[EM] Problems with finding the probable best governor
Bart Ingles
bartman at netgate.net
Sat Jul 22 21:30:40 PDT 2000
Markus Schulze wrote:
>
> Dear Bart,
> dear Mike,
>
> Bart wrote (20 July 2000):
> > My position was that while I didn't consider it necessary to always
> > maximize SUE, I did think it important to avoid electing candidates
> > with a very low SUE. If I were to try to put it into the form of a
> > criterion today, I would say that the winner's SUE should never be
> > lower than about 1/2 or 3/4 of the maximum SUE for a given election.
> > Satisfying that, Condorcet or Smith Set would be next in importance
> > (at least in terms of outcome-based criteria). Lower priority still
> > might be to maximize SUE among multiple Smith Set members.
>
> Mike wrote (20 July 2000):
> > I don't think there's need for concern about SU vs CW. It
> > seems to me that our partisan political elections are close
> > to being 1-dimensional, though not entirely. Under those
> > conditions there's always a sincere CW, who will also be the
> > SU maximizer, if the voters' distribution density increases with
> > decreasing distance from the voter-median position. Maybe that
> > relation between sincere CW & SU maximizer extends to more
> > dimensions, to some degree. Besides, my interest in electing
> > the sincere CW is based on my concerns about defensive strategy
> > need, and the methods that do well by criteria that measure
> > that standard tend to do well by SU anyway.
>
> 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.
>
> But the von Neumann-Morgenstern utilities are defined only on a
> relative scale and not on an absolute scale. Example: If voter
> V has a von Neumann-Morgenstern utility of candidate A of
> N(V,A)=8000$ and a von Neumann-Morgenstern utility of candidate
> B of N(V,B)=10000$ then this means (1) that this voter would
> spend N(V,B)-N(V,A)=2000$ if he could change the winner from
> candidate A to candidate B and (2) that if this voter got a
> compensation of N(V,B)-N(V,A)=2000$ when candidate A was elected
> instead of candidate B then he wouldn't care about the winner
> any more. Therefore it is the same whether voter V has a von
> Neumann-Morgenstern utility of candidate A of 8000$ and a von
> Neumann-Morgenstern utility of candidate B of 10000$ or whether
> voter V has a von Neumann-Morgenstern utility of candidate A of
> 3000$ and a von Neumann-Morgenstern utility of candidate B of
> 5000$.
>
> 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.
Sorry, I misused the term SUE -- I should have said SU. SUE doesn't
apply to individual elections. Rephrasing my statement, in any given
election, the winner's social utility should not be lower than about 1/2
or 3/4 of that of the candidate with the highest social utility. I
didn't mean to imply that the utilities were not relative. And yes, I
was assuming von Neumann-Morganstern utilities.
To measure performance relative to that standard, you could either count
the percentage of elections that fail the test, or you could get fancy
and calculate the average magnitude of the failures.
I disagree with your definition of vN-M Utility, though. First, with
only two candidates, the vN-M utility of the lower one would be 0, and
that of the higher one would be the difference, but that is not really
unique to vN-M utilities. Second, using an absolute dollar amount to
describe the difference between two candidates is meaningless, since we
don't know how the voter values the $2000.
The way I understand vN-M utility, you also need to consider risk. The
utility of a choice is equal to the utility of success times the
probability of achieving it.
Suppose you have two alternatives, one worth a sure $5000, and the other
a gamble worth either $10,000 or $0, with a 50% probability of success.
If we say that $10,000 has a utility of 1.0, then the 50-50 lottery for
$10,000 has a utility of 0.5.
It would be tempting to say that the sure $5000 also has a utility of
0.5, but that depends on the person making the choice. If you are risk
averse, and would rather have the $5000, then its utility is actually
greater than 0.5.
If you needed $5000 for a life-saving operation, and would choose the
safe alternative even if there were a 90% chance of success at winning
$10,000, then the utility of the $5000 choice is actually > 0.9.
Conversely, if the operation costs $10,000, and you have no choice but
to take the gamble, then the utility of the sure $5000 is close to zero.
The point is that the von Neumann-Morgenstern utility of a choice is
defined by the willingness of someone to make that choice, as opposed to
a ratings system where someone assigns values (sincere or otherwise).
This means that even sincere ratings would not necessarily be accurate
for a real voting system -- the only way for a real voting system to
reflect vN-M utilities is to force the voter to weigh risks and benefits
of each choice.
Of course, that's not a problem in simulations, since we don't have to
measure utilities -- we create them.
> Also Mike's statement that the Condorcet winner usually has a
> high SUE seems to me to be not justified. Especially in a divided
> society, the SUE of the Condorcet winner and even of the majority
> winner can be very small. The reason why I support the majority
> 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.
It seems to me to be manipulable by the way that candidates position
themselves, though, at least in theory. A candidate just has to look
for races where the existing candidates are on opposite sides of the
voter median on important issues, and split the difference -- more a
concern if Markus is correct rather than Mike regarding the likelihood
of low SU winners.
Bart
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