The Complexity of Fair Allocation under Ternary Valuations

Core Concepts

The author explores the complexity of fair allocation under ternary valuations, showing APX-hardness for maximizing Nash welfare and egalitarian welfare.

Abstract

The content delves into the intricacies of fair allocation problems under ternary valuations. It discusses the challenges in computing exact fair allocations and presents results on maximizing Nash welfare and egalitarian welfare. The analysis involves detailed proofs and reductions from graph theory problems to establish the computational complexity of these allocation scenarios.

Stats

We show that for any distinct non-negative a, b, and c, maximizing Nash welfare is APX-hard — i.e., the problem does not admit a PTAS unless P = NP. We also show that for any distinct a, b, and c, maximizing egalitarian welfare is APX-hard except for a few cases when b = 0 that admit efficient algorithms. These results make significant progress towards completely characterizing the complexity of computing exact MNW allocations and MEW allocations. When agents have {a, b, c}-valuations with 0 ≤ a < b < c and 2b < c, computing an MNW allocation is APX-hard. Assume agents have {0, 1, 3}-valuations. It is impossible to approximate MNW by a factor smaller than 1.00013 unless P = NP.

Quotes

"We study the problem of fair allocation of indivisible items when agents have ternary additive valuations." "Our objective is to find an allocation of items to agents satisfying certain natural justice criteria." "These results make significant progress towards completely characterizing the complexity of computing exact MNW allocations."

Key Insights Distilled From

On the Hardness of Fair Allocation under Ternary Valuations

by Zack Fitzsim... at arxiv.org 03-05-2024

https://arxiv.org/pdf/2403.00943.pdf

Deeper Inquiries

Can fairness criteria be universally defined across different valuation types

Fairness criteria cannot be universally defined across different valuation types due to the inherent differences in how agents value items. The concept of fairness is subjective and can vary based on individual preferences, cultural norms, and societal values. Different valuation types, such as binary valuations, additive valuations, or submodular valuations, may require distinct fairness criteria to ensure equitable resource allocation. For example, what constitutes a fair distribution under binary valuations (where items are valued at 0 or 1) may not apply to additive valuations (where values are integers) or submodular valuations (where interactions between items matter).

What implications do these findings have on real-world resource allocation systems

The findings on the hardness of fair allocation algorithms under various valuation types have significant implications for real-world resource allocation systems. Understanding the computational complexity of computing fair allocations helps in designing efficient algorithms for distributing resources among agents with diverse preferences. By knowing which scenarios are computationally hard to solve optimally, decision-makers can adapt their strategies and possibly simplify the problem by imposing constraints or approximations. In practical terms, these findings can influence policy-making in areas such as course seat assignments for students, shift scheduling for workers in hospitals or factories, and even resource distribution in humanitarian aid efforts. By being aware of the computational challenges involved in achieving fairness under different valuation models, organizations can tailor their allocation processes to balance efficiency with equity.

How can computational economics benefit from advancements in fair allocation algorithms

Advancements in fair allocation algorithms offer several benefits to computational economics by enhancing decision-making processes and optimizing resource utilization. By developing efficient algorithms that consider various fairness criteria like Nash welfare or egalitarian welfare under different valuation settings (such as ternary valuations), researchers can provide valuable tools for economists and policymakers. These advancements enable better modeling of real-world scenarios where resources need to be allocated fairly among multiple agents with complex preferences. By incorporating insights from algorithmic research into economic models, computational economics can improve its predictive accuracy and generate more informed policy recommendations. Furthermore, advancements in fair allocation algorithms contribute to interdisciplinary collaborations between computer scientists and economists. This cross-pollination of ideas fosters innovation and leads to novel approaches for addressing challenging problems related to resource allocation within economic systems.