[email protected] wrote:
If we had a method that chose the winner on the basis of all possible
candidates, rather than just the actual candidates, then the method would
satisfy the IIAC, because any loser that was removed would automatically be
replaced with a virtual candidate at the same position in issue space.
Eight or nine years ago Richard Moore, a former contributor to this list,
suggested this idea in conjunction with Copeland or Borda. If we could populate
the issue space with a uniformly distributed dense set of virtual candidates,
and infer from the given ballots how the same voters would rank them, then we
could apply Copeland or Borda to the entire set of candidates (actual and
virtual), and the Clone problems would go away because all of the potential
clones for all of the candidates would always automatically be included. In
Richard's method, the real candidate with the highest Copeland or Borda score
would be elected.
However, we know from Arrow's theorem that if the method satisfies IIAC,
it loses at least one of the other criteria we want. Is the idea then to
try to weaken IIAC the least possible to get the other criteria?
The idea also reminds me of one I had for proportional representation
methods. That idea was to somehow reconstruct a probability distribution
function of the random voter over opinion space, then just sample it
regularly. For instance, on the 1D political line, while a single-winner
method would pick the candidate closest to the half-point (median
voter), a two-winner PR method would pick the candidates closest to 1/3
and 2/3.
Is there a more practical way of doing this?
Recently it dawned on me that when we use cardinal ratings (aka Range ballots)
we have a simple, natural, ready-made way of specifying positions of virtual
candidates as well as knowing how the voters would rate them relative to the
other candidates:
We can get virtual candidates by taking affine combinations of actual
candidates, and we can get their respective ratings on the ballots by computing
the same affine combinations of the respective ratings for the candidates on the
ballots.
With cardinal ballots, you could also construct opinion space by using a
synthetic coordinate system. Say that all the voters vote based on
difference of opinion alone (rather than some objective quality of the
candidate). Then the difference between voter A's vote and voter B's
vote (by some metric) is related to their distances in opinion space.
Feed the distances into an algorithm to determine objective coordinates
so that the distances match as closely as possible, and those could be
used as opinion space coordinates.
However, people don't vote based on difference of opinion alone. Some
candidates can be, objectively speaking, worse than others; and there is
also the problem of anchoring the candidates themselves, who don't
submit any ballots. Still, the concept may be useful. If IIAC failure
appears in a method based on this, I think it would do so because of the
metric or synthetic coordinate algorithm, either of which makes
assumptions (about difference of opinion and dimensions of opinion
space, respectively) that we can't test.
Is there another way to get a metric on the voters besides the one suggested
above?
See above :-)
Of course, we could have them fill out a questionnaire including questions about
the potential candidates, and then use the differences between their responses
to construct a metric.
But how do we make sure that "clone questions" on the questionnaire don't bias
the metric?
Once we solve that problem, i.e. once we have a metric on the questionnaires, we
can consider each point (in some dense uniformly distributed set of points in
the metric space of voters) to be a virtual candidate.
Apply your favorite method to the set of virtual candidates, and then let the
questionnaire closest to the virtual winner pick the actual winner.
This approach reminds us of the "dictator theorem" where one voter gets to pick
the winner, except that in this case one questionnaire has all power, the
questionnaire closest to the virtual winner as determined by your favorite method.
That would fail universal/unrestricted domain, I suppose, because it
would require more information than just the voter preferences.
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