A counter example to the theorems of social preference transitivity and social choice set existence under the majority rule

TL;DR

This paper presents a counterexample showing that existing theorems on social preference transitivity and social choice set existence under majority rule are not universally valid, challenging current assumptions in social choice theory.

Contribution

It provides a specific example that violates the conditions required by established theorems, highlighting gaps in the current theoretical framework.

Findings

Counterexample invalidates some social choice theorems

Existing conditions are not exhaustive or necessary

Challenges current assumptions in social preference theory

Abstract

I present an example in which the individuals' preferences are strict orderings, and under the majority rule, a transitive social ordering can be obtained and thus a non-empty choice set can also be obtained. However, the individuals' preferences in that example do not satisfy any conditions (restrictions) of which at least one is required by Inada (1969) for social preference transitivity under the majority rule. Moreover, the considered individuals' preferences satisfy none of the conditions of value restriction (VR), extremal restriction (ER) or limited agreement (LA), some of which is required by Sen and Pattanaik (1969) for the existence of a non-empty social choice set. Therefore, the example is an exception to a number of theorems of social preference transitivity and social choice set existence under the majority rule. This observation indicates that the collection of the conditions listed by Inada (1969) is not as complete as might be supposed. This is also the case for the collection of conditions VR, ER and LA considered by Sen and Pattanaik (1969). This observation is a challenge to some necessary conditions in the current social choice theory. In addition to seeking new conditions, one possible way to deal with this challenge may be, from a theoretical prospective, to represent the identified conditions (such as the VR, ER and LA) in terms of a common mathematical tool, and then, people may find more.