[EM] Extend Myerson-Weber equilibrium to more methods?

[MIKE OSSIPOFF]

EM list--

In the _American Political Science Review_, vol. 87, No. 1,
page 103 (March 1993), Roger Myerson & Robert Weber published an
article as fascinating & compelling as Myerson's article on
corruption-encouragement. It's entitled "A atheory of Voting
Equilibria".

It's best to start with an example. Say the method is Plurality,
and the media only devote airtime & printspace to 2 parties, and
keep referring to those 2 parties as "The 2 choices". And say
people believe that, and vote accordingly. Of course everyone will
vote for whichever of those 2 is his favorite, or almost everyone
will. The result is easy to predict: Those 2 parties will be the
only ones to get any significant number of votes. That will confirm
the media's claim that they're the only viable choices. The electorate
could be stuck forever on a suboptimal outcome if those aren't really
the 2 most preferred parties. We'd never know.

Myerson & Weber call that a "voting equilibrium". They point out
that, with Plurality, _any_ 2 parties, as long as at least one of
them isn't in Condorcet loser position, can win at voting equilibrium,
if voters are led to believe that they're the frontrunners.

They also point out that, with Approval, if there's a candidate at the
voter-median position, he's the only candidate who can win at
voting equibrium.

A simplified definition of voting equilibrium might be:

An outcome, including the officially reported & recorded count
results, that is consistent with the prediction(s) that led voters
to vote as they did, producing that outcome.

It's assumed that everyone believes the same prediction(s), and
that everyone votes to get the best outcome for themselves.

For this simplified definition, I'd also add that the predictions
are certain enough that people feel certain that one candidate-pair
is the one that definitely should vote between, but not so certain
that the other candidate-pairs have no importance. (Of course with
all methods, one votes between more than 1 candidate pair, but
only one candidate pair will be the one that is in closest contention,
and the one for which you can affect the outcome, if you can affect
any outcome).

***

Voting Equilibrium Criterion (VEC (tentative)):

If candidates & voters are positioned in a 1-dimensional policy-space,
where voters prefer near candidates to farther ones, and if there's
a candidate at voter-median position, then he should be the only
candidate who can win at voting equilibrium.

***

Well that's the awkward part, about the certainty of the predictive
beliefs. That's the simplification.

Myerson & Weber deal with it in a detailed way, which involves the
use of candidates' "expected scores". In point systems, of course
scoring is simple, and the candidate with highest score wins.
They relate the Pij, the frontrunner probabilities, to the expected
scores of the candidates. I can't imagine how to apply that to
non-point systems, and so I've tried the simplification that I wrote
above. Let me know if you notice a problem with it. Especially, let
me know if you can improve it. If it seems unsound, then how can
it be written more soundly? And how does Condorcet do? My impression
is that Condorcet doesn't pass this one, because all rank methods
have so many ways things can happen that unintended results can
be possible.

I don't expect all methods to meet the same criteria, of course.
Approval meets some that Condorcet doesn't meet, and Condorcet meets
some that Approval doesn't meet. The important thing for me is that
both meet a number of important criteria, something that sets them
both above all the other methods.

Mike Ossipoff


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