Incoherent majorities: The McGarvey problem in judgement aggregation

Authors:
Klaus Nehring;Marcus Pivato
Affiliations:
Department of Economics, University of California at Davis, USA;Department of Mathematics, Trent University, 1600 West Bank Drive, Peterborough, Ontario, K9J 7B8, Canada
Venue:
Discrete Applied Mathematics
Year:
2011

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Abstract

Judgement aggregation is a model of social choice where the space of social alternatives is the set of consistent truth-valuations ('judgements') on a family of logically interconnected propositions. It is well known that propositionwise majority voting can yield logically inconsistent judgements. We show that, for a variety of spaces, propositionwise majority voting can yield any possible judgement. By considering the geometry of sub-polytopes of the Hamming cube, we also estimate the number of voters required to achieve all possible judgements. These results generalize the classic results of McGarvey (1953) [13] and Stearns (1959) [22].