INTRODUCTION

Kenneth J. Arrow (see Arrow 1950) provided a striking answer to a basic problem of democracy: how can the preferences of a set of individuals be aggregated into a social ordering? The answer, known as Arrow’s impossibility theorem, was that every conceivable aggregation method has some flaw. That is, a handful of reasonable-looking axioms, which one thinks an aggregation procedure should satisfy, lead to an impossibility. This impossibility theorem created a large literature and the major field called social choice theory, which is one of the main subject matters of this book. Specifically, Arrow’s impossibility theorem or the general possibility theorem or Arrow’s paradox is an impossibility theorem stating that when agents face three or more distinct alternatives (options), every social welfare function that can convert the preference ordering of all the individuals into a social ordering while also meeting unrestricted domain, weak Pareto, and independence of irrelevant alternatives must be dictatorial. Unrestricted domain (also called universality) is a property in which all possible preferences of all individuals are allowed in the domain. Weak Pareto requires that the unanimous preferences of individuals must be respected, that is, if every agent strictly prefers one alternative over the other, then society must also strictly prefer the same alternative over the other. Social preference orderings satisfy the independence-of-irrelevant-alternatives criterion if the relative societal ranking of any two alternatives depends only on the relative ranking of those two alternatives of all the individuals and not on howthe individuals rank other alternatives.Adictatorial social welfare function is one in which there is a single agent whose strict preferences are always adopted by the society. In this chapter, we present a detailed analysis of Arrow’s impossibility theorem and then provide two proofs of the theorem.

THE FRAMEWORK

Consider a society with n agents and we denote this society by N = ﹛1, … , n﹜. Each agent i ∊ N has an ordering ≿i defined on A and assume that |A| ≥ 3. The assumption that ≿i is an order on A has implications for the strict preference relation ≿i and the indifference relation ≿i. These implications are summarized in the next proposition.