Editing Arrow's impossibility theorem (section)

==== Part two: The pivotal voter for B over A is a dictator for B over C ====
In this part of the argument we refer to voter ''k'', the pivotal voter for '''B''' over '''A''', as the ''pivotal voter'' for simplicity. We will show that the pivotal voter dictates society's decision for '''B''' over '''C'''. That is, we show that no matter how the rest of society votes, if ''pivotal voter'' ranks '''B''' over '''C''', then that is the societal outcome. Note again that the dictator for '''B''' over '''C''' is not a priori the same as that for '''C''' over '''B'''. In part three of the proof we will see that these turn out to be the same too.
[[File:Diagram_for_part_two_of_Arrow's_Impossibility_Theorem.svg|right|thumb|Part two: Switching '''A''' and '''B''' on the ballot of voter ''k'' causes the same switch to the societal outcome, by part one of the argument. Making any or all of the indicated switches to the other ballots has no effect on the outcome.]]
In the following, we call voters 1 through ''k − 1'', ''segment one'', and voters ''k + 1'' through ''N'', ''segment two''. To begin, suppose that the ballots are as follows:

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

Then by the argument in part one (and the last observation in that part), the societal outcome must rank '''A''' above '''B'''. This is because, except for a repositioning of '''C''', this profile is the same as ''profile k − 1'' from part one. Furthermore, by unanimity the societal outcome must rank '''B''' above '''C'''. Therefore, we know the outcome in this case completely.

Now suppose that pivotal voter moves '''B''' above '''A''', but keeps '''C''' in the same position and imagine that any number (even all!) of the other voters change their ballots to move '''B''' below '''C''', without changing the position of '''A'''. Then aside from a repositioning of '''C''' this is the same as ''profile k'' from part one and hence the societal outcome ranks '''B''' above '''A'''. Furthermore, by IIA the societal outcome must rank '''A''' above '''C''', as in the previous case. In particular, the societal outcome ranks '''B''' above '''C''', even though Pivotal Voter may have been the ''only'' voter to rank '''B''' above '''C'''. [[Condorcet paradox|By]] IIA, this conclusion holds independently of how '''A''' is positioned on the ballots, so pivotal voter is a dictator for '''B''' over '''C'''.