[EM] Resume: Proportional multi-winner ranked voting methods -

[Juho Laatu]

Here are some more thoughts on grant allocation methods with varying grant sizes.

First one very simple method. All trustees propose an ideal allocation of grants. The total sum of money that is to be allocated is divided in n parts of equal size (n = number of trustees). Each part is allocated to projects as proposed by the corresponding trustee. These allocations are summed together to find the total amount of grants to each project.

Then a method that provides results that are quite similar (not the same) but is very "Condorcet style". Let's take CPO-STV as our starting point. In Electorama (http://wiki.electorama.com/wiki/CPO-STV) the comparison of a pair of options is described as follows.

1. Eliminate all candidates who are not in either outcome.
2. Transfer excess votes from candidates who are in both outcomes.
3. The number of pairwsie votes for an outcome is equal to the sum of votes for the candidates in that outcome.

Let's replace that part with the following algorithm.

1. Each vote (Vi) allocates some sum of money (Vij) to each candidate/project (Cj)
2. Both compared options (allocations), O1 and O2, are similarly allocations of money (O1j, O2j) to each candidate
3. For each vote and each candidate, the amount of guaranteed money is Gij = min(Vij, O1j, O2j)
4. The remaining strength of each vote is ( 1 - (sum over j of Gij) / all_money )
5. Vote Vi supports O1 (with the remaining strength) if ( sum over j of abs( Vij - O1j ) ) < ( sum over j of abs( Vij - O2j ) )
6. Vote Vi supports O2 (with the remaining strength) if ( sum over j of abs( Vij - O2j ) ) < ( sum over j of abs( Vij - O1j ) )

There may be any kind of restrictions on what kind of allocations are "legal". Only such options (O) will be considered. There could be e.g. limitations on the smallest and largest sum that can be allocated to a project, or on the number of supported projects, or maybe only amounts divisible by 1000 would be allowed. Votes need not, but could follow the same rules.

The method uses some Condorcet method to pick the best option (O), just like CPO-STV. Since the number of different options is large, it may be good to use some Condorcet method that can evaluate options locally, like Minmax that can evaluate one option by comparing it to some other strong options. The proposed method is quite straight forward, so it is not difficult to find local optimums (best options when compared to some known other option). Simple heuristics/algorithms that can search local optimums and compare only them (without listing and comparing all possible options) may provide good results. An alternative approach is to compare all those options that different people / interest groups propose as possible outcomes of the method (within agreed time limits, e.g. 1h after latest best identified option). I'm not sure yet on how efficient one can be with respect to finding the global optimum.

In addition to using a Condorcet method to find the best option (as in CPO-STV), the method is Condorcet-like also e.g. so that if all money is to be allocated to a single project, the method picks the project that is the Condorcet winner, when rankings are derived from the given ratings (= money allocations).

I didn't cover any strategic voting related aspects. I instead assumed that although the trustees disagree on how the grants should be allocated, they are not so competitive that they would try to cheat the method (in order to force an outcome that they personally like more than the fair result produced by the algorithm).

Juho



P.S. I made a small Excel (actually Mac Numbers) file to test the behaviour of the comparison algorithm (available via email on request).


> On 05 Jun 2017, at 02:18, Mat Danaher <mat at organise.win> wrote:
> 
> Hi everyone thanks for the useful and interesting responses. 
> 
> Juho in particular raises an interesting point - yes the grants can be different amounts and you're right we could factor magnitude of the grant into the initial calculation - historically the trustees have all known each other and have used consensus on everything - however last year they couldn't reach consensus so agreed some form of majority voting could be implemented for when consensus can't be achieved, they know I am an organisation and decision making "geek" so asked me to find solutions...
> 
> I've got some reading to do!
> 
> Mat 
> 
> 
> On Mon, 5 Jun 2017 at 01:17, Armando <pitocco.ma at anche.no <mailto:pitocco.ma at anche.no>> wrote:
> 
>> El 2 jun 2017, a las 16:41, steve bosworth <stevebosworth at hotmail.com <mailto:stevebosworth at hotmail.com>> escribió:
>> 
>> In response to your kind thanks below,  I would still like to point out that in contrast to  Associational Proportional Representation (APR), the "proportional multi-winner Condorcet [method] in very large magnitude constituencies" which you currently seem to prefer would still "waste" some citizen's votes both quantitatively and qualitatively.  
> 
> I’ve found your explication of APR
> http://election-methods.electorama.narkive.com/61PGaAbe/em-no-wasted-votes <http://election-methods.electorama.narkive.com/61PGaAbe/em-no-wasted-votes>
> Maybe I didn’t understand, however the main feature is that MPs’ votes have different weight. It implies a different view of assembly. In Italy you should change Constitution. 
> Actually I prefer that all MPs have the same dignity, and would like to find a system where assemblies represent globally all electors.
> ----
> Election-Methods mailing list - see http://electorama.com/em <http://electorama.com/em> for list info
> -- 
> Sent from my iPhone apologies for spelling and brevity
> ----
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