[EM] Voter Space Methods, and Antidote to Clone Sensitivity

[Forest Simmons]

Here's an example of what I have in mind:

Consider the (rather defective) Above Median Approval (AMA) method which
takes ranked ballots as input.  A candidate's score under AMA is the
difference between the number of ballots on which the candidate appears
above the median and the number of ballots on which she appears below the
median. Candidates ranked equally in first place are considered to be
above median, and candidates ranked equally in last place are always
considered to be below median.

AMA does not satisfy clone independence as the following example clearly
shows:

6 ABCD
4 BDCA

Under AMA the respective candidates get scores of
  2, 10, -10, and -2, so B wins with a score of 10.

We see that the "majority winner" A lost to candidate B who was propped up
artificially by clones C and D.

However, if we project this example into Voter Space we get

6 a1=a2=a3=a4=a5=a6>b1=b2=b3=b4
4 b1=b2=b3=b4>a1=a2=a3=a4=a5=a6

AMA applied to this case gives each A voter a score of 6-4 and each B
voter a score of 4-6.  The A's are tied for first place.  But all of the A
voters agree that A should win.

The majority winner A wins in the Voter Space version of AMA because
projection into Voter Space neutralizes clones.

The defective AMA becomes a rather nice method when projected into Voter
Space.

Now, just for fun, let's forget about Voter Space and approach the same
example

6 ABCD
4 BDCA

from the point of view of Joe Weinstein's weighted median approval
strategy.

Joe says the weights should be (in proportion to) the winning
probabilities.  What should we use for winning probabilities?

Well if the method were random ballot the winning probabilities would be
60% and 40% for candidates A and B, respectively.

Let's use 6 and 4 as the respective weights. To find the weighted medians
we replicate A six times and B four times on each ballot (and zero out the
zero probability candidates):

6 AAAAA|ABBBB
4 BBBBA|AAAAA

The vertical bar marks the median position.

According to Joe, candidate A gets approved on six ballots and candidate B
gets approved on four ballots. So A beats B six to four.

If you compare this method (suggested to me by Joe) with the Voter Space
example above, you will see that Joe's method is equivalent to Voter Space
AMA.

In fact, while working on Joe's method, and having Richard's issue space
ideas in the back of my mind, the inspiration for projection of any method
into Voter Space came to me.

I hope this example clarifies the connection.

Forest


On Tue, 3 Feb 2004, Forest Simmons wrote:

> If you have a favorite method that is not independent of clones, you might
> try the Voter Space version of that method to neutralize the clone
> sensitivity.
>
> If you could guarantee that candidates would be evenly distributed among
> the voters, then you would neutralize all clone problems.  This is what
> Voter Space accomplishes, because in Voter Space every voter is a virtual
> candidate.
>
> Here's how it works: every voter ranks all of the other voters according
> to how well they like the other voters' ballots. [This can be done
> automatically, believe it or not.]
>
> Then your favorite method is applied to these voter rankings instead of
> the original ballots.
>
> Finally, the ballot of the winning voter is used to pick the election
> winner.
>
> Enjoy!
>
> Forest
>
>
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