# Distributed Multi-Voting

Distributed Multi-Voting (DMV) is a Single-Winner and Multi-Winner voting system.

This voting method consists of evaluating all possible elections (subset of candidates) to find out which candidate loses the most and then eliminate him; by repeating the procedure several times, 1 or more winners (candidates left) are obtained.

## Procedure

Each voter has 100 points to distribute among the candidates according to his preferences. All candidates in the vote have 0 points by default.

1. For each single vote, get the normalized votes on all subsets containing at least 2 candidates. Add up the points for each candidate of the normalized votes, obtaining the converted original vote.
2. After obtaining all the converted original votes, the candidate with the lowest sum, of the converted votes, loses.
3. Eliminate the loser from all the original votes, and setting the candidate with the lowest score in each vote to 0. Repeat the whole process from the beginning, leaving as many winner as you like.

% of victory: got the winners, eliminate the losers from all the original votes and normalize. The % of victory are obtained from the sum of the points for each candidate.

### Normalization

Given a vote like this: A,B,C,D to normalize it to the subset of candidates A,B,C you have to:

• set the candidate (s) with the lowest score between A,B,C to 0.
• apply the following formula on the other candidates:
S = sum of points of the candidates in the subset.
v0 = value of candidate X, before normalization
v1 = value of candidate X, after normalization.
$\begin{equation} v1=\frac{v0}{S} \cdot 100 \end{equation}$ In normalization for the % of victory, use the same formula without setting the candidate with the lowest score to 0.

If the candidates of the subset, in a certain vote, all have the same score different from 0 then, before normalization, don’t set the lowest score to 0.

## Tactical vote resistance

The original vote of the voter through point 1 of the procedure is converted, and the vote obtained is in part of the type:

• ranking (Borda), because the points tend to be distributed linearly in the converted vote (see all cases).
• range (Score), because by distributing the points in quite different quantities, the candidates tend to keep their score in the converted vote (see A,B,C in cases , , ,  ).
• cumulative, because the points distributed in the converted votes are however limited and fixed (700 in the case , 1000 in the cases  and , 1100 in the cases , ,  based the number of candidates evaluated).

Meets Honesty criterion under the same assumptions as the Distributed Voting.

Under the same assumptions as the DV, this system also tries to satisfy the Perfect Honesty criterion, by applying a conversion of the vote which serves to drastically reduce the deviation from the honest one.

  Example
Candidates:             [A    B   C   D   E]
Honest vote:            [50   30  15  5   0]
Tactical vote:          [90   6   3   1   0]
Honest converted vote:  [1118 820 486 176 0]
Tactical coverted vote: [1434 622 390 153 0]
Error in %:             [22%  24% 20% 13% 0]   Avg error = 20%
633 out of 2600 is around 25%.


On average about 80% of the score, expressed in the tactical vote, is honest at the start of the counting, even if the voter used a tactical vote. A good representation of interests is also provided.

Other properties

In cases  and  it’s noted that the addition of 1 point on B, left the score of A practically unchanged in the converted vote, but in case  it obtained 303 points for B (same speech observing the case  or even  ); this means that the voter has an interest in expressing his preference towards B. At the same time, the voter doesn’t even have the interest of giving his limited points to candidates he doesn’t really support (reduced Vote splitting) .

## Systems variants (Multi-Voting method)

In general, starting from a vote, all the votes are obtained for each subset of the candidates (some normalization is used), and then they are added together obtaining the converted vote. Subsequently, the count on the converted votes is applied.

### Score Multi-Voting (SMV)

The following formula is used to normalize the votes:

[MIN, MAX] = [0,9] = range of the voting system.
[min, max] = [3,6] = lower and higher value among the candidates of the subset, in a vote.
Cnew = new value of the candidate, in a vote.
C = old value of the candidate, in a vote.

 $\begin{equation} C_{new} \ =\ MIN\ +\ ( C\ -\ min) \cdot \frac{MAX\ -\ MIN}{max\ -\ min} \end{equation}$ If max = min, then Cnew = MAX


Example of normalized voting for a subset of candidates:

Original vote:    A B C D F G H
Subset {A,C,F}:   A C F
Subset {A,B,C}:   A B[2.25] C
Subset {C,D,F}:   C D[5.4] F
Subset {B,C,D}:   B C D
Subset {B,C,D,F}: B C[7.5] D[4.5] F
Subset {G,H}:     G H
Subset...


By adding the converted votes, the candidate with the highest sum wins.

### STAR Multi-Voting (STAR-MV)

The converted votes are calculated, as in the Score Multi-Voting, on which the STAR count is then applied.