[EM] MultiGroup voting method

[Juho]

Here's one method for PR multi-winner elections. This is to some  
extent a derivation of tree based methods that I have discussed  
earlier, but since this one has also interesting use cases with flat  
structures, and tree based inheritance of votes is not explicitly  
used in the calculation process, I have to use word "group" instead  
of "tree" to name it.

Candidates (or parties) are free to form any kind groups. Some  
typical groups are parties and regions. The groups are allowed to be  
hierarchical and to overlap. Also mandatory groups can be covered  
within the method (all candidates might e.g. be mandated to represent  
one of the regions).

It is possible to provide (close to) optimal PR between parties and  
it is possible to combine this with (close to) optimal PR between  
regions (etc.). One reason behind the interest in richer methods is  
to provide the possibility to have additional proportionality  
criteria in the elections. It is for example possible to guarantee  
proportionality between the left and right wing of a party, or to  
identify some specific interest group within the party (e.g.  
"greenish ones" or "pro nuclear power"). It is also possible to  
provide more fine grained proportionality between different  
geographical parts of the basic region. Parties or regions might  
establish alliances. In addition to these it is also possible to  
offer proportionality to groups that consist of candidates from  
multiple parties or multiple regions. The target is to allow all kind  
of hierarchical and overlapping groups to be used (at least in  
principle, maybe all not supported by all parties even if candidates  
would have interest).

Use of groups gives the voters ability to give "more exact" votes  
than just voting a party. Votes to a candidate of some group will  
contribute to having exact PR of this group (when compared e.g. to an  
open list method). The voters may this way also better influence the  
policy of the party as a whole. (This is to counter claims that the  
parties and politicians never change. More detailed granularity  
provides better feedback to the parties. And voters may gain better  
control of what the policies of the parties will be.)

In the basic version we may assume that each voter simply votes for  
one of the candidates. Instead of trying to draw the possibly tree  
like, possibly overlapping structure of the groups it is maybe  
easiest to imagine that we will list next to each candidate name also  
to which groups that candidate belongs to (not necessarily all the  
groups if the structure has e.g. lots of hierarchical groups that can  
be already derived from belonging to some of the listed groups).  
Voters are then able to see what values their candidates support. We  
could have e.g. the following candidates (C=Candidate, P=Party,  
S=Subgroup, Co=Coalition, I=Interest group).

C1: CoP1P2, P1
C2: CoP1P2, P1
C3: CoP1P2, P1, I1
C4: CoP1P2, P2, I1
C5: CoP1P2, P2, P2S1
C6: CoP1P2, P2, P2S1
C7: CoP1P2, P2

Here we have two parties that are in coalition with each others  
(there may be other parties too in addition to these two). The second  
party has a subgroup. And there is one interest group that has  
members from both parties. All other groups form a tree like  
hierarchical structure but I1 is a separate structure.

There is a function s(g,e) that indicates the strength of the will of  
the voters to elect one additional candidate from some group g when e  
is the set of candidates that are considered to be elected. The rough  
idea is that value 1 (or one quota) means that the voters have the  
strength to get one candidate elected. Value 10 (or 10*quota) would  
mean strong enough support to elect 10 candidates.

If in the (partial) example above the votes to different candidates  
could be C1:13, C2:12, C3:11, C4:10, C5:14, C6:15, C7:16. Then (with  
one very basic s function) the s values for the groups when no  
candidates are considered elected could be I1:21, P2S1:29, P1:36,  
P2:55 and CoP1P2:91 (=sum of the votes of the candidates belonging to  
each group).

The optimal outcome of the election can be defined e.g. as that  
subgroup e (of agreed size) of candidates with which the highest s 
(g,e) value of all the groups (and individual candidates) is lowest.  
If the highest value is the same for more than one group, then the  
second highest value will be compared and so on. It is also possible  
to allocate priorities or other more complex functions to the  
different groups. One trick is to break e.g. country level regions to  
party internal regional groups to get "less strict" PR.

It is possible to adjust e.g. the number of given votes at some  
region to be in proportion to number of inhabitants in that region.  
This is to make the number of elected candidates correspond to the  
number of inhabitants instead of the number of active voters (that is  
quite typical today in elections where a country is divided in  
regions). It is of course also possible to do the same with respect  
to sexes, age groups, ethnic groups etc or to make the votes favour  
some set of voters.

Finding the optimal outcome as defined above is computationally  
complex. Therefore the calculation process could be approximate. It  
could find the elected candidates e.g. sequentially in some kind of a  
preference order. This method may not always produce a good enough  
result, and therefore the result could be further optimised after the  
first round of calculation. Also other options like finding the  
candidates in parallel (e.g. for each party or for each region) and  
then optimising, or just to use a generic optimisation algorithm from  
the start are equally ok. One interesting approach is to allow any  
interested entity to demonstrate a better outcome than the first  
official outcome is within a week after the elections and make that  
outcome the final outcome instead of the first approximation  
(assuming that the preference order of possible outcomes is well  
defined and can be calculated easily).

In the example above it seems that election of three candidates would  
lead to allocating two seats to party P2 and one to P1 (the s  
function could be e.g. largest reminder or d'Hondt). Within the P2  
party subgroup P2S1 should get one seat (=> C6). The remaining two  
seats should go to the I1 group or to the individual candidates with  
most votes (C1 and C7). One of the seats should go to I1 since it has  
21 votes, but should we elect C3 (from party P1) or C4 (from party  
P2)? If we elect C3, then it seems that the third elected candidate  
should be C7. If we elect C4, then it seems that the third elected  
candidate should be C1. Maybe the first  option provides the optimal  
outcome since then the "16 votes strong" support to C7 can be  
satisfied (and the highest remaining unsatisfied opinion strengths  
would be lower than in the second option).

When compared to pure "individual candidate based" methods like STV  
"MultiGroup style" methods (if the votes are to one candidate only  
and the possibility to "cancel" links to groups is not supported)  
have the limitation that it is not possible to support some set/chain  
of individual candidates without supporting also their groups/party.  
One benefit of the MultiGroup approach is that candidates need to  
openly declare their policy. It is thus not possible for the  
candidates to tell to all voter groups "yes, I support especially you  
and your targets". The elected candidates are (morally) bound to the  
groups/targets that they have indicated to support and it is not as  
easy to forget all the (vaguely) given promises after the election  
day. The messages to the parties are also clear. If large part of the  
voters indicate support to policy X, the party can not ignore that  
wish. There will of course also be a corresponding (proportional) set  
of representatives supporting policy X.

The most typical use cases where this voting method would provide  
additional benefits could maybe be party internal overlapping  
groupings in different dimensions. Individual candidates could e.g.  
support independently "green values", "initiative to build a new  
library" etc. Also different options/balances (more or less strict)  
to combine party and regional PR are interesting.

I hope the presented concepts are in reasonable shape although I  
didn't yet verify all of this by programming it (which often reveals  
some gaps in thoughts). Any feedback on potential weaknesses will be  
appreciated.

Juho



		
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