On the Ranking of Social Alternatives.

Abstract

Being better than the others has been an aspiration of every competitive individual whether there are economic or simply prestigious benefits. It is common that the decision of who is better than whom is based on either voting or diverse criteria, which after having chosen the appropriate methodology we combine in order to obtain a highly regarded ranking of the concerned individuals. This dissertation is about these methodologies and these criteria. We study how different factors interact within a particular index, we examine what are the conditions on the set of voters such that, following a supermajority decision rule, obtain a transitive ranking, and finally we investigate how a group of peers can rank themselves in a way that each individual is impartial on his rank. Each of these topics is elaborated in a different chapter of this dissertation. We devote the first one to the Human Development Index (HDI), which is a measure of a country's development level that considers components that go beyond income. In this chapter first we revise all past and present HDI together with their corresponding marginal rates of substitution (MRS). We find that according to the existing HDI the implementation of a certain health policy in a country is independent of its education level and vice versa. On the other hand, following a literature indicating that education and health are relevant in the production of the other we propose an alternative HDI that complies with these findings. In chapter 2 we establish a necessary and sufficient single-profile condition for obtaining a transitive relation under a supermajority decision rule. On the beginning we suppose that the individual preferences are linear orders. We start by reducing the population by disregarding the individuals who have inverse preferences over the alternative set, and then we consider an equivalent supermajority rule that depends on the agents in the reduced population only. Our condition applies to the reduced population and it is both necessary and sufficient to guarantee transitive social preferences. Our profile condition is composed out of two distinct properties. The first states that whenever there is an alternative that is preferred to the other two by a sufficient number of individuals then transitivity is guaranteed. The second one discards the possibility of a cycle in the supermajority social preference relation when no alternative is preferred by a qualified majority. Then we extend our result to the case in which individual preferences are weak orders by introducing an equivalent supermajority rule relation on a profile constructed by linear orders only. Then we apply the appropriate balancedness condition to the latter and the generalization of the result trivially follows. In the last chapter we consider a situation when we have to rank a set of individuals when each agent provides an ordered list of the others, representing his opinion about the position of the others. We call the rule that determines the final ranking a social ranking function and we say that it is impartial if a change of a single agent's preference does not influence his rank. Here we construct several ranking functions indicating the result, which we later prove, that there is no impartial social ranking function that is fully Paretian. Accepting this debility, we further introduce several properties and demonstrate that there are impartial ranking functions that satisfy them. At the end, we characterize the impartial ranking functions by the use of three axioms, which we prove that are sufficient and necessary condition for impartiality.

Date of Award	12 Jul 2013
Original language	English
Awarding Institution	Universitat Autònoma de Barcelona (UAB)
Supervisor	Salvador Barberà (Director)