[EM] equlibrium issues

[MIKE OSSIPOFF]

Alex pointed out:

If the players are
individual voters than any result with a margin greater than 1 vote is a
Nash equilibrium. That isn't a very useful result, however.

I reply:

Yes, that seemed a bit unfair to me. Surely not the intent of the
idea of the Nash equilibrium.

In my example for wv & margins, the players were the 3 factions.
It seems that the original Nash equilibrium definition is for when
there are a few players each with different interest and strategies.

I suggest that that intent is best served, for voting systems, with
a different wording. No doubt someone has already written such a
definition, but until I find out what he called it, I'll call it

Many Voter Equilibrium:

A strategies-configuration & outcome from which no set of voters
whose utilities and strategy are the same can gain by changing their
strategy.

[end of definition]

Again, I make no claim that that's original.

I understand that game theorists have defined many kinds of equilibrium.
Maybe that's one of them. Maybe some of the others will turn out to
have relevance to voting systems too.

>From now on, when I say equilibrium, I mean many-voter equilibrium.

Alex continued:

It could still turn out that in a 3-way race all Nash equilibria elect the
CW. However, I haven't proven it.

I reply:

I've been wondering the same thing. I read that Riker proved that
when voters have complete information about eachother's preferences,
and act to optimize their immediate outcome, the sincere CW will
win, no matter what (nonprobabilistic?) voting system is used.

If so, does that mean that the sincere CW is the only candidate who
can win at equilibrium?

This subject is new to me, but it seems important for evaluating
voting systems.

How about this classification of voting systems' sincerity:

(Again, by equilibrium, I mean many-voters equilibrium)

If there's a sincere CW, and voters have complete information about
eachothers' preferences then:

A voting system that has situations where the only equilibria have
order-reversal is a falsifying method.

Any method that isn't falsifying is nonfalsifying.

A voting system in which there are always equilibria in which no
one reverses a preference or votes a less-liked candidate equal to
a candidate he's voted for is an expressive method.

A voting system in which, if falsification of preferences is
ruled out as a strategy, there are always eqilibria in which
everyone sincerely ranks all of the candidates is a
conditionally completely expressive method.

[end of definitions]

Since, as I said, this subject is new to me, all I can say now is
that Plurality, IRV, RP(m), & BeatpathWinner(m) are falsifying methods;
and Approval, PC(wv), BeatpathWinner(wv), CSSD(wv), & RP(wv) are
nonfalsifying methods.

The wv Condorcet versions are expressive and conditionally
completely expressive.

Of course, with anything new (to me) there's always the possibility
of an erroneous statement, or something overlooked when writing
a definition. But right now, the definitions seem good, and the
statements seem correct.

Mike Ossipoff



Alex




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