At 09:51 AM 1/23/2009, Jobst Heitzig wrote:
I did not mean to say the voter has no opinion. He may well hold the
opinion that, say, A is much better than B in some respect, and B is
much better than A in another respect, so that neither is A
preferable to B nor B to A nor are they equivalent (equally
preferable). This is just an ordinary case of what some people
pejoratively call "incomplete" preferences. Or the voter may hold
the opinion that A is better than B in two of three respects, B is
better than C in two of three respects, and C is better than A in
two of three respects, so that A is strictly preferable to B, B to
C, and C to A. This would be a case of "complete" but cyclic
preferences. Or, even more simple, A and B may just be completely
equivalent, so that neither is preferable to the other. In all these
cases, a "favourite" is inexistent, not just unknown.
Yes. However, this problem actually doesn't afflict Range Voting, it
is a *voter* problem. In the end, the voter risks his or her vote,
spends it. How much of the voter's vote will the voter risk?
Put it another way. Suppose the voter can decide the outcome of the
election by bidding. The voter still has the problem of choosing
between A, B, and C, but the choice *between* them isn't forced.
I.e., the voter can bid equally for them. Now, in real elections, the
voter has something that the voter can "spend" as a bid. It's a vote,
one single vote. How the voter will spend this depends not only on
the voter's preferences and preference strengths, but also on
probabilities of success. That's what VNM utilities are: bids in
lotteries, determined not only by absolute utilities, but also by
estimated relevance. The voter doesn't normally want to spend the
vote discriminating between moot candidates. But if the voter doesn't
care that much about which of the frontrunners is elected (perhaps
they are all equally bad, or equally good, in the voter's eyes), then
the voter may indeed invest the vote in moot pairwise races.
To understand what I mean by investing the vote, imagine a Range
election with three candidates, A, B, and C. The voter has one full
vote to invest in influencing the outcome. How does the voter vote?
It's a fairly straightforward problem in game theory. Let R(A) be the
range rating of A, similarly with B and C. Arrange the candidates in
preference order, and we'll assume that they've been named in that
order, i.e., A>B>C. A "strategic" vote is, first of all, a normalized
one, so R(A) equals 100% and R(C) equals 0%. Actual votes are V(A),
V(B), V(C). The vote is distributed as follows: V(A:B) = R(A) - R(B),
V(B:C) = R(B) - R(C). The Range constraint is that the votes are all
in the range of 0 - 100%, and V(A:B) + V(B:C) is not greater than 1
full vote, i.e., 100%. If the vote is normalized, the sum of
preference strengths is 100%. V represents preference strengths
between adjacent candidates in the Range spectrum.
If there is a cycle as described by Jobst, how does the voter express
the votes? We face this problem all the time with choices; in the
end, a particular choice is worth something to us; the worth is not
cyclic; we do not, in fact, do Condorcet analysis, we do Range
analysis. With VNM utilities; we don't invest our resources in moot
choices. We might not even consider them, but if we do, we don't
shift our non-moot votes, or if we shift them, we shift them only a little.
The voter determines the vote to invest by effectively multiplying
the value of a pairwise election, then multiplying it by the
probability that the vote is effective. (It's a relative probability,
the actual probability is very low in public elections unless there
are very few voters.) Voters already do all this; voters don't
actually do the math, or at least such voters would be rare.
We vote this way in plurality; complicating it is that we have other
values to satisfy besides election results. However, that's covered
in the "calculations," if they are complete. It's all about how much
the voter cares about an election result. If the voter is indifferent
between A and B, but strongly prefers C to either of them, and A and
B are frontrunners, that a vote in the A/B pair is the only
"effective vote," as we would normally look at it, i.e., only A or B
can realistically win, that the probability of an A result is close
to one can't overcome the value of the election pair, which I just
expressed as zero.
But wouldn't this work on the other pair, but in reverse, the value
is 100% for C, but the probability of "success" is zero?
Sure it would, if the voter doesn't have a value for simply having
contributed to the vote count for C. Voters who don't have a high
value for that *don't vote.* However, voters *do* value voting for
candidates who can't win, we know that. So insignificant value
between A and B, and high value in voting for C, leads to a vote of 0, 0, 100.
And the vote is sincere. Further, most voters will probably vote
similarly for frontrunners. They may bullet vote, or, alternatively,
if they value minor candidates, they, again, may consider the vote
itself to have value, and in this case, they are likely to vote more
or less accurately, or even approval style, i.e., we will see some
voters who prefer Gore, say, voting 100% also for Nader. In doing
this, they risk electing Nader; but if they'd be adequately pleased
by that result, even though Nader isn't their favorite, they might
take that risk. Small risk, I must assume, plus the loss is
relatively small. They have cast a full vote in the critical frontrunner pair.
Basically, excluding consideration of election probabilities makes
voting into an exercise in pure abstraction, rather than an efficient
substitute for full deliberative process with many rounds of voting
being possible. Attempting to abstract from voters sincere absolute
utilities may not be desirable at all, besides the detail of it being
extraordinarily difficult and not even clearly defined and not
necessarily commensurable unless we make some simplifying (and
inaccurate) assumptions. Systems where voters are actually bidding
something of real value, something that costs voters something of
significant value to them, will encourage more sophisticated VNM
utility voting, perhaps, but doesn't change the need for strategic
consideration, a consideration which is part of ordinary, everyday
decision-making.
> For a voter that doesn't have a sincere
> opinion it is also difficult to vote in any
> way (not just sincerely).
Again, I talk about voters who *do* have sincere opinions which
however happen do not fall into the narrow set of possible opinions
the voting method's designer cared to take serious. The problem is
on the designer's side, not on the voter's. One must not assume that
such thiings as "favourites" always exist or that preferences are
complete or transitive as long as one cannot prove that this is
indeed the case for all voters. And by "prove" I don't mean "show
its validity in some arbitrary narrow-minded economic model of utility".
One does not have all these problems when one avoids to speak of
"sincere" votes!
Yours, Jobst
> --- On Wed, 21/1/09, Jobst Heitzig <[email protected]> wrote:
>
> > From: Jobst Heitzig <[email protected]>
> > Hi Juho!
> >
> > > What is the problem with
> > > sincerity in Plurality?
> >
> > Well, that's simple: Any voter who does not have a
> > unique favourite option (whether that is because of
> > indifference or uncertainty or because of cyclic
> > preferences) cannot vote "sincerely" in Plurality!
> >
> > Yours, Jobst
>
>
>
> .... and the older mail ...
>
>
> --- On Fri, 16/1/09, Jobst Heitzig <[email protected]> wrote:
>
> > To determine how I should vote, is that quite complicated
> > or does it depend on what I think how others will vote?
> >
> > Or is my optimal way of voting both sufficiently easy to
> > determine from my preferences and independent of the other
> > voters?
> >
> > If the latter is the case, the method deserves to be called
> > "strategy-free". The whole thing has nothing to do
> > with "sincerity". Refering to
> > "sincerity", that concept in itself being
> > difficult to define even for methods as simple as Plurality,
> > complicates the strategy discussion unnecessarily.
>
> Are you looking for the English language
> meaning of sincerity or some technical
> definition of it (e.g. some voting related
> criterion)? What is the problem with
> sincerity in Plurality?
>
> Juho
>
>
>
>
>
>
>
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