[EM] When will Approval Voting defeat a majority candidate

[Bart Ingles]

Two points to consider:

(1) When examining actual ballots, if only one candidate has a majority,
that candidate will be the Approval winner.  In other words, Approval
Voting cannot fail to elect a first choice majority *as expressed in
actual ballots*.

(2) If concerned about sincere preferences rather than ballots-as-cast,
then any voting system can fail to elect the sincere first choice
majority.  A Hare (IRV) example:

          Voter acceptability scale (sincere)
     100        70                            0
     ------------------------------------------
26%   A                       B               C
23%   B                                      AC
51%   C          B                            A

Assuming the actual percentages are not known in advance, or only
approximately known, then the polls may predict a runoff with A and C in
a dead heat.  If so, then the C voters' expected outcome has an expected
utility of only 0.5 with sincere voting.  If instead they reverse the
order of their top two choices, they can guarantee that B wins with an
expected utility of 0.7 (assuming the group is cohesive enough to carry
out the strategy).  So the actual votes-as-cast will be:

26%  A  B
23%  B     
51%  B  C

(so of course the candidate with a majority of actual votes wins -- the
same as under approval voting).  I'm fairly sure you can show the same
with any voting system.  This may not be a common situation, but neither
is the Approval equivalent.



Alexander Small wrote:
> 
> 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:  [...]