Aggregation with Variable Probabilities

Abstract

Condorcet (1785) initiated the statistical approach to vote aggregation. Two centuries later, Young (1988) showed that a correct application of the maximum likelihood principle leads to the selection of rankings called Kemeny orders, which have the minimal total number of disagreements with those of the voters. The Condorcet-Kemeny-Yoiung approach is based on the assumption that the voters have the same probability of comparing correctly two alternatives and that this probability is the same for any pair of alternatives. We relax the second part of this assumption by letting the probability of comparing correctly two alternatives be increasing with the distance between two alternatives in the allegedly true ranking. This leads to a rule in which the majority in favor of one alternative against another one is given a larger weight the larger the distance between the two alternatives in the true ranking, i.e. the larger the probability that the voters compare them correctly. This rule is not Condorcet consistent. Thus, it may be different from the Kemeny rule. Yet, it is anonymous, neutral, and paretian. However, contrary to the Kemeny rule, it does not satisfy Young and Levenglick (1978)'s local independence of irrelevant alternatives. Condorcet also hinted that the Condorcet winner or the top alternative in the Condorcet ranking is not necessarily most likely to be the best. Young confirms that indeed with a constant probability close to 1/2, this alternative is the Borda winner while it is the alternative whose smallest majority is the largest when the probability is close to 1. We extend his analysis to the case of variable probabilities. Young's result implies that the Kemeny rule does not necessarily select the alternative most likely to be the best. A natural question that comes to mind is whether the rule obtained with variable probabilities does better than the Kemeny rule in this respect. It appears that this performance imporves with the rate at which the probability increases.Vote Aggregation, Kemeny Rule, Maximum Likelihood, Variable Probabilities