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Each member of a committee ranks applicants A, B, C in some order. It is given that the majority of the committee ranks A higher than B, and also that the majority of the commitee ranks B higher than C. Does it follow that the majority of the committee ranks A higher than C?

This question came into one of our test. But I have some doubts in the problem. First of all, let's say a committee member ranks A higher than B, and then he ranks B higher than C. Would it be automatically counted as that he ranked A higher than C? If yes, then it is possible that the majority of the committee ranks A higher than C right? Let's say 60% of the committee ranked A higher than B, and then the same 60% ranked B higher than C, then the majority indeed ranked A higher than C. However it is not necessary. Let's say that 60% ranked A higher than B , and another 55% ranked B higher than C, but of that 55% not all of them ranked A higher than B, hence not part of that 60%, in this case it's not necessary that the majority automatically ranked A higher than C right? Am I right on this argument?

轻型八神
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  • More simply, say we have exactly three voters, whose preferences are $A>B>C$ and $ B>C>A$ and $C>A>B$ respectively. Easy to verify that the assumptions are met, but that the voters prefer $C$ to $A$. – lulu Dec 06 '22 at 12:30
  • Usually in these problems the assumptions of Arrow's Theorem are meant to hold. In particular it is assumed that each individual's preferences are transitive. That is, if a given voter prefers $A$ to $B$ and $B$ to $C$ then that voter must prefer $A$ to $C$. The question is whether this transitivity extends to group voting. – lulu Dec 06 '22 at 12:50
  • @lulu yes this is what I assumed as well. Assuming it's transitive, my argument, it is then correct right? – 轻型八神 Dec 06 '22 at 13:09
  • Your argument is needlessly complex, as my first comment shows. I didn't read your argument carefully, but reading it quickly, you need to demonstrate that those numbers can be realized by a given list of (valid) voters' preferences. – lulu Dec 06 '22 at 13:12
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    So basically I need to provide an example of the scenario I'm talking about? – 轻型八神 Dec 06 '22 at 13:23
  • You need to show that the numbers you describe can be realized by a valid slate of voters' preferences. In my first comment I gave you all the voters' preferences. Easy to read off that $\frac 23$ prefer $A$ to $B$, $\frac 23$ prefer $B$ to $C$, but $\frac 23$ prefer $C$ to $A$, so group transitivity has failed. You could easily modify my example if you wanted to, but a single counterexample is all that was called for. – lulu Dec 06 '22 at 13:26
  • Yes, you should provide an example to prove that the scenario is possible. Your observations are correct in identifying a problem with a certain hypothetical proof of impossibility, but that doesn't prove that it's possible. – Karl Dec 06 '22 at 13:36

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