[Paper Review] Fair Division with Minimal Sharing

Fedor Sandomirskiy, Erel Segal-Halevi|arXiv (Cornell University)|Aug 1, 2019

Game Theory and Voting Systems13 citations

TL;DR

This paper addresses fair division of indivisible goods and bads among agents with heterogeneous preferences, focusing on minimizing object sharing to achieve Pareto-optimal, envy-free, or proportional allocations. It proves that for generic instances (non-degenerate valuations), a fair division with the minimal number of shared objects can be computed in polynomial time when the number of agents is fixed, though the problem becomes NP-hard in degenerate cases where valuations are highly aligned.

ABSTRACT

A set of objects, some goods and some bads, is to be divided fairly among agents with different tastes, modeled by additive utility-functions. If the objects cannot be shared, so that each of them must be entirely allocated to a single agent, then fair division may not exist. What is the smallest number of objects that must be shared between two or more agents in order to attain a fair division? We focus on Pareto-optimal, envy-free and/or proportional allocations. We show that, for a generic instance of the problem --- all instances except of a zero-measure set of degenerate problems --- a fair and Pareto-optimal division with the smallest possible number of shared objects can be found in polynomial time, assuming that the number of agents is fixed. The problem becomes computationally hard for degenerate instances, where the agents' valuations are aligned for many objects.

Motivation & Objective

• To determine the minimum number of objects that must be shared among agents to achieve a fair allocation when objects cannot be divided.
• To identify conditions under which fair and Pareto-optimal allocations exist with minimal sharing.
• To develop an efficient algorithm for computing such allocations in generic instances of the fair division problem.
• To analyze the computational complexity of the problem, especially in degenerate cases where agent valuations are highly aligned.

Proposed method

• Models agent preferences using additive utility functions over a set of goods and bads.
• Defines fairness via envy-freeness, proportionality, and Pareto optimality.
• Identifies 'generic instances' as those outside a zero-measure set of degenerate problems where valuations are not aligned across agents.
• Applies algebraic geometry and optimization techniques to show that a fair allocation with minimal sharing exists and can be computed in polynomial time for fixed agent counts.
• Uses a perturbation-based approach to avoid degeneracies and ensure algorithmic tractability.
• Employs a reduction to convex optimization problems to find the minimal sharing solution efficiently in generic cases.

Experimental results

Research questions

• RQ1 What is the smallest number of objects that must be shared to achieve a fair allocation when all objects must be fully assigned to one agent?
• RQ2 Can a Pareto-optimal and envy-free allocation be computed efficiently when the number of shared objects is minimized?
• RQ3 Under what conditions does the problem remain tractable, and when does it become computationally hard?
• RQ4 How do degenerate valuation structures—where agents value objects similarly—affect the complexity of finding minimal-sharing fair allocations?

Key findings

• For generic instances (non-degenerate valuations), a fair and Pareto-optimal allocation with the minimal number of shared objects can be computed in polynomial time when the number of agents is fixed.
• The minimal number of shared objects is bounded and can be determined efficiently in the generic case.
• The problem becomes computationally hard (NP-hard) in degenerate instances where agent valuations are highly aligned across many objects.
• The algorithmic approach relies on avoiding degeneracies through perturbation, ensuring that the solution space remains tractable.
• The existence of a fair allocation with minimal sharing is guaranteed for generic instances, even with mixed goods and bads.
• The results hold for envy-free and proportional allocations as well as Pareto-optimal ones, under the same computational conditions.