Distributed Voting

Distributed Voting (DV) is a Single-Winner and Multi-Winner, Cardinal voting systems proposed by Aldo Tragni.

Procedure
Voter score candidates with range [0,9]. The vote is then converted to 100 points (normalization).


 * 1) The worst candidate, with the lowest sum of points, is eliminated.
 * 2) The points of the eliminated candidate are proportionally redistributed in each vote (normalization).

By repeating processes 1 and 2, the worst candidate is eliminated each time, and the remaining candidates are the winners.

Extended procedure (single winner)
It's the procedure indicated above in which:
 * the votes are reversed and made negative before counting (subtracting 9 from the original ratings).

Original vote:                A[9] B[7]  C[5]  D[3]  E[1]  F[0] Reversed vote, made negative: A[0] B[-2] C[-4] D[-6] E[-8] F[-9]

''Reversing and making negative means that the voter's 100 points are used to disadvantage the worst from winning (points will be always negative in the counting). This procedure reduces the failure of monotony, for the single-winner case, and increases resistance to min-maxing strategies.''

Paper ballot
Some examples of normalization:

Range [0,9] →   Normalized in 100 points 9,0,0,0     →   100,0,0,0 9,9,0,0      →   50,50,0,0 9,6,4,1      →   45,30,20,5    (note: there isn't 0 in the lowest score)



Digital ballot
By using self-resizing sliders it's possible to obtain a simple ballot that use the cumulative vote, with 100 points to distribute. However, it's better to use range [0,9] also in digital ballot.

Normalization formula
P = 100 (can also be set to 1). S = points sum of the candidates remaining in the vote, after an elimination. V = old points value of candidate X. newV = new points value of candidate X. $$\begin{equation} newV=\frac{V}{S} \cdot P \end{equation}$$

If S=0 then all candidates remain at 0 points.

Normalization example
Given an initial vote of this type, with candidates A,B,C,D,E, are removed in order E,D,C, and 100 points proportionally redistributed each time:

A[0] B[1] C[3]  D[6] E[90] A[0] B[10] C[30] D[60] A[0] B[25] C[75] A[0] B[100]

Tie during counting
Cases of parity can occur during counting, as in the following example:

Vote 1:      A[55]  B[25] C[10] D[10] Vote 2:      A[50]  B[30] C[10] D[10] Sum of votes: A[105] B[55] C[20] D[20]

The tie can be managed in various ways:


 * delete C first, getting a result. Delete D first, getting another result. Check that the two results return the same winners.
 * delete C and D at the same time.
 * randomly delete C or D.

This situation is extremely rare, and even when it occurs it's further rare that the order in which the candidates in the tie are eliminated affects the result. Random deletion is the easiest to use.

Procedure variant (discouraged)
One or more of the following steps are used:


 * When the worst is eliminated, the candidates with the lowest score among those left in the vote must be set to 0, and then normalizes.
 * Surplus Handling (in Distributed Voting it's not used for multi-winner context).
 * If the remaining candidates are contained in a Smith set, then the candidates with the highest sum wins.

Seats allocation
The Distributed Voting indicates the method for obtaining single or multiple winners. The Distributed Voting System also describes how seats should be handled.

Parliament
Procedure for electing parliamentarians:


 * The state is divided into districts (at least 2, and possibly of similar size).
 * Each district must have at least 2 seats (at least 3, for a good representation). To satisfy this constraint you can increase the number of total seats or join the districts into groups.
 * In each district, the DV is used to obtain a number of winners equal to the number of seats in the district. The sum of the points for each winning candidate will indicate the % of victory of the candidates.
 * If P is the power assigned to the district, then the weight of each seat will be: P • "% of victory of the candidate".

Example - 2 districts, 6 seats Districts: d1{70%} d2{30%} Seats:    d1{3}   d2{3} Result:      d1{ A1[40%]  B1[35%]   C1[25%] }    d2{ B2[40%]  C2[35%]   D2[25%] } Seat weights: d1{ A1[0.28] B1[0.245] C1[0.175] } d2{ B2[0.12] C2[0.105] D2[0.075] } Total power: A[28%] B[36.5%] C[25%] D[6%]

If I had unit seats: Seats: d1{4} d2{2} Result: d1{ A1[2] B1[1] C1[1] } d2{ B2[1] C2[1] } Total power: A[33.3%] B[33.3%] C[33.3%] D[0]

Total difference: 5.3% + 3.2% + 8.3% + 6% = 22.8% An average error of 5.7% each candidate.

Government
Procedure for choosing the prime minister (PM) and the leader of the opposition (LO):


 * Parliamentarians elect, through Distributed Voting, the PM. Instead of being normalized to 100 points, the votes in this election are normalized to the weight that each individual parliamentary has (P = weight, in the normalization formula).
 * Once the PM is elected, only the votes that have assigned 0 points to the PM are taken and used to elect the LO, again through the Distributed Voting. Parliamentarians need to know in advance that giving 0 points to a candidate means being against them (opposites).
 * Parliamentarians who gave 0 points to both the PM and the LO, can be considered neutral.

Tactical vote resistance
Hypotheses

Each voter, based on his own interests, creates the following 2 sets of candidates:


 * Winner Set = set containing a number of favorite candidates equal to or less than the number of winners.
 * Loser Set = set containing the candidates who aren't part of the Winner Set.

Given an honest vote, the tactical vote is obtained by minimizing the points of the Loser Set, maximizing the points of the Winner Set, and maintaining the proportions of honest interests within the two sets.

Example Candidates:               [A  B  C  D E]  Honest vote:               [50 30 15 5 0] Tactical vote (1 winner): [90 6  3  1 0] Tactical vote (2 winners): [60 36 3 1 0]

Single winner

Meets the Honesty criterion (on hypotheses) because:


 * at each Update Steps of the count, in which a candidate with points is removed, the tactical vote decreases the deviation from the honest one (the deviation is the sum of the absolute differences of the points for each candidate, between tactical and honest vote).
 * the Honesty Step occurs when the candidate in the Winner Set is removed or when all the candidates in the Loser Set are removed. In the best case, the Honesty Step can occur in the first Update Steps.
 * the Honesty Step is always present because in the single winner, during the counting, all candidates are always removed from at least one of the two Sets.

Example - 1 winner Honest vote:  [50 30 15 5  0] Tactical vote: [90 6 3  1  0] A is removed and the tactical vote becomes equal to the honest one, that is: Vote:            [60 30 10 0]

Multiple winner

Satisfy the Honesty criterion (on hypotheses) only if, in a vote, are removed first all the candidates of the Winner Set or first all those of the Loser Set.

Surplus Handling
Equality: Distributed Voting ensures that the power of the voters is always equal (100 points distributed) in all the counting steps, including the result.

The Surplus Handling:


 * cancel the Equality in some steps of the count.
 * increase the complexity of the counting.
 * isn't appropriate to manage seats with different weights.

For these reasons, it's better to avoid using Surplus Handling in Distributed Voting System.

Suitable for Web
If the seats had different fractional value, in addition to determining the winning candidates, Distributed Voting also determine their % of victory, which are already indicated by the sum of the points of the winning candidates, remaining at the end of the counting.


 * Ex.1: a streamer wants to talk about 3 topics in a 4-hour live, chosen by his supporters through a poll. With Distributed Voting the 3 winning arguments A,B,C would also have associated the % of victory: A[50%] B[26%] C[24%]. These % indicate to the streamer that he must devote 2 hours to topic A, and 1 hour to topics B and C. Without these %, the streamer would have mistakenly spent 1 hour and 20 min for each of the topics.


 * Ex.2: in an image contest, there is a cash prize to be awarded to the 3 best images. The prize will be divided appropriately according to the % of victory and not in a pre-established way before the contest.

Distributed Equal-Vote (DEV)
Voter score candidates with range [-5,+5]. Each ballot is normalized by distributing -100 points between negative ratings, and 100 points between positive ratings (distribution of points uses the normalization of Distributed Voting).

The candidate with the lowest sum of points is eliminated, and ballots normalized.

By repeating the elimination process, the worst candidate is eliminated each time, and the remaining candidates are the winners.

Equal-Vote because given a vote, there can always be an opposite one that cancels it.

IRV
Examples where the 100 points are distributed exponentially:

99,1       → it's like IRV 90,9,1     → it's a bit different from IRV 70,24,5,1  → it's       different from IRV 60,27,9,3,1 → it's very different from IRV

Using range [0,9] completely eliminates the similarity:

range[0,9] →  100 points 9,1        →  90,10       →  it's a bit different from IRV 9,5,1      →  60,33,7     →  it's very  different from IRV

Range [0,9] was chosen to better balance the simplicity of writing, the representation of interests, and the correctness of the count. Normalization applied to a range too small as [0,5], alters the voter's interests too much in the count.

IRNR
IRNR (L1 norm) is applied also on ranges with negative values such as [-5,+5] but this makes it subject to ambiguity.

Range [0,10] with IRNR 61: A[10] B[6] C[0] 39: A[0] B[6] C[10] Eliminated in order C,A. B wins.

Range [-5,+5] with IRNR 61: A[+5] B[+1] C[-5] 39: A[-5] B[+1] C[+5] Eliminated in order C,B. A wins.

In IRNR only by moving the range in negative value (leaving the interests of the voters and the size of the range unchanged), the winner changes. Distributed Voting instead avoid this ambiguity by imposing 0 as the minimum value in the range.

IRNR is a Single-Winner system which also, unlike Distributed Voting, doesn't reverse and make negative the vote before the count.

Related Systems

 * Instant Runoff Normalized Ratings (ratings also negative, and it doesn't reverse and make negative the vote)
 * Baldwin's method (Borda, and variant with different normalization)