[EM] Re: Alex: Nash Equilibrium

[MIKE OSSIPOFF]

Alex--

You wrote:

It's been a while since I made that post, but as I recall I felt that your 
definition was too vague because it contained the phrase "no set of voters" 
rather than the more restrictive phrase "No set of voters with identical 
ordinal preferences."

I reply:

Some definitions use more restrictive language than others. But that isn't 
what "vague" means.

So a you could more accurately have said, "The voting equilibrium that 
you've defined is different from the one that I have defined".

In general, a definition is vague if it could mean more than one thing, or 
if it's in doubt whether or not some particular thing fits that definition.

In particular, from that meaning, the definition of a criterioin is vague if 
it can't be definitely established whether some particular method passes 
that criterion. And, also from that meaning, the definition of an 
equilibrium is vague if it can't be established whether or not a particular 
outcome is an equilibrium of the kind defined.

Now, you forgot to tell us what two different meanings my definition could 
have. Or what outcome can't be determined to be, or not be, a voting Nash 
equilibrium, as I defined that term.

Or are we a being a little vague about what we mean by "vague"? :-)

You continued:

Using the phrase "no set of voters" rather than "no set of voters with 
identical preferences" is too vague and can lead to situations with no Nash 
equilbrium.  T
I reply:

Here you repeat your claim about vagueness, and I refer you to my reply to 
it, above.

As for situations with no Nash equilibrium, it's easier for a situation to 
be an equilibrium by your definition, because you restrict what voters are 
referred to. That means only that your definition, and your equilibrium, is 
different.

Could there be times when your more-easily-found equilibrium suits a problem 
or project better? Sure. And there could just as easily be times when it 
would suit as well. That's the advantage of there being more than one 
equilibrilium, you know.

We have 2 equilibria that are extensions of Nash equilibrium now, and we 
have two names for them. Hey, that works out just right! :-)

This may come as a surprise to you, but game theorists actually use even 
more than two kinds of equilibrium. In fact, they have a whole taxonomy of 
them. If you can just forgive and tolerate there being more than one :-)

You continued:

his would violate Nash's Nobel-winning discovery that (with suitable 
definitions) there are always equilibria.

I reply:

"Violate a discover"y. It isnt entirely clear what it means to "violate a 
discovery".  I'm not claiming that there isn't a Nash equilibrium, as Nash 
defined it, for individuals chaning their strategy alone, under the 
conditions under which Nash said that there's always such an equilibrium.

If trhere isn't always a voting Nash equilibrium, as I've defined it, that 
doesn't violate Nash's discover, which is about the equilibrium that he 
defined, for idivuals chaning their strategy alone.

Nash spoke of situations that no one individual can improve (for himself) by 
changing his strategy alone.

I speak of situations that no set of voters can improve for themselves by 
changing their vote if no one else does.

That sounds like a reasonalbe, natural, and obvious extension of Nash 
equilibrium to voting.

It's an obvious generalization of Nash equilibrium as Nash defined it.

Yes it's more general than your equilibrium definition.

A more complete, but wordier statement of my definition would be:

A voting Nash equilibrium is an outcome such that no set of voters, by 
changing their vote if no one else does, can change that outcome to make an 
outcome that all of them prefer to it.

You continued:

Suppose that we use Approval Voting.  If we don't define the factions 
carefully, and we have a Condorcet Cycle, then it will always be the case 
that a majority of the voters prefer some candidate to the current winner, 
and acting in concert that majority could elect somebody that it prefers to 
the current winner simply by approving that preferred candidate and no other 
candidate.  (They might have to vote insincerely to do it, but the point is 
that they could do it.)

I reply:

As I said, there will be outcomes that are Nash equilibria as you define it 
for voting, and not as I define it for voting. That's ok.

And there may be situations where there's an equilibrium of the kind that 
you derfine, but there isn't a voting Nash equilibrium. That's ok too.

You continued:

I'll give my definition of Nash equilibrium in elections, and a rationale 
for why I prefer that definition to various other possible definitions.

I reply:

You seem to believe that there can be only one Nash-derived voting 
equilbrium, and that this town isn't big enough for your Nash-derived voting 
equilibrilum and mine. Actually, as I said, game theorists use even more 
than two kinds of equilibrium. I aplogize if telling you that makes you 
angry about or bothered by the use of more than one Nash-dereived voting 
equilibrium
by game theorists. :-)

You continued:

Also, notice that at no point did I make reference to whether or not the 
voting method uses ranked ballots.

I reply:

But could you have been expected to? Why should a definition of a voting 
equilibrium specify what kind of balloting is used?

You continued:

Even in non-ranked methods (e.g. Approval Voting) your strategic incentives 
depend only on your ordinal preferences, not your cardinal preferences.

I reply:

Wrong. Cardinal ratings have a large part in Approval strategy, even though 
whether you'd prefer A winning to B winning only depends on your ordinal 
prefernce between them.

By the way, I've always heard "preference" used in the ordinal sense. You 
prefer Smith to Jones. For cardinality, we speak of ratings or utilities, 
etc.
Maybe you can find a reference somewhere in which someone spoke of cardinal 
preferences, but that isn't how we've been using "preference" on EM.

You continued:

If you prefer A to B, changing your strategy to elect A rather than B is 
always in your best interests, regardless of whether you think the 
difference between the candidates is minor or huge.

I reply:

But you often don't know whether denying an Approval vote to B will let C 
win instead of B, when you like B better than C. In situations like that, 
your cardinal utility ratings of A, B, & C affect your Approval strategy.

But, yes, with only 2 candidates, your statement is true.

You continued:

Above is my definition, and if it matches yours then I guess we agree on 
everything.

I reply:

No your definition doesn't match mine, but I wasn't aware that that meant 
that we disagree on something. It isn't entirely clear what we disagree on,  
because it isn't clear what you're saying.

If you're saying that there can be only one correct definition of a 
Nash-derived voting equilibrium, then I disagree with you. You're wrong.

A little history:

  Steve & Alex have called the definition of voting Nash equilibrium "Mike's 
definition".  Maybe I should let you believe that, and take credit for being 
the first to define it. But I'm going to be honest, and admit that Blake 
first defiined it on EM.

One person, who shall remain anonymous, because his correspondence was 
offline, wants to call voting Nash equilibrium "Ossipoff equililbrium". As 
much as I'd like to agree with that name for it, I'm going to be honest and 
admit that Blake first defined it on EM.

  I had earlier defined a different Nash-equilibrium-derivative for voting, 
one that somewhat resembles yours. Then later I found that the one that 
Blake defined worked better for what I was using it for. Of course it's also 
much more general.

Because of its generality,  voting Nash equilibrium is the equilibrium that 
most deserves to be called the obvious extension of Nash equilibrium to 
voting.

But, Alex, do you really have nothing better to do than to jealously try to 
establish that your kind of Nash-derived voting equilibrium should be the 
only one?

Mike Ossipoff

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