La abstención y otros problemas en los métodos de votación

Abstract

The main objective of this work is to explore and study the Abstention Paradox in voting methods, that is to say, the Participation properties of those methods. This study includes the definition of new concepts and new Participation properties. It also includes the extension or generalization of some important results appeared in the literature on this paradox, and, particularly, affects the Condorcet voting methods, such as the Moulin and Young and Levenglick theorems. A further objective is to compare the performance, with respect to this paradox, that Condorcet and Positional methods display, and build a Generating Scheme based on distances from which a family of voting methods with good performance is obtained. Describing with more detail the intended objectives, this work aims: a) To contribute to the extension of the Moulin theorem (which states that any Condorcet voting function fails to satisfy the Participation property) to the context of voting correspondences. To achieve this target, we will define new Participation properties and identify which Condorcet voting correspondences fail to satisfy these properties. b) To contribute to the extension of the Moulin theorem to the context of k-voting functions and k-voting correspondence. To achieve this, we will adapt the concepts involved in the Moulin theorem (including Condorcet candidate) to this new context. c) To contribute to the expansion and deepening of the concept of Positive Involvement, defining more demanding Participation properties and identifying what voting correspondences, both Positional and Condorcet, violate these properties. d) To contribute to the generalization of Young and Levenglick theorem, which states that all Condorcet voting correspondences fail to satisfy the Consistency property. To achieve this, we will identify and explore the properties that are somewhere in between Consistency property and Positive Involvement property. e) To contribute to a better understanding of the relationships and differences between Condorcet and Positional methods, building a Generating Scheme based on distances defined over preference matrices, and obtaining from it a single parameter family of methods which all have, even if they are Condorcet, the best possible Participation properties.