On the computability of binary social choice rules in an infinite society and the halting problem

This paper investigates the computability problem of the Arrow impossibility theorem [K.J. Arrow, Social Choice and Individual Values, second ed., Yale University Press, 1963] of social choice theory in a society with an infinite number of individuals (infinite society) according to the computable calculus (or computable analysis) by Aberth [O. Aberth, Computable Analysis, McGraw-Hill, 1980, O. Aberth, Computable Calculus, Academic Press, 2001]. We will show the following results. The problem whether a transitive binary social choice rule satisfying Pareto principle and independence of irrelevant alternatives (IIA) has a dictator or has no dictator in an infinite society is a nonsolvable problem, that is, there exists no ideal computer program for a transitive binary social choice rule satisfying Pareto principle and IIA that decides whether the binary social choice rule has a dictator or has no dictator. And it is equivalent to nonsolvability of the halting problem. A binary social choice rule is a function from profiles of individual preferences to social preferences, and a dictator is an individual such that if he strictly prefers an alternative to another alternative, then the society must also strictly prefer the former to the latter. 2007 Published by Elsevier Inc.