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[[File:DV Procedure.svg|alt=DV procedure|351px|thumb|DV procedure]]
Distributed Voting (DV) is a [[Single Member system|Single-Winner]] and [[Multi-Member System|Multi-Winner]] [[Cumulative voting|Cumulative voting system]].
Distributed Voting (DV) is a [[Single Member system|Single-Winner]] and [[Multi-Member System|Multi-Winner]], [[Cardinal voting systems]] proposed by [[User:Aldo Tragni|Aldo Tragni]].
 
==Procedure==
[[File:DV Procedure.svg|alt=DV procedure|350px|thumb|DV procedure]]
 
Voter score candidates with range [0,9]. The vote is then converted to 100 points (normalization).
Each voter has 100 points to distribute among the candidates according to his preferences.
 
# The point for eachworst candidate are summed and the one, with the lowest sum of points, is eliminated.
# In each individual vote, theThe points of the eliminated candidate are removedproportionally andredistributed thein each vote is normalized, so that it has 100 points again(normalization).
 
By repeating theprocesses process1 fromand the beginning2, athe worst candidate is eliminateeliminated each time, and the remaining candidates are the winners.
 
==Extended procedure (single winner)==
The remaining candidates are the winners.
 
It's the procedure indicated above in which:
==Procedure specification==
* 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]
===Example normalization of a single vote===
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.''
Given an initial vote of this type, with candidates A,B,C,D,E:
 
==Ballot==
A[0] B[1] C[3] D[6] E[90] : E is removed
 
===Paper ballot===
A[0] B[10] C[30] D[60] : D is removed
 
Some examples of normalization:
A[0] B[25] C[75] : C is removed
 
Range [0,9] → Normalized in 100 points
A[0] B[100]
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)
 
[[File:Digital ballot DV.gif|320px|thumb|DV digital ballot (cumulative 100 points)]]
===Normalization of the vote===
===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.
e := value of the candidate eliminated from the vote.
 
==Procedure specification==
v0 := old value of candidate X.
 
===Normalization formula===
v1 := new value of candidate X.
 
P = 100 (can also be set to 1).
<math>\begin{equation}
S = points sum of the candidates remaining in the vote, after an elimination.
v1=\frac{v0}{1-\frac{e}{100}}
V = old points value of candidate X.
\end{equation}</math>
newV = new points value of candidate X.
<math>\begin{equation}
newV=\frac{V}{S} \cdot P
\end{equation}</math>
 
If S=0 then all candidates remain at 0 points.
In an electronic system it’s possible to divide by 100 all the points present in the initial votes, and use the following simplified formula throughout the counting process:
 
===Normalization example===
<math>\begin{equation}
v1=\frac{v0}{1-e}
\end{equation}</math>
 
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:
During counting, points can be represented in decimal form.
 
A[0] B[1] C[3] D[6] E[90]
===Managing votes without 0 points===
A[0] B[10] C[30] D[60]
A[0] B[25] C[75]
A[0] B[100]
 
===Tie during counting===
If the only candidate C with 0 points is eliminated from a vote like this A[80] B[20] C[0], there are 2 forms that the vote can take:
 
Cases of parity can occur during counting, as in the following example:
# honest form: A[80] B[20]
# tactical form: A[100] B[0]
 
Vote 1: A[55] B[25] C[10] D[10]
It's recommended to use the honest form, also because the vote from the beginning may not have candidates with 0 points.
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:
===Managing votes with 0 points===
 
*delete C first, getting a result. Delete D first, getting another result. Check that the two results return the same winners.
If the only candidate C with points is eliminated from a vote like this A[0] B[0] C[100], you can proceed in 2 ways:
*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.
# The vote is excluded from the count: A[0] B[0].
# The points are divided equally between the remaining candidates with 0 points: A[50] B[50].
 
===Procedure variant (discouraged)===
Using procedure 2 you get a vote that:
 
One or more of the following steps are used:
* cannot affect the victory of candidates who received the same points.
* reduces the distance between the candidates present in it, and this can affect a possible process of assigning seats.
* it can be considered not in accordance with the interests of the voter who, to those remaining candidates, had not awarded points.
 
* 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.
The two procedures return the same winners, but in the [[Multi-Member System|multi-winner]] case the winners can have different % of victory; in this case it's better to use procedure 1 for the reasons indicated above.
* [[Surplus Handling]] (in Distributed Voting it's not used for [[Multi-Member System|multi-winner]] context).
* If the remaining candidates are contained in a [[Smith set]], then the candidates with the highest sum wins.
 
==Seats allocation==
===Managing tie during the count===
 
The Distributed Voting indicates the method for obtaining single or multiple winners. The Distributed Voting System also describes how seats should be handled.
Cases of parity can occur during counting, as in the following example:
 
===Parliament===
* 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]
 
Procedure for electing parliamentarians:
In this case, the worst candidate is both C and D so you have to eliminate them simultaneously. The amount of points to be redistributed will be the sum of the points that had C and D (40 in the example).
 
* 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.
 
==Other properties==
 
===ResistanceTactical tovote tactical votesresistance===
 
'''Hypotheses'''
In the Distributed Voting, given an honest vote with this distribution of points [50 30 15 5 0], a tactical vote generally takes the following form [90 6 3 1 0].
 
Each voter, based on his own interests, creates the following 2 sets of candidates:
* If the first candidate to be eliminated were the first (the one with the most points), the two votes would both become like this [60 30 10 0], so the tactical vote would disappear.
 
* Winner Set = set containing a number of favorite candidates equal to or less than the number of winners.
* If instead the second and third candidates were eliminated, the two votes would become [91 9 0] (honest) and [99 1 0] (tactical). They are different but they are very similar, comparing them to their initial state.
* Loser Set = set containing the candidates who aren't part of the Winner Set.
 
InGiven thean Distributedhonest Votevote, it'sthe validtactical that,vote duringis theobtained counting,by minimizing the more points areof redistributedthe afterLoser Set, maximizing the eliminationpoints of the worstWinner candidateSet, theand moremaintaining the votesproportions becomeof honest interests within the two sets.
 
Example
===Fractional seats - Suitable for Web===
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'''
If the seats had fractional value (instead of unitary), in addition to determining the winning candidates, the voting method should also determine the % of victory of the winning candidates. In the Distributed Voting the % of victory are already indicated by the sum of the points of the winning candidates, remaining at the end of the counting.
 
Meets the [[Honesty criterion]] (on hypotheses) because:
Eg: 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.
 
* at each [[Honesty criterion|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).
Eg: on a crowdfunding platform, fans can have a different weight in the vote, based on how much money they have donated. In Distributed Voting you can manage directly this difference in power by assigning fans different amounts of points to distribute.
* the [[Honesty criterion|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 criterion|Honesty Step]] can occur in the first [[Honesty criterion|Update Steps]].
* the [[Honesty criterion|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
===Simplified voting writing===
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'''
To make the writing of the vote more comprehensible and simple, the voter can be left with almost complete freedom in the use of numerical values or only X.
 
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.
Before the counting process, the grades will be normalized to 100-point grades, where the Xs are considered as equal weight values.
 
===[[Surplus Handling]]===
Examples of how a vote can be written by the voter and subsequently, in the counting, converted into 100 points:
 
Equality: Distributed Voting ensures that the power of the voters is always equal (100 points distributed) in all the counting steps, including the result.
X,0,0,0,0        →        100,0,0,0,0
 
The [[Surplus Handling]]:
X,X,X,X,0       →        25,25,25,25,0
 
* cancel the Equality in some steps of the count.
4,3,2,1,0         →        40,30,20,10,0
* 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.
40,6,3,1,0       →        80,12,6,2,0
 
===Suitable for Web===
101,0,0,0,0     →        100,0,0,0,0
 
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.
999,99,9,1      →        89.17, 8.83, 1, 1
 
* 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.
The complexity in writing the vote adapts to the voter, and it’s also noted that, if 101 or 99 points are mistakenly distributed, the vote will still be valid.
 
* 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.
In the last example they are set to 1, the decimal values which should be less than 1, and the remaining points are divided proportionally among the other candidates (it serves to prevent Distributed Voting from becoming like [[IRV]]).
 
===AboutSystems [[IRV]]=Variations==
 
===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.''
 
==Systems comparison==
 
===[[IRV]]===
 
Examples where the 100 points are distributed exponentially:
 
100 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.
99,1 → it's like [[IRV]]
 
===[[IRNR]]===
90,9,1 → it's a bit different from [[IRV]]
 
[[IRNR]] (L1 norm) is applied also on ranges with negative values such as [-5,+5] but this makes it subject to ambiguity.
70,24,5,1 → it's different from [[IRV]]
 
Range [0,10] with IRNR
60,27,9,3,1 → it's very different from [[IRV]]
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
By distributing points between 3 or more candidates, the Distributed Voting becomes increasingly different from the IRV, because of normalization in the counting.
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.
===About Equality===
 
IRNR is a [[Single Member system|Single-Winner system]] which also, unlike Distributed Voting, doesn't reverse and make negative the vote before the count.
By "Equality" means "one person, one vote (100 points)".
 
* In the Distributed Voting the voters at the beginning all have 100 points to distribute according to their preferences, therefore Equality is satisfied.
* During all the counting steps, through the use of normalization, it ensures that all voters continue to have 100 points each, always distributed according to their interests, therefore Equality is satisfied.
* The result is one of the counting steps, in which Equality continues to be satisfied.
 
==Related Systems ==
There is no passage in the Distributed Voting where Equality doesn’t met.
* [[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)
 
==Forum Debate==
Aldo Tragni
206

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