<p>No, single-peakedness is not necessary for majority rule to be transitive.  For example, "single-dipped" preference profiles (the vertical mirror image of "single-peaked" profiles) also produce transitive majority preferences.   A much-more general condition is "triple-wise value restriction", which Sen (1966) showed generates a transitive majority preference relation.  Here is a good summary of these and related concepts, with some references to the primary literature:</p>

<p><a href="https://plato.stanford.edu/entries/social-choice/#NonDicPreAggRul" rel="nofollow noreferrer">https://plato.stanford.edu/entries/social-choice/#NonDicPreAggRul</a></p>

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2019-11-17T11:06:18.710Z
2019-11-17T11:06:18.710Z
Marcus Pivato
<p>I understand that when individuals have single-peaked preferences, majority rule will be transitive and the choice set will be non-empty. So single peaked preferences is a sufficient condition.</p>

<p>Is it also necessary? If not, what other domain restriction can given majority rule a non empty choice set? </p>

<social-choice>
Is single-peaked preferences necessary for majority rule to be transitive and yield non-empty choice set?
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# Is single-peaked preferences necessary for majority rule to be transitive and yield non-empty choice set?

(Asked by: username_0 on 2019-11-17)

I understand that when individuals have single-peaked preferences, majority rule will be transitive and the choice set will be non-empty. So single peaked preferences is a sufficient condition.

Is it also necessary? If not, what other domain restriction can given majority rule a non empty choice set?

**Tags:** social-choice

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## 1 answers

### Answer by username_1 on 2019-11-17. Score: 2

No, single-peakedness is not necessary for majority rule to be transitive. For example, "single-dipped" preference profiles (the vertical mirror image of "single-peaked" profiles) also produce transitive majority preferences. A much-more general condition is "triple-wise value restriction", which Sen (1966) showed generates a transitive majority preference relation. Here is a good summary of these and related concepts, with some references to the primary literature:

<https://plato.stanford.edu/entries/social-choice/#NonDicPreAggRul>


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