Resumen de Aportaciones al estudio de los sistemas electorales
Vicenç Sales Ingles
An important question that modern societies have to decide is the election of some people who represent them and can also make some decisions. Mechanisms to do it are called Electoral Systems. In fact, there are a lot of them and they are very different. In this work, we present some ideas to carry out a mathematical analysis of them. The first chapter is a global analysis of electoral systems. It begins with the definition of electoral system --helped by Probability Theory-- and the introduction of most important simple examples found in practice. Next we define two operations on electoral systems and, in particular, we obtain two different generalizations of each one of the examples. We also introduce some important properties that electoral systems may have -- superadditivity, monotonicity, increasing and stability-- and we study which examples enjoy them and the behavior of these properties with respect to the operations defined before. Finally, stability give us the possibility to define majority and proportional electoral systems. In the second chapter we study electoral systems individually. In this sense, we introduce the electoral expectatives, obtained when the vote vector of an arbitrary list of candidates is fixed. Then we study their connection with the operations before defined and we finish the chapter by introducing some individual properties that electoral systems may have and analysing what happens when we consider again the operations defined before. The third chapter refers to a parameter introduced to assess whether an arbitrary list of candidates gets profit or not from an electoral system. The way used to do it consists in considering the expected value of electoral expectatives introduced in the second chapter. We obtain in this form the concept of average electoral expectative. And we finish the chapter by studying the behavior of this concept with respect to the operations and examples introduced above. In the fourth chapter we analyse three questions of majority weighted games that we will use in the next chapter: another form to define them, a new operation between them and their convergence. Finally, in the fifth chapter we replace the number of seats of each litst of candidates by its Shapley-Shubik power index and we study the electoral systems using this new indicator. In this form, we obtain the concept of power system and, analogously, the concepts of power expectative and average power expectative. We finish by introducing some new properties of each one of these concepts and we also analyse their relation with the analogous properties of electoral systems.
