Hallo,

Well, you could calculate the Schwartz set and then
eliminate all those candidates who are not in the
Schwartz set and then apply the Schulze method with
"pairwise opposition" to the remaining candidates.
But this would violate monotonicity because it could
happen that, by ranking the candidate A higher, some
other candidate B, who was in the strongest path from
candidate A to some other candidate C, is kicked out
of the Schwartz set.

You could use the Schulze method with "pairwise
opposition" to calculate a complete ranking of all
candidates and then declare the highest ranked
candidate elected who is in the Schwartz set.
This would satisfy monotonicity, but violate
independence of Smith-dominated alternatives.

Markus Schulze

Thanks for all the suggestions. I'll briefly describe my method to make
my goal clear:

I want to avoid aggregating all votes in a tally, because I think that
loses information. Specifically, I would like a method that can find a
representative ranking. I.e. if I want the top 10 songs, it should
reflect the preferences of everyone, but not only contain electronic
music because a majority likes that.
When I you do the simple Schulze ranking (not STV) I find it
over-represents certain groups because they essentially can vote more
than once.
I think once their preference is (partially) included their voting
strength should lessen, but with a tally their vote can not be removed
(reduced) since it's all agglomerated.

My idea is to combine Quadratic Voting with ranking. The setup is this:

• Everyone has a voting budget of 1.0
• They spread their budget over their preferences: So if you have A > B

C. With quadratic voting this means spending .333 on each preference,
each gets a voting strength of sqrt(.333)

• All votes are averaged.
• Everyone has a policy, which is to have an equal avg. voting strength
  for their preference, and taking the resulting average into account,
  they adjust their votes (using a simple gradient step).
• This is repeated until convergence.

The result is a preference matrix, which has to be turned into a
ranking. I think this is possible by eliminating the weakest preference,
and re-vote and rebalance, until a ranking is obtained. But it is not
clear that this works, it could result in a broken graph, so I'm not
sure about what to do there.

The advantage would be that no structure is lost by tallying. It also
avoids the problem of Quadratic Voting where it assumes people have an
adequate estimate of what others will vote (unrealistic in my opinion),
instead the policy is enacted by the algorithm and the user just
presents a ranking.

I'm not sure if the 'averaging policy' is the best approach, maybe there
are other policies that are more fair. Also, instead of voting on
preferences, the votes can be done directly on candidates, but it is
unclear to me how to take ranking preferences into account there.

Any thoughts?

Marijn Stollenga

On 01/10/15 22:14, Markus Schulze wrote:

Hmm, how do you envision that? You start by selecting the strongest one
and fill a list like that?
How would you prevent people from reusing their voting budget in that case?

Marijn

On 05/10/15 23:52, Toby Pereira wrote:

On 10/05/2015 11:38 AM, Marijn Stollenga wrote:

Schulze, like other single-winner Condorcet methods, is intended for
determining a best single choice given the preferences of the
electorate. This means that if a majority all place the candidates in a
particular preference order, then that order will show up as the social
order. Clearly, that's not a good outcome if you want proportional
representation.

For that, you could use STV or Schulze STV. Or, if you have continuous
ratings (like it seems you have), I would suggest Monroe's method:

• Let each ballot be fractionally or fully assignable to a candidate.
  When the ballot is assigned to a candidate, that candidate gets a number
  of points equal to his rating on the ballot.
• Let the system assign ballots to candidates so that the sum of scores
  is maximized, subject to that each candidate either gets zero ballots or
  v/s ballots, where v is the number of voters and s is the number of seats.

For small elections, this can be solved by integer programming. For
larger elections, approximations exist: see
http://arxiv.org/pdf/1301.6400.pdf for example.

I would prefer no quadratic constraint, but it should be relatively easy
to fit it to Monroe if you want it. When the voter submits his (rated)
ballot, just check that the sum of the squares of the ratings is less
than or equal to his budget.

Of course, that suggests a DSV approach where the system rescales the
ratings so that the voters don't waste budget points on candidates they
can't get elected anyway. Doing that might be trickier; you could
probably make a quadratically constrained programming version of
Monroe's method to handle it, but finding a PTAS for that is... well,
harder.

What do the numbers in the matrix denote?