Fair Allocation with Initial Utilities

TL;DR

This paper explores fair allocation of indivisible resources considering agents' initial utilities, introducing new fairness notions and algorithms to ensure equitable outcomes despite initial inequalities.

Contribution

It extends the classic resource allocation model by incorporating initial utilities and proposes a new fairness concept, minimum-EF1-init, with a polynomial-time algorithm for its computation.

Findings

Complete EF1-like allocations may not exist with initial utilities.

Finding EF1-like allocations is computationally hard in this setting.

The proposed minimum-EF1-init algorithm guarantees fair allocations in polynomial time.

Abstract

The problem of allocating indivisible resources to agents arises in a wide range of domains, including treatment distribution and social support programs. An important goal in algorithm design for this problem is fairness, where the focus in previous work has been on ensuring that the computed allocation provides equal treatment to everyone. However, this perspective disregards that agents may start from unequal initial positions, which is crucial to consider in settings where fairness is understood as equality of outcome. In such settings, the goal is to create an equal final outcome for everyone by leveling initial inequalities through the allocated resources. To close this gap, focusing on agents with additive utilities, we extend the classic model by assigning each agent an initial utility and study the existence and computational complexity of several new fairness notions following the principle of equality of outcome. Among others, we show that complete allocations satisfying a direct analog of envy-freeness up to one resource (EF1) may fail to exist and are computationally hard to find, forming a contrast to the classic setting without initial utilities. We propose a new, always satisfiable fairness notion, called minimum-EF1-init and design a polynomial-time algorithm based on an extended round-robin procedure to compute complete allocations satisfying this notion.