Majoritarian Assignment Rules

TL;DR

The paper addresses fair object assignment under majority comparisons by transferring majoritarian social-choice concepts to the assignment domain. It introduces and leverages the majority graph $G_P$ to characterize central notions such as Pareto-optimality, unpopularity, and mixed popularity, and proves that many majoritarian rules are determined solely by $G_P$. A key contribution is the complete characterization of the top cycle for $n\ge 5$, showing it can only take the five sizes $1$, $2$, $n!-2$, $n!-1$, or $n!$, along with a sublinear-time method to represent TC, and it proves TC contains all Pareto-optimal assignments. The uncovered-set concepts are also adapted, with three variants providing stronger discrimination within the top cycle, and extensive discussion highlights computational and structural questions distinct from general social choice. Overall, the work establishes a graph-centric framework for majoritarian assignment rules, enabling efficient representation and guiding design of fair allocation mechanisms when popularity fails.

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.

Figures (3)

• Figure 1: Size distributions of UCs for $n = 5$. The high peak is at size $2$ for all of them. In total, there are $9,078,630$ profiles (up to symmetries) and there are $5!=120$ different assignments.
• Figure 2: Size distributions of UCs for $n = 7$ in $1,000$ profiles sampled via the impartial culture model. The high peak is at size $2$ for McKelvey and Bordes. Gillies-$\mathit{UC}$ has an even higher peak at size $4$.
• Figure 3: Cumulative distributions of the ratio of UC and PO sizes for $n = 5$. The plot shows the relative difference, i.e., the ratio. In total, there are $9,078,630$ profiles up to symmetry. We note that the plots for McKelvey and Bordes almost perfectly align.

Theorems & Definitions (28)

• Example 1
• Example 2
• Theorem 1
• Remark 1
• Remark 2
• Corollary 1
• Example 3
• Proposition 1
• Proposition 2
• Theorem 2