[EM] IRV UUCC badexample outline

[MIKE OSSIPOFF]

EM list--

I found my March posting about IRV & an extended Myerson test
(extended to allow for more than 2 policy positions).

I was going to post it, because Blake asked for an IRV UUCC
failure example. But I decided that, because the wording of that
message is about Myerson's test, it would be better to just
state the badexample outline. Then, tonight or tomorrow, I'll
put numbers to the outline, making it a specific concrete
example. But it's still useful as-is, as an outline.

But first, maybe my definition of UUCC is a little to brief for
clarity. Let me write it a little more completely:

No one should be able to contrive a configuration of candidates,
voters, voters' sincere ratings, and voters' actual votes, in
which everyone prefers X to Xc, Xc wins, and if one voter
changed his ballot so as to no longer vote Xc over X, that would
cause the election of someone whom that voter likes less than
Xc.

Example outline:

Four candidates: A, B, C, & Bc.

Though all that's necessary are the voters' sincere ratings of
the candidates, let's say that the candidates & voters are in
a 1-dimensional policy space, and that there are the A, B, & C
positions in that space, with B between A & C. Bc is at B's
policy space position, differing from B only by having a
nonpositional disutility, equal for all voters, in addition to
the disutility that he might have to the various voters due to
his policy position. Bc's nonpositional disutility could be,
for instance, the result of his corruption, which is why I call
him Bc.

B's corruption disutility is less than the utility difference
between the B & C policy positions, as judged by all the voters.

The B position is closer to the C position than to the A position.

The combined number of voters at the A & B positions is greater
than the number at the C position.

The combined number of voters at the B position and the C position
is greater than the number at the A position.

There are more voters at the A position than at the B position.

There are more voters at the C position than at the B position.

Many voters at the A & B positions vote Bc in 1st place (in IRV).

The number of voters at the B policy position plus the number of
A-position voters who vote for Bc in 1st place is greater than the
number of A-position voters who don't vote Bc in 1st place.

Because some A-position & B-position voters vote for Bc, and
some don't, say B & Bc are the 2 candidates with fewest 1st choice
votes. B & Bc are the "frontrunners" for immediate elimination.

About equal percentages of A-position voters & B-position voters
rank Bc in 1st place.

Say Bc has one more 1st choice vote than B has. B gets eliminated
in the first IRV round. Because A & B voters like Bc better than
Bc (because Bc's corruption disutility is less than the utility
difference between the B & C positions), Bc's votes transfer to B.

Now Bc has more votes than A does (because the number of B-position
voters plus the number of A-position voters who voted Bc 1st is
greater than the number of A-position voters who didn't vote Bc
1st). A has fewest votes, and gets eliminated, transfering to Bc
who now has more votes than the only other remaining candidate, C.
Bc wins.

Then one B-position voter who'd voted Bc 1st & B 2nd changes
his ballot to rank B 1st & Bc 2nd. Now Bc gets eliminated 1st,
and his votes transfer back to A & B, and now B has fewest votes.

B gets eliminated, and his votes transfer to C, since the B
position is closer to the C position than to the A position.

So C wins.

When that voter changed his ballot so as to not vote Bc over B,
he caused C to win instead of Bc. As a voter at the B position,
he likes Bc better than C, because Bc's corruption disutility is
greater than the utility difference between the B & C positions,'
for all voters.

IRVies, so far as I'm aware, have never agreed to require that
voters be allowed to rank candidates equal, and so anyone not
ranking Bc over B has to rank B over Bc.

So IRV fails UUCC.

Now, tomorrow, or within a few days, I'll post the same example
with actual numbers. But the outline shows the failure.

Someone might say that if the A voters who rank Bc 1st also rank
B 2nd, instead of A, that would save B from elimination, and
C wouldn't win. Fine, but all it takes to show a method's failure
of a criterion is to show _one_ example where it fails. My
failure example is one in which some A voters vote ABBc, and
some vote BcAB. That isn't an implausible example. It isn't
implausible to say that A voters vote A in 1st or 2nd place and
not in 2nd to last place.

Say it's an A-position voter instead of a B-position voter who
reverses the positions of Bc & B, so that B is over Bc. Again,
Bc gets eliminated, giving his votes back to A & B, and then C
wins, as before.

Mike Ossipoff







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