Maximum Likelihood Approach to Vote Aggregation with Variable Probabilities

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

Evaluation Measures for Hierarchical Classification: a unified view and novel approaches

Hierarchical classification addresses the problem of classifying items into a hierarchy of classes. An important issue in hierarchical classification is the evaluation of different classification algorithms, which is complicated by the hierarchical relations among the classes. Several evaluation measures have been proposed for hierarchical classification using the hierarchy in different ways. This paper studies the problem of evaluation in hierarchical classification by analyzing and abstracting the key components of the existing performance measures. It also proposes two alternative generic views of hierarchical evaluation and introduces two corresponding novel measures. The proposed measures, along with the state-of-the art ones, are empirically tested on three large datasets from the domain of text classification. The empirical results illustrate the undesirable behavior of existing approaches and how the proposed methods overcome most of these methods across a range of cases.Comment: Submitted to journa

Coordination and Learning with a Partial Language

This paper explores how efficiency promotes the use of structure in language. It starts from the premise that one of language’s central characteristics is to provide a means for saying novel things about novel circumstances, its creativity. It is reasonable to expect that in a rich and changing environment, language will be incomplete. This encourages reliance on structure. It is shown how creative language use emerges form common knowledge structures, even if those structures are consistent with an a priori absence of a common language. ZUSAMMENFASSUNG - (Koordination und Lernen mit einer Partialsprache) In diesem Beitrag wird die Anwendung von Strukturen in einer Sprache aus Effizienzsicht begründet. Der Artikel geht davon aus, daß eines der wichtigsten Merkmale der Sprache in ihrer Kreativität zu sehen ist, d. h. als Mittel, um Neues über neue Sachverhalte auszusagen. Es ist deshalb zu erwarten, daß in einer vielfältigen und sich verändernden Umwelt die Sprache unvollständig bleiben wird. Dies fördert die Anwendung von Strukturen. Es wird gezeigt, wie die kreative Sprachanwendung aus allgemeinen Wissensstrukturen entsteht, auch dann, wenn diese Strukturen a priori noch keine gemeinsame Sprache bilden.Language; Coordination; Optimal Learning; Common Knowledge

Statistical Comparison of Aggregation Rules for Votes

If individual voters observe the true ranking on a set of alternatives with error, then the social choice problem, that is, the problem of aggregating their observations, is one of statistical inference. This study develops a statistical methodology that can be used to evaluate the properties of a given or aggregation rule. These techniques are then applied to some well-known rules.Vote aggregation, ranking rules, figure skating, maximum likelihood, optimal inference, Monte Carlo, Kemeny, Borda

Uniqueness and minimal obstructions for tree-depth

A k-ranking of a graph G is a labeling of the vertices of G with values from {1,...,k} such that any path joining two vertices with the same label contains a vertex having a higher label. The tree-depth of G is the smallest value of k for which a k-ranking of G exists. The graph G is k-critical if it has tree-depth k and every proper minor of G has smaller tree-depth. We establish partial results in support of two conjectures about the order and maximum degree of k-critical graphs. As part of these results, we define a graph G to be 1-unique if for every vertex v in G, there exists an optimal ranking of G in which v is the unique vertex with label 1. We show that several classes of k-critical graphs are 1-unique, and we conjecture that the property holds for all k-critical graphs. Generalizing a previously known construction for trees, we exhibit an inductive construction that uses 1-unique k-critical graphs to generate large classes of critical graphs having a given tree-depth.Comment: 14 pages, 4 figure