On the Existence of Competitive Equilibrium with Chores
We study the chore division problem in the classic Arrow-Debreu exchange setting, where a set of agents want to divide their divisible chores (bads) to minimize their disutilities (costs). We assume that agents have linear disutility functions. Like the setting with goods, a division based on competitive equilibrium is regarded as one of the best mechanisms for bads. Equilibrium existence for goods has been extensively studied, resulting in a simple, polynomial-time verifiable, necessary and sufficient condition. However, dividing bads has not received a similar extensive study even though it is as relevant as dividing goods in day-to-day life.
In this paper, we show that the problem of checking whether an equilibrium exists in chore division is NP-complete, which is in sharp contrast to the case of goods. Further, we derive a simple, polynomial-time verifiable, sufficient condition for existence. Our fixed-point formulation to show existence makes novel use of both Kakutani and Brouwer fixed-point theorems, the latter nested inside the former, to avoid the undefined demand issue specific to bads.
Fair Division
Competitive Equilibrium
Fixed Point Theorems
Theory of computation~Exact and approximate computation of equilibria
41:1-41:13
Regular Paper
http://jugal.ise.illinois.edu/itcs22.pdf
Bhaskar Ray
Chaudhury
Bhaskar Ray Chaudhury
University of Illinois at Urbana Champaign, IL, USA
Jugal
Garg
Jugal Garg
University of Illinois at Urbana Champaign, IL, USA
NSF Grant CCF-1942321 (CAREER).
Peter
McGlaughlin
Peter McGlaughlin
University of Illinois at Urbana Champaign, IL, USA
Ruta
Mehta
Ruta Mehta
University of Illinois at Urbana Champaign, IL, USA
NSF Grant CCF-1750436 (CAREER).
10.4230/LIPIcs.ITCS.2022.41
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Bhaskar Ray Chaudhury, Jugal Garg, Peter McGlaughlin, and Ruta Mehta
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