It is well known that the majority rule may not be transitive for some configurations of individual preferences. Domain restrictions are possible ways out. But what is known about maximal such domains (i) with respect to the cardinality? (ii) via set inclusion?
A good starting point may be a paper by Dasgupta and Maskin, published in the Journal of the European Econ. Association. Vol. 6, 2008
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Abstract
We show that simple majority rule satisfies five standard and attractive axioms—the Pareto property, anonymity, neutrality, independence of irrelevant alternatives, and (generic) decisiveness—over a larger class of preference domains than (essentially) any other voting rule. Hence, in this sense, it is the most robust voting rule. This characterization of majority rule provides an alternative to that of May (1952). (JEL: D71)
[1] J. Abello and C. R. Johnson, How Large are Transitive Majority Domains?, SIAM J. Alg. Disc. Meth. Vol. 5(1984), pp. 603-618.
[2] J. Abello, The Weak Bruhat Order, Consistent Sets and Catalan Numbers, SIAM J. Alg. Disc. Meth. Vol. 1.(1991), pp. 1-16.
J. Abello, "The Majority Rule and Combinatorial Geometry (via the Symmetric Group)", Annales Du Lamsade, No. 3, pp. 1- 13, October 2004.
Wulf's reply holds only if one assumes that all voters vote and vote sincerely. Majority rule is infallible only in a one-dimensional policy space.
Dear professor William, do you mean that single-peakness guarantees the transitivity of the majority rule only in the case of a one dimensional policy space? I am not getting clearly what is the immediate link between the transitivity problem and sincere votes. Thank you for your reaction.
Yes, that's what I mean. If there are three or more voters and three or more options on the ballot, simple majority rule can exhibit "collective irrationality" (indecisiveness, vote "cycling" ) even if every voter is rational and his/her preferences are transitive.Single-peakness is necessary for Black's Median Voter Theorem to hold. That result tanslates in mutlidimensional policy space only if voter's preferences are "radially symmetric" -- a very strong assumption that is unlikely to hold in actual practice. Unless we have single peakness in one-dimensional space or radial symmetry in multidimensional space, anything can happen (McKelvey's "chaos theorem") under any voting rule. Chaos is magnified with strategic (insincere) voting under ALL voting rules, none of which been shown to be strategy-proof. Voting insincerely simply means that some voters vote for lower ranked options in order to avoid an even worse outcome.So, voters' preferences may be transitive, but those preferences are mis-represented at the ballot box for strategic reasons.
That is very instructive. Are there some references for further readings?
I have just seen this discussion by chance. Franz Dietrich and I have a recent paper on an analogue of your question in judgment-aggregation theory (concerning domain restrictions of varying strength under which majority voting on multiple, interconnected issues is consistent). Some of the questions we discuss were inspired by Dasgupta and Maskin's paper. Here is the link:
http://personal.lse.ac.uk/list/PDF-files/MajorityPaper.pdf
The paper was published in JET 145(2), 2010: 512-543
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