[EM] Myerson's Corruption Test--Borda fails big.

[MIKE OSSIPOFF]

EM list--

Roger Myerson described an interesting & revealing test
of how much a method encourages corruption.

I'll describe it as well as I can, but if there's anything wrong
with this, it's my error only.

The problem is that if some corrupt incumbant is the traditional
coming-together point for voters of a certain issue position,
then they might keep electing him for fear that individual
desertion will result in the election of someone with a worse
issue position.

Say there are just 2 issue positions. There's a Y/N issue.
Each party publishes their position on that issue. Some parties
are corrupt.

For every corrupt candidate there's at least one uncorrupt one
at the same issue positon.

Every candidate of a party shares that party's Y/N position and
its corruption level. Every voter knows those positions &
corruption levels.

If a Yes candidate wins, then that counts as a utility of 1
for "Yes" voters, and as -1 for "No" voters. If a "No" candidate
wins, that counts as zero for all voters.

Additionally, if a corrupt candidate wins, that adds a
negative utility for every voter, equal for every voter,
depending on that candidate's corruption level.

***

A "Nash" equilibrium is an outcome in which no person can
improve his utility expectation by unilaterally changing his
strategy. (i.e. By changing his strategy if no one else does).

***

A method is completely effective against corruption if there's
no situation that has a Nash equilibrium where a corrupt
candidate wins.

A method is partly effective if there's no situation that doesn't
have a Nash equilibrium where an uncorrupt candidtate wins.

A method is completely ineffective if there's a situation that
doesn't have a Nash equilibrium where an uncorrupt candidate wins.

***

Approval is completely effective.

Plurality (FPTP) is partly effective.

Borda is completely ineffective.

***

I was mistaken the other day when I said that because Borda
will never be enacted, that means that it isn't worthwhile to
argue against it. Borda's threat isn't that it will be enacted.
Its threat is that it will be used to muddy the water.
Every silly method, along with its wild & wacky justsification,
will eventually be trotted out when single-winner reform is
proposed.

Some people will then say "Hey, even the professors disagree,
so let's not change the voting system."

That's why it's desirable to shoot Borda down before it even
gets close, like now when it's advocated in an international
magazine. And when Saari or someone drags it into a sw reform
discussion, that's why it's desirable to dispose of it as quickly
as possible, before it can discourage people about single-winner
reform.

***

Now you might say that it isn't realistic, a model where there's
just a Y/N issue. Usually there are more than 2 issue positions.
Sure, and that means that passing that test doesn't mean much.
But failing it means plenty. It means that Plurality, and
especially Borda, fail so easily that it can happen even with
just 2 issue positions.

***

And so, whenever Borda is advocated, with Saari calling it
"BC" (Borda Count), then it should be pointed out to people that
BC stands for "Best for Corruption".

***

With more than 2 issue positions, I don't know if any method
passes strictly. But Approval fails only if we assume some
improbably inconsistent frontrunner probabilities. The conditions
needed for Approval to fail, with more than 2 issue positions,
have been said by some authors to be so unreasonable that they
shouldn't be considered in strategy calculations.

With 3 issue positions, IRV fails for practical purposes.
It isn't a Nash equilibrium, but it's a situation where the
people who upranked the corrupt candidate to keep a worse
issue position from winning can only worsen their utility
expectation by withdrawing that giveaway to the corrupt
candidate. That's really all that's needed to make a practical
failure. What keeps that situation from being a Nash equilibrium
is that the supporters of someone at that worse issue position
have incentive to try to steal the election by offensive order-
reversal of the kind well-known in IRV. I'm not saying
offensive order-reversal is likely, but I'm saying that that
incentive hardly counts in IRV's favor.

This has been said sketchily. More detail tomorrow when there's
more time.

Mike Ossipoff
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