[EM] Problems with finding the probable best governor

[MIKE OSSIPOFF]

Dear Markus--

You wrote:

>There seems to me to be a discrepancy about what the SUE is.
>
>As far as I remember correctly, the social utility expectation
>of a given candidate is the sum of the von Neumann-Morgenstern
>utilities of the voters about this candidate.

You mean his vN-M utility. Sure. But a the voter also
has _sincere_ utility ratings for the candidates.

If the method that we're going to use in some future election
has a good average _sincere_ SU, then your sincere utility
expectation for that future election is good.

Of course, for the purpose of calculating voting strategy, the
voter wants to maximize his utility expectation for the election,
but that's another subject.

vN-M utility is sometimes used for calculating one's best
voting strategy. Not always, though. For instance, I just start
with the fact that I want to maximize my _sincere_ utility
expectation, and that means that my sincere utilities for the
candidates are also my vN-M utilities for them.

But vN-M utility isn't what we want for SU. We're interested
, for a particular election outcome, in the combined or average
_sincere_ utility of the winning candidate, combined or averaged
over all the voters. And, for a particular method, we're interested
in its average sincere SU, averaged over many elections.
By sincere SU, I mean the combined or average sincere utility
of the winner, combined or averaged over all the voters.

vN-M utilities are for the strategy calculations of a voter
who doesn't necessarily want to maximize his sincere utility
expectation. Maybe he's a little risk-averse, for instance, and
a chance of electing a candidate with sincere utility of +U
isn't enough to justify an equal chance of electing someone with
sincere utility of -U. So he calculates his vN-M utilities based
on how he values lotteries among the candidates. But I feel that
if a voter is really rational about his strategy, he'll just want
to maximize his sincere utility, making it unnecessary to bother
with determining vN-M utilities.

Well, it's just occurred to me that a person might also
be risk-averse when choosing a voting system too. In that case,
for that person, maybe vN-M utility would be more meaningful,
even for the problem of choosing a voting system. But I believe
that it's the universal practice to go by sincere utility,
when comparing voting systems by their average SU. For one thing,
of course, determining or arbitrarily assigning vN-M utilities
for all the candidates, for each voter, is would be a problem
and a nuisance. Anyway, isn't there a strong case for saying that
what you really want, when it comes to SU, is a voting system
that maximizes your _sincere_ utility expectation for that
future election? If you suggest using vN-M utility for determining
average SU, then should we estimate, what the statistical
distribution of such things as risk-aversion are, and assign
those attributes to the voters in the simulation? Maybe that would
be for if we're using the simulation to show the public as
a whole which method is overall best for them. But what if you're
trying to convince a particular person, Joe, which method is best?
Then you use Joe's vN-M utilities. Maybe find out the
personal characteristics of the people whom you're trying to
convince, and find out how they value various kinds of risks
& lotteries, and then use that information to calculate the
simulated voters' vN-M utilities for the candidates, based on
their sincere utilities and your personal information about
the people you're trying to convince, and your assumption that
the simulated voters have the same attributes as the people whom
you're trying to convince. A lot of trouble, and not usually
worth it. That's why sincere utility is always what's used
for comparing methods average SU.

***

Even if I thought that, in November 2000, I'd maximize
my sincere utility by voting for Gore instead of Nader, I
would still vote for Nader, because in our elections the
candidates can usually be divided into acceptables & unacceptables--
sometimes known as "sleazes".

So for me, it's "dichotomous preferences" pretty much in our
elections. If there were several acceptables, with Plurality,
I'd vote for whichever acceptable has the best chance of winning.
With Approval, I'd vote for all of the acceptables.
With Condorcet, I'd rank only the acceptables, as a matter of
principle, and to send a message.

Under better conditions, though, I'd vote to maximize my
sincere utility expectation, though I don't do that in our
elections now.

>
>But the von Neumann-Morgenstern utilities are defined only on a
>relative scale and not on an absolute scale. Example: If voter

Sure, but do you want to try to estimate those for all the
simulated voters? It's easier to use sincere utilities,
calculated in the simulation, based on distances in issue-space.

>But in so far as the von Neumann-Morgenstern utilities are
>defined only on a relative scale and not on an absolute scale,
>it isn't feasible to say that a given candidate has a SUE of
>1/2 or 3/4 of the maximum SUE.

For candidates, you only designate an SU. A voting system's
average SUE is the average, for that method, over a large series
of elections, of:

(Um-Uav)/(Umax-Uav)

...where Um is the social utility of the winner by that method,
         Uav is the average SU of all the candidates,
         and Umax is the SU of the candidate in that election who
         has highest SU of all the candidates in that election.

Maybe SUE is a more helpful figure than Um/Umax, because it
calculates a ratio of how much that method improves on
RandomCandidate, compared to how much improvement is possible.
Instead of comparing the SU itself, it compares how much of it
_the method_ can take credit for as an improvement over random
selection of the winner.


>Also Mike's statement that the Condorcet winner usually has a
>high SUE seems to me to be not justified. Especially in a divided

I didn't say that the sincere CW (SCW) usually has a high SU
in an absolute sense, only that he has a higher SU than any
other candidate has.

I can demonstrate why, in one dimensional elections, the
sincere CW is the SU maximizer. At least I can demonstrate that
a candidate at the voter-median point is always the SU maximizer,
and is always the sincre CW. And that means that the sincere CW
is always the SU maximizer.

So that statement's accuracy isn't in doubt. I'll send a
demonstration of my claim in a subsequent message.

>society, the SUE of the Condorcet winner and even of the majority
>winner can be very small. The reason why I support the majority

I'm not sure how well my CW/SU statement extends to more
dimensions, but it's such a strong statement that one would
expect it to survive, to some extent, with more dimensions.
What is for sure is that, with one dimension, the sincere CW
is _always_ the SU maximizer.

>criterion (resp. the Condorcet criterion) is not that the
>majority winner (resp. the Condorcet winner) usually has a high
>SUE but that a method that meets the majority criterion (resp.
>the Condorcet criterion) is less manipulable.

That's true. But, at least in 1 dimension, the sincere CW
always has more SU than any other candidate, and surely that
remains true, to some significant degree, with more dimensions.
Even if, in 3 dimensions, the SCW isn't always the SU maximizer,
he's still most likely someone who does very well by SU. It's worth
checking to find out how well my CW/SU claim extends to
more dimensions.

By the way, it was Steve Eppley who told me that that statement
doesn't depend on the voters' distribution. At first I misunderstood
and thought that his demonstration contained an error, but
then I realized that it's as he says.

But I don't believe that the Condorcet criterion is enough.
Sure, Norm's simulation, & that of Merrill, show that any
method meeting the Condorcet Criterion does well by average SU,
under sincere voting. That, by the way, since it holds true
when the number of dimensions is increased, shows that the
CW/SU claim probably holds true pretty well under those conditions,
since it's the Condorcet Criterion methods that always choose
the sincere CW under sincere voting.

When information is available to the voters, some Condorcet
Criterion methods will do much more poorly than others.
Methods other than Schulze, and the wv Condorcet versions
fail to offer the majority rule and defensive strategy
guarantees that the better methods offer. The Condorcet Criterion
isn't enough.

Mike Ossipoff

p.s. Maybe my way of voting in the elections we have nowadays
, when it doesn't maximize my sincere utility expectation, could
be explained (alternatively) in terms of vN-M utilities.
I just explain it in terms of sleaze. A small chance of
electing Nader is worth more to me than a big chance of electing
Gore instead of Bush, even if, hypothetically, there were a
difference between those two.


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