Distributed Voting

Distributed Voting (DV) is a Single-Winner and Multi-Winner, Cardinal voting systems.

This system, in the Single-Winner context, avoids the ambiguity of the IRNR (L1 norm). More information in the dedicated section.

Procedure

DV procedure

DV paper ballot (range [0,10])

DV digital ballot (cumulative 100 points)

Voter score candidates with range [0,10]. The vote is then converted to 100 points (normalization).

1. The worst candidate, with the lowest sum of points, is eliminated.
2. 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

Paper ballot

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)

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, it's better to use range [0,10] also in digital ballot.

Procedure specification

Normalization formula

   P = 100 (can also be set to 1).
   S = points sum of the candidates remaining in the vote.
   V = old value of candidate X.
newV = new value of candidate X.

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{equation}   newV=\frac{V}{S} \cdot P   \end{equation}}

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]

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:

1. A[0] B[0] : the vote is excluded from the count.
2. 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.

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 Distributed Voting it's not used in the multi-winner context).
• 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.

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

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.

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.

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 satisfaction (he's already 99% satisfied with the victory of A).
• isn't appropriate to manage seats with different weights.

For these reasons it's better to avoid using Surplus Handling in Distributed Voting System.

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, through the cumulative vote.

• 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:

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 [0,10] completely eliminates the similarity:

range[0,10]  →  100 points
10,1         →  91,9        →  it's a bit different from IRV
10,5,1       →  63,31,6     →  it's very  different from IRV

Range [0,10] 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 on rating ballots, also on ranges with negative values such as [-5,+5]. Distributed Voting, in the Single-Winner context, is a subcategory of IRNR, which binds the minimum value of the rating ballots to 0 (doesn't accept ratings with negative values). This constraint is important because it avoids the ambiguity of the IRNR:

Range [0,10] with IRNR and Distributed Voting
61: A[10] B[6] C[0]
39: A[0] B[6] C[10]
Eliminated in order C,A.
B wins.
IRNR and Distributed Voting are equivalent in this case.

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 changing the range, 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 rating.

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.