Complete Characterization of Functions Satisfying the Conditions of Arrow's Theorem

TL;DR

This paper fully characterizes social choice functions satisfying Arrow's conditions, especially when non-strict preferences are involved, revealing new classes of functions beyond traditional dictatorial ones.

Contribution

It provides an exact characterization of functions satisfying Arrow's conditions with non-strict preferences and derives formulas for counting such functions.

Findings

Constructed examples of dictatorial functions not satisfying Transitivity and IIA.

Established an if-and-only-if condition for functions satisfying Transitivity and IIA.

Derived formulas for the number of functions meeting these conditions.

Abstract

Arrow's theorem implies that a social choice function satisfying Transitivity, the Pareto Principle (Unanimity) and Independence of Irrelevant Alternatives (IIA) must be dictatorial. When non-strict preferences are allowed, a dictatorial social choice function is defined as a function for which there exists a single voter whose strict preferences are followed. This definition allows for many different dictatorial functions. In particular, we construct examples of dictatorial functions which do not satisfy Transitivity and IIA. Thus Arrow's theorem, in the case of non-strict preferences, does not provide a complete characterization of all social choice functions satisfying Transitivity, the Pareto Principle, and IIA. The main results of this article provide such a characterization for Arrow's theorem, as well as for follow up results by Wilson. In particular, we strengthen Arrow's and Wilson's result by giving an exact if and only if condition for a function to satisfy Transitivity and IIA (and the Pareto Principle). Additionally, we derive formulas for the number of functions satisfying these conditions.