[Paper Review] Almost Envy Freeness and Welfare Efficiency in Fair Division with Goods or Bads

Martin Aleksandrov|arXiv (Cornell University)|Aug 1, 2018

Game Theory and Voting Systems32 references3 citations

TL;DR

This paper introduces new fairness notions—1EF and XEF—for fair division with bads (undesirable items), generalizing existing proxies like EF1 and EFX for goods. It proposes novel algorithms (Lipton–, leximax–, Alg-Identical–) that achieve almost envy-freeness and welfare efficiency (Pareto efficiency, Nash, and egalitarian diswelfare) under additive and identical valuations, proving that Nash diswelfare maximization yields XEF and PE allocations with identical valuations, while also establishing impossibility results for combining 1EF and efficiency under anti-monotone valuations.

ABSTRACT

We consider two models of fair division with indivisible items: one for goods and one for bads. For goods, we study two generalized envy freeness proxies (EF1 and EFX for goods) and three common welfare (utilitarian, egalitarian and Nash) efficiency notions. For bads, we study two generalized envy freeness proxies (1EF and XEF for goods) and two less common diswelfare (egalitarian and Nash) efficiency notions. Some existing algorithms for goods do not work for bads. We thus propose several new algorithms for the model with bads. Our new algorithms exhibit many nice properties. For example, with additive identical valuations, an allocation that maximizes the egalitarian diswelfare or Nash diswelfare is XEF and PE. Finally, we also give simple and tractable cases when these envy freeness proxies and welfare efficiency are attainable in combination (e.g. binary valuations, house allocations).

Motivation & Objective

• To extend fairness and efficiency concepts from goods to bads in fair division, addressing limitations of existing algorithms that fail for bads.
• To define new fairness proxies—1EF and XEF—for bads that are robust to zero marginal valuations and more general than prior notions.
• To design new algorithms (Lipton–, leximax–, Alg-Identical–) that achieve both almost envy-freeness and welfare efficiency for bads.
• To establish conditions under which almost envy-freeness and welfare efficiency can be simultaneously achieved, including tractable cases like 0/-1 valuations.
• To prove impossibility results, showing that 1EF and efficiency cannot be combined under anti-monotone identical valuations.

Proposed method

• Proposes 1EF and XEF as generalized envy-freeness proxies for bads, defined via marginal disutility and bundle removal.
• Introduces the Lipton– algorithm, a variant of Lipton’s algorithm, to compute 1EF allocations under anti-monotone valuations.
• Develops the leximax– solution, a maximin-based approach that guarantees XEF and MEDW (egalitarian diswelfare) with identical or 2-agent distinct valuations.
• Designs the Alg-Identical– algorithm to mirror the Alg-Identical algorithm on the negated problem, ensuring XEF and welfare efficiency under identical valuations.
• Uses duality between goods and bads: an allocation maximizing Nash diswelfare in the negated bads problem corresponds to an XEF and PE allocation in the original bads setting.
• Employs reduction techniques and problem negation to transfer results from goods to bads, especially for identical valuation cases.

Experimental results

Research questions

• RQ1 Can existing fairness proxies for goods (EF1, EFX) be generalized to bads in a way that avoids trivial satisfaction in low-value scenarios?
• RQ2 Are there new algorithms that can achieve both almost envy-freeness and welfare efficiency (Pareto efficiency, Nash, egalitarian) in fair division with bads?
• RQ3 Under what conditions (e.g., 0/-1 valuations, identical valuations) can 1EF and XEF be combined with welfare efficiency for bads?
• RQ4 Is it possible to achieve 1EF and welfare efficiency simultaneously under anti-monotone identical valuations, and if not, why?
• RQ5 How do the axiomatic properties of fairness and efficiency in goods generalize to bads, especially in terms of duality and algorithmic transfer?

Key findings

• An allocation maximizing Nash diswelfare (product of disutilities) is guaranteed to be XEF and PE when agents have identical valuations for bads.
• The Lipton– algorithm computes a 1EF allocation with arbitrary anti-monotone valuations and bounds maximum envy, extending Lipton’s original algorithm to bads.
• The leximax– solution produces an XEF and MEDW (egalitarian diswelfare) allocation for any number of agents with identical valuations or 2 agents with possibly distinct valuations.
• With 0/-1 valuations, the responsive draft algorithm computes an XEF, PE, MUW, MEW, and mNDW allocation in linear time.
• For anti-monotone identical valuations, no 1EF allocation can be Pareto efficient, as shown by counterexamples with 2 agents and 3 or 4 items.
• An MEW (maximin egalitarian welfare) allocation may fail to be 1EF or MEDW under anti-monotone valuations, even with 2 agents, demonstrating a fundamental incompatibility.