[EM] Intermediate RV rating is never optimal

[Abd ul-Rahman Lomax]

A huge amount of smoke has been blown over the issue of optimal 
voting in Range. Quite a few writers have asserted, with apparent 
confidence, what Ossipoff asserted in the Subject header of the post 
to which I'm writing. And it is blatantly false.

Remarkably, I've never seen anything resembling a proof of the 
statement. To refute it is, however, made a bit easier by the rash 
use of "never" in it.

Let's start with an obvious problem. What is "optimal"? There are a 
number of ways of determining what is optimal, and they may generate 
differing conclusions about optimality.

How do we measure optimality? I haven't seen Ossipoff making a 
proposal, other than claiming that electing the candidate "approved" 
by the most people is optimal. While that is an interesting question, 
"approval" has no fixed meaning and it is difficult to use it with a 
method that doesn't specify an approval cutoff.

And, in this case, "optimal" is referring, not to the election 
outcome as it relates to society as a whole, but to an individual 
voter's expected outcome from the vote. And to know that expected 
outcome, we must have utilities for each outcome. But Ossipoff denies 
utility as being useful for measuring election outcomes, leaving us 
with ... nothing.

However, the denial of utility as a measure of outcome, and thus of 
the whole field of game theory, which involves the interaction of 
utility and probabilities, is idiosyncratic. Most writers, asserting 
that approval-style voting is always optimal, *do* have expected 
utility in mind, the expected utility for the individual voter. It is 
being asserted, by them, that the voter maximizes his or her personal 
expectation of outcome by voting full strength, i.e., "Approval 
style," where the voter votes either the maximum or minimum rating 
for all candidates, presumably based on whether or not the voter 
"approves" of the candidate. How is "approval" determined? Well, the 
real answer is that the voter has a set of utilities for the 
candidates, i.e., there is some kind of internal "Range Vote." While 
it is beyond the scope of this post to go deeply into neuroscience, 
this is probably a reasonable first-pass model of how we make 
decisions, though, certainly, the reality is more complex than that. 
Still, we routinely can sort lists in order of preference, and we 
likewise have varying preference strengths. (We know that preference 
strength is small when it is difficult to choose between two 
candidates, and that it is large when it is easy to choose.) All this 
points to some kind of internal scale or figure of merit, I would 
assume roughly related to some kind of neural activity level 
connected with attraction and aversion potentials.

In any case, depending on context, a voter would select some kind of 
"approval cutoff" and then determine whether or not to approve of 
each candidate. If the candidate's utility to the voter is higher 
than the cutoff, the voter would approve of the candidate; if not, 
then not. But, quite clearly, the approval cutoff is not a fixed 
thing. The "approval" of a candidate is not an absolute; whether or 
not a voter will actually approve of a candidate depends on the 
election context, i.e., the candidate set, as well as the estimated 
probabilities of election, for some voters. Others may not consider 
the probabilities.

But we are not studying Approval here, we are studying Range Voting. 
What *is* the optimal Range vote? Range is just like Approval in that 
the optimal vote will depend on the election context; while we may 
imagine (and in simulations use) absolute utilities for each 
candidate which are independent of other candidates, we expect that 
in real elections most voters (though not all) will normalize their 
votes to give themselves full voting strength and, by the measure of 
optimality I propose here, this never hurts the voter personally. It 
*might* hurt society, by losing utility information and thus failing 
to optimize overall utility. While there is a way to ameliorate this 
effect, it, again, is beyond the scope of this post.

What I propose to do is to study the game matrix for a voter in a 
particular situation. I find this the only way to answer the question 
of optimality, and I find it remarkable that writers on this subject, 
Warren Smith excepted, seem to have avoided doing this. (Of course, 
they may have done it and I haven't seen it.) What seems to have been 
done, instead, is to propose some scenario where voting Approval 
style helps the voter, and from this they make the claim that it is 
always optimal. Surely this is deficient analysis.

Now, I proposed a simple example as a counterexample, the situation 
where the voter prefers A>B>C with equal preference strength, 
examining how the voter should vote for B in Range. An objection was 
raised that in this situation, whether the voter should approve B or 
not is a tossup, and voting max for B or min might as well be chosen 
at random. But nobody showed a game matrix.

Okay, this is Range 2. The sincere, accurate utilities are 2, 1, 0. 
We know that if the probable pairwise election is between A and C 
(top two), the vote for B is moot; however, for reasons independent 
of determining the specific election outcome, the voter may consider 
it optimal to vote sincerely; but this does not improve outcome, so 
this situation does not serve as a clear counterexample. Likewise if 
the probable pairwise election is between A and B, the voter 
optimizes outcome by voting 2, 0, 0, and if the probable pairwise 
election is between B and C, the voter optimizes outcome by voting 2, 
2, 0. This inspires a general theorem:

When it is certain that the winner of an election will come from a 
set of only two candidates, the optimal vote maximizing an individual 
voter's expected satisfaction is Approval style. While we have not 
examined the situation of less than certainty, it is not unreasonable 
to allow that Approval style voting remains optimal at least until 
the probability of election of some other candidate is above some threshold.

But what happens in the zero-knowledge situation? Now, because of the 
complication introduced by the equal preference strengths -- if I had 
more time I'd go more deeply into that particular situation -- I'm 
now proposing to examine a somewhat different scenario.

There is a Range 3 election with three candidates, and the voter 
prefers A>B>C, with utilities of 3,2,0. The voter has no knowledge of 
how others will vote, and so we will assume that all combinations of 
other votes are equally likely. I'm going to take two approaches, one 
very simple, one quite a bit more complex.

The simple approach is to vast simplify the calculations by noting 
that the voter could simply consider that the voter's vote is moot, 
in the end, unless the election is, without the voter's vote, tied or 
nearly tied, such that the vote of the voter may swing the election. 
Since the voter may only vote a maximum vote of 3 for any candidate, 
before the voter's vote is considered, the gap between the winner and 
loser must be 3 or less.

Now, the voter is going to vote one of two votes: 3,3,0 -- Approval 
style -- or 3,2,0 -- sincere Range. Which of these votes has the 
better expected outcome?

My first approach to this is to consider that there is only one other 
voter. Thus, the gap between winner and loser is at most 3.

There are sixteen possible votes of the other voter, being the base-4 
numbers from 000 to 333. If I consider each of these votes as equally 
possible, and if I consider that ties are going to be resolved by 
random choice between the tied candidates, what is the expected 
outcome for each of the two votes above, the Approval-style vote and 
the sincere Range vote?

I'm not posting the matrix immediately, and, indeed, I have not 
prepared it yet, beyond looking at a few cells in it. There are those 
here who seem certain that Approval-style voting will be optimal, so 
they will have an opportunity to predict what the outcome will be. If 
they dare. Or, of course, perhaps one of them has already done 
similar work and will merely need to point us to it.

I have written before that the optimal vote, zero-knowledge, was the 
sincere vote, but I have never proven it beyond using a inference 
approach, which can be flawed. So I, too, am being tested by this example.

Now, I can already expect another smokescreen. It will be claimed, I 
expect, by at least one writer, that real elections will have 
strategic voters, and if other voters vote strategically, then our 
subject voter would be a sucker to vote sincerely. Well, is that 
true? We could look at this by considering that the other voter only 
votes "approval style," thus restricting the subset. But this would 
be a flawed analysis, because we have substituted that single voter 
for a whole mass of voters, looking simply at the "dregs," i.e., the 
difference between all those votes, in the case where our subject 
voter's vote will matter, and the original vote will be spread across 
the range, since votes from various factions will cancel out, and it 
only takes two voters not voting max or min to create the full range 
of vote possibilities for what remains before this voter votes. It is 
simply not believable that, with a large number of voters, Range 3 
total votes would only exist in multiples of 3.

In any case, knowledge that other voters will vote strategically, 
i.e., Approval style, perhaps limiting Approvals to favor some 
personal outcome, is not only not "zero-knowledge," it is not 
relevant to the voter's expectations.

It will also be argued, I might expect, that in real elections the 
chance that the voter's vote will turn the outcome is so vanishingly 
small that such calculations are irrelevant, they don't properly 
affect real behavior. However, by this argument, nobody would vote, 
yet people *do* vote. (And many don't, of course.)

What voters know that some analysts apparently don't is that people 
assume that other people will think like them, and that if they think 
that their vote is useless, others like them will think the same way. 
And the effect of many people thinking this way, and acting on it, is 
*real* disempowerment for the people who think like this. While it 
would be a kind of magical thinking to assume that our private 
decision as to whether or not to vote is going to cause others to 
vote or not vote, there is an intuitive satisfaction in behaving in a 
way that, if everyone were to behave that way, would be beneficial, 
and an intutitive dissatisfaction in behaving in a way which would be 
harmful if everyone behaves that way. This explains why the tragedy 
of the commons is not a common as one might expect from pure, 
self-interest game theory analysis.

I'm suggesting that, in fact, we should vote as if there were only 
one or two other voters. We should vote as if our vote *probably* counts.

(And I could do the analysis with two other voters, which, while it 
would be interesting, I simply don't have time for. Range creates 
complex voting patterns compared to other systems. For the simple 
case given above, there are 64 possible votes from the other voter; 
each one must then be compared with the two possible votes from the 
subject voter and the value of the outcome assigned. In any case, 
Warren Smith has done simulations with large numbers of voters and we 
are only doing this simple example here as an exercise and possible 
counterexample to the claim being made about optimal voting in Range.)