[EM] Some chance for consensus (was: Buying Votes)

[Raph Frank]

On Thu, Oct 30, 2008 at 12:16 AM,  <fsimmons at pcc.edu> wrote:
> Jobst,
>
> That was a great exposition of how to find the consensus candidate when there is one.
>
> Not quite as important, but still valuable, is achieving partial cooperation when that is the best that can be
> done:
>
> 25  A1>A>>A2
> 25  A2>A>>A1
> 25  B
> 25  C
>
> Here there isn't much hope for consensus, but it would be nice if the first two factions could still cooperate
> on gettiing A elected, say 25% of the time. (50% seems too much to hope for)
>
> It seems to me that if we require our method to accomplish the potential cooperation in this scenario while
> achieving consensus where possible, the ballots would have to have more levels, and there would have to be
> an intermediate fall back between the consensus test and the random ballot default.
>
> What do you think?

I was thinking of a 'stable marriage problem' like solution.

Each voter rates all the candidates.

Each voter will assign his winning probability to his highest choice
(probably split equally if he ties 2 candidates for first).

If 2 voters 'marry', then the candidate with the highest score sum is
the compromise candidate.

Solve the stable marriage problem.  It might be necessary to randomly
split the ballots into 2 'genders' to guarantee that a stable solution
exists.

Using the above example:

G1:  A1(100) A(70) A2(0)
G2:  A1(0) A(70) A2(100)
G3: B(100)
G4: C(100)

(unnamed options are rated zero)

If a member of G1 'marries', then the compromises are
G1: A1 (+0)
G2: A (+40), i.e. 100->70 (-30) and 0->70 (+70)
G3: A1 and B tie (+0) .. effectively not a 'marriage'
G4: A1 and C tie (+0) .. effectively not a 'marriage'

Thus rankings are
G1: G2>G1=G3=G4

Similarly
G2: G1>G2=G3=G4
G3: all equal
G4: all equal

Thus the 25 G1s will 'marry' the 25 G2s and compromise on A.

The result being

A: 50%
B: 25%
C: 25%

Also, what about an iterative method.  If the candidate with the
lowest probability has less than 1/3 probability, eliminate him and
re-run the calculations (and probably rescale the ratings).  This is
kind of similar to the requirement that a candidate has 1/3 approval
before being considered.

As an added complication, in the above, it might be worth doing a
second pass.  Once all the marriages are stable, you could have
'suitors' propose to 'engaged' voters and make an offer with a
different compromise candidate.

For example, if two voters has ratings,

A1(100) A2(90) A3(75) A4(55) A5(0)
A1(0) A2(55) A3(75) A4(90) A5(100)

The possible compromises are A2, A3 and A4.  However, A2 favours the
first voter and A4 favours the 2nd voter.  It might be the case that
after being refused, a 'suitor' could sweeten the deal by offering a
better option.