# [EM] Using Schulze Election Method to elect a flexible amount of winners

Juho Laatu juho.laatu at gmail.com
Fri Oct 7 02:10:55 PDT 2016

```> On 07 Oct 2016, at 07:29, VoteFair <ElectionMethods at VoteFair.org> wrote:
>
> On 10/5/2016 2:16 AM, Kristofer Munsterhjelm wrote:
> > On 10/03/2016 04:29 PM, Paul Smits wrote:
> >> ...
> >> Now the question arose how we could use the Schulze method in a decision
> >> where the amount of winners is also up for debate.
> ...
> > The way I'm reading this is that you're wondering how you might extend a
> > Condorcet method to a situation where you want "how many winners to
> > elect" to also be up to vote, and not just to a situation where you know
> > the number of winners to elect.
>
> In my previous reply I focused on the importance of using a proportional method rather than a single-winner method, and I forgot about this part of the question.
>
> The answer for this part depends on the answers to these questions:
>
> * What is the range for the number of seats?
>
> * Is the number voted on at the same time as the candidates are ranked?  Or are there two rounds of voting such that the voters know which candidates will get elected when they vote on increasing the number of winners?
>
> As an extreme case, the transition from 1 to 2 seats likely would not get majority support -- because the majority chooses the first winner, who gets total control.

I was wondering why should the voters decide what the number of seats will be. It is not a positive goal to do it for strategic reasons, like in the example above. Alternatively we could assume that they want the number of seats to be such that it gives a fair result. One could derive that result from their votes. Let's study my earlier example (just as a random example).

40: A>B>C
10: B>A>C
10: B>C>A
40: C>B>A

- If we elect one representative, that could be the centrist candidate B. But 80% of the voters would not have their own representative.
- If we elect two representatives, maybe A and C. No representative for 20%.
- Three representatives, A, B and C. A and B voters will be somewhat under-represented.
- Four representatives, A1, A2, B and C (assuming that A, B and C are parties with multiple candidates). A gets more representatives than C.
- Five representatives, A1, A2, B, C1 and C2. Perfect match.

The conclusion is that if voters want to be represented fairly, obviously they prefer having five representatives to having four representatives. There could well be a range of possible sizes for the representative body, and the method would pick the size that gives the most accurate proportionality. A simple algorithm could be one that minimizes the difference between proportions on representatives and corresponding proportions of first preference votes.

If the voters can in addition vote directly on the size of the representative body, these approaches could be combined so that the voters could pick whatever range, but the width of the range would be predetermined. Alternative ranges could be e.g. 10-15, 11-16, 12-17 etc., but the final number of seats would be chosen automatically by the method, from the range that was chosen directly by the voters. This would eliminate some strategic intent, although this effect could in many cases be marginal. The effect of getting more accurate proportionality os more important here.

In general it makes sense to the supporters of large parties to vote for a small size of the representative body (to cut some of the smallest parties out). The small party supporters would prefer a large body. The best strategy depends on the used allocation method (some method favour large parties). And on if there is a cutoff threshold. And on the activity of incumbent party voters vs. voters of those parties that have no seats. I can't give clear rules on what the balance would be, but what I'm trying to point out is that it is possible that for example in time (after say ten rounds of elections) this kind of ability of voters to freely decide the number of the seats, would lead to minimal number of seats if the balance is in favour of the large parties, or to a very high number of seats if the balance favours small parties.

Juho

>
> A similar but more complex effect can occur going from 2 to 3 candidates, or going from 3 to 4 candidates.
>
> At about 5 candidates the proportionality of the results stabilizes -- if a good proportional method is chosen.  At about this number or higher there is less strategic advantage (mathematically anyway) to support or oppose an increased number of seats.
>
> Without knowing the answers to the above questions it's difficult to further answer this question beyond Kristofer's thorough reply.
>
> Richard Fobes
>
>
> On 10/5/2016 2:16 AM, Kristofer Munsterhjelm wrote:
>> On 10/03/2016 04:29 PM, Paul Smits wrote:
>>> Dear election enthusiasts,
>>>
>>> First of all I would like to congratulate you on the great wealth of
>>> works and ideas you brought into the world of voting/election methods.
>>> Even though it may be out of the scope of your focus, I have a
>>> consideration I would like to consult you on. If I came to the wrong
>>> place, let me know.
>>>
>>> In my organisation we are implementing the Schulze method to all
>>> situations where a single winner or sorted list of winners has to be
>>> chosen from more than two options. We basically did a straight
>>> implementation from the wikipedia pseudocode into our own online voting
>>> system.
>>>
>>> Now the question arose how we could use the Schulze method in a decision
>>> where the amount of winners is also up for debate. We used to make this
>>> decision by conducting an approval vote with a certain threshold for
>>> threshold. Now some colleagues wish to again see some value by which the
>>> quantity of support for all the candidates can be understood.
>>
>> The way I'm reading this is that you're wondering how you might extend a
>> Condorcet method to a situation where you want "how many winners to
>> elect" to also be up to vote, and not just to a situation where you know
>> the number of winners to elect. If it's just an ordinary multiwinner
>> situation (you know the assembly will be, say, 12 seats), see below.
>>
>> If your scenario can handle repeated balloting, I would suggest
>> something like this:
>>
>> Produce a list by the Schulze proportional ranking
>> (http://m-schulze.9mail.de/schulze2.pdf).
>>
>> Then, starting with one winner, repeatedly hold a majority vote of
>> whether the assembly size should be increased to two seats, three seats,
>> etc. When the motion fails, the assembly size is set.
>>
>> If you can't do repeated balloting, something like this should work:
>>
>> On the ballot, have two questions. The first asks for the voter's
>> ranking of the candidates, and the second asks for how large an assembly
>> he wants to have.
>>
>> Once you have the ballots, determine the median value of the second
>> question: this is the value where just as many voters want a larger
>> assembly as a smaller one, so in a sense there's a balance there.
>>
>> Then use either a proportional ranking method (like the Schulze
>> proportional ranking) or an STV method (like Schulze STV) to elect that
>> many candidates.
>>
>> The greatest problem with these proposals, I think, is that both Schulze
>> STV and the Schulze proportional ranking are very complex systems. You
>> might have a problem getting them accepted to begin with.
>> ----
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>
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