[EM] When will Approval Voting defeat a majority candidate

[Alexander Small]

The fact that Approval Voting can fail to elect somebody who is the first
choice of a majority is often cited as a defect.  The circumstances under
which this can happen, however, are rare and worth examining, at least in
a race with 3 candidates:

Suppose that candidates A and B have wide support, and C is polarizing
(i.e. either love him or hate him).  Suppose that a very narrow majority
supports C as their first choice, and everybody else considers him their
last choice, and that the election will be held with Approval Voting.

If all of the B supporters give a "crossover" vote to A, in hopes of
defeating C, C will still win if C's supporters have access to polling
data and see that C has a majority.  They know that their candidate will
win so they have no need to back any other horses in the race.  Even if
everybody who doesn't support C votes for both A and B, C will still win
if C's camp is faithful and has access .

If, however, the race is exceptionally close, well within the margin of
error of the polls, so it isn't clear which of the three will win, then
and only then will supporters of the majority candidate (C) have reason
to consider casting an additional vote for whichever of A and B they
consider the "lesser evil."

3 things to note:
1)  Supporters of A and B had to unite to defeat C, putting aside loyalty
to their own candidate in favor of beating C at any cost.  C probably did
something to anger the other factions of the electorate.  Admittedly,
this is a political judgement, not a technical judgement, but elections
have both political and technical aspects.

2)  Victory due to inaccurate polling data may seem cheap and unfair, but
C's majority had to be very slim in order for statistical error to work
against him.  We aren't talking about a resounding first choice losing
the election (although that admittedly does not matter in a technical
sense).

3)  C's supporters _consented_ to the victory of somebody other than C.
The essence of democracy is "consent of the governed."

4)  C's supporters would never give cross-over votes if it were only a
two- way race, no matter how close it might be.  The opposition had to
unite _and_ offer two choices to C's supporters.  C's supporters had the
upper hand here.  If they considered both A and B to be equally bad they
would have refused to give an additional vote to one of them, and C would
still win.

5)  C lost to somebody with an even larger majority of votes.

In other words, majority candidate will only lose if their margin is
slim, the opposition is united, and the opposition offers a reasonable
alternative.  The "majority loser" objection needs to be put in
perspective as a rare occurence with mitigating political circumstances.