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

[Jobst Heitzig]

Dear Raph,

you wrote:
> I was thinking of a 'stable marriage problem' like solution.

Good idea! If it works, the main difficulty will be to make the whole process monotonic, I guess...

Yours, Jobst


> 
> 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.
>