# Sequential Elimination systems

I am breaking this off from another thread and taking a step back. @Essenzia recently proposed a system called distributed voting. It is a multi member cardinal system. Unlike the standard systems like RRV, SMV, SSS, ect which select a winner then reweights to find the next, distributed voting eliminates a candidate then reweights. This would be done down to the desired number of winners. This was not the exact formulation it was given in so I will redefine it.

1. Voters score candidates like in other score systems
2. Each ballot is normalized by the total score given to all remaining candidates by that voter [divide by sum(score)]
3. The candidate with the lowest sum over the normalized scores is eliminated
4. Return to step 2 until the number of candidates is reached.

The difference to what was originally proposed is that @Essenzia wanted voters to give the normalized ballot at voting time. There are two reasons why I think doing it this way is better. The first is that this does not require the voters to do any math. The math may not be hard but it does complicate things. It would make it harder than ranking and that is one things that cardinal systems have over ordinal systems. The second reason is that the normalized scores can be any real number. The original proposition was for the normalization to be done to 100 and the scores given would be integers only. It would have some edging and binning effects which are undesirable. In the end these are the same system so it should not make a mathematical difference.

The big advantage I see in this system is that it is closer to STV in design. STV is a sequential elimination method since you canâ€™t really select with ranking. The institutional power is behind STV so a better system which is similar might be better for advocacy.

There is also no explicit quotas so it might be better for free riding.

A potential complication is that it might be non-monotonic. This is just my intuition based on the similarity to STV

This method is philosophically from the Vote Unitarity school of Proportional Representation. I always thought of that as a subclass of the Monroe school but this does not seem very Monroe. There are likely â€śsequential elimination systemsâ€ť based on the Phragmen or Theile schools. For the optimal metric you could eliminate the candidate which contributes least.

Excellent explanation, I only add my philosophical vision so that itâ€™s clear which school it belongs to:

• I start from a set of candidates (e.g. 10), among whom I donâ€™t know who is the best.
• I make an election to get the first results (sums) with the DV, which clearly show who is the worst of all (the one who receives the least points). This is quite evident as it is evident that the worst cannot be the best. The original set, in which the best could be contained, is reduced by 1 unit (from 10 to 9).
• At this point, in theory, the worst should be removed and the elections redone on the new reduced set of candidates.
In the DV, however, itâ€™s possible to know exactly what a personâ€™s vote would have been, if a candidate had not been there from the beginning, through the proportional redistribution of points in the votes.
In the DV, eliminating the worst candidate and redistributing the points in the votes (normalizing the votes) is equivalent to redoing the elections with 1 less candidate (all while maintaining the representation of interests and equal weight of voters).
A new worst will be found, reducing the set by 1 again, and so on.

In the end the set of candidates that could contain the best will contain only 1 candidate who will therefore inevitably be the best.
By stopping the process first, you can get a set containing more candidates who will be the winners (to whom a specific percentage will also be associated indicating how much better they are than each other).

Observation
Obviously all this works well if the starting vote honestly represent the interests of the voters (that is, if they arenâ€™t tactical).
Be careful though, because while in DV only tactical votes could be the cause of a misrepresentation of interests, in other methods (SNTV, AV, IRV, Ranking, etc) itâ€™s the way you are forced to write the vote, which represents bad the interests in addition then also to the tactical votes.
A method that forces you to roughly represent your interests is worse than a tactical vote (because you can choose not to vote tactically while you canâ€™t choose the form in which to write the vote that depends only on the system).

It makes vote management easier, because voters can be given uniform instructions; rather than have to be divided into baliwicks, as voters can divide the points themselves. If I want to win 2 seats and have 100 voters, with STV, Iâ€™d be most likely to get both seats if I divide them into two groups of 50. But with DV, I can just instruct each voter to score both candidates 50. However, this still has the effect of limiting voter choice within a party.

Also, there is no surplus distribution, which strongly encourages free riding.

Moving the discussion into this thread, per the request of @Keith_Edmonds

This does not need surplus handling as that is done in the normalization step. Can you give an example where that is bad?

3 seats; 2 parties contesting the seats. Each run 2 candidates; Party A outnumbers B 53-47. Party B runs a vote management strategy where each voter gives each of Bâ€™s candidates 50 points. Party A does not, and one candidate winds up averaging 55 points on A ballots, while the other averages 45. Party B wins 2 seats. To ensure election, only a Droop quota of points is necessary (and possibly less), which candidate A1 has exceeded. However, the surplus is never transferred to A2, so A2 finishes in last place.

Note that the system is very closely related to STV. In fact, an election held under DV can be transformed into an STV election where each ballot transforms into several ranked choice ballots, which have a weight that sums to 1. If C1,â€¦,Cn are the candidates whom a ballot gives points to, the weight given to the ranked ballot C1>â€¦>Cn will be (Points given to C1/100)(Points given to C2/(100-Points given to C1)â€¦*(Points given to Cn/[100-(Points given to C1+â€¦+Points given to Cn-1)]). However, unlike normal STV, there are no surplus transfers.

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Favorite Betrayal

You can take a clearer example by writing the honest interests of the groups, how many people are in those groups and what tactical voting consists of.
Ex, 4 candidates in the order A,B,C,D:
â€śG1-3: 54,46,0,0â€ť
Which means â€śgroup of 3 voters giving 54 points to A, 46 to B and 0 to the othersâ€ť.
From what little I understood, I point out that any party could try to convince the disputants to give them a few more points so everything on average is balanced.

Voters Understanding the System

â€śFree Votingâ€ť variant, ie the voter can write the vote in various ways (based on what he prefers); the vote will then be converted (not by the voters) into a 100 point vote, example (0 equals empty space):
X,0,0,0,0 --> 100,0,0,0,0
X,X,X,X,0 --> 25,25,25,25,25
4,3,2,1,0 --> 40,30,20,10,0
40,6,3,1,0 --> 80,12,6,2,0
101,0,0,0,0 --> 100,0,0,0,0
This thing is possible only in the DV and allows the vote to adapt to the complexity required by the voter so it will always be, not only understood, but also appreciated (Iâ€™m referring only to the problem of writing the vote).

3 seats; 2 parties contesting the seats

(EDIT)
A,B,C of the â€śrightâ€ť party.
D,E,F of the â€śleftâ€ť party.
The voting form is separated into 3 parts (3 forms):

1. â€śrightâ€ť vs â€śleftâ€ť
2. A vs B vs C
3. D vs E vs F

The DV is used on every single part (100 points for each part; total 300 points).
Fight n.1 determines how many seats to give to each party.
There are now two ways to proceed:

1. everyoneâ€™s votes are used to determine which candidates are the best in fights n.2,n.3
2. in fight n.2 only the votes of the people who in combat n.1 claimed more â€śrightâ€ť than â€śleftâ€ť are used.
If a voter claims â€śrightâ€ť and â€śleftâ€ť equally (50-50), his vote in fights n.2,n.3 will be considered half (as if it were 50 points instead of 100).

The number of winners to be considered in fights n.2,n.3 depends on the number of seats assigned to each party.

Candidates are A1, A2, B1, B2:

```G1-55: 60, 40,    0,    0
G2-45:  0,  0, 49.8, 50.2```

A1 gets 3300, A2 gets 2200, B1 gets 2241, B2 gets 2259. A2 is eliminated, so the minority B party wins 2 seats.

However, it seems easy to fix this method by just saying that since A1 has a Droop quota (2500), the excess 800 votes can simply be given to A2, who takes the second seat with 3000 points.

Yes this is the flaw that @Marylander was talking about. The need for surplus redistribution is a major flaw. Luckily in score systems they are pretty easy to do.

In the case of parties with candidates and multiple winners, I have exposed this method (I insert it as my quote):

A,B,C of the â€śrightâ€ť party.
D,E,F of the â€śleftâ€ť party.
The voting form is separated into 3 parts (3 forms):

1. â€śrightâ€ť vs â€śleftâ€ť
2. A vs B vs C
3. D vs E vs F

The DV is used on every single part (100 points for each part; total 300 points).
Fight n.1 determines how many seats to give to each party.
There are now two ways to proceed:

1. everyoneâ€™s votes are used to determine which candidates are the best in fights n.2,n.3
2. in fight n.2 only the votes of the people who in combat n.1 claimed more â€śrightâ€ť than â€śleftâ€ť are used.
If a voter claims â€śrightâ€ť and â€śleftâ€ť equally (50-50), his vote in fights n.2,n.3 will be considered half (as if it were 50 points instead of 100).

The number of winners to be considered in fights n.2,n.3 depends on the number of seats assigned to each party.

Applying this method in your example you would get:
â€śleftâ€ť 55% (with A1, A2 candidates)
â€śrightâ€ť 45% (with B1, B2 candidates)
These percentages would be used to distribute the seats.
I assume that 2 seats are given to â€śleftâ€ť and 1 to â€śrightâ€ť.

A1, A2 candidates will then be evaluated only by G1.
Candidates B1, B2 will then be evaluated only by G2.

There is a reason why I evaluate A1, A2 even if â€śleftâ€ť has 2 seats, but I will explain it in another post, here itâ€™s not necessary.