Barter Exchange with Shared Item Valuations

TL;DR

This paper studies centralized barter exchanges with shared item valuations, proving NP-hardness of perfect reallocation and proposing a randomized algorithm that maximizes expected utility while ensuring fairness.

Contribution

It introduces a randomized algorithm that optimally maximizes collective utility in barter exchanges with shared valuations, despite NP-hardness of perfect reallocation.

Findings

Finding a reallocation with equal total given and received values is NP-hard.

The proposed randomized algorithm achieves optimal expected utility.

The algorithm guarantees each agent's received value is close to their given value with high probability.

Abstract

In barter exchanges agents enter seeking to swap their items for other items on their wishlist. We consider a centralized barter exchange with a set of agents and items where each item has a positive value. The goal is to compute a (re)allocation of items maximizing the agents' collective utility subject to each agent's total received value being comparable to their total given value. Many such centralized barter exchanges exist and serve crucial roles; e.g., kidney exchange programs, which are often formulated as variants of directed cycle packing. We show finding a reallocation where each agent's total given and total received values are equal is NP-hard. On the other hand, we develop a randomized algorithm that achieves optimal utility in expectation and where, i) for any agent, with probability 1 their received value is at least their given value minus $v^*$ where $v^*$ is said agent's most valuable owned and wished-for item, and ii) each agent's given and received values are equal in expectation.