[Paper Review] Weighted Maxmin Fair Share Allocation of Indivisible Chores

Haris Aziz, Hau Chan+1 more|arXiv (Cornell University)|Jun 18, 2019

Auction Theory and Applications22 references4 citations

TL;DR

This paper introduces weighted maxmin share (WMMS) fairness for indivisible chore allocation among agents with asymmetric shares, proposing a polynomial-time 4-approximation algorithm for OWMMS fairness. It also provides exact or improved constant-approximation algorithms for two special cases: binary valuations and the 2-agent setting, where agents achieve at least 3/2 of their WMMS value.

ABSTRACT

We initiate the study of indivisible chore allocation for agents with asymmetric shares. The fairness concept we focus on is the weighted natural generalization of maxmin share: WMMS fairness and OWMMS fairness. We first highlight the fact that commonly used algorithms that work well for the allocation of goods to asymmetric agents, and even for chores to symmetric agents do not provide good approximations for allocation of chores to asymmetric agents under WMMS. As a consequence, we present a novel polynomial-time constant-approximation algorithm, via linear program, for OWMMS. For two special cases: the binary valuation case and the 2-agent case, we provide exact or better constant-approximation algorithms.

Motivation & Objective

• To study fair allocation of indivisible chores when agents have asymmetric shares, extending maxmin share fairness to weighted settings.
• To address the lack of algorithms and theoretical guarantees for chore allocation under asymmetric shares, unlike the well-studied case of goods.
• To design efficient algorithms that provide constant-factor approximations to WMMS fairness in the presence of asymmetric entitlements.
• To identify limitations of existing greedy algorithms in this setting, showing they can perform arbitrarily poorly.

Proposed method

• Introduces OWMMS (optimal weighted maxmin share) as a relaxation of WMMS to ensure existence of fair allocations.
• Develops a novel polynomial-time algorithm combining greedy allocation for identical valuations and linear programming with rounding techniques.
• Applies a divide-and-choose protocol variant for the 2-agent case, ensuring each agent receives at least 3/2 of their WMMS value.
• Uses a greedy algorithm (EgalGreedy) for identical valuations, proving it achieves exact WMMS when all agents value chores uniformly.
• Proves that for binary valuations (0 or -1), a full WMMS allocation always exists and can be computed efficiently.
• Employs linear programming to compute approximate allocations under asymmetric shares, with rounding to ensure integral allocations.

Experimental results

Research questions

• RQ1 Can existing algorithms for goods or symmetric chores allocation be adapted to achieve good approximations in the asymmetric chore allocation setting?
• RQ2 What is the best possible approximation factor for WMMS fairness in chore allocation with asymmetric agents?
• RQ3 Does a constant-factor approximation algorithm exist for OWMMS fairness in polynomial time?
• RQ4 Can exact or improved approximations be achieved in special cases such as binary valuations or two agents?
• RQ5 Is it possible to guarantee each agent receives at least a constant fraction of their WMMS value under asymmetric shares?

Key findings

• No algorithm can simultaneously guarantee each agent’s value to be higher than 4/3 of their weighted maxmin share in the 2-agent case.
• Many widely used greedy algorithms for goods or symmetric chores perform arbitrarily poorly in the asymmetric chore setting.
• A polynomial-time 4-approximation algorithm exists for OWMMS fairness, providing a constant-factor guarantee for approximate fairness.
• For the 2-agent case, a variant of the divide-and-choose protocol ensures each agent receives at least 3/2 of their WMMS value.
• In the binary valuation case (where each chore is valued as 0 or -1), a full WMMS allocation exists and can be computed efficiently.
• For identical valuations, the EgalGreedy algorithm achieves an exact WMMS allocation, proving the existence of optimal solutions in this special case.