[EM] Correction. Big CS fault?
Forest Simmons
fsimmons at pcc.edu
Wed Dec 18 18:02:52 PST 2002
Here's the version of Candidate Space (CS) that I like the best now:
The ballots must have some way of determining favorite, so must have at
least the expressivity of Majority Choice ballots.
[The favorite on the expressive side of the ballot must have maximal
positive instrumentality in the instrumental aspect of the ballot, as
well.]
Step 1. Each candidate has a voter profile vector; the ith component is
the rating given this candidate by ballot number i.
[Convert rankings to ratings if necessary.]
Step 2. The distance from each candidate to every other candidate is
measured by comparing their voter profile vectors.
Step 3. These distances are used to generate preference ballots for each
candidate. The closer candidate B to candidate A, the higher the
preference ranking of candidate B on candidate A's preference ballot.
Step 4. If a candidate is the favorite of 7526 voters, then that
candidate's preference ballot is counted 7526 times in a Condorcet
election.
Step 5. The winner of the Condorcet election applied to those ballots is
declared to be the CS winner.
Addendum: If a voter lists two favorites, then he contributes 1/2 to the
count of each in the Condorcet Step (4).
Remark: There are many possible distances that could be used in Step 2.
I prefer using the angles between the profile vectors, which is related
monotonically to the Euclidean distance between the normalized profile
vectors.
[If theta is the angle between the vectors, then 2*sin(theta/2) is the
Euclidean distance between the normalized vectors.]
The angles are most easily computed by taking the inverse cosines of the
dot products of the normalized vectors.
The bigger the dot product, the smaller the angle, so the inverse cosine
step can be left out if you remember that the bigger the dot product the
closer the two candidates are to each other.
The dot product of two approval profile vectors is just the number of
voters that approve both candidates. Dividing this result by the
geometric mean of the magnitudes of the two vectors normalizes the result.
[In general, the dot product of two profile vectors is the sum of the
products of corresponding components. A vector is normalized by dividing
by its Euclidean magnitude, i.e. the square root of the sum of its squared
components.]
With this dot product measure of closeness, the shape of the profile
determines the closeness.
Candidates without any positive overlap in their voter profiles are at
maximum distance in candidate space, 90 degrees apart [on the positive
octant of a unit sphere centered at the origin].
[When negative ratings are allowed, then if two profiles are opposites,
they are at maximum distance, 180 degrees apart on the surface of sphere
of unit radius.]
In the preference ballots generated in Step 4, these maximum distances
show up as common truncations.
By the way, it turns out that in the three candidate case, if the
preference ballots are generated in this way, regardless of the metric
used in step 2, a CW is assured; there can be no cycle.
So somewhere in this 5 step process the cyclical
"contradiction" is eliminated automatically.
All Condorcet methods give the same result!
Whether the same can be said for four or more candidates remains to be
seen.
Remark: Suppose that candidates C1 and C2 each had exactly one (not the
same) supporter in an election with a million voters and twenty
candidates. If we used the un-normalized vectors to measure distance it
would appear that the two unpopular candidates were extremely close,
differing in only two components out of a million.
Yet they could be Hitler and Stalin, at opposite extremes.
Normalizing the vectors before subtraction, or (equivalently) using
angles, solves the problem. They end up at maximal distance apart.
Forest
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