Competitive Equilibrium with Equal Incomes for Allocation of Indivisible Objects

TL;DR

This paper investigates the computational complexity of finding fair allocations of indivisible objects using CEEI and introduces a stronger, efficiently verifiable fairness notion called CEEI-FRAC.

Contribution

It proves that computing a discrete CEEI assignment is strongly NP-hard and introduces CEEI-FRAC, a stronger, polynomial-time verifiable fairness concept.

Findings

Computing a discrete CEEI assignment is strongly NP-hard.

CEEI-FRAC is a stronger fairness notion than CEEI.

CEEI-FRAC can be computed in polynomial time when utilities are zero or one.

Abstract

In AAMAS 2014, Bouveret and Lemaitre (2014) presented a hierarchy of fairness concepts for allocation of indivisible objects. Among them CEEI (Competitive Equilibrium with Equal Incomes) was the strongest. In this note, we settle the complexity of computing a discrete CEEI assignment by showing it is strongly NP-hard. We then highlight a fairness notion (CEEI-FRAC) that is even stronger than CEEI for discrete assignments, is always Pareto optimal, and can be verified in polynomial time. We also show that computing a CEEI-FRAC discrete assignment is strongly NP-hard in general but polynomial-time computable if the utilities are zero or one.