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

[Michael Ossipoff]

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, 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.

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