[EM] Why the concept of "sincere" votes in Range is flawed.

[Abd ul-Rahman Lomax]

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.