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Wikipedia claims for the proof of Arrow's Impossibility Theorem that when

  • Every voter in segment one ranks B above C and C above A.
  • Pivotal voter ranks B above A and A above C.
  • Every voter in segment two ranks A above B and B above C.

(B and C may switch for segments one and two but not pivot)

by IIA the societal outcome must rank A above C, as in the previous case

Why is this true?

Number Basher
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    The pivotal voter and segment two both rank A above C (using transitivity for segment two). – Henry May 10 '23 at 14:42

1 Answers1

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The given profile is defined such that A beats B, but if the pivotal voter switches their rank, now B beats A. Since A beats B in this profile and B beats C in the profile by unanimity, A must beat C by transitivity.

In the modified profile, the only thing that changes is where the voters rank B relative to A. By independence of irrelevant alternatives, this can't change the relative ranking of A and C. So A must still beat C in the modified profile.

eyeballfrog
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