Arrow, Hausdorff, and Ambiguities in the Choice of Preferred States in Complex Systems

TL;DR

This paper extends Arrow's impossibility theorem to a mathematical setting using Hausdorff measures, exploring how preferences among complex objects can lead to inherent ambiguities and irreversibility in collective choice.

Contribution

It introduces a novel mathematical framework for preference aggregation using set distances, generalizing Arrow's theorem beyond sociological contexts.

Findings

Preferences modeled with Hausdorff measures reveal inherent ambiguities.

Reversibility notions are linked to configurations of set patterns.

The framework highlights fundamental limitations in preference aggregation.

Abstract

Arrow's `impossibility' theorem asserts that there are no satisfactory methods of aggregating individual preferences into collective preferences in many complex situations. This result has ramifications in economics, politics, i.e., the theory of voting, and the structure of tournaments. By identifying the objects of choice with mathematical sets, and preferences with Hausdorff measures of the distances between sets, it is possible to extend Arrow's arguments from a sociological to a mathematical setting. One consequence is that notions of reversibility can be expressed in terms of the relative configurations of patterns of sets.