On May 21, 2010, at 4:13 AM, [email protected] wrote:
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.
In some sense Condorcet is a DSV method for Range. If X is about to
win then voters would like to cast a full one vote for all candidates
that I prefer over X and cast a full vote for X against all candidates
that I like less than X. In a Range ballot (and DSV) that could mean
rating X at 0 or max depending on if the second strongest is liked
less or more than X. If there is a Condorcet winner (X) then there is
a "stable" strategy in the sense that no matter who the second
strongest candidate is considered to be, X will win. Many voters (all
but first preference X supporters) still have an interest to rate X at
0 and all the more preferred candidates at max, but if that would make
one of the more preferred candidates win then there would be more
voters with (DSV) interest to reverse that change. Condorecet methods
would do all this automatically.
We lost all the preference strength information in the Condorcet
process but maybe this is unavoidable (in a highly competitive
environment). Or could there be a declared strategy (or some other
approach) where a voter would be happy to accept election of a
slightly less liked candidate if other voters have clearly stronger
preferences supporting some other outcome. Then we could make use of
the Range ratings (or of indicated strengths of the rankings, e.g.
A>B>>>C>D). Voters could voluntarily give up some of their voting
strength. Maybe something like one indicated weak pairwise opinion to
be canceled (tie) for each corresponding strong opinion in the reverse
direction. (Such weak opinions could be used also to defend clones (or
a grouping or a wing). There was some discussion on this on the EM
list long time ago.)
Juho
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
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