[EM] Approval & Condorcet are same (if approval-style ballots) but IRV isn't

[Warren Smith]

I can answer one of my recently-posted questions easily:

THEOREM:
Any Condorcet voting method with rank-order(equalities-permitted) ballots, if
ONLY approval-style ballots are fed to it (i.e. A=B>C=D=E, involving at most ONE
">" and the rest are "=", truncation not permitted) will deliver the
SAME winner as approval voting.   (We assume the absence of a perfect
tie.)

PROOF:
Consider two candidates A,B.   Possible votes on them are 11, 10, 01, 00
with respective counts x,y,z,t.   Then A is ahead of B in approvals if
and only if
y>z.   And A is pairwise ahead of B if and only if y>z.   So the approval winner
is automatically a Condorcet (pairwise-beats-every) winner.
Q.E.D.

However

THEOREM
In instant runoff voting with rank-order(equalities-permitted) ballots, if
ONLY approval-style ballots are fed to it (i.e. A=B>C=D=E, involving at most ONE
">" and the rest are "=", truncation not permitted) under the rules that
a ballot ranking K candidates coequal top is regarded as 1/K votes for each
and the candidate with the fewest top rank votes then is eliminated...
the winner will NOT necessary be the same as approval voting.
(We again assume there are no ties, not even at any IRV stage.)

PROOF
In this election
#voters   their vote
4  A=B=C=D>E
2  B>all
2  C>all
2  D>all
3  E>all
3  A=E>B=C=D
A is the approval winner with 4+3=7 approvals, but
has only 4/4 + 3/2 = 2.5 top-rank votes, which is the fewest, hence
A is eliminated in the first IRV round.
Q.E.D.

-- 
Warren D. Smith
http://RangeVoting.org  <-- add your endorsement (by clicking
"endorse" as 1st step)
and
math.temple.edu/~wds/homepage/works.html