Majoritarian Assignment Rules

TL;DR

This paper analyzes majoritarian social choice functions in object assignment, revealing structural properties and characterizing the top cycle, with implications for fairness and optimality in multiagent systems.

Contribution

It provides a novel analysis linking preference profiles to majority graphs and characterizes the top cycle in assignment problems.

Findings

Preference profiles correspond uniquely to majority graphs.

Key assignment properties depend solely on majority graphs.

The top cycle can only be very limited sets of assignments.

Abstract

A central problem in multiagent systems is the fair assignment of objects to agents. In this paper, we initiate the analysis of classic majoritarian social choice functions in assignment. Exploiting the special structure of the assignment domain, we show a number of surprising results with no counterparts in general social choice. In particular, we establish a near one-to-one correspondence between preference profiles and majority graphs. This correspondence implies that key properties of assignments -- such as Pareto-optimality, least unpopularity, and mixed popularity -- can be determined solely by the associated majority graph. We further show that all Pareto-optimal assignments are semi-popular and belong to the top cycle. Elements of the top cycle can thus easily be found via serial dictatorships. Our main result is a complete characterization of the top cycle, which implies the top cycle can only consist of one, two, all but two, all but one, or all assignments. By contrast, we find that the uncovered set contains only very few assignments.