Editing Arrow's impossibility theorem (section)

==== Part one: There is a pivotal voter for A vs. B ====
[[File:Diagram_for_part_one_of_Arrow's_Impossibility_Theorem.svg|right|thumb|Part one: Successively move '''B''' from the bottom to the top of voters' ballots. The voter whose change results in '''B''' being ranked over '''A''' is the ''pivotal voter for'' '''B''' ''over'' '''A'''.]]
Consider the situation where everyone prefers '''A''' to '''B''', and everyone also prefers '''C''' to '''B'''. By unanimity, society must also prefer both '''A''' and '''C''' to '''B'''. Call this situation ''profile[0, x]''.

On the other hand, if everyone preferred '''B''' to everything else, then society would have to prefer '''B''' to everything else by unanimity. Now arrange all the voters in some arbitrary but fixed order, and for each ''i'' let ''profile i'' be the same as ''profile 0'', but move '''B''' to the top of the ballots for voters 1 through ''i''. So ''profile 1'' has '''B''' at the top of the ballot for voter 1, but not for any of the others. ''Profile 2'' has '''B''' at the top for voters 1 and 2, but no others, and so on.

Since '''B''' eventually moves to the top of the societal preference as the profile number increases, there must be some profile, number ''k'', for which '''B''' ''first'' moves ''above'' '''A''' in the societal rank. We call the voter ''k'' whose ballot change causes this to happen the ''pivotal voter for '''B''' over '''A'''''. Note that the pivotal voter for '''B''' over '''A''' is not, [[A priori knowledge|a priori]], the same as the pivotal voter for '''A''' over '''B'''. In part three of the proof we will show that these do turn out to be the same.

Also note that by IIA the same argument applies if ''profile 0'' is any profile in which '''A''' is ranked above '''B''' by every voter, and the pivotal voter for '''B''' over '''A''' will still be voter ''k''. We will use this observation below.