[EM] Proxy voting questions

[mrouse1 at mrouse.com]

(I apologize for this long post -- I thought it would be a  brief couple
of questions but it grew from there.)

I've been looking into proxy voting, and I've been impressed with many of
the features of it, especially how it reduces wasted votes and eliminates
gerrymandering in legislative elections. I do have a couple of questions
about how to handle ties, truncation, and cutoffs, for those who have
looked into proxy voting in more detail.

First of all, let's assume the following:
1. At some time prior to the election, all candidates publicly announce
their own preference profiles. This can range from bullet-voting for
themselves to a complete profile listing all candidates in the race.
2. During the election, voters choose their own preference profile, which
can be as brief as just their top choice to a complete ranking of all
candidates. Ties and truncation (which is simply a tie in the lower
rankings) is fine.
3. Voters with truncated ballots will have incomplete preferences match
their first-place candidate as closely as possible. For example, if I rank
Candidate A>B and leave the rest blank, and Candidate A ranks A>C>B>D,
then my preference will become A>B>C>D.
4. All candidates are given a proxy equal to their number of first place
votes.
5. In case of a tie (ranking two or more candidates in the top position),
a selection rule determines how the vote is distributed.
6. Since the number of seats is limited, there is a cutoff score, and
votes under the cutoff are distributed to winning candidates.

(Steps 1 and 3 give important information about a candidate -- how he
views other candidates --and complete truncated preference orders. A
person could cast a single vote for a candidate he trusts, and know that
he has a better chance of getting someone who matches his views -- even if
his primary candidate loses -- than picking other candidates randomly or
leaving his choices blank.)

In step 5, I can see three major ways of distributing first-place ties in
rankings:

1. Ties go to the higher-ranked candidate (using Borda, Condorcet,
Kemeny-Young, or whatever).
     Pros: Easy to calculate, gives the most popular candidate a stronger
voice in government. Might result in fewer circular ties.
     Cons: Can eliminate candidates under the cutoff, leading to more
wasted votes. Might also give top candidate too much power.

2. Ties are split evenly among candidates.
    Pros: The most natural split -- "If you like both A and B, both get
half a vote."
    Cons: Can eliminate candidates under the cutoff, leading to wasted
votes. Fractional votes would act as full votes, which could be
problematic in close races -- a 100.5 score would win over a 100 score
-- plus the fractions can become quite ugly.

3. Tie votes are given to the lowest ranking candidate.
    Pros: Fewest number of wasted votes, since borderline candidates are
more likely to be pushed above the cutoff. Lower-ranked candidates
also receive a small boost in power.
    Cons: Can be complicated to figure out -- if the number of tie votes
received by a candidate is greater than the difference between two
candidates, the lower-ranked candidate would change. Can make a nice
Condorcet order look like a wacky tangle of preferences, since
pairwise votes get closer together.

Despite its problems, I'm personally leaning toward method 3 (I like the
idea of underdog candidates getting a break and dislike wasted votes), but
I'd be interested in hearing other people's thoughts.

For step 6, the question is how to distribute the votes underneath the
cutoff to minimize wasted votes. Candidates that cannot possibly win with
the remaining votes should transfer their proxy to the next candidate in
their preference order. My question is, would it be best to use a
Condorcet order (or Kemeny-Young order) on the remaining candidates, or
some other method?

Anyway, I find proxy voting an interesting solution to several problems in
our standard winner-take-all district-oriented voting, and I'd be
interested in any thoughts on the subject. I'll probably describe a
unicameral legislature idea I have (Nebraska, anyone? :) ) in a future
post.

Michael Rouse
mrouse1 at mrouse.com