# Set-Rationalizable Choice and Self-Stability

## TL;DR

This paper introduces set-rationalizability and self-stability as new frameworks for understanding choice functions, addressing limitations of traditional rationalizability in social choice and encompassing well-known Condorcet extensions.

## Contribution

It proposes set-rationalizability and self-stability as novel concepts, providing a unified approach to rationalizing choice functions and extending the theory of social choice.

## Key findings

- Set-rationalizability characterized by lpha
- Self-stability equivalent to lpha and mma conditions
- Includes Condorcet extensions like minimal covering set

## Abstract

A common assumption in modern microeconomic theory is that choice should be rationalizable via a binary preference relation, which \citeauthor{Sen71a} showed to be equivalent to two consistency conditions, namely $\alpha$ (contraction) and $\gamma$ (expansion). Within the context of \emph{social} choice, however, rationalizability and similar notions of consistency have proved to be highly problematic, as witnessed by a range of impossibility results, among which Arrow's is the most prominent. Since choice functions select \emph{sets} of alternatives rather than single alternatives, we propose to rationalize choice functions by preference relations over sets (set-rationalizability). We also introduce two consistency conditions, $\hat\alpha$ and $\hat\gamma$, which are defined in analogy to $\alpha$ and $\gamma$, and find that a choice function is set-rationalizable if and only if it satisfies $\hat\alpha$. Moreover, a choice function satisfies $\hat\alpha$ and $\hat\gamma$ if and only if it is \emph{self-stable}, a new concept based on earlier work by \citeauthor{Dutt88a}. The class of self-stable social choice functions contains a number of appealing Condorcet extensions such as the minimal covering set and the essential set.