Distributed Voting: Difference between revisions
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[[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 has 100 points to distribute among the candidates.
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\end{equation}</math>
===
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:
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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.
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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:
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* 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]]
===Tie during counting===
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* [[Surplus Handling]] (in the standard Distributed Voting it's not used, also in [[Multi-Member System|multi-winner]]).
* If the remaining candidates are contained in a [[Smith set]], then the candidates with the highest sum wins.
==Seats assignment==
The Distributed Voting indicates the method for obtaining single or multiple winners. The Distributed Voting System also describes how seats should be handled.
===Candidates===
In an election between candidates (with at least 3 winners), proceed as follows:
* the value S (threshold) is obtained, using the following formula: <math>\begin{equation} S=\frac{50\%+\frac{100\%}{\#seats}}{2} \end{equation}</math>
* 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.
* starting from the best candidate, the [[Surplus Handling]] is applied using S as threshold, that is the points that exceed the threshold are redistributed among other winning candidates, based on the interests expressed in the votes.
* the seats will have a fractional weight equal to the % of victory of the candidates.
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{ A[40%] B[35%] C[25%] } d2{ B[40%] C[35%] D[25%] }
Seat weights: d1{ A[0.28] B[0.245] C[0.175] } d2{ B[0.12] C[0.105] D[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{ A[2] B[1] C[1] } d2{ B[1] C[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.
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).
==Other properties==
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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 the [[Surplus Handling]]:
* cancel the [[Distributed Voting#Equality|Equality]] in some steps of the count.
* increase the complexity of the counting.
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For these reasons it's better to avoid using Surplus Handling in Distributed Voting.
===Suitable for Web===
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