Polynomial Time Algorithms to Find an Approximate Competitive Equilibrium for Chores
Abstract
Competitive equilibrium with equal income (CEEI) is considered one of the best mechanisms to allocate a set of items among agents fairly and efficiently. In this paper, we study the computation of CEEI when items are chores that are disliked (negatively valued) by agents, under 1homogeneous and concave utility functions which includes linear functions as a subcase. It is wellknown that, even with linear utilities, the set of CEEI may be nonconvex and disconnected, and the problem is PPADhard in the more general exchange model. In contrast to these negative results, we design FPTAS: A polynomialtime algorithm to compute $\epsilon$approximate CEEI where the runningtime depends polynomially on $1/\epsilon$. Our algorithm relies on the recent characterization due to Bogomolnaia et al.~(2017) of the CEEI set as exactly the KKT points of a nonconvex minimization problem that have all coordinates nonzero. Due to this nonzero constraint, naive gradientbased methods fail to find the desired local minima as they are attracted towards zero. We develop an exteriorpoint method that alternates between guessing nonzero KKT points and maximizing the objective along supporting hyperplanes at these points. We show that this procedure must converge quickly to an approximate KKT point which then can be mapped to an approximate CEEI; this exterior point method may be of independent interest. When utility functions are linear, we give explicit procedures for finding the exact iterates, and as a result show that a stronger form of approximate CEEI can be found in polynomial time. Finally, we note that our algorithm extends to the setting of unequal incomes (CE), and to mixed manna with linear utilities where each agent may like (positively value) some items and dislike (negatively value) others.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.06649
 Bibcode:
 2021arXiv210706649B
 Keywords:

 Computer Science  Computer Science and Game Theory;
 Computer Science  Data Structures and Algorithms