[EM] Arrow's Theorem.

[Alex Small]

> The point is that if Plurality is covered by Arrow's Theorem, then so
> is Approval since if in an Approval election the voters only vote for  a
> single option, that election would be equivalent to a Plurality
> election.
> Therefore, if Plurality fails Arrow's Theorem, so would Approval.

In the formal derivations of Arrow's Theorem that I've seen, an election
method is defined as a mapping from the set of voter preferences to the
set of candidates.  Show me the preference order of each individual voter,
and (barring the case of ties) I'll show you who the winner is.  No
ambiguity.

Approval Voting is not a preferential method because the voters'
strategies cannot be uniquely determined from their preferences.  Show me
their preferences and I'll say "Well, if they all decide to just vote for
one then A wins.  If some of them vote for 2 candidates then B or C might
win.  If all vote for 2 candidates then C would win."

For Approval to be equivalent to plurality, we'd have to know that voters
would ALWAYS vote for just one.  But since voters are free to pick their
strategy, Approval is not equivalent to plurality, nor is it equivalent to
any other ranked method.

There is one important exception, proved in Brams and Fishburn:  Suppose
that every voter's preferences are binary:  Every voter lumps the
candidates into 2 groups, and every voter is truly, sincerely indifferent
among the candidates in that group, i.e. their preferences are of the form

c1=c2=...cn>cn+1=cn+2=...cz

Then there is always a candidate who defeats all other candidates pairwise
(barring the case of ties, which some careful mathematics can show to be
almost irrelevant).  Brams and Fishburn use this result in their book to
prove that, in such a case, Approval always picks the CW.

More important is that the case of binary preferences is the ONLY case
where voters' Approval strategies are uniquely determined by their
preference orders.  In that case, Approval is equivalent to a ranked
method, and it also satisfies IIAC.  You might be thinking "Arrow was
wrong!" but every proof I've ever seen of Arrow's Theorem assumed strict
preference orders (i.e. nobody is indifferent between two candidates).  So
that narrow case doesn't satisfy Arrow's assumptions.



Alex