Arrow-Sen theory simplified

TL;DR

This paper simplifies Arrow--Sen Social Choice Theory by reformulating higher-order axioms into first-order axioms, making the theory more accessible while preserving key theorems like Arrow's and Sen's impossibility results.

Contribution

It introduces a simplified first-order version of TSCT, called SSCT, which retains core principles and theorems of the original higher-order theory.

Findings

Derived first-order axioms from higher-order ones

Proved key theorems like Arrow's impossibility within SSCT

Made social choice theory more accessible for teaching

Abstract

The traditional Arrow--Sen Social Choice Theory $\bf{TSCT}$ is a mathematical theory built apparently on higher--order formal language. In this paper, we propose a reformulation and reclassification of the $\bf{TSCT}$ axioms in order to obtain a simpler theory based on the first--order language axioms, keeping the spirit of original ideas. This new theory, called Simplified Social Choice Theory, denoted by $\bf{SSCT}$, presents a sub--theory of $\bf{TSCT}$. Roughly speaking, we extract all quatifications over $n$--tuples of binary relations from the axioms of $\bf{TSCT}$ and move them to the meta--level obtaining a sub--theory $\bf{SSCT}$ of $\bf{TSCT}$. More accurately, we assign to each traditional higher--order axiom $\bf{TA}$ its simplified first--order version $\bf{SA}$ such that $\bf{TA}\vdash\bf{SA}$, i.e. $\bf{SA}$ can be logically derived from $\bf{TA}$. In this way we define a simpler and more accessible set of principles that provide a context in which we can prove many propositions analogous to well--known theorems including the Arrow's impossibility of Paretian non--dictatorship and Sen's impossibility of Paretian liberal. These simplifications are the result of decades of lecturing by the author with the aim of bridging the barriers between this beautiful complex theory and his students.