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

by   Bhaskar Ray Chaudhury, et al.

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.


page 1

page 2

page 3

page 4


Fair Chore Division under Binary Supermodular Costs

We study the problem of dividing indivisible chores among agents whose c...

Fair Division with Bounded Sharing

A set of objects is to be divided fairly among agents with different tas...

Competitive Equilibrium with Chores: Combinatorial Algorithm and Hardness

We study the computational complexity of finding a competitive equilibri...

Fair Division with Minimal Sharing

A set of objects, some goods and some bads, is to be divided fairly amon...

Approximation Algorithms for Envy-Free Cake Division with Connected Pieces

Cake cutting is a classic model for studying fair division of a heteroge...

Best of Both Worlds: Agents with Entitlements

Fair division of indivisible goods is a central challenge in artificial ...

Fair Cake Division Under Monotone Likelihood Ratios

This work develops algorithmic results for the classic cake-cutting prob...

Please sign up or login with your details

Forgot password? Click here to reset