Competitive Equilibrium with Chores: Combinatorial Algorithm and Hardness

We study the computational complexity of finding a competitive equilibrium (CE) with chores when agents have linear preferences. CE is one of the most preferred mechanisms for allocating a set of items among agents. CE with equal incomes (CEEI), Fisher, and Arrow-Debreu (exchange) are the fundamental economic models to study allocation problems, where CEEI is a special case of Fisher and Fisher is a special case of exchange. When the items are goods (giv-ing utility), the CE set is convex even in the exchange model, facilitating several combinatorial polynomial-time algorithms (starting with the seminal work of Devanur, Papadimitriou, Saberi and Vazirani [DPSV08]) for all of these models. In sharp contrast, when the items are chores (giving disutility), the CE set is known to be non-convex and disconnected even in the CEEI model. Further, no combinatorial algorithms or hardness results are known for these models. In this paper, we give two main results for CE with chores: To the best of our knowledge, these results show the first separation between the CEEI and exchange models when agents have linear preferences, assuming PPAD (cid:54) = P. Furthermore, this is also the first separation between the two economic models when the CE set is non-convex in both cases. Finally, we show that our new insight implies a straightforward proof of the existence of an allocation that is both envy-free up to one chore (EF1) and Pareto optimal (PO) in the when have factored bivalued preferences. EPS22] involved