Set systems related to a house allocation problem

TL;DR

This paper investigates the maximum number of distinct house sets in Pareto optimal allocations, improving previous bounds by linking the problem to extremal set theory.

Contribution

It establishes a tighter upper bound on the number of possible house sets in Pareto optimal allocations through novel connections to extremal set theory.

Findings

Improved upper bound on the size of family of house sets

Connection established between house allocation and extremal set theory

Enhanced understanding of Pareto optimal allocations

Abstract

We are given a set $A$ of buyers, a set $B$ of houses, and for each buyer a preference list, i.e., an ordering of the houses. A house allocation is an injective mapping $\tau$ from $A$ to $B$, and $\tau$ is strictly better than another house allocation $\tau'\neq \tau$ if for every buyer $i$, $\tau'(i)$ does not come before $\tau(i)$ in the preference list of $i$. A house allocation is Pareto optimal if there is no strictly better house allocation. Let $s(\tau)$ be the image of $\tau$ (i.e., the set of houses sold in the house allocation $\tau$). We are interested in the largest possible cardinality $f(m)$ of the family of sets $s(\tau)$ for Pareto optimal mappings $\tau$ taken over all sets of preference lists of $m$ buyers. We improve the earlier upper bound on $f(m)$ given by Asinowski, Keszegh and Miltzow by making a connection between this problem and some problems in extremal set theory.