[EM] IRV with extended test

[MIKE OSSIPOFF]

EM list--

With IRV, unlike Approval, it's necessary to describe a specific
situation. We want a configuration of conditions where a
corrupt candidate wins, and if anyone who helped make him win
unilaterally decided not to, it would worsen his expectation.

***

Say there are 4 candidates, A, B, C, & Bc. Bc is the corrupt
counterpart of B, sharng B's policy position. So these candidates
are at policy positions A, B, & C.

These positions are in a 1-dimensional policy space. Utility
has the obviously natural property that, since position B is
between positions A & C, an A voter likes position B better than
position C.

Bc's corruption disutility is less than the positional difference
between the utilities of positions B & C, as judged by all the
voters.

***

When I say something is "expected", I mean that it's considered
almost a sure thing.

The combined support of the A & B positions is expected to be
more than that of position C.

The support of position A is expected to be greater than that
of position B.

***

Because Bc is a longstanding incumbant, of a traditional old
party, a party where people of the A & B persuasions have
traditionally been coming together for a long time, it's expected
that lots of A & B position voters will vote Bc in 1st place
(That's explained by saying that they've had an early bad
experience, after the switch to IRV, where someone of C's party
won because they didn't do that).

The B voters plus those A voters who vote Bc 1st are expected
to be more numerous than the A voters who don't vote Bc 1st.

But it's also expected that, since A & B have a definite appeal,
the vote will be split (as it has tended to be), and that
therefore, B & Bc will be the candidates with fewest 1st choice
votes. B & Bc are expected to be the "frontrunners" for immediate
elimination.

About equal percentages of the A & B voters are expected to
rank Bc in 1st place.

***

These are plausible ordinary conditions.

***

When Bc wins (by assumption), or would win if one of those
A or B voters didn't change his mind at the last minute and
unilaterally withdraw his 1st choice support for Bc, and
one of those voters does withdraw it, that worsens the
utility expectation for that voter.

First, the way Bc wins is: Bc has one more vote than B does
(because we're talking about a situation where 1 A or B voter
changing his vote can change the outcome). B gets eliminated, and,
because Bc's corruption disutility is less for B voters than
the disutility of other policy positions, B's votes transfer to
Bc as 2nd choice. Now Bc has more votes than A, and A has fewest,
and his votes transfer to Bc, who now has more than C.

But if one B position voter moves Bc from 1st place to 2nd place,
putting B in 1st place, that reverses things, and now Bc gets
eliminated first. His votes transfer back to A & B, and
B now has fewest votes, and gets eliminated. I forgot to say
that C position is closer to B position than A position is.
When B is eliminated, his votes transfer to C, who wins.

You might say that maybe the A voters who rank Bc 1st might
rank B 2nd, for the same strategic reason, and that that would
save B from elimination. But it's far from obvious that they
would, and the person making up the example can say they don't
or that they aren't expected to.

The same thing happens if one of the A voters replaces B with
Bc in 1st place: Again Bc gets eliminated, giving votes back to
A & B, and B then gets eliminated, transferring to C.

If an A voter puts A in 1st place, where he formerly had Bc in
1st place, then B & Bc now are tied for elimination. There's
a 1/2 chance that B will win, and a 1/2 chance that C will win.
Previous to that change, C would have won. That A voter too has
worsened his expectation by withdrawing his giveaway to Bc.

***

So if any one of those voters who ranked Bc 1st decides unilaterally
to change his vote at the last minute, he worsens his utility
expectation. Even though the situation isn't a Nash equilibrium,
it fully makes those A & B voters afraid to be the one to
defect from the good old boy incumbant of the old corrupt
traditional party.

It isn't a Nash equilibrium because, if a C voter gives B 1st
place position, and it changes the outcome, it makes B & Bc
tie. 1/2 probability that Bc wins, and 1/2 probability that C
wins. The C voter, then, improves his expectation by changing
his strategy, and so the situation isn't a Nash equilibrium.
But surely what we're concerned about is that the A & B voters
are afraid to withdraw their Bc giveaway. Not only is C voters'
offensive order-reversal incentive not a mitigation of IRV's
failure there, but it's an additional bad thing about IRV.
So I claim that a Nash equilibrium where a corrupt candidate
wins is really too demanding a requirement for a failure. That
IRV situation is a failure.

***

Mike Ossipoff
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