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).
Paper ballot
In the paper ballot is used the ranges, that are easier to understand for a voter. In the procedure, a cumulative vote was used only to simplify the explanation.
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)
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 vote)
Failed to parse (unknown function "\begin{equation}"): {\displaystyle \begin{equation} v1=\frac{v0}{1-\frac{e}{P}} \end{equation}}
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-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
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.
- Negative points can also be awarded, however the absolute quantity of points distributed must be 100.
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 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 Candidates: [A B C D E] 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.
Equality
By "Equality" means "equal power (100 points) to each person".
- 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.
There is no passage in the Distributed Voting where Equality doesn’t met.
Free Riding
Given an honest vote of this type A[50] B[30] C[15] D[5], Free Riding can have the following consequences:
- 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.
Using the Surplus Handling:
- 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 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.
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.
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)
- Cumulative voting (no insta-runoff)
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.