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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.jpg|alt=DV procedure|thumb|DV procedure]]
 
Voter score candidates with range [0,9]. The vote is then converted to 100 points (normalization).
===Voting===
 
# The worst candidate, with the lowest sum of points, is eliminated.
Each voter has 100 points to distribute among the candidates according to his preferences.
# 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.
All candidates in the vote have 0 points by default.
 
==Extended procedure (single winner)==
===Counting the votes===
 
It's the procedure indicated above in which:
# The point for each candidate are summed and the one with the lowest sum is eliminated.
* the votes are reversed and made negative before counting ''(subtracting 9 from the original ratings)''.
# In each individual vote, the points of the eliminated candidate are removed and the vote is normalized, so that it has 100 points again.
 
Original vote: A[9] B[7] C[5] D[3] E[1] F[0]
By repeating the process from the beginning, a candidate is eliminate each time.
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.''
The remaining candidates are the winners. The sum of the points of the remaining candidates indicates the % of victory.
 
==Ballot==
==Procedure specification==
 
===Paper ballot===
===Example normalization of a single vote===
 
Some examples of normalization:
Given an initial vote of this type, with candidates A,B,C,D,E:
 
Range [0,9] → Normalized in 100 points
A[0] B[1] C[3] D[6] E[90] : E is removed
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)]]
A[0] B[10] C[30] D[60] : D is removed
===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.
A[0] B[25] C[75] : C is removed
 
==Procedure specification==
A[0] B[100]
 
===Normalization of the voteformula===
 
P = 100 (can also be set to 1).
e := value of the candidate eliminated from the vote.
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.
v0 := old value of candidate X.
 
===Normalization example===
v1 := new value of candidate X.
 
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:
<math>\begin{equation}
v1=\frac{v0}{1-\frac{e}{100}}
\end{equation}</math>
 
A[0] B[1] C[3] D[6] E[90]
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:
A[0] B[10] C[30] D[60]
A[0] B[25] C[75]
A[0] B[100]
 
===Tie during counting===
<math>\begin{equation}
v1=\frac{v0}{1-e}
\end{equation}</math>
 
Cases of parity can occur during counting, as in the following example:
During counting, points can be represented in decimal form.
 
Vote 1: A[55] B[25] C[10] D[10]
===Managing votes 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:
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 first, getting a result. Delete D first, getting another result. Check that the two results return the same winners.
# The vote is excluded from the count: A[0] B[0].
*delete C and D at the same time.
# The points are divided equally between the remaining candidates with 0 points: A[50] B[50].
*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.
Using procedure 2 you get a vote that:
 
===Procedure variant (discouraged)===
* 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.
 
One or more of the following steps are used:
For the reasons indicated above, it’s strongly discouraged to use procedure 2.
 
* 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.
===Managing votes without 0 points===
* [[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==
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:
 
The Distributed Voting indicates the method for obtaining single or multiple winners. The Distributed Voting System also describes how seats should be handled.
# honest form: A[80] B[20]
# tactical form: A[100] B[0]
 
===Parliament===
It's recommended to use the honest form, also because the vote from the beginning may not have candidates with 0 points (possible case).
 
Procedure for electing parliamentarians:
===Managing tie during the count===
 
* The state is divided into districts (at least 2, and possibly of similar size).
Cases of parity can occur during counting, as in the following example:
* 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):
* Vote 1: A[50] B[25] C[25]
* Vote 2: A[50] B[25] C[25]
* Sum of votes: A[100] B[50] C[50]
 
* 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).
In this case, the worst candidate is both B and C so you have to eliminate them simultaneously. The amount of points to be redistributed will be the sum of the points that had B and C (100 in the example).
* 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==
 
===SimplifiedTactical votingvote writingresistance===
 
'''Hypotheses'''
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.
 
Each voter, based on his own interests, creates the following 2 sets of candidates:
Before the counting process, the grades will be normalized to 100-point grades, where the Xs are considered as equal weight values.
 
* Winner Set = set containing a number of favorite candidates equal to or less than the number of winners.
Examples of how a vote can be written by the voter and subsequently, in the counting, converted into 100 points:
* 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.
X,0,0,0,0        →        100,0,0,0,0
 
Example
X,X,X,X,0       →        25,25,25,25,0
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'''
4,3,2,1,0         →        40,30,20,10,0
 
Meets the [[Honesty criterion]] (on hypotheses) because:
40,6,3,1,0       →        80,12,6,2,0
 
* 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).
101,0,0,0,0     →        100,0,0,0,0
* 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
999,99,9,1      →        89.17, 8.83, 1, 1
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'''
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.
 
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.
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 DV from becoming like IRV).
 
===About[[Surplus Equal VoteHandling]]===
 
GivenEquality: aDistributed scoreVoting withensures that the power of the voters is always equal (100 points distributed) in all the followingcounting order:steps, A[50]including B[30] C[15] D[5]the E[0]result.
 
The [[Surplus Handling]]:
you can always get a counter-balancing by exchanging the order of the points: A[0] B[5] C[15] D[30] E[50]
 
* cancel the Equality in some steps of the count.
Applying the count only considering the 2 votes defined above, you will always obtain equality between the two candidates to the extremes of the vote (in this example, equality between A and E).
* 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.
These 2 votes in the end support only 2 candidates out of the initial 5, so adding those 2 votes to a tally with other votes, can change the results of the tally.
 
===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.
 
==Systems 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:
 
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 Member system|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)
 
==Forum Debate==
 
*{{cite web | title=Sequential Elimination systems | website=The Center for Election Science | date=2020-01-27 | url=https://forum.electionscience.org/t/sequential-elimination-systems/583 | ref={{sfnref | The Center for Election Science | 2020}} | access-date=2020-02-19}}
*{{cite web | title=Best single-winner Voting System (in full honest context) | website=The Center for Election Science | date=2020-01-29 | url=https://forum.electionscience.org/t/best-single-winner-voting-system-in-full-honest-context/585 | ref={{sfnref | The Center for Election Science | 2020}} | access-date=2020-02-19}}
*{{cite web | title=Distributed Voting (DV) vs Range Voting (RV) | website=The Center for Election Science | date=2020-05-12 | url=https://forum.electionscience.org/t/distributed-voting-dv-vs-range-voting-rv-extended/647 | ref={{sfnref | The Center for Election Science | 2020}} | access-date=2020-05-15}}
*{{cite web | title=Sequential Elimination systems | website=The Center for Election Science | date=2020-01-27 | url=https://forum.electionscience.org/t/sequential-elimination-systems/583 | ref={{sfnref | The Center for Election Science | 2020}} | access-date=2020-02-19}}
 
[[Category:Single-winner voting methods]]
Aldo Tragni
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