[Election-Methods] A "sincere" ballot is suboptimal in Range Voting.

[Abd ul-Rahman Lomax]

At 03:15 PM 7/25/2007, Stephane Rouillon wrote:
>This is my opinion, I have no proof yet.

Thanks for both, for the opinion and for the acknowledgement of no 
proof. Good starting point.

>But the more I read and the more I am convinced that
>"sincere" voting is a suboptimal strategy for any Range voting method.
>Thus Approval should be preferred as an election method to replace it.

First of all, the second does not follow from the first. It can 
occur, and indeed I have indications that it does occur, that 
individual strategy in a Range election is maximized by 
approval-style voting, but not only is overall satisfaction maximized 
by sincere voting (this result, if the votes are fully sincere, i.e., 
not normalized, is a tautology), but restricting the election method 
to Approval results in lower expected return for the voter. That is, 
suppose the voter could vote on the question, should this election, 
we are going to have after this vote, be Range 2 (0,1,2) or Approval 
(0,1), and the voter maximizes expected return by voting for Range 2 
and then voting either sincere or Approval strategy, as appropriate 
-- it seems to depend on the utility pattern -- in the election.

Secondly, I have only studied two special cases, and the results were 
not what I expected, and, it seems, not what others expected, in 
detail, as well. The first of these cases is truly artificial, with 
only two voters.
But the second is far more general: many voters (sufficiently large 
that three-way ties are essentially impossible), but, still, 
zero-knowledge, with all presenting vote patterns being equally likely.

This latter assumption can be challenged, but my guess is that we 
aren't going to do better. We'll see.

>M. Lomax (or please Kevin or anyone else promoting Range Voting),
>
>if you disagree with this statement, please show me a clear
>counter-example,
>with the number of candidates, the number of available values,
>the sincere optimal ballot and all others ballots subset with their
>probability
>(I suppose they would be equals from what you said before).

Yes. There is a spreadsheet at 
http://beyondpolitics.org/OptimalRangeVote.xls. It's my current 
source of information on this. I keep updating it, and parts of it 
have been entered manually, that is, the winners and utilities have 
been determined for each case manually. I made errors the first time 
I entered these, and there might still be some. And it only takes one 
error to change the result.

The election has three candidates, and our voter has utilities for 
ABC of 210. We have no knowledge of how the other voters, and, with 
many voters, and restricting our examination to situations where the 
vote of the voter has no effect on the outcome, and excluding certain 
situations where there is a three-way tie or near-tie before our 
voter votes, there are 27 possible presenting votes. These are the 9 
possible votes in each candidate pair, and there are three pairs, all 
equally likely. I've expressed each vote in the form x10, which 
represents a vote where the vote for A is below the threshold where A 
could win with a totally optimal vote from our voter, shown by the x, 
the voter votes 1 for B, and 0 for C.

(These votes have had an offset subtracted from them. The offset 
reduces them to, as it were, the vote of a single voter, voting in 
each pairwise election. The offset taken from the condorcet loser 
reduces that vote to -3 or lower, and, in the spreadsheet, I now use 
-3 instead of X. It guarantees that the candidate getting this vote 
cannot win or tie, no matter what our voter does).

>I bet Chris or myself can find another unsincere ballot that produces
>an equal or better expected result relative to the average social
>utility of the outcome.

And the stakes are?

My trial shows that *in this narrow case*, of three candidates, Range 
2 (also called CR-3), the optimal strategy is the normalized sincere 
vote of 210. The utility for 220 and 200 are identical, 39/27, 
compared to 1.0 for not voting, and the utility for 210 is 40/27. Check it out.

I highly doubt you will find a ballot that will improve the outcome 
over these, on average. There are ballots that approve the outcome 
with a specific presenting vote, but those same ballots reduce the 
outcome for other presenting votes, and, it turns out, there are more 
of the latter cases, five vs. three. Since the utility reduction in 
the extra cases is 0.5 each, the net utility loss by voting Approval 
style is 1/27. Small. But there. And given the other values of voting 
sincerely, which would be a good idea even if there were some *small* 
loss of best-strategic-vote utility, sincere voting is starting to 
look quite a bit better than before this study was done.

We already knew that maximally sincere voting (tricky to do, as well 
as tricky to manage politically) produces the best possible outcomes 
in overall social utility, they are, in fact, the *definition* of 
maximum outcome, the very measure of it.

The debate ranges over the question of whether or not other voting 
patterns optimize the voter's return from the election -- and most 
have thought that some do, under some common circumstances -- and, 
more importantly, what damage is done to a sincere voter by this. If, 
though, the sincere vote is the best strategy, or at least as good as 
the best in most cases, there is no damage to the sincere voter *from 
voting sincerely*. That's an important result, if true.

The other discovery here is something which, as far as I can tell, 
nobody even suspected. There is apparently a Range N, probably, which 
is optimal from the point of view of individual voter strategy. If 
you were a voter, and you could, before the election, *choose* the 
value of N, not knowing who the candidates are, even, what would you 
choose? What choice will optimize your personal expectation of 
utility. One might notice that this choice will, by definition, 
maximize overall social utility as long as other voters realize, like 
this voter, the optimal strategy to use.

It may be objected that the optimal strategy is Approval of some 
kind, but even if that were true -- and this is the very argument we 
are having, is it always true -- so the voter is maximizing his 
return by setting up the "suckers," in the delicate language of 
Ossipoff. However, the method doesn't discriminate, and even if it 
were true that individual voters who voted intermediate votes were 
suckers, *it only takes one in the entire election to make the 
presenting vote cover the full range.*

I need to repeat again that I've only studied a very narrow case. I 
was motivated to do so by what seemed to me to be an outrageous 
claim, that voting anything other than approval was suboptimal. This 
original statement is clearly false, made as a general statement; for 
the best vote does apparently depend on presenting conditions. I pick 
an extreme to work with: zero knowledge. If I know the presenting 
probabilities, if, for example, I know that a particular pair is 
going to include the winner, my vote indeed becomes Approval style 
for those candidates. But this says nothing about how I rate the 
third candidate! And there are very good reasons for wanting to be 
able to rate certain losers, it is socially useful!

Voting "approval style" means that *all* votes, not just effective 
ones, be cast as max or min.