[EM] How to get closer to the impossible ideal of the IIAC

[fsimmons at pcc.edu]

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.

We never really followed up on this idea before getting decoyed away from it
into other questions.

How do we construct the dense uniformly distributed set of virtual candidates? 
If we had an objective way of describing the issue space for any given election,
and a way of knowing where all the voters resided in that issue space, then the
task would be easy.

One way to do this would be to have each voter sketch the two most important
dimensions of the issue space and self identify where they considered themselves
in that diagram. Then the distance between two voters would be computed from how
dissimilar their perceptions of the issues were and how far apart they placed
themselves relative to the issues at hand.  It isn't hard to come up with
metrics that compute distances between diagrams or locations within diagrams or
a combination of both.

Once we have an appropriate metric on the issue space and the positions of the
voters within that space, then that issue space can be filled in with virtual
candidates uniformly distributed according to the metric.  Then any voting
method can be used to pick the best virtual candidate according to that method.

Finally the actual candidates are located within the issue space, and the actual
candidate closest to the virtual winner is elected.  If any of the actual losers
are removed, then the actual winner is still closest to the virtual winner, and
so still wins.  In other words, the IIAC is satisfied.

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.

So if A, B, and C are actual candidates with respective ratings a, b, and c on
some ballot,  and p, q, and r are numbers that add up to one, then
 X=pA+qB+rC 
is a virtual candidate with a rating of
 x=pa+qb+rc.

In this context removing any real candidate would not (in theory) change the set
of virtual candidates as long as the remaining candidates still contained a
basis for the same affine hull.  So to this extent the method partially
satisfies the IIAC.

In other words, the set of "irrelevant alternatives" would include all real
candidates not needed to maintain the affine hull in which the virtual
candidates are distributed.

This idea is not quite good as the more elaborate one sketched above, because it
requires the real candidates to set the stage for the virtual ones, so not all
losers are irrelevant; we have to keep enough to maintain the affine hull.

If we could use the affine hull of the voters instead of the candidates, then we
would be in a better situation.  It is unlikely that a decent election method
would choose a virtual candidate outside the convex hull of the voters.  We
could make that an axiom.

Is there another way to get a metric on the voters besides the one suggested 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.