[Paper Review] Algorithmic Fair Allocation of Indivisible Items: A Survey and New Questions

Haris Aziz, Bo Li+2 more|arXiv (Cornell University)|Feb 17, 2022

Game Theory and Voting Systems12 citations

TL;DR

This survey provides a comprehensive overview of algorithmic fair allocation for indivisible items, focusing on approximation techniques to achieve fairness in settings where exact fairness (e.g., envy-freeness, proportionality) is unattainable. It presents key algorithmic approaches, relaxations of fairness criteria like EF1 and MMS, and identifies open problems in efficiency, truthfulness, and complex valuation structures.

ABSTRACT

The theory of algorithmic fair allocation is within the center of multi-agent systems and economics in the last decade due to its industrial and social importance. At a high level, the problem is to assign a set of items that are either goods or chores to a set of agents so that every agent is happy with what she obtains. Particularly, in this survey, we focus on indivisible items, for which absolute fairness such as envy-freeness and proportionality cannot be guaranteed. One main theme in the recent research agenda is about designing algorithms that approximately achieve the fairness criteria. We aim at presenting a comprehensive survey of recent progresses through the prism of algorithms, highlighting the ways to relax fairness notions and common techniques to design algorithms, as well as the most interesting questions for future research.

Motivation & Objective

• To provide a systematic survey of algorithmic approaches to fair allocation of indivisible items, emphasizing approximation techniques due to the impossibility of exact fairness.
• To identify and analyze common algorithmic techniques used to achieve relaxed fairness notions such as EF1, MMS, and PROP1.
• To highlight recent advances in handling complex valuation types (e.g., submodular, XOS, subadditive) and mixed goods/chore settings.
• To explore the compatibility of fairness with efficiency (Pareto optimality) and truthfulness in strategic environments.
• To identify and frame open research questions in fairness approximation, algorithmic complexity, and mechanism design for indivisible allocations.

Proposed method

• Categorizes fairness notions into exact (e.g., envy-freeness, proportionality) and relaxed forms (e.g., EF1, MMS, PROP1) suitable for indivisible items.
• Reviews algorithmic techniques such as round-robin allocation, maximum Nash welfare maximization, and greedy-based methods for achieving fairness approximations.
• Analyzes the use of valuation models including additive, submodular, XOS, and subadditive valuations, with focus on approximation guarantees.
• Examines the integration of fairness with efficiency via Pareto optimality (PO), particularly through Nash welfare maximization.
• Investigates truthfulness in strategic settings, including mechanisms with and without monetary transfers, and equilibria in sequential allocation algorithms.
• Introduces advanced settings such as mixed goods and chores, group fairness, and dynamic allocation, highlighting new challenges and open problems.

Experimental results

Research questions

• RQ1 What are the most effective algorithmic techniques for approximating fairness in indivisible item allocation when exact fairness is unachievable?
• RQ2 How can fairness notions like EF1, MMS, and PROP1 be efficiently computed for complex valuation functions such as subadditive or XOS?
• RQ3 To what extent can fairness be simultaneously achieved with Pareto optimality and truthfulness in strategic allocation settings?
• RQ4 What are the fundamental limits of approximation for fairness criteria in mixed goods and chores environments?
• RQ5 How do strategic behaviors in algorithms like Round-Robin affect fairness outcomes under true agent preferences?

Key findings

• For subadditive valuations, algorithms exist that achieve O(n)-approximation to maximum Nash welfare while being approximately EFX.
• For bi-valued valuations, the maximum Nash welfare allocation is both EFX and Pareto optimal, and can be computed in polynomial time.
• EF1+PO allocations exist and can be computed efficiently for bi-valued instances, but their existence remains open for general subadditive or three-valued settings.
• For chores, PROP1+PO allocations can be computed in polynomial time even in mixed goods/chore settings, but EF1 and PROPX compatibility with PO is still unresolved.
• In lexicographic valuation settings, EFX+PO allocations exist and can be computed in polynomial time.
• For two agents, truthful approximation algorithms for EF1 and MMS are completely characterized with tight bounds, but such characterizations remain elusive for larger agent sets.