Distributed Voting
Distributed Voting (DV) is a Single-Winner and Multi-Winner, Cardinal voting systems.
This system is a specific type of Instant Runoff Normalized Ratings (L1 norm), which also deals with the Multi-Winner System and which doesn't accept the case of negative ratings.
Procedure
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, a worst candidate is eliminate each time.
The remaining candidates are the best (winners).
Ballot
Digital 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
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 [0,10] → Normalized in 100 points 10,0,0,0 → 100,0,0,0 10,10,0,0 → 50,50,0,0 10,5,5,0 → 50,25,25,0 10,6,3,1 → 50,30,15,5 (note: there isn't 0 in the lowest score)
Real 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.
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.
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.
Procedure specification
Normalization example
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:
A[0] B[1] C[3] D[6] E[90] A[0] B[10] C[30] D[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)
Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} v1=\frac{v0}{1-\frac{e}{P}} \end{equation}}
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-winner procedure 2 generates more similar % of victory and this can be positive in the seats assignment.
Tie during counting
Cases of parity can occur during counting, as in the following example:
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]
The tie can be managed in various ways:
- delete C first, getting a result. Delete D first, getting another result. Check that the two results return the same winners.
- 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.
Procedure variant (discouraged)
One or more of the following steps are used:
- 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 in multi-winner).
- If the remaining candidates are contained in a Smith set, then the candidates with the highest sum wins.
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.
Candidates
In an election between candidates (with at least 3 winners), proceed as follows:
- the value S (threshold) is obtained, using the following formula: Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} S=\frac{50\%+\frac{100\%}{\#seats}}{2} \end{equation}}
- 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 50% 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 All 0 points section.
Parties
In an election between parties, proceed as follows:
- the value S (threshold) is obtained, using the following formula: Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} S=\frac{100\%}{\#seats\cdot 2} \end{equation}}
- 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 D'Hondt method).
- 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 can increase.
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
Tactical vote resistance
Hypotheses
Each voter, based on his own interests, creates the following 2 sets of candidates:
- Winner Set = set containing a quantity 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.
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.
Example 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
Meets the Honesty criterion (on hypotheses) because:
- at each 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).
- the 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 Step can occur in the first Update Steps.
- the 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 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
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.
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 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 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.
For these reasons it's better to avoid using Surplus Handling in Distributed Voting.
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: 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.
- 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.
Systems comparison
IRV
Examples where the 100 points are distributed exponentially:
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
- Distributed Multi-Voting (particular vote conversion)
- Instant Runoff Normalized Ratings (ratings also negative)
- Baldwin's method (Borda, and variant with different normalization)
Forum Debate
- "Distributed Voting (DV) vs Range Voting (RV)". The Center for Election Science. 2020-05-12. Retrieved 2020-05-15.
- "Sequential Elimination systems". The Center for Election Science. 2020-01-27. Retrieved 2020-02-19.