# Arrovian Aggregation of Convex Preferences

## TL;DR

This paper characterizes the conditions under which convex preferences and Arrow's axioms allow for a unique, anonymous social welfare function suitable for aggregating divisible resources like probability, time, or money.

## Contribution

It provides a complete characterization of domains and social welfare functions that enable Arrovian aggregation with convex preferences, extending Arrow's framework to divisible resources.

## Key findings

- Unique social welfare function characterized under convex preferences
- Domains allow arbitrary preferences over alternatives
- Applicable to settings with divisible resources such as probability, time, or money

## Abstract

We consider social welfare functions that satisfy Arrow's classic axioms of independence of irrelevant alternatives and Pareto optimality when the outcome space is the convex hull of some finite set of alternatives. Individual and collective preferences are assumed to be continuous and convex, which guarantees the existence of maximal elements and the consistency of choice functions that return these elements, even without insisting on transitivity. We provide characterizations of both the domains of preferences and the social welfare functions that allow for anonymous Arrovian aggregation. The domains admit arbitrary preferences over alternatives, which completely determine an agent's preferences over all mixed outcomes. On these domains, Arrow's impossibility turns into a complete characterization of a unique social welfare function, which can be readily applied in settings involving divisible resources such as probability, time, or money.