# Distributed 2-Voting

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Distributed 2-Voting (D2V) is a single- and multi-winner cardinal voting method.

## Procedure

Voter score candidates with range [-5,+5] or [-9,+9] without the value 0. Each vote is split into two votes, which are then normalized to 100 points:

• positive vote: it contains positive scores (all other scores set to 0).
• negative vote: it contains negative scores, to which the maximum range value is added (all other scores set to 0).

### Counting

1. Sum of all positive votes. If a positive vote has all values equal to 0, then the negative vote is added to the sum instead of the positive one.
2. The candidate with the lowest sum is removed, and all votes normalized.

By repeating processes 1 and 2, a worst candidate is eliminate each time.

The remaining candidates are the best (winners).

## Procedure specification

### Normalization formula

   P = 100 (can also be set to 1).
S = points sum of the candidates remaining in the vote.
V = old value of candidate X.
newV = new value of candidate X.

$\displaystyle newV=\frac{V}{S} \cdot P$



### Example of vote conversion

Original vote:   [-5|-4|-3|-2|-1|1 |2 |3 |4 |5 ]
Voting split
Positive vote:   [0 |0 |0 |0 |0 |1 |2 |3 |4 |5 ]
Negative vote:   [0 |1 |2 |3 |4 |0 |0 |0 |0 |0 ]
100-point normalization
Positive vote N: [0 |0 |0 |0 |0 |7 |13|20|26|34]
Negative vote N: [0 |10|20|30|40|0 |0 |0 |0 |0 ]


### No 0 in range

1. In the range [-9,+9] the value 0 is given to a candidate considered to be best of negative candidates and worst of positive ones. To satisfy this condition, it's necessary to convert 0 to +9 in the negative vote.
2. The value 0 however, in the original vote, is assigned by default to the candidates who aren't evaluated (the unknown ones), and it should therefore remain 0 both in the positive and in the negative vote, because an unknown candidate must not be favored in any way.

The two points indicated above are in contradiction, and this problem can be solved by forcing the voter to assign at least -1 or +1 to all candidates, even those to whom the voter would like to give 0. This is also consistent with the philosophy of the method that requires you to create 2 distinct votes, and the 0 isn't clear on which side it should stand.

## Properties

### Unknown candidates

Unrated candidates (unknown ones) receive 0 points. The vote, after normalization, consists of 100 points distributed, and starting from this observation:

1. in the positive vote, it's better to favor the approved candidates, instead of using limited points on unknown candidates.
2. in the negative vote, it's better to favor the less disapproved candidate (compared to the most disapproved), instead of using limited points on unknown candidates.
3. if there was only one disapproved candidate in the negative vote, the voter could use the 100 negative points on unknown candidates favoring them unfairly. This case is rare because if the candidates are few, then there are no unknown candidates; if there are many candidates then it's very likely that at least two differently disapproved candidates are present.

Overall D2V doesn't favor unknown candidates. D2V supports that unknown candidates must not be favored in any way.

### Independence from results

Between the 2 probable winning candidates known by the voter, considering range [-5,+5]:

• the worst candidate will receive the lowest score, that is -5 points (which with normalization becomes 0 points).
• the best candidate receives a score greater than -5.

By voting in this way, in the vote at the end, the best candidate will have 100 points while the worst will have 0 points.

Knowing the likely winners can only reduce the points given to a candidate who, for the voter, isn't the best of all.

## Other sections

They refer to Distributed Voting (DV) and are also valid for Distributed 2-Voting (D2V):

## Systems comparison

### Distributed Voting

The D2V will ensure that the 100 points of the vote are always used against negative candidates (all at 0 points), as long as there is at least one positive candidate left. Only if all positive candidates are eliminated, then the 100 points will be used to favor the least negative candidates among those left.

In the Distributed Voting instead, when all the positive candidates (with scores other than 0) are eliminated, the vote becomes irrelevant in the clash between the negative candidates.

### IRV

D2V use the IRV process with 2-position.

• in 1st position the positive vote is used.
• when the positive vote loses all the supported candidates, move to the 2nd position where the negative vote is used.

Also the comparison made in the DV is also valid for the D2V.

## Systems variants (2-Voting method)

In general, it consists of dividing the vote into two parts, positive and negative, so that the voter can concentrate first on evaluating the candidates he approves and then those he disapproves. One by one the candidates with the least sum of points are eliminated (instant-runoff is used). Only the positive vote will be considered of the voter, as long as there are approved candidates in it; when they finish, the negative vote is considered.

This general process applies to each following method, it only changes the way in which the two, positive and negative, votes are obtained starting from the original one.

### Approval 2-Voting (A2V)

A range with values {-1,0,1} is used. Unrated candidates (unknown ones) receive -1 by default.

• positive vote: set 0 where there is -1 in the original vote.
• negative vote: set everything to 0, then set 1 where there is 0 in the original vote.

The candidate with the smallest sum of points is eliminated from time to time.

Original vote: [-1|-1|-1|0 |0 |0 |1 |1 |1]
Positive vote: [0 |0 |0 |0 |0 |0 |1 |1 |1]
Negative vote: [0 |0 |0 |1 |1 |1 |0 |0 |0]


### Borda 2-Voting (Borda 2V)

A fixed number of candidates is classified (generally all), then half of the ranking is used for the positive vote and the other half is used for the negative vote.

Original vote: [10th|9th|8th|7th|6th|5th|4th|3rd|2nd|1st]
Positive vote: [0   |0  |0  |0  |0  |5th|4th|3rd|2nd|1st]
Negative vote: [5th |4th|3rd|2nd|1st|0  |0  |0  |0  |0  ]
or
Negative vote: [10th|9th|8th|7th|6th|0  |0  |0  |0  |0  ]
Then assign a certain score based on the position (depending on the type of Borda).


### Score 2-Voting (S2V)

It works like the D2V but without normalizing the votes (i.e. when the candidate with the worst sum is eliminated, his points become 0 in all votes).

### STAR 2-Voting (STAR2V)

It works like the S2V in which however stop eliminating candidates when there are only 2. Among the 2 remaining candidates, the candidate who has less points than the other, in more than half of the votes, loses (only the votes where the 2 remaining candidates have different scores are considered).

### Property of 2-Voting method

• The positive and negative votes always have ratings >=0, in the count. The concept of "disadvantaging a candidate by favoring all others" doesn't exist in this category of methods.
• if all negative votes are null at start (all values equal to 0), then the method becomes equivalent to that in which 2 votes are not used (ex. S2V without negative ratings in the original vote, is equivalent to SV).