Relaxing Arrow’s Axioms

Abstract

In this chapter, we explore the extent to which we can escape the dictatorship result if we relax some of Arrow’s axioms. If we relax independence of irrelevant alternatives, then we get the Borda Count which is a well-defined social welfare function satisfying unrestricted domain and weak Pareto. This we have already discussed is Example 4.5 of Chapter 4. In Section 5.2, we relax weak Pareto and replace it by a weaker axiom of non-imposition and then we get the result due to Wilson (1972) that stipulates that a social welfare function satisfying unrestricted domain, independence of irrelevant alternatives, and non-imposition axioms must be null or dictatorial or inverse-dictatorial, given that the number of alternatives is not less than three. Both inverse-dictatorial and null social welfare functions are quite uninteresting. Inverse-dictatorship requires that there exists an agent i whose strict preference over every pair of alternatives is reversed for the society under all profiles in the domain. Like inverse-dictatorial social welfare function, the null social welfare function is also uninteresting since the society is always indifferent across all alternatives for all possible profiles in the domain.