Approval discussion

[MIKE OSSIPOFF]

I'd said:

>Sometimes a number of same-voting voters change their strategy
>in the same way, or in a few same ways. Writing definitions that
>way, then, doesn't seem unrealistic or unreasonable. After all,
>when only one voter changes his strategy, there's rarely any result.

Richard replied:

I'm not disagreeing with the idea of defining concepts in terms of
deterministic strategies and/or block strategies, even though actual
voters will not use those strategies.

I reply:

But isn't it true that sometimes a same-voting group of voters
vote in a different way in a subsequent election? Definitions about
what could happen when a set of voters voted differently don't
go against any notion of how people vote.

And as for "deterministic", though it would be interesting to study
the game we spoke of, where voters choose their strategy based on
probability-info about the other voters, there's nothing unrealistic
about simply asking if an outcome could be improved on by some voters.

For the purpose of defining a game, it's necessary to say what the
voters know, and, for that purpose I said that they have complete
information about eachothers' preferences. But it isn't even necessary
to go into that question of what voters know in a game.

It's meaningful & useful to just speak of whether an outcome can
be improved on by anyone. If it can, then it's obviously unstable.

We can accurately say that without talking about what kind of a game
we propose, with voters having a certain kind of information.

But one thing that we could suppose about voters' information is that
it's based on past elections. Say, for instance, that the method
is Ranked-Pairs(margins), and we get this set of rankings:

100: ABC
49:  BAC
75:  CBA

This is known to voters, because these results are published after
the election.

In the next election these candidates (or maybe ones identical to them)
are running again.

Can anyone improve on that result above? It elects B. But if the A
voters order-reversed against B, A would win. Some voters can improve
on that result. Bloc voting? I've often voted based on the effect if
everyone having some political preferences in common with me voted the
same way. It's obvious that if you prefer A, you could do better.

So in that next election, the A voters order-reverse, and A wins.

What happens in the next election? The election that A won isn't
an equilibrium either. Someone can improve on it too. The C voters
can improve on it by voting BCA. So they do that, and B wins. But
now B wins at equilibrium. Now no set of same-voting voters can
improve on the outcome by changing their strategy in the same way.

So, when the system arrives at equilibrium, it's an equilibrium in
which many voters are burying their favorite, voting defensive
order-reversal strategy. In that, and many other common ordinary
situations, that's the only kind of equilibrium that the margins
methods have.


Blake's defintion of electoral equilibrium doesn't have as many
requirements as mine. Maybe at least some of my requirements aren't
needed; I don't know right now.


Richard continues:

But then we are no longer
talking about what the best strategy is based on information known
to an individual at voting time, but rather about what would have
been the best strategy for a block of voters had they known in
advance the actual ballots that would be cast by the other blocks.

I reply:

In my scenario above, they've just observed what happened when they
voted a certain way, and they notice that they can improve on the result.

Or it can be regarded as a game in which voters know eachother's
preferences, but not eachother's votes. Or it can be looked at as
a series of elections in which voters change their strategy to
improve on the outcome.

Or we could just say that certain strategy-configurations can
be improved on by someone, and certain ones can't. That, of itself,
is meaningful, useful information, without getting into the question
of exactly what voters know. An improvable outcome is unstable.


Richard continued:

It is interesting that your equilibrium-based definitions don't
really require voter utilities but only the ranked values, since
these definitions relate to whether a group of voters could have
gotten a better (for them) result than the actual result, which
only requires a knowledge of relative rankings. Probabilistic
strategies for Approval necessarily require utilities.

I reply:

True, maybe, if we look at it as a game in which voters have
complete knowledge about eachother, maybe it would be better to
say that they have complete information about eachother's utilities,
instead of just preferences. But didn't I say it that way earlier
a few times?

Richard continues:

Well, to be precise, I should have said that real-world Approval
voting won't have the same equilibria that are found modeling the
election with the block-vote, deterministic method. So if it finds
an equilibrium, it won't be what that model predicts.

I reply:

A strategy configuration and outcome that some voters can improve on
will be improved on, no matter what we say about bloc voting or
determinism. I've shown that with the margins methods, there are
many typical situations where all the outcomes that don't have
favorite-burial strategy are improvable by someone.

The meaning of equilibrium that I use when I say that in many
common typical situations Margins' only equilibria are ones that
include favorite-burial strategy by lots of people isn't a meaning
that requires special assumptions about bloc voting or
determinism. Either a strategy-configuration & outcome is improvable
by someone or it isn't, and that's all I'm talking about.

Richard continues:

However, if there are multiple equilibrium points, then unless
the voters (or voter blocks) know ahead of time which point the
method will converge on, they do not know which strategy to play.

I reply:

To use an example, the 100,49,75 example, there are several equilibria,
but all of them have the C voters, or many of them, voting BCA.
It doesn't matter which of those equilibria the system ends up at.
They're all undesirable; they all have widespread favorite-burial.

My point, you remember, was that in many situations, Margins'
only equilibria are ones with favorite-burial.

Richard continues:

Which incidentally leads back to probabilistic strategies. BTW,
unless I'm mistaken, those dice-rolling strategies also have
Nash equilibria, in the sense that each player has an optimum
way of setting the probabilities of selecting each of his
possible strategies.

I reply:

That's how I looked at it too. Voters can choose among probabilistic
& nonprobabilistic strategies.

I'm going to reply to the other half of the message tomorrow.

Mike Ossipoff


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