Dear folks,
I haven't followed this long thread, so perhaps this has been mentioned
before. If so, sorry...
Abd ul-Rahman Lomax wrote:
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."
In my view, the main question in the whole strategy-proofness debate
should be this:
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.
Applied to Approval and Range Voting, this clearly renders them "not
strategy-proof", since optimal strategy does heavily depend on what I
think others will do. Random Ballot, on the other hand, is clearly
"strategy-free" since my optimal strategy is always to tick my favourite.
Yours, Jobst
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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