Dividing Bads is Harder than Dividing Goods: On the Complexity of Fair and Efficient Division of Chores

TL;DR

This paper investigates the computational complexity of dividing chores fairly and efficiently among agents with linear disutility, revealing that unlike goods, the problem is computationally hard and lacks simple polynomial-time solutions.

Contribution

It establishes the NP-hardness and PPAD-hardness of finding fair and efficient chore divisions, providing the first complexity results for chore division with linear preferences.

Findings

Finding a no-huge-dislike division is NP-hard.

Existence of such divisions is guaranteed under a simple condition.

Computing competitive divisions is PPAD-hard under that condition.

Abstract

We study the chore division problem where a set of agents needs to divide a set of chores (bads) among themselves fairly and efficiently. We assume that agents have linear disutility (cost) functions. Like for the case of goods, competitive division is known to be arguably the best mechanism for the bads as well. However, unlike goods, there are multiple competitive divisions with very different disutility value profiles in bads. Although all competitive divisions satisfy the standard notions of fairness and efficiency, some divisions are significantly fairer and efficient than the others. This raises two important natural questions: Does there exist a competitive division in which no agent is assigned a chore that she hugely dislikes? Are there simple sufficient conditions for the existence and polynomial-time algorithms assuming them? We investigate both these questions in this paper. We show that the first problem is strongly NP-hard. Further, we derive a simple sufficient condition for the existence, and we show that finding a competitive division is PPAD-hard assuming the condition. These results are in sharp contrast to the case of goods where both problems are strongly polynomial-time solvable. To the best of our knowledge, these are the first hardness results for the chore division problem, and, in fact, for any economic model under linear preferences.