On Interim Envy-Free Allocation Lotteries

TL;DR

This paper explores interim envy-freeness (iEF) in lotteries for fair division, presenting polynomial-time algorithms for matching instances and analyzing tradeoffs with efficiency, extending to payments and rent division scenarios.

Contribution

It introduces a polynomial-time method for computing iEF lotteries in matching problems and analyzes the relationship between iEF, efficiency, and payments.

Findings

Polynomial-time algorithms for iEF in matching instances.

Tradeoffs identified between iEF and efficiency.

Extension of iEF concepts to rent division with payments.

Abstract

With very few exceptions, recent research in fair division has mostly focused on deterministic allocations. Deviating from this trend, we study the fairness notion of interim envy-freeness (iEF) for lotteries over allocations, which serves as a sweet spot between the too stringent notion of ex-post envy-freeness and the very weak notion of ex-ante envy-freeness. Our analysis relates iEF to other fairness notions as well, and reveals tradeoffs between iEF and efficiency. Even though several of our results apply to general fair division problems, we are particularly interested in instances with equal numbers of agents and items where allocations are perfect matchings of the items to the agents. We show how to compute iEF allocations in matching allocation instances in polynomial time, while also maximizing several efficiency objectives, even though this proves to be considerably more challenging than computing envy-free allocations. Our algorithms use efficient solutions to a novel variant of the bipartite matching problem. We also study the extension of iEF when payments to or from the agents are allowed. We present a series of results on two optimization problems, including a generalization of the classical rent division problem to random allocations.