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Let say the electorate consists of three segments of voters: 1, 2, and 3 with corresponding weak preference relations defined over the candidates. Let the preferences be given by --

Segment 1: Biden > Warren > Sanders; Biden > Sanders.

Segment 2: Warren > Sanders > Biden; Warren > Biden.

Segment 3: Sanders > Biden > Waren; Sanders > Warren.

Question: Derive the preference relation of the electorate, decided by majority rule. As a preferences of the electorate as a whole rational? Why or why not.

For this question, I feel like it would not be possible to derive a preference relation of the entire electorate because it seems that no one candidate is strongly/weakly preferred. Therefore, it would seem that the preferences of the entire electorate would not be rational because it does not satisfy the transitive and completeness principles. Would I be correct in thinking this way and that a net preferece relation may not be derived? Any help would be greatly appreciated. Thank you.

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Have you tried writing down your intuition? Maybe the preference relationship of the electorate as a whole is transitive, but not complete, or complete but not transitive, etc.

I would start, for example asking the electorate, which one is preferred Biden or Warren? 2 voters would prefer Biden and one would prefer warren, so using the majority rule, we can conclude that for the electorate as a whole: Biden $>_{elect}$ Warren. Keep going with all the pairs and conclude whether the preferences are complete or transitive.

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