[EM] A modified Random Ballot that supports compromising

Jobst Heitzig heitzig-j at web.de
Wed Jun 28 14:27:47 PDT 2006

Dear Forest!

Another modification of Random Ballot that encourages election of good
compromise candidates, using ranked ballots (instead of approval ballots
as in AP-RB):

1. Ranked ballots with truncation.
2. Draw two ballots at random.
3. If no candidate is ranked on both, elect the one ranked top on the
first ballot.
4. Otherwise, elect from those candidates ranked on both ballots the one
ranked topmost on the first ballot.

In the example with true preferences and utilities

a times  A:1 > C:alpha > B:0
b times  B:1 > C:beta  > A:0

and ballots

a-d times  A
d   times  A>C
e   times  B>C
b-d times  B

the winning probabilities are A:a-de, B:b-de, C:2de, so that the
expected utilities are

a+(2alpha-1)de for the A-supporters and
b+(2beta -1)de for the B-supporters.

That is, no matter how many of the A-supporters rank C 2nd, it is
advisable to do so for all B-supporters if and only if beta>1/2, and
similarly for all A-supporters is alpha>1/2. In particular, no
prisoners' dilemma arises! Since it encourages *full* cooperation
instead of only partial cooperation, this method leads to even better
social welfare than AP-RB.

Like AP-RB, this method is monotonic and cloneproof, and any proportion
of p% of the voters can distribute p% of the probability as they like
(by bullet-voting accordingly). On the other hand, the approval winner
need not get a winning probability proportional to her approval
proportion as it was the case with AP-RB.

Again, this method can be safeguarded against dangerous candidates by
elimination all who are stricken-out by more than 75%.

Yours, Jobst

I wrote:

> Dear Forest,
> You wrote:
>>Martin suggested listing all of the candidates in order of approval
>>count, and then on each ballot circle the candidate approved on that
>>ballot that is highest on the list.  Each ballot has one candidate
>>circled, so each voter ends up supporting exactly one candidate.
> That's very similar indeed and even simpler.
>>Do a plurality count of circled candidates.  It's not hard to show
>>that the Plurality winner is always the highest candidate on the
>>list, i.e. the Approval winner.
> So his was not a new method but a new interpretation of Approval Voting?
>>Your method is an improvement in some ways, but how to eliminate
>>dangerous candidates w/o destroying the nice properties???
> First, why don't I just circle the most approved candidate on each
> ballot? Because I want to ensure that each group of p% of the voters can
> distribute their share of p% of the probability independently from the
> other voters, as in Random Ballot, but with an easy way to cooperate and
> support compromise candidates without having to order reverse.
> Second, how can we make sure no dangerous candidate can be elected? My
> conjecture is that in any system that does not give a mere majority 100%
> of the power, dangerous candidates can only be avoided when they are
> considered dangerous by a very broad share of the voters, say 2/3. The
> easiest mechanism would be to simply let voters mark candidates they
> consider really dangerous with a minus sign and exclude all who receive
> more than 75% minus signs. In the very unlikely event that no candidate
> remains, the election should be repeated, thereby disencouraging misuse
> of the minus sign somewhat. So my suggested method is this:
> ------------------------------------------------------------
> 1. Approval style ballots with the option of striking out candidates
> which one considers really dangerous for society (not just unwanted).
> 2. Eliminate all candidates stricken out by more than 75% of the voters.
> If no-one remains, repeat the election.
> 3. Determine the approval winner among the remaining candidates, using
> all ballots (in the first iteration of this step) or only the remaining
> ballots (in later iterations of this step). On all of these ballots
> which approve this candidate, mark this candidate and put these ballots
> aside.
> 4. Repeat step 3 as long as there are some remaining ballots.
> 5. From all ballots, draw one at random. The winner is the candidate
> marked on that ballot.
> This method is still monotonic and cloneproof and gives the largest
> probability to the most approval candidate not considered dangerous by
> 75% of society. Only very large majorities can oppress the rest of
> society by using the safeguarding option strategically.
> Jobst
> ----
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