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

[Abd ul-Rahman Lomax]

At 01:38 PM 12/5/2008, Kristofer Munsterhjelm wrote:
>Abd ul-Rahman Lomax wrote:
>>
>>Ballots do not ask for the voter's sincere opinion. They ask voters 
>>to make a choice or choices.
>
>I think that is incorrect. Ranked methods ask for the sincere opinion of
>the voter, and that opinion can be well defined.

Understand that before writing this, I actually looked at some real 
ballots. Quite possible, some real ballots ask for ranking in order 
of preference. But the RCV ballots in the U.S., just like the name 
implies (Ranged Choice Voting) asked for "First Choice," "Second 
Choice," etc. Which leaves it up to the voter how to vote.

In my view, to ask for "First Preference" on a ranked ballot would be 
just as offensive as asking voters on a Plurality ballot to vote for 
"their favorite." It would generate a political bias, making "Core 
Support" the standard in elections. Plurality tends to choose a 
Condorcet winner because voters strategize and make their compromises 
based on information about each other. "Strategic voting" is how 
voters can improve the performance of a voting system. The problem, 
of course, is that when the voting system gets very good, strategic 
voting can sometimes reduce the overall voter satisfaction. However, 
my contention is that when it does this, it does not do it to any 
great extent, and I have yet to see an example, of any 
reasonableness, that contradicts this.

>Now, you may say that only order reversal is insincere. This sounds 
>a bit like a ranked vote advocate saying that only altering your 
>first preference is insincere, and therefore, ranked methods that 
>pass FBC are strategyproof because altering your subsequent 
>preferences is mere optimization.

That's correct, about order reversal. It's a reasonable statement, 
because equality is a judgement that does not require any specific 
precision. "Equality" means "below some threshold of difference 
considered significant." "Sounds like" is a personal statement, a 
subjective impression.

(We focus on "exaggeration," but it's equally valid to focus on 
"minimization," which in Range is rigidly correlated with 
exaggeration. In ranked methods which allow or require truncation, 
"exaggeration" by ranking in the presence of little or no preference 
takes place, and minimization of preference takes place by 
truncation, whether voluntary or forced.

>Election methods in general are thus algorithms that take individual
>opinions as input and returns a good common choice, or a social
>ordering. What is a good common choice may be defined by criteria (e.g
>Condorcet) or by utility.

Okay. However, the definition by criteria was never widely accepted, 
beyond certain simple ones, such as the Majority Criterion, and Arrow 
blew the whole thing out of the water. Paradoxically, Arrow rejected 
ordering by utility, based on alleged indeterminacy or other 
incomplete consideration; but it turns out that it's possible to 
satisfy the Arrovian criteria, that are allegedly incompatible, with 
a minor tweak to IIA, which has often been considered the weakest 
link in Arrow's chain. The substance of IIA is preserved. See Dhillon 
and Mertens, Relative Utilitarianism, Econometrica, May 1999, pp 
471-498. See http://rangevoting.org/DhillonM.html

>As for Range, either Range, the method, has a well defined input or 
>it has not. If it has, then incentives to misrepresent the input is 
>bad, and would count as strategy. If it has not, then how can it 
>make sense of the input to find a good output (choice or social ordering)?

The input is well-defined, it's provided by the voters. What cannot 
be done is to define the input exclusively by a preference profile, 
additional information my be incorporated. "Preference profile" is 
clearly inadequate information to allow utility optimization; 
preference strength information is also necessary, and, for practical 
decision making (even by individuals, and voting systems involve 
individual decision-making), lottery probabilities are also 
necessary. The voter is making choices in a lottery, and to do this 
intelligently requires estimating the lottery probabilities for each option.

A strategic Range vote does not "misrepresent the input," because the 
input to the system is a product, in the voter's estimation, of the 
relative utility of the outcome and the probability that the outcome 
is relevant.

I.e., the full preference profile consists of a rank order, with a 
preference strength in each adjacent pair in the order. The 
preference strength is then adjusted according to the probability 
that this pair is a relevant one, that there is some finite 
probability that the vote in that pair will improve the outcome. 
Further, there is a constraint: the sum of all the adjacent pairwise 
preferences must equal one full vote.

In this social welfare function, preference order is preserved with 
two exceptions: where the probability of relevance is zero, the 
preference strength in the pair goes to zero, thus equating the two 
outcomes; as long as there is nonzero probability, no matter how 
small, the preference order is maintained. It will be noticed, I'm 
sure, that this sets up independence from irrelevant alternatives, 
IIA, because if an irrelevant alternative appears in the full 
candidate set (which Dhillon/Mertens also define), it does not have 
any effect on the other preference strengths. Only relevant 
alternatives can do that.

Further, if the preference strength is below the resolution of the 
voting system, the preference may be lost. In pure RU, there is no 
resolution limit. I don't know of any significant opinion that a 
resolution beyond 1/100 of a vote is needed in practical systems.

The problem here is that estimating probabilities is "strategy." Thus 
what we may call "strategy" is part of the system.

Saari got it wrong. The system is not indeterminate, but the input is 
far more complex than we can readily predict. Nevertheless, in most 
elections, the estimation of probability is quite easy.

Saari gives a preposterous example. Since he considers the allege 
indeterminacy of Approval to be a total disaster, which he repeats 
over and over in one of the most badly-written (academically 
speaking) papers I've seen in a peer-reviewed journal, he gives an 
example, as if it were telling, of a totally extreme and impossible situation.

10,000 voters are choosing from three candidates, A, B, and C. 9,999 
voters consider A to be excellent, B to be mediocre, but barely 
acceptable, and C to be awful. One voter considers C to be superior, 
A to be awful, and B to be greatly preferable to A. He doesn't state 
them, but we could give the accurate estimation of utilities as

9999: A, 100; B, 51; C, 0
    1: A, 0;   B, 51; C, 100

This brings us face to face with the question, "What is a strategic vote?"

Saari imagines that the 9,999 voters follow what he claims is the 
advice of Brams et al: vote for any candidate better than the mean 
utility. While this is a reasonable voting strategy under some 
conditions, and if the 9,999 voters have no idea at all as to how 
others will vote -- which is actually not true even under 
zero-knowledge situations -- and if we assume that they all think 
*exactly* alike, not only in terms of utility but in terms of how to 
respond to utilities, then, yes, they could all approve both A and B, 
in Open Voting, i.e., Approval.

Leaving the only sane voter, the C supporter, to determine the 
outcome by choosing B, the "mediocre" candidate. Note the failure: it 
is not the worst outcome, it is a mediocre one, which *every voter* 
accepted. Saari is waving his hands to distract us; I'm sure he's 
sincere, he is himself distracted.

Does the average voter know that he is average? Usually, we have an 
idea. Imagine a society in which 99.99% of people think one way and 
0.01% of people think another. Do most of the 99.99% know that they 
are "normal," and do the 0.01% know that the way they think is unusual?

Leaving aside the question of how C even came to be on the ballot, 
and how C was then included in the estimation of mean utility, when 
sane people would exclude C from this as irrelevant, the A voters 
will expect that *most people* will vote the same way that they will, 
on average. So, even without knowing anything specific about the 
election, they will trust that their own vote will be common and 
might even be very common or even unanimous.

Consider what this thinking would do to the A votes. By voting as 
Saari thinks Brams et all are recommending, they are setting up a 
tie, as if the society has no preference between A and B. But the 
society, with only an isolated exception (Saari calls this voter a 
"maverick." Is that a good thing? I think so! We need more mavericks, 
not less, though they should be well-behaved mavericks) unanimously 
prefers, with very significant strength, A to B.

Saari has set up a situation and has presented with an apparently 
straight face, where nearly everyone votes in an insane way, based on 
a foolish interpretation of a supposedly recommended strategy, and 
they get, as a result, not an awful result, but a mediocre one. We 
could actually consider that the system performed well and the 
problem was not with the system, but with the voters. And, of course, 
this problem disappears immediately if we simply add one rating level 
to make Open Voting, Range 1, into Range 2.

If we do that, and if voters simply vote sincerely and accurately -- 
pretty easy to do in this scenario -- we'd get exactly the optimal 
social ordering:

A: 19,998
B: 10,000
C:    100

This is, in fact, a Borda result. That's because with utility 
profiles like this, Borda with N candidates and Range (N-1) are identical.

What Saari has done is to collapse a set of utility profiles and, 
given the alleged indeterminacy of the votes, based on Saari's total 
neglect of the voters' understanding of the social situation, he 
considers it legitimate to propose a preposterous outcome as an 
indictment of the system, even though, were we to hold an election 
every second for the lifetime of the universe, we would never see 
such an outcome. Or did I estimate the math on that correctly. Maybe 
we would! No, forget it. I think I got it right.

In fact, the votes are not indeterminate. They are simply determined 
based on other variables besides the ones that Saari -- and Borda -- look at.

One sound zero-knowledge strategy: assume that other voters are like 
you, and vote that way. If you have a strong preference, vote it. If 
you have a weak preference, don't vote it. *You will usually be right*.

The 9,999 voters, doing this, would have assumed that *most* voters 
would have utilities like themselves; therefore they would only 
approve A, because the mean utility, given this estimation, is above 
that for B. Only the C voter using the same strategy, would bullet 
vote for C. Results in Approval:

A: 9,999
B: 0
C: 1

Pretty good for such a simple method! (Naturally, in this case the 
method defaults to Plurality. Plurality is, in fact, a more complex 
method than Open Voting; it seems simpler, but it requires an 
additional rule that discards some votes considered, by the rule, improper.)

However, the C voter is either insane or knows that his or her view 
is isolated. The C voter may be C! After all, *somebody* had to think 
C was a good choice! So the C voter will know that the mean outcome 
is at or close to that of A. And thus the C voter will vote for B as 
well, making the outcome have a tie for B in second place. The social 
ordering is then fully correct.

The problem is that Saari probably misunderstood the concept of "mean 
utility." I haven't looked at the original Brams paper, it's 
conceivable that they improperly presented it. I've always heard it 
as being a voter's estimation of the mean outcome, and I've only seen 
it suggested that this be a simple average of all the candidates 
under certain narrow conditions; generally it only includes the 
frontrunners. And voter can estimate the frontrunners as being those 
they would accept themselves, as a first approximation. In this case 
that would yield an expected outcome of 0.75, leading to the correct 
(optimizing) bullet vote for A.

The large majority of people with Open Voting will, under anything 
resembling current conditions, Bullet Vote. *That is a strategic vote*.

Strategic voting generally *improves* outcomes, in real voting 
situations. When the voting system gets very good, such that "fully 
sincere and accurate" voting will choose the optimal winner, this 
becomes untrue, but it never goes to the point of serious damage.

And we have to remember that "fully sincere and accurate" voting with 
a Condorcet method can sometimes choose a very poor winner, *in 
realistic social choice situations.* It's not common, but it happens.