[Paper Review] Settling the Price of Fairness for Indivisible Goods

Siddharth Barman, Umang Bhaskar+1 more|arXiv (Cornell University)|Jul 13, 2020

Game Theory and Voting Systems27 references4 citations

TL;DR

This paper resolves the price of fairness for indivisible goods by proving that the worst-case social welfare loss due to EF1 and 1/2-MMS fairness constraints is O(√n), using efficient algorithms. The result holds for subadditive valuations (EF1) and additive valuations (1/2-MMS), closing an open problem posed by Bei et al. (2019).

ABSTRACT

In the allocation of resources to a set of agents, how do fairness guarantees impact the social welfare? A quantitative measure of this impact is the price of fairness, which measures the worst-case loss of social welfare due to fairness constraints. While initially studied for divisible goods, recent work on the price of fairness also studies the setting of indivisible goods. In this paper, we resolve the price of two well-studied fairness notions for the allocation of indivisible goods: envy-freeness up to one good (EF1), and approximate maximin share (MMS). For both EF1 and 1/2-MMS guarantees, we show, via different techniques, that the price of fairness is $O(\sqrt{n})$, where $n$ is the number of agents. From previous work, it follows that our bounds are tight. Our bounds are obtained via efficient algorithms. For 1/2-MMS, our bound holds for additive valuations, whereas for EF1, our bound holds for the more general class of subadditive valuations. This resolves an open problem posed by Bei et al. (2019).

Motivation & Objective

• To quantify the social welfare loss due to fairness constraints in the allocation of indivisible goods.
• To resolve the price of fairness for two prominent fairness notions: EF1 and 1/2-MMS.
• To close an open problem posed by Bei et al. (2019) regarding the tightness of the price of fairness for these fairness criteria.
• To establish tight O(√n) upper bounds on the price of fairness for both EF1 and 1/2-MMS under different valuation classes.
• To develop efficient algorithms that achieve these bounds, ensuring practical applicability of the theoretical results.

Proposed method

• Developed a novel analytical framework to bound the worst-case social welfare loss under EF1 and 1/2-MMS constraints.
• Applied distinct proof techniques for EF1 (subadditive valuations) and 1/2-MMS (additive valuations), reflecting different valuation structures.
• Designed efficient algorithms that achieve the derived O(√n) price of fairness bounds, ensuring computational feasibility.
• Used combinatorial and approximation techniques to relate fair allocation guarantees to social welfare loss in the worst case.
• Leveraged known lower bounds from prior work to establish tightness of the O(√n) upper bounds.
• Formalized the price of fairness as the ratio of optimal social welfare to the maximum social welfare achievable under fairness constraints.

Experimental results

Research questions

• RQ1 What is the worst-case social welfare loss due to enforcing EF1 fairness in the allocation of indivisible goods?
• RQ2 What is the worst-case social welfare loss due to enforcing 1/2-MMS fairness in the allocation of indivisible goods?
• RQ3 Can tight O(√n) upper bounds on the price of fairness be established for both EF1 and 1/2-MMS, and are these bounds achievable via efficient algorithms?
• RQ4 How do the price of fairness bounds differ across valuation classes, particularly for subadditive versus additive valuations?
• RQ5 Is the O(√n) bound tight for both EF1 and 1/2-MMS, and does it resolve the open problem posed by Bei et al. (2019)?

Key findings

• The price of fairness for EF1 is O(√n) for subadditive valuations, and this bound is tight.
• The price of fairness for 1/2-MMS is O(√n) for additive valuations, and this bound is also tight.
• The upper bounds are achieved via efficient algorithms, ensuring practical feasibility of fair allocations under these constraints.
• The results resolve an open problem posed by Bei et al. (2019) regarding the tightness of the price of fairness for indivisible goods.
• The analysis confirms that the worst-case social welfare loss grows as O(√n), which is the best possible asymptotic bound under the given fairness constraints.
• The findings establish a clear quantitative trade-off between fairness and social welfare in indivisible good allocation, with the trade-off scaling as √n in the worst case.