Two's Company, Three's a Crowd: Consensus-Halving for a Constant Number of Agents

Argyrios Deligkas, Aris Filos-Ratsikas, Alexandros Hollender

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We consider the ε-Consensus-Halving problem, in which a set of heterogeneous agents aim at dividing a continuous resource into two (not necessarily contiguous) portions that all of them simultaneously consider to be of approximately the same value (up to ε). This problem was recently shown to be PPA-complete, for n agents and n cuts, even for very simple valuation functions. In a quest to understand the root of the complexity of the problem, we consider the setting where there is only a constant number of agents, and we consider both the computational complexity and the query complexity of the problem.For agents with monotone valuation functions, we show a dichotomy: for two agents the problem is polynomial-time solvable, whereas for three or more agents it becomes PPA-complete. Similarly, we show that for two monotone agents the problem can be solved with polynomially-many queries, whereas for three or more agents, we provide exponential query complexity lower bounds. These results are enabled via an interesting connection to a monotone Borsuk-Ulam problem, which may be of independent interest. For agents with general valuations, we show that the problem is PPA-complete and admits exponential query complexity lower bounds, even for two agents.
Original languageEnglish
Title of host publicationProceedings of the 22nd ACM Conference on Economics and Computation
Place of PublicationNew York, NY, USA
PublisherAssociation for Computing Machinery, Inc
Number of pages22
ISBN (Electronic)9781450385541
Publication statusPublished - 18 Jul 2021
EventThe 22nd ACM Conference on Economics and Computation, 2021 - Online
Duration: 19 Jul 202123 Jul 2021
Conference number: 22

Publication series

NameEC '21
PublisherAssociation for Computing Machinery


ConferenceThe 22nd ACM Conference on Economics and Computation, 2021
Abbreviated titleEC 21
Internet address


  • fair division
  • generalised robertson-webb
  • monotone borsuk-ulam
  • query complexity
  • PPA


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