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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]]
 
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.''
 
==Ballot==
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.
 
===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===
Line 51 ⟶ 60:
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>
 
===Vote without 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.
 
===Vote with only 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]] case the winners can have different % of victory; in this case it's better to use procedure 1 for the reasons indicated above.
 
===Tie during counting===
Line 113 ⟶ 82:
 
* 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.
 
* Negative points can also be awarded, however the absolute quantity of points distributed must be 100.
==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.
 
==Other properties==
Line 125 ⟶ 129:
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.
 
Line 154 ⟶ 158:
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.
 
===Equality[[Surplus Handling]]===
 
By "Equality": meansDistributed "equalVoting ensures that the power of the voters is always equal (100 points distributed) toin all the counting steps, including eachthe person"result.
 
The [[Surplus Handling]]:
* 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 initial interests, therefore Equality is satisfied.
* The result is one of the counting steps, in which Equality continues to be satisfied.
 
* cancel the Equality in some steps of the count.
There is no passage in the Distributed Voting where Equality doesn’t met.
* 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.
===[[Free Riding]]===
 
===Suitable for Web===
Given an honest vote of this type A[50] B[30] C[15] D[5], [[Free Riding]] can have the following consequences:
 
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.
#increase the points given to the most preferred candidates who probably lose. The vote becomes similar to A[90] B[6] C[3] D[1].
#decrease the points given to candidates who probably win. The vote, with a decreasing probability of candidates' victory from left to right, becomes similar to A[25] B[25] C[35] D[15].
#the candidates' chances of winning aren't known enough. In this case, [[Free Riding]] doesn't occur and the voter tends to vote honestly.
 
* 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.
Using the [[Surplus Handling]]:
 
* 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.
* reduces the tactic number 2 and greatly increases the tactic number 1, to the point that this would be used even when the voters don't know enough the chances of victory of the candidates.
* cancel the [[Distributed Voting#Equality|Equality]] in some steps of the count.
* increase the complexity of the counting.
* 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.
 
==Systems Variations==
For these reasons it's better to avoid using Surplus Handling in Distributed Voting.
 
===Distributed Equal-Vote (DEV)===
===[[Independence of Worst Alternatives|IWA]] example===
 
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]]).
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]
 
The candidate with the lowest sum of points is eliminated, and ballots normalized.
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.
 
By repeating the elimination process, the worst candidate is eliminated each time, and the remaining candidates are the winners.
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.
 
''Equal-Vote because given a vote, there can always be an opposite one that cancels it.''
===Suitable for Web===
 
==Systems comparison==
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.
 
===[[IRV]]===
* 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.
 
Examples where the 100 points are distributed exponentially:
* 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.
 
99,1 → it's like [[IRV]]
* 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.
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:
==Systems comparison==
 
range[0,9] → 100 points
===[[IRV]]===
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.
Examples where the 100 points are distributed exponentially:
 
===[[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.
100 → it's like [[IRV]]
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]]
 
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)
 
Aldo Tragni
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