Fair congested assignment problem

TL;DR

This paper introduces a fair and efficient method for assigning agents to posts considering congestion, ensuring fairness and efficiency through a competitive assignment approach, with solutions applicable to fractional and weighted models.

Contribution

It develops a novel framework for fair congestion-aware assignment, including ex ante and ex post fairness, with existence and efficiency results, and extends to weighted congestion scenarios.

Findings

Unique competitive congestion profiles exist under certain conditions.

Fractional models always admit a unique competitive congestion profile.

Approximate deterministic assignments can closely replicate welfare profiles.

Abstract

We propose a fair and efficient solution for assigning agents to m posts subject to congestion, when agents care about both their post and its congestion. Examples include assigning jobs to busy servers, students to crowded schools or crowded classes, commuters to congested routes, workers to crowded office spaces or to team projects etc... Congestion is anonymous (it only depends on the number n of agents in a given post). A canonical interpretation of ex ante fairness allows each agent to choose m post-specific caps on the congestion they tolerate: these requests are mutually feasible if and only if the sum of the caps is n. For ex post fairness we impose a competitive requirement close to envy freeness: taking the congestion profile as given each agent is assigned to one of her best posts. If a competitive assignment exists, it delivers unique congestion and welfare profiles and is also efficient and ex ante fair. In a fractional (randomised or time sharing) version of our model, a unique competitive congestion profile always exists. It is approximately implemented by a mixture of ex post deterministic assignments: with an approxination factor equal to the largest utility loss from one more unit of congestion, the latter deliver identical welfare profiles and are weakly efficient. Our approach to ex ante fairness generalises to the model where each agent's congestion is weighted. Now the caps on posts depend only upon own weight and total congestion, not on the number of other agents contributing to it. Remarkably in both models these caps are feasible if and only if they give to each agent the right to veto all but (1/m) of their feasible allocations.