[Election-Methods] Why extreme ratings are optimal in RV

[Abd ul-Rahman Lomax]

At 05:13 AM 7/26/2007, Michael Ossipoff wrote:


>Suppose that the method is 0-10 RV. Suppose that everyone but you has voted,
>and now you're going to cast the final ballot. As in actual elections, you
>don't know how others have voted, though you have some sort of probability
>estimates, such as pair-tie probabilities.

Now, my study is of the zero-knowledge case. Because this is simpler 
to calculate, etc., so far that's all I've looked at. It sets a kind 
of baseline.

And I also studied Range 2, which also makes it extremely simple. 
What I'm now noting is that it is entirely possible that optimal 
strategy in Range N is different from optimal strategy in Range M. I 
have indications of that, but have not yet exhaustively studied any 
other Range resolution than Range 2.

There are indications that Range 999999 has an optimal strategy of 
extreme voting, though it is clear that sincere voting is totally 
harmless, and perhaps better than that, *with some kinds of 
preference and probability patterns.* Simulations, generally, have 
not answered this question, the strategies, quite simply, may not be adequate.

There are indications, for example, that optimal strategy could start 
out as Sincere with exact equal preference ratings (2, 1, 0), zero 
knowledge, but drift toward Approval with increase of knowledge, or 
with shift of preference. The simulations looked at, apparently, 
*all* preference gaps, so the results are averaged over many. So with 
*some* kinds of preference gaps, an advantage of one over the other 
could disappear. Perhaps it hurts more than it helps, sincere voting, 
averaged over all preference strengths. But would help with some.

I've seen Warren note an optimal Range vote that was neither extreme, 
nor sincere. It had been shifted. So an optimal strategy may be 
neither sincere nor extreme, but determined in some way. And this has 
not, to my knowledge, been exhaustively tested.


>Now, suppose that you consider the points that you're awarding
>one-at-a-time, as if it were a series of 10 Approval elections. Who are all
>the candidates whom you would give a point to in the first Approval
>election. Then, who are all the candidates whom you'd give a vote to in the
>2nd Approval election? And so on, for 10 Approval elections. In each round
>(Approval election), you vote to maximize the amount of expectation good
>that your Approval votes in that round will give you, just as you would if
>your were maximizing your expectation in an actual Approval election.

You have a series of actions you can take. You are *assuming* that 
the actions aren't interrelated, that the first action does not 
affect the utility of the second. So it is not surprising that you 
would conclude that, if in an Approval election, your optimal vote is 
Approval style, that you should, in each one of them, vote the same.

Reductio ad absurdem: Suppose that you were the only voter, voting in 
a Range election. But your preferences shift and change depending on 
what is in front of you. Over 10 days, you experience a range of 
considerations, and so your ratings of the candidates vary from day 
to day, but you assume that your experience, averaged out, will be 
constant. You can hold these 10 elections, one per day. What voting 
method would you prefer, Approval or Range? Which one will maximize 
your *personal* utility, *overall*.

If you are smart, I'd suggest, you would not simply record your 
favorite each day (let's make it simple, there are two candidates). 
You would record your *rating* each day, for both candidates. You'll 
make a better decision, overall, because you will be integrating your 
utilities over many days and facing different conditions.

We would do this with society, integrating our preferences over many 
voters, thereby likely increasing the wisdom of the result. Now, the 
fly in the ointment is sincerity. If voters distort their votes, what 
is the effect.

However, I made this discovery with my study: it does not matter what 
the other voters do, the utility of the sole voter without knowledge 
is maximized by voting sincerely. If it is an approval *election*, 
forcing all to vote in that style, the voter's expected utility is 
reduced no matter what strategy he uses. If all the other voters vote 
approval style, I forget what result I got, it may be that Approval 
is the best strategy.

But *one* other voter can convert the case, creating one point gaps. 
No matter how many voters in the election. So it does not matter if 
99% of the voters vote the extremes, if a few voters vote otherwise, 
it is as if all had voted with full resolution, the vote 
distributions are the same.

Interesting result, eh?


>We're assuming that it's a public election, so that there are so many voters
>that your own votes have no significant effect on the probabilities.
>
>Your Approval strategy is based on two things: The candidates' utility to
>you, and the probabilities that you estimate. The pair-tie probabilities are
>the most widely-recognized probabilities that an Approval voter would
>ideally use if s/he had estimates for those probabilities. (I don't use the
>pair-tie probabilities in the strategies that I recommend, because I don't
>believe that people have a feel for estimating those probabilities). The
>probability that I'm referring to is the probability (when deciding whether
>to vote for i) that either i and j are in an exact tie before you cast the
>last vote, or that j has one more vote than i. In other words, the
>probability that your vote can make or break a tie between i and j.
>
>Your utilities don't change during that series of 10 Approval elections that
>you vote. The probability estimates don't change either, because we're
>assuming that it's a public election, with so many voters that your own
>votes have no significant effect on the probability estimates.
>
>Therefore you vote the same way in all 10 of the Approval elections. If you
>give to a candidate any points at all, you give hir 10 points.
>
>And that maximizes your utility expectation in 0-10 RV. Since you're
>maximizing the good that your votes do for you in each of the 10 Approval
>elections, then the result must maximize the sum of that good.
>
>Obviously, I could have said "N" instead of "10". I said "10" for clarity.
>
>That completes the demonstration that Approval strategy, voting only top and
>bottom ratings, is optimal in RV.
>
>As I said, that's true if it's a public election, whenever it makes any
>difference whether give a candidate an extreme vote or an intermediate vote.
>
>The details of what your strategy would be are irrelevant to this
>discussion. What matters is that it's based on utilities and probabilities,
>and that those do not change during the series of 10 Approval elections that
>you vote, in making out your 0-10 RV ballot.
>
>Mike Ossipoff
>
>
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