Chapter 2 Categories of arrovian voting schemes

Abstract

Within the framework of the axiomatic approach three types of voting schemes are investigated according to the form in which the individual opinions about the alternatives are defined, as well as to the form of desired social decision. These types of rules are Social Decision Rules, Functional Voting Rules, and Social Choice Correspondences. Consideration is given to local rules, i.e., to the rules which satisfy some analogue of Arrow's Independence of Irrelevant Alternatives condition. A general description of the problem of axiomatic synthesis of local rules, and various formalizations of voting schemes are given. The notion of “rationality” of individual opinions and social decision is described. Various types of binary relations (preferences) are introduced. The characteristic conditions (Expansion-Contraction Axioms) on choice functions are defined, and the interrelations between them are established. Two types of Social Decision Rules (transforming individual preferences to social ones) are studied. The explicit forms of those rules are investigated. The rules restricted by rationality constraints, i.e., by the constraints on domains and ranges of the rules, are studied as well. Functional Voting Rules are investigated which transform individual opinions defined as choice functions into a social choice function. In doing so, a rationalizability of those choice functions is not assumed. The explicit form of these rules is obtained, and the rules which satisfy different rationality constraints are studied. Social Choice Correspondences deal with the case when the individual opinions are formalized as binary relations, and the collective decision that we look for is a choice function. The explicit form of rules is studied. The obtained classes comprise the rules such as the generalized Pareto rules. Several new classes of the rules are introduced and analyzed. The explicit form of the Nash-implementable rules is found. The analysis of publications on the axiomatic synthesis of the local aggregation rules is made.