Describing Sen's Transitivity Condition in Inequalities and Equations

TL;DR

This paper reformulates Sen's transitivity condition in social choice theory using inequalities and equations, providing a mathematical characterization based on preference sets and their properties.

Contribution

It introduces a novel set-theoretic and algebraic description of Sen's transitivity condition in terms of inequalities and equations.

Findings

Preference maps represent individual preferences over triples of alternatives.

Sen's transitivity condition can be expressed through set unions and cardinalities.

The condition is further characterized using set membership functions and equations.

Abstract

In social choice theory, Sen's value restriction condition is a sufficiency condition restricted to individuals' ordinal preferences so as to obtain a transitive social preference under the majority decision rule. In this article, Sen's transitivity condition is described by use of inequality and equation. First, for a triple of alternatives, an individual's preference is represented by a preference map, whose entries are sets containing the ranking position or positions derived from the individual's preference over that triple of those alternatives. Second, by using the union operation of sets and the cardinality concept, Sen's transitivity condition is described by inequalities. Finally, by using the membership function of sets, Sen's transitivity condition is further described by equations.