[Paper Review] Algorithms for Competitive Division of Chores

Simina Brânzei, Fedor Sandomirskiy|arXiv (Cornell University)|Jul 3, 2019

Economic theories and models7 citations

TL;DR

This paper presents a strongly polynomial-time algorithm to compute all competitive equilibrium allocations for divisible chores among agents with additive utilities, when either the number of agents or chores is fixed. The method combines enumeration of Pareto optimal consumption graphs, explicit formulae for candidate utility profiles, and maximum flow computations to verify competitiveness, resolving an open problem in fair division of bads.

ABSTRACT

We study the problem of allocating divisible bads (chores) among multiple agents with additive utilities when monetary transfers are not allowed. The competitive rule is known for its remarkable fairness and efficiency properties in the case of goods. This rule was extended to chores in prior work by Bogomolnaia, Moulin, Sandomirskiy, and Yanovskaya (2017). The rule produces Pareto optimal and envy-free allocations for both goods and chores. In the case of goods, the outcome of the competitive rule can be easily computed. Competitive allocations solve the Eisenberg-Gale convex program; hence the outcome is unique and can be approximately found by standard gradient methods. An exact algorithm that runs in polynomial time in the number of agents and goods was given by Orlin (2010). In the case of chores, the competitive rule does not solve any convex optimization problem; instead, competitive allocations correspond to local minima, local maxima, and saddle points of the Nash social welfare on the Pareto frontier of the set of feasible utilities. The Pareto frontier may contain many such points; consequently, the competitive rule's outcome is no longer unique. In this paper, we show that all the outcomes of the competitive rule for chores can be computed in strongly polynomial time if either the number of agents or the number of chores is fixed. The approach is based on a combination of three ideas: all consumption graphs of Pareto optimal allocations can be listed in polynomial time; for a given consumption graph, a candidate for a competitive utility profile can be constructed via an explicit formula; each candidate can be checked for competitiveness, and the allocation can be reconstructed using a maximum flow computation. Our algorithm gives an approximately-fair allocation of indivisible chores by the rounding technique of Barman and Krishnamurthy (2018).

Motivation & Objective

• Address the open problem of computing competitive allocations for divisible chores when monetary transfers are not allowed.
• Overcome the non-convexity of the chore allocation problem, which prevents standard convex optimization techniques from being applied.
• Provide a polynomial-time algorithm for computing all competitive utility profiles when either the number of agents or the number of chores is fixed.
• Enable approximately-fair allocation of indivisible chores via rounding techniques.
• Extend algorithmic insights from two-agent cases to general n-agent settings with non-trivial structural complexity.

Proposed method

• Enumerate all possible consumption graphs of Pareto optimal allocations in polynomial time using the structure of the Pareto frontier.
• For each consumption graph, construct a candidate competitive utility profile via an explicit formula based on agent values and budgets.
• Verify competitiveness of each candidate using maximum flow computation to check budget and price consistency conditions.
• Use the algorithm of Devanur et al. [DPSV02] to reconstruct feasible allocations from verified utility profiles.
• Leverage the fact that competitive allocations for chores correspond to critical points (saddle points, local extrema) of Nash social welfare on the Pareto frontier.
• Apply a rounding technique from Barman and Krishnamurthy [BK19] to obtain approximately-fair allocations for indivisible chores.

Experimental results

Research questions

• RQ1 Can all competitive allocations for divisible chores be computed in strongly polynomial time when the number of agents or chores is fixed?
• RQ2 What structural properties characterize the set of competitive allocations in the case of chores, given that they are not captured by convex optimization?
• RQ3 How can the non-uniqueness of competitive outcomes for chores be algorithmically addressed despite the absence of a convex program?
• RQ4 Can the approach used for chores be extended to mixed economies with both goods and bads?
• RQ5 Is there a general method to compute competitive allocations in constrained economies where agents face caps on consumption?

Key findings

• All competitive utility profiles for chores can be computed in strongly polynomial time when either the number of agents or the number of chores is fixed.
• The algorithm enumerates all Pareto optimal consumption graphs and checks each for competitiveness using maximum flow, ensuring correctness and completeness.
• The competitive rule for chores is not captured by any convex program; instead, competitive allocations are critical points of the Nash social welfare on the Pareto frontier.
• The method yields an approximately-fair allocation of indivisible chores via rounding, matching the performance of existing techniques but with a new algorithmic foundation.
• For two agents, the structure of the Pareto frontier allows efficient computation, but the extension to more agents required novel insights into graph enumeration and flow-based verification.
• The approach generalizes to fair assignment problems and constrained economies with consumption caps, suggesting broader applicability beyond the base model.