Average Envy-freeness for Indivisible Items

TL;DR

This paper introduces the concept of average envy-freeness (AEF) for fair division of indivisible items, analyzing its computational complexity and proposing algorithms for specific cases with quotas and valuations.

Contribution

It defines a new fairness notion suitable for unequal entitlements, studies its computational complexity, and provides algorithms for special cases with quotas and binary valuations.

Findings

Deciding AEF existence is NP-complete.

AEF-1 allocations always exist and are computable in polynomial time.

Algorithms are proposed for fixed number of agents with binary valuations and approximate solutions for general valuations.

Abstract

In fair division applications, agents may have unequal entitlements reflecting their different contributions. Moreover, the contributions of agents may depend on the allocation itself. Previous fairness notions designed for agents with equal or pre-determined entitlement fail to characterize fairness in these collaborative allocation scenarios. We propose a novel fairness notion of average envy-freeness (AEF), where the envy of agents is defined on the average value of items in the bundles. Average envy-freeness provides a reasonable comparison between agents based on the items they receive and reflects their entitlements. We study the complexity of finding AEF and its relaxation, average envy-freeness up to one item (AEF-1). While deciding if an AEF allocation exists is NP-complete, an AEF-1 allocation is guaranteed to exist and can be computed in polynomial time. We also study allocation with quotas, i.e. restrictions on the sizes of bundles. We prove that finding AEF-1 allocation satisfying a quota is NP-hard. Nevertheless, in the instances with a fixed number of agents, we propose polynomial-time algorithms to find AEF-1 allocation with a quota for binary valuation and approximated AEF-1 allocation with a quota for general valuation.