INTRODUCTION

In this chapter, we introduce some basic concepts that will be used throughout this book. In Section 2.2, we start by defining the “at least as good as” relation ≿ that describes the preferences of the individuals over the set of alternatives that they face. We specify certain properties associated with these preference relations. In Section 2.3, we introduce the notion of maximal sets (or choice sets) and link it with the properties of ≿. Finally, in Section 2.4, we discuss social orderings, that is, given a set of agents in a society along with their preferences, how do we aggregate them into social preferences. Specifically, in Section 2.4, we discuss some well-known social aggregation rules like plurality rule, Borda count, anti-plurality rule, oligarchy, and pairwise majority rule.

RELATIONS

Let A = ﹛x,y, z,w …﹜ be the set of alternative states of affairs (alternatives, for short). A relation ≿ on A is a subset of A × A. We shall write x ≿ y if (x, y) ∊ ≿. We say that x and y are unordered by ≿ if neither x ≿ y nor y ≿ x. They are ordered by ≿ if they are not unordered, that is, either x ≿ y holds or y ≿ x holds. We will call ≿ as the “at least as good as” relation defined on the set of alternatives A. Given ≿, let ≻ and ∼ be the asymmetric and the symmetric parts of ≿. That is, x ≻ y if and only if x ≿ y and ￢(y ≿x), where ￢ (y ≿ x) means that y ≿ x is not true. Moreover, x ∼ y if and only if x ≿ y and y ≿ x. We will also refer to ∼ as the strict preference part of ≿ and we will also refer to ∽ as the indifference part of ≿. In words, for any person (society) with the preference relation ≿, between any two alternatives x and y, x ≻ y means that the person (society) strictly prefers x to y and x ∼ y means that the person (society) is indifferent between x and y.