[EM] Range Voting in presence of partial information of a certain character

[fsimmons at pcc.edu]

Thanks for the comments Kevin and Lomax.

Let me start over in the same vein:

Suppose that candidate X was just announced as the winning candidate, and no
indication was given of how the other candidates fared in the range style election.

How would you wish that you had voted your range ballot?

Personally, I would be mostly satisfied with my ballot if I had given max
support to everybody that I preferred over X, and no support for anybody I liked
less than X. 

But what about X ?  I could say that since X was the winner, X probably didn't
need my support, so I would wish that I had not supported X at all.  But if
everybody took this attitude, then everybody would regret their support for X,
except those that preferred X over all of the other candidates.  And most likely
X could not have won with support only from those who considered X as favorite.

Suppose that due to some technicality the election had to be repeated.  Would
you give any support to X this time around (still not knowing anything about how
the other candidates fared) ?

In my last message under this topic I suggested that perhaps the thing to do in
the case of a sure or almost sure winner (when you know nothing about the
chances of the other candidates) is to just give them your sincere rating.  

Sincere ratings can be constructed by asking questions like this:  Would I
prefer X to a lottery of 31%favorite+69%worst?  Suppose that the answer is yes,
but when the same question is put with the percentages changed to 29 and 71, my
answer changes to no.  Then my natural rating for X would be about 30 percent.

What is the point of all of this?   I'm looking for a DSV (Declared Strategy
Voting) method that takes sincere natural ratings and converts them into
strategic range ballots in such a way that when the winner is announced, the
voters will be as satisfied as possible with the way the DSV handled their ballots.

It turns out that it is impossible to do this in such a way that everybody is
perfectly satisfied with the handling of their ballot.  So what I am trying to
do is to minimize the number of disgruntled voters or minimize something like
the total or maximum disgruntlement of the voters.

How do we define "disgruntlement" in this context?

Here's another stab at this problem:

Let  r  be the highest rating in the allowed range such that for some
alternative X ...

If 
the DSV approves all alternatives rated above X on all of the ballots, none of
the alternatives rated below X on any of the ballots, and all of the
alternatives rated equal to X only at level r and above ...

Then
alternative X is the approval winner.

At this level r there may be several alternatives that would qualify as the
alternative X in the above statement (just as in Bucklin there may be several
alternatives with the same median rank or rating).

Of the alternatives X that fulfill the above condition for the level of r
defined above, which one should we choose?

Should it be the one with the greatest approval total under the above
conditions? Or how about the greatest approval margin?  Or should it be the one
that needed the fewest approvals at the level r to which we stooped in order to
satisfy the above condition?

In other words, requiring some ballots to approve X when rated at level r (below
the topRating value) is the thing that is most likely to cause disgruntlement. 
And that is what we want to minimize.

Note that this does not mean that the level r alternatives will be approved on
every ballot (unless r happens to be the maxrange value).  When r is below the
toprange value, the only ballots that approve the alternatives at level r are
those that rate the winner X at level r or below.

Suppose, for example, that for both alternatives X1 and X2 we have to stoop to
r=85% in order to get enough support to sustain a win, but in the case of X1
thirteen approvals are required at the 85% level, whereas for X2 only seven
approvals are required at the 85% level.  Then X2 requires less disgruntlement
than X1 in order to be a range winner.  If all of the other alternatives require
stooping to approve X at a level strictly below 85%, then X2 is the winner by
this DSV method.

More comments?

Thanks,

Forest