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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]], [[Cardinal voting systems]].
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]].
This system is a specific type of [[Instant Runoff Normalized Ratings]] (L1 norm), which also deals with the [[Multi-Member System|Multi-Winner System]] and which doesn't accept the case of negative ratings.
==Procedure==
[[File:DV Procedure.svg|alt=DV procedure|350px|thumb|DV procedure]]
[[File:Digital ballot DV.gif|320px|thumb|DV digital ballot (cumulative 100 points)]]
[[File:DV paper ballot.svg|320px|thumb|DV paper ballot (range [0,10])]]
Voter score candidates with range [0,9]. The vote is then converted to 100 points (normalization).
Voter has 100 points to distribute among the candidates.
# The worst candidate, with the lowest sum of points, is eliminated.
# The points of the eliminated candidate are proportionally redistributed in each vote (normalization).
By repeating processes 1 and 2, athe worst candidate is eliminateeliminated each time, and the remaining candidates are the winners.
==Extended procedure (single winner)==
The remaining candidates are the best (winners).
It's the procedure indicated above in which:
==Ballot==
* 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]
===Digital ballot===
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.''
By using self-resizing sliders it's possible to obtain simple ballot that use the cumulative vote, with 100 points to distribute. However, the ranges [0,10] can also be used for digital ballot, as described below.
==Ballot==
===Paper ballot===
Some examples of normalization:
In the paper ballot is used the ranges, that are easier to understand for a voter. Ballots using ranges will be normalized to 100-point votes, and then apply the Distributed Voting procedure. Some examples of normalization:
Range [0,109] → Normalized in 100 points
109,0,0,0 → 100,0,0,0
109,109,0,0 → 50,50,0,0
109,56,54,01 → 5045,2530,2520,5 (note: there isn't 0 in the lowest score)
10,6,3,1 → 50,30,15,5 (note: there isn't 0 in the lowest score)
[[File:Digital ballot DV.gif|320px|thumb|DV digital ballot (cumulative 100 points)]]
===Real ballot===
===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.
The way to vote in Distribute Voting is, in theory, to assign 1 point to the least preferred candidate, and then assign points to the other candidates proportionally to the appreciation towards the less preferred candidate. Non-preferred (or unknown) candidates will remain with 0 points.
==Procedure specification==
A vote like this: A[1] B[2] C[4] D[0] means that voter likes B 2 times A, and likes C 4 times A (or 2 times B). Vote like this: A[1] B[0] C[0] D[0] means that the voter likes only A. Both votes are then normalized to 100 points so that they have the same power.
===Normalization formula===
This way of voting has no restrictions on the rating, therefore it offers the best representation of interests, but it's the most complex to understand and subject to tactical votes (in which certain candidates are awarded more points than necessary). To avoid such complexity and tactical votes, it's best to use range [0,10], by accepting a reduction in interest representation.
P = 100 (can also be set to 1).
==Procedure specification==
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.
<math>\begin{equation}
newV=\frac{V}{S} \cdot P
\end{equation}</math>
If S=0 then all candidates remain at 0 points.
===Normalization example===
A[0] B[25] C[75]
A[0] B[100]
===Normalization formula===
e = value of the candidate eliminated from a vote.
v0 = old value of candidate X.
v1 = new value of candidate X.
P = 100 (total points used in a normalized vote)
<math>\begin{equation}
v1=\frac{v0}{1-\frac{e}{P}}
\end{equation}</math>
===No 0 points===
If the only candidate C with 0 points is eliminated from a vote like this A[80] B[20] C[0], there are 2 procedures you can use:
# A[100] B[0] : set the candidate with the least points to 0.
# A[80] B[20] : having eliminated C (0 points), there aren't points to redistribute.
Eg. given the following 2 votes to count: V1-A[55] B[45] C[0] and V2-A[0] B[100] C[0] then:
*using procedure 1, a tie is obtained between A and B.
*using procedure 2, B would win.
V1 likes A and B almost in the same way, so the victory of B would make both V1 and V2 happy. For this reason it's recommended to use procedure 2, which keeps the voter's initial interests even in the counting.
===All 0 points===
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:
# A[0] B[0] : the vote is excluded from the count.
# A[50] B[50] : the points are divided equally between the remaining candidates with 0 points.
Using procedure 2 you get a vote that:
* 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.
The two procedures return the same winners, but in the [[Multi-Member System|multi-winner]] procedure 2 generates more similar % of victory and this can be positive in the [[Distributed_Voting#Seats_assignment| seats assignment]].
===Tie during counting===
* 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 the standard Distributed Voting it's not used, also infor [[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 assignmentallocation==
The Distributed Voting indicates the method for obtaining single or multiple winners. The Distributed Voting System also describes how seats should be handled.
===CandidatesParliament===
Procedure for electing parliamentarians:
In an election between candidates (with at least 3 winners), proceed as follows:
* The state is divided into districts (at least 2, and possibly of similar size).
* the value S (threshold) is obtained, using the following formula: <math>\begin{equation} S=\frac{50\%+\frac{100\%}{\#seats}}{2} \end{equation}</math>
* 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.
* the worst candidates are eliminated, leaving a quantity of winners equal to the seats. The sum of points for each winning candidate is used to derive the % of victory.
* startingIn fromeach the best candidatedistrict, the [[Surplus Handling]]DV is appliedused usingto Sobtain asa threshold,number thatof iswinners equal to the pointsnumber thatof exceedseats in the thresholddistrict. areThe redistributedsum amongof otherthe points for each winning candidates,candidate basedwill onindicate the interests% expressedof invictory of the votescandidates.
* If P is the seatspower willassigned haveto athe fractionaldistrict, then the weight equalof toeach theseat will be: P • "% of victory of the candidatescandidate".
Example - 3 winners
Result: A[51%] B[27%] C[22%] S = 41,7% ≈ 40% (rounded for simplicity)
Redistribute A points that exceed 40%
Result: A[40%] B[35%] C[25%]
Seats weight: A[0.4] B[0.35] C[0.25]
Fractional seats offer better proportionality than unit seats, but there is a risk that a candidate alone will gain more than 51% of the power. The formula indicated for S serves to ensure that a single candidate cannot have a majority on his own, while maintaining the benefits of fractional seats. The effectiveness of these properties is noted with increasing seats.
Example - 10 winners
Result: A[30%] B[20%] ... L[1%] S = 30%
Seats weight: A[0.3] B[0.2] ... L[0.01]
A's seat is worth 30 times that of L, in respect of the % of victory obtained by the candidates.
Assigning A and L a seat with the same weight would be unfair.
The difference between the % of victory is reduced in a fair way, through the procedure indicated in the [[Distributed_Voting#All_0_points| All 0 points]] section.
===Parties===
In an election between parties, proceed as follows:
* the value S (threshold) is obtained, using the following formula: <math>\begin{equation} S=\frac{100\%}{\#seats\cdot 2} \end{equation}</math>
* the worst parties are eliminated until both of the following 2 conditions are met:
# the number of winners is less than or equal to the number of seats (if a party has more than 50%, then the number is considered "seats-1").
# all candidates have a % of victory greater than or equal to S.
* 1 seat is assigned to each party (if there is a party that has obtained more than 50%, it will receive 2 seats).
* If seats remain to be filled, they distribute according to % of the party victory, using a method of your choice (as %%%, %%%, ...).
* dividing the % of victory of the parties by the number of seats they have, the fractional weight of each seat is obtained.
Example - 5 seats
Result: A[39%] B[25%] C[15%] D[9%] E[2%] S = 10% exclude E
Result: A[40%] B[25%] C[15%] D[10%] S = 10%
Seats: A[2] B[1] C[1] D[1]
Seats weight: A[0.2] B[0.25] C[0.15] D[0.1]
if had been used unit seats and S = 20%:
Unit seats: A[2] B[2] C[1]
The fractional seats, through the formula of S, allow for greater proportionality and representation than the unit seats.
A certain party will always have total power equal to the % of victory in the elections, regardless of how many seats are divided by that power. This property solves problems related to the [[Alabama paradox]].
===Districts===
The winning candidates and the fractional weight of the seats are obtained using the methods described above. To ensure representation, the district must be large enough to have at least 2 seats available (at least 3 for a good representation).
Example - 2 districts, 6 seats
Districts: d1{70%} d2{30%}
Seats: d1{3} d2{3}
Result: d1{ AA1[40%] BB1[35%] CC1[25%] } d2{ BB2[40%] CC2[35%] DD2[25%] }
Seat weights: d1{ AA1[0.28] BB1[0.245] CC1[0.175] } d2{ BB2[0.12] CC2[0.105] DD2[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{ AA1[2] BB1[1] CC1[1] } d2{ BB2[1] CC2[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. The more seats and districts increase, the more the error increases.
===Government===
The size of the district is represented only by the power it possesses and which will be assigned proportionally to the seats, therefore it's not strange that two districts of different sizes can still have the same number of seats (with different weight).
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==
Each voter, based on his own interests, creates the following 2 sets of candidates:
* Winner Set = set containing a quantitynumber 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.
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.
===[[Independence of Worst Alternatives|IWA]] example===
35 A[0] B[1] C[99]
33 A[99] B[0] C[1]
32 A[1] B[99] C[0]
Sum A[3299] B[3203] C[3498]
Head-to-head: A beats C beats B beats A. Distributed Voting in the first step eliminates candidate B, considered the worst, and between A and C, wins A.
Distributed Voting satisfies the [[Independence of Worst Alternatives|IWA]], so if candidate B (the worst) is added to the AvsC context (with A winner), it makes sense that A continues to be the winner.
===[[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.
Using theThe [[Surplus Handling]]:
* cancel the [[Distributed Voting#Equality|Equality]] in some steps of the count.
* increase the complexity of the counting.
* isn't appropriate to manage seats with different weights.
* if a voter votes A[99] B[1] C[0], in case A wins by getting double the threshold, the voter would be very satisfied with A's victory, then move half the points from A to B would mean giving the voter extra unjustified power.
For these reasons, it's better to avoid using Surplus Handling in Distributed Voting System.
===Suitable for Web===
* 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.
* Ex.2: 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.
==Systems Variations==
===Distributed Equal-Vote (DEV)===
* Ex.3: 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.
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==
Examples where the 100 points are distributed exponentially:
100 99,1 → it's like [[IRV]]
99 90,9,1 → it's likea bit different from [[IRV]]
90 70,924,5,1 → it's a bit different from [[IRV]]
70 60,2427,59,3,1 → it's very 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 Member system|Single-Winner system]] which also, unlike Distributed Voting, doesn't reverse and make negative the vote before the count.
By distributing points between 3 or more candidates, the Distributed Voting becomes increasingly different from the [[IRV]], because of normalization in the counting.
==Related Systems ==
* [[Instant Runoff Normalized Ratings]] (ratings also negative, and it doesn't reverse and make negative the vote)
* [[Distributed Multi-Voting]] (particular vote conversion)
* [[Instant Runoff Normalized Ratings]] (ratings also negative)
* [[Baldwin's method]] (Borda, and variant with different normalization)
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