Your College Dorm and Dormmates: Fair Resource Sharing with Externalities

TL;DR

This paper investigates fair resource sharing among agents with externalities, focusing on dorm assignments with capacity constraints, and introduces polynomial algorithms for Pareto envy-free solutions when capacity is two.

Contribution

It establishes the existence of Pareto envy-free dorm assignments for capacity two and provides a polynomial-time algorithm, highlighting the complexity increase at capacity three.

Findings

Pareto envy-free assignments always exist for capacity 2.

A polynomial-time algorithm is provided for capacity 2.

Existence fails for capacity 3 and above.

Abstract

We study a fair resource sharing problem, where a set of resources are to be shared among a group of agents. Each agent demands one resource and each resource can serve a limited number of agents. An agent cares about what resource they get as well as the externalities imposed by their mates, who share the same resource with them. Clearly, the strong notion of envy-freeness, where no agent envies another for their resource or mates, cannot always be achieved and we show that even deciding the existence of such a strongly envy-free assignment is an intractable problem. Hence, a more interesting question is whether (and in what situations) a relaxed notion of envy-freeness, the Pareto envy-freeness, can be achieved. Under this relaxed notion, an agent envies another only when they envy both the resource and the mates of the other agent. In particular, we are interested in a dorm assignment problem, where students are to be assigned to dorms with the same capacity and they have dichotomous preference over their dormmates. We show that when the capacity of each dorm is 2, a Pareto envy-free assignment always exists and we present a polynomial-time algorithm to compute such an assignment. Nevertheless, the result breaks immediately when the capacity increases to 3, in which case even Pareto envy-freeness cannot be guaranteed. In addition to the existential results, we also investigate the utility guarantees of (Pareto) envy-free assignments in our model.