And now that rarity from me, an original post....
Approval Voting is a special case of Range, with
rating values restricted to 0 and 1. When Brams
proposed Approval, it was as a method free of
vulnerability to "tactical" or "strategic"
voting, i.e., voting with reversed preference in
order to produce a better outcome. And, indeed,
both Range and Approval are immune to that, i.e.,
there is no advantage to be gained by it, ever
(at least not in terms of outcome).
The proponents of other methods attacked this by
redefining -- without ever being explicit about
it -- the meaning of strategic voting. Because
the concept was developed to apply to methods
using a preference list, whether explicit on the
ballot or presumed to exist in the mind of the
voter, a strategic vote was one which reversed
preference, simple. But with Approval and Range,
it is possible to vote equal preference. Is that
insincere if the voter has a preference? The
critics of Range and Approval have claimed so,
and thus they can claim that Range and Approval
are "vulnerable to strategic voting."
Arrow, in explaining why he did not study
cardinal rating methods (like Range and
Approval), methods that allow equal ranking,
wrote that they offended him because there is no
single sincere vote. I.e., a whole set of votes
could be considered sincere. If the voter prefers
A>B>C, the voter could vote for A or for A and B
(and, for that matter, for A and B and C), and still be sincere.
(A side-note: unless a preferential ballot allows
ranking all candidates, it does allow equal
ranking *at the bottom,* indeed it requires it.
But we've tended to focus on the winner only.)
The critics, I've seen, will consider a vote for
A only, with Approval, when the voter supposedly
"approves" both A and B, to be "strategic." It
certainly is strategic in the sense of "smart,"
under some conditions. However, this is where
preference strength comes in, and a strange twist
of the definitions takes place. We must assume
that if the voter votes only for A, the voter
does, indeed, prefer A. So with a preferential
method, as with Plurality, the vote for A alone
is sincere, and a vote for B alone would be
insincere. In other words, Approval voting is
"vulnerable" to a voter voting what would be
considered a sincere vote in a method that does
not allow equal ranking. This is having the critical cake and eating it too.
So what are "sincere" Range and Approval votes?
Should voters in Range vote "sincerely?" Or
should they vote "strategically," which means
that their vote is different depending on their
perception of the election probabiities. The
voter votes only for A in the example above, even
though the voter supposedly "approves" of B as
well, because the voter perceives the important
choice as being between A and B, with C being
unimportant. If the voter sees C as possibly
winning, with significant probability, the voter
is much more likely to vote for both A and B.
The root of the critical problem is that votes
have been considered expressions of preference
alone, and the goal has been to find a voting
method that works, even in the presence of voter
knowledge of the election probabilities, just
like a zero-knowledge election. The problem is
that this is a strange and artificial creation,
when we look at it carefully. It doesn't exist in
the real world, and there are many obstacles in
the way of it, including Arrow's theorem and how
people will always behave. When we ask people
what they want, they will *always* modify the
answers according to how they perceive the probabilities of each possibility.
Which would you prefer, $10 or $100? Seems
simple, eh? And I've argued that any good ballot
design will allow you to express that preference.
However, suppose there are three alternatives, $0
or $10 or $100. We can easily rank these, but
suppose that these are personal utilities for the
three alternatives, and they are not identical
for all the voters, and some voters will prefer
the outcomes in a different order. And if we vote
100>10>0, the probabilities, in our judgement,
are that we'll get 0. While the 100 outcome is
obviously preferable to us, we consider it
unlikely. So how do we vote in an Approval
election? I've set it up to be obvious. We vote
for 100 and for 10. Now, how do we vote in Range?
The supposed sincere vote, based on true personal
utilities, which we've made obvious, would be, in
Range 100, to vote the dollar values. Yet that
would be almost as foolish as to vote for 100 only.
I came across the following piece, at
http://cepa.newschool.edu/het/essays/uncert/vnmaxioms.htm
In the von Neumann-Morgenstern hypothesis,
probabilities are assumed to be "objective" or
exogenously given by "Nature" and thus cannot be
influenced by the agent. However, the problem of
an agent under uncertainty is to choose among
lotteries, and thus find the "best" lottery in D
(X). One
of
<http://cepa.newschool.edu/het/profiles/neumann.htm>von
Neumann and
<http://cepa.newschool.edu/het/profiles/morgenst.htm>Morgenstern's
major contributions to economics more generally
was to show that if an agent has preferences
defined over lotteries, then there is a utility
function U: D (X) ® R that assigns a utility to
every lottery p Î D (X) that represents these preferences.
Of course, if lotteries are merely
distributions, it might not seem to make sense
that a person would "prefer" a particular
distribution to another on its own. If we follow
<http://cepa.newschool.edu/het/essays/uncert/bernoulhyp.htm>Bernoulli's
construction, we get a sense that what people
really get utility from is the outcome or
consequence, x Î X. We do not eat
"probabilities", after all, we eat apples! Yet
what von Neumann and Morgenstern suggest is
precisely the opposite: people have utility from
lotteries and not apples! In other words,
people's preferences are formed over lotteries
and from these preferences over lotteries,
combined with objective probabilities, we can
deduce what the underlying preferences on
outcomes might be. Thus, in von
Neumann-Morgenstern's theory, unlike
Bernoulli's, preferences over lotteries
logically precede preferences over outcomes.
How can this bizarre argument be justified? It
turns out to be rather simple actually, if we
think about it carefully. Consider a situation
with two outcomes, either $10 or $0. Obviously,
people prefer $10 to $0. Now, consider two
lotteries: in lottery A, you receive $10 with
90% probability and $0 with 10% probability; in
lottery B, you receive $10 with 40% probability
and $0 with 60% probability. Obviously, the
first lottery A is better than lottery B, thus
we say that over the set of outcomes X = ($10,
0), the distribution p = (90%, 10%) is preferred
to distribution q = (40%, 60%). What if the two
lotteries are not over exactly the same
outcomes? Well, we make them so by assigning
probability 0 to those outcomes which are not
listed in that lottery. For instance, in Figure
1, lotteries p and q have different outcomes.
However, letting the full set of outcomes be (0,
1, 2, 3), then the distribution implied by
lottery p is (0.5, 0.3, 0.2, 0) whereas the
distribution implied by lottery q is (0, 0, 0.6,
0.4). Thus our preference between lotteries with
different outcomes can be restated in terms of
preferences between probability distributions
over the same set of outcomes by adjusting the set of outcomes accordingly.
But is this not arguing precisely what
<http://cepa.newschool.edu/het/profiles/bernoulli.htm>Bernoulli
was saying, namely, that the "real" preferences
are over outcomes and not lotteries? Yes and no.
Yes, in the sense that the only reason we prefer
a lottery over another is due to the implied
underlying outcomes. No, in the sense that
preferences are not defined over these outcomes
but only defined over lotteries. In other words,
von Neumann and Morgenstern's great insight was
to avoid defining preferences over outcomes and
capturing everything in terms of preferences
over lotteries. The essence of von Neumann and
Morgenstern's expected utility hypothesis, then,
was to confine themselves to preferences over
distributions and then from that, deduce the
implied preferences over the underlying outcomes.
"Preferences" in Range Voting is preferences over
lotteries, not preferences over outcomes, as
such. I, of course, support voting methods which
allow the expression of both, hybrid methods, and
which resolve the occasional conflict between a
sum-of-votes approach and a pairwise winner
approach, using not the original ballot, but a
new one, i.e., a runoff that turns the choice
involved back to the voters. Some supporters of
Range are disturbed by this, because, supposedly,
the Range Votes, summed, elect the social utility
winner, which, they argue is the best winner for
society. However, they've neglected the overall
process in favor of resolving it in a single
ballot. If a single ballot *must* be used, no
matter what the cost, the Range outcome is indeed
the closest we can get to ideal, I suspect. But
we are not limited to that, and we can go back to
the voters -- a different set of voters, usually!
-- and ask them. The exact details of that
additional election I'll leave for another paper;
parliamentary procedure would suggest that it be
an entirely new election, informed by the results
of the first one, plus additional campaigning,
but practicality may suggest something different.
Regardless, it's apparent to me that two ballots
is better than one, when one doesn't come up with
a clear majority choice or better.
Economics. It seems to be a field that is
disreputable to political scientists. But this
is, of course, a field where substantial
theoretical expertise has been applied to the
problem of making decisions. A voting system is,
obviously, such a problem. Warren Smith found
this paper and pointed it out to us:
http://ideas.repec.org/a/ecm/emetrp/v67y1999i3p471-498.html
This is a paper by Dhillon and Mertens. An abstract:
In a framework of preferences over lotteries,
the authors show that an axiom system consisting
of weakened versions of Arrow's axioms has a
unique solution, 'relative utilitarianism.' This
consists of first normalizing individual von
Neumann-Morgenstern utilities between zero and
one and then summing them. The weakening
consists chiefly in removing from IIA the
requirement that social preferences be
insensitive to variations in the intensity of
preferences. The authors also show the resulting
axiom system to be in a strong sense independent.
Relative Utilitarianism is an analytical method
which takes as input Range Votes; as Warren Smith
has stated he prefers, the Votes are rational
numbers (I think) in the range of 0-1, with no
restriction on resolution. I.e., practical Range
Voting uses some specified resolution; I define
Range N as being Range with N+1 choices, so Range
1 is Approval (with two choices, 0 and 1), and we
can express Range votes as 0-N; i.e, Range 100
may vote as 0-100, but is really 0-1 in steps of 1/100 vote.
In pure relative utilitarianism, then, unless the
voter is indifferent to a choice, the vote in
that choice will always show preference, but the
magnitude of the preference will vary according
to perceived probabilities. In practical Range
Voting, if the von Neumann-Morgenstern utilities
get rounded off, thus showing equal preference
when the reality is that there is an underlying
preference, but with a combination of absolute
magnitude and relative probability that brings it
within the resolution of the Range method.
Now, voters don't sit down with a calculator, but
what was claimed in the first paper above is that
this is, in fact, how we make decisions. It's
much simpler than one might think, and in real
elections, the normal procedure for determining
these utilities is quite simple in *most*
elections: Pick two frontrunners (which depends
on probabilities only, not personal preferences).
Then use preferences to rate one of them at
maximum and one at minimum. If one has
preferences of significance outside this set and
this range (i.e., one has a candidate preferred
over the best frontrunner and one over the worst
frontrunner, then one might consider, if the
method has sufficient resolution, pulling the
frontrunner down a notch or the worst up a notch,
to preserve preference expression. Alternatively,
perhaps the method allows expression of preference independently of rating.
Only when there are three candidates considered
possible winners does it get more complicated.
But the point of all this is that voters will
always consider election probabilities, and thus
pure Independence of Irrelevant Alternatives is a
real stumbling block if insisted upon. The voting
power I assign to the pairwise preference of $10
to $100 must depend on, not only my pure
intensity of preference, but on my perception of
the probabilities. Voters in Range are choosing
lotteries with specified prizes, with values and
probabilities as estimated by the voter, and setting their votes accordingly.
And they sincerely choose them, i.e., they
attempt to maximize their personal expected
return, and this is *exactly* what we want them
to do. It's not "greedy" or "selfish," it's
"intelligent." Now, the shift in votes due to the
probability perceptions can be mistaken. A dark
horse candidate may not receive the full vote
strength that the candidate would receive in a
zero-knowledge election, with all candidates
being considered equally likely. Thus, we'd need
runoffs to fix problems, which might be detected
through preference analsyis. But the method is
theoretically ideal, as Dhillon and Mertens show,
it is a unique solution to Arrow's theorem (with
a minimal tweak, one that is utterly necessary;
the requirement of absolute Independence of
Irrelevant Alternatives was, quite simply, a
mistaken intuition. Relative Utilitarianism
doesn't require -- actually does not allow, in
its pure form, -- the suppression of preferences,
but the *magnitude* of the expressed preference
varies with the alternatives, and that
technically violates IIA, as it was understood.
This brings us to a problem with Range Voting. If
voters expect that their task, with Range, is to
give "sincere ratings," regardless of the effect
on results, Range will suffer badly from IIA and,
indeed, as claimed by critics, voters who pay no
attention to irrelevant candidates, in
determining how they vote for two frontrunners,
will have an advantage, through "bullet voting"
or through "exaggerating." Range Voting is still
*voting,* but merely with fractional votes
allowed, not required, when N is greater than 1.
Approval is still voting. It is not about
"approving" the candidates, except in the sense
that by voting for a candidate, one is approving
the election of that candidate, *compared to the
likely alternatives*. It is not a sentiment, it's
an action, adding weight to an outcome, choosing
to effectively participate in it. Add weight to
an irrelevant alternative, it doesn't matter, by
definition. In almost all elections, there are
two frontrunners, and, this is why Plurality
usually works, and only breaks down through the
related spoiler or center squeeze effects,
because of the restriction against voting for
more than one. All advanced voting methods --
with one exception, not applied anywhere that I'm
aware of, Asset Voting -- allow voting for more
than one, but through various procedures. (Even
Asset would normally allow voting for more than
one, the original form was proposed for an STV
ballot with optional preferential voting, to deal
with the very common problem of exhausted ballots.)
Comments invited.
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