Distributed Voting (DV) vs Range Voting (RV) - Extended

The purpose of the voting system is to resolve the disagreements in the fairest way feasible.

and the fairest way is the one that satisfies everyone, not the one that “suppresses” the others (through negative vote, which cancel each other out).

I favor A and oppose B and C, but you have the opposite opinion, it takes two million (double) who agree with you to balance just one million voters who agree with me

This is wrong, in the DV:
V1 (50%): A[100] B[0] C[0]
V2 (50%): A[0], B[51], C[49]
I add the votes, I eliminate the worst ( C ), I normalize the votes (i.e. proportionally redistribute the points in each vote) and I get:
V1 (50%): A[100] B[0]
V2 (50%): A[0] B[100]
V2 is opposed V1, without the need to have double the votes and without having to cancel the V1 vote.

My definition of Opposing Votes: votes that if counted among themselves, generate a situation of equality between some candidates (i.e. only among candidates supported by someone in the votes considered; in the example parity only between A and B, C correctly excluded).

I wonder if what you meant was that a voter’s 100 points are shared between positive and negative votes, so you could give out 80 positive points and 20 negative, or 60 negative and 40 positive, but NOT 100 of each. (You could give out 50 of each if you wanted.)

Then, in the “unknown candidate” scenario, the voters would have to choose among several honest options. For example, if I am in the A camp and think this B is particularly bad, I might vote A[20] B[-80] C[0]. But if instead I was feeling cautious against the unknown C then I might vote A[99] B[-1] C[0].

Probably in real life most voters will either vote 50,-50,0 or 100,0,0. Then C does not win.

I just realized something else:

According to the bottom of the page it seems like I could in theory cast a 5-candidate exponential vote by intending to cast:
A[999] B[99] C[9] D[1] E[0]
Then all of them are divided by 1108.

In the presence of negative votes, it’s practically always valid that:
the more 2 candidates get damaged (through the negative points), the more all the other candidates benefit from it.

The method proposed by you reduces the advantage to unknown candidates because you assume that voters vote on average using few negative points, that is: 50,-50 or 100,0 and obviously, the more the use of negative points is reduced, the less advantage have unknown candidates.
Having said that, it’s worth removing the negative points, or no?

It’s my mistake of ambiguity that I will correct immediately, thank you very much for the warning!

The correction is the following process:
If there is any value <1 in the conversion, then:

  • set 1 to the lowest value, reducing the total points by 1 (from 100 to 99, at the beginning).
  • recalculate the values, considering that now there are 99 points to distribute (not 100).
  • repeat, until the initial condition is false.

Staying on RV vs DV , regarding the negative vote :

RV

V1 only loves A, V2 hates A and wants to put him at a disadvantage.
V1: A[10] B[0] C[0] D[0]
V2: A[0] B[10] C[10] D[10]
The negative vote in the case of RV coincides with counter-balancing.

New ex.: V1 hates A, V2 hates B
V1: A[0] B[10] C[10] D[10]
V2: A[10] B[0] C[10] D[10]
Sum: A[10] B[10] C[20] D[20]

Note that in the RV it’s possible to create a negative vote with the consequence that C and D (candidates unknown to V1 and V2) are very favored (they receive 2 votes even though nobody really supports them).

In practice, RV makes those who would be abstentionists, to cast negative votes against the candidates they hate most, drastically favoring unknown candidates.

DV

V1 only loves A, V2 hates A.
The only way V2 has to disadvantage A is to benefit someone of the other candidates (the one that is the best for him). There is no other alternative to disadvantage A. This pushes people who vote to give a representation of their interests, even if their initial goal would be only to disadvantage A.
V1: A[100] B[0] C[0] D[0]
V2: A[0] B[100] C[0] D[0] or
V2: A[0] B[0] C[100] D[0] or
V2: A[0] B[0] C[0] D[100] or
V2: A[0] B[50] C[30] D[20] or … etc

In all cases, the counting of V1 and V2 votes would end with a tie between A and another candidate (preventing A from winning).

In the DV it isn’t possible to create a negative vote and this also drastically reduces the negative consequences (unknown candidates aren’t favored as could happen in the RV).

Yes, that’s what I had in mind.

I would oppose unknown candidates as hard as I would hated candidates, while kicking myself for not researching all the candidates. This is one reason I like Score, or some kind of multiround application of Score. I can oppose my unknowns at no cost in terms of my support of my favorites or opposition to my hateds. And I find it necessary to oppose unknown candidates, because they could be the worst. Minimizing risk is what to value, when government officials can take decisions that result in extinction for all our families.

That example worked well. I don’t know whether there is a proof that all the other possible examples would meet the balance constraint.

Definition of the "Balance Constraint"

Given a set of votes V and that the election would have a certain outcome with those votes and a different outcome with V U {a} the addition of one other vote from voter a, then a strategy must be available to voter b (not from set V) to assure that Oe(V) = Oe(V U {a, b}) i. e. the election outcome is restored when b votes to cancel a’s vote.

In the example I have clearly shown that a negative vote (in Score Voting), which nullifies another’s preferences, drastically favors all unknown candidates, so using a similar vote is wrong.
A real balance between voters, in my opinion, is when two votes that support different candidates, if counted together, return a tie. However, these 2 votes support only 2 specific candidates so if they were added to an election, they would affect the outcome.

In short, satisfying the “Balance Constraint” is more negative than positive, since satisfying it means that there is the possibility of creating negative votes that favor unknown candidates.

That’s not possible. If you can negate my vote, you can negate the portion of my vote that pertains to the candidates that are unknown by me, unknown by you, or the intersection or union of those sets. Since our votes cancel, they do not favor the unknown candidates.

This discourse relates to the form of interests, in which I say that unknown candidates are (theoretically and on average) better than negative (hated) candidates, but they should be disadvantaged as hated.
RV as it’s made tends to favor unknown candidates very much when the voter focuses on hated candidates rather than on those who might like him.

If you want a democratic republic, you can only support, for the single-winner context, voting systems that meet the balance constraint. Here’s why. We who want a democratic republic need assurance that a proposed system doesn’t replicate FPtP’s characteristic of weighing votes that join a bandwagon at greater weight than other votes. That is because if the system rewards bandwagon following, the voters will remain sensitive to how much money support a given candidacy is receiving. That sensitivity puts money in charge of the election. Consequently the certain result is a government full of kleptocrats. It’s “one dollar, one vote”. To counter that, we must implement “one person, one vote”, which means that all the votes count equally. And this requires balance. Lack of balance is a blatant exposure of unequal weighting of votes. Even if some readers want to cast doubt on the idea that the balance constraint assures equality, you have to admit that the lack of it exposes inequality. Since the bandwagon-following phenomenon depends fundamentally on an inequality of power, any evidence that increases our confidence in the equality provided by a given system under consideration, helps to work against the bandwagon-following phenomenon and thereby to work against the influence of money in politics.

To counter that, we must implement" one person, one vote ", which means that all the votes count equally.

Perfect, exactly what the DV does in which a voter is always worth 100 points. Even when the worst candidate is eliminated, normalization ensures that the vote continues to be worth 100 points (distributed according to the preferences of the voter).

Even if some readers want to cast doubt on the idea that the balance constraint assures equality, you have to admit that the lack of it exposes inequality.

No, I admit that the presence of the “Balance Constraint” means equality in theory, but in practice it can generate other problems. The problems of the RV that I described in this post also derive from the fact that you want to satisfy the “Balance Constraint”.
Furthermore, I also say that the lack of “Balance Constraint” doesn’t necessarily mean inequality (the DV doesn’t satisfy this constraint but still seems to respect equality well).

Didn’t I already cover this? Consider an election in which 300 million voters have already cast their ballots. If no one else voted, the election would have a specific outcome. Let us call that Oe(A), the outcome of the election given A the set of 300 million votes. Now you and I are on our way to the polls to cast the last two votes. I cast my vote and it changes the outcome. You arrive and are unable to cancel my vote with yours. Clearly the system accorded me more power than it did you, since I was able to move the needle and you were not able to move it back. You are a slave, and I am your master. How does the logic escape you that this scenario shows that you and I were not granted equal votes by the system responsible for tallying the election? Does any other reader stand where Essenzia does in rejecting this logic?

I am not quite sure which position to take.

The Balance Criterion states that (assume number of candidates is fixed):

For every set of votes A there exists another equal-size set of votes B such that given any prior election E, Oe(E) = Oe(E + A + B).

But I think the contention arises from this weaker criterion:

For every election E and every set of additional votes A, there exists an equal-size set of votes B which may depend on E such that Oe(E) = Oe(E + A + B).

Here the cancel votes are allowed to be different depending on how the election plays out.
FPTP does not satisfy this, but it only seems to break in the case of a tie. (Example: A:40 B:30 C:25, add 11 votes to B, then just add 11 votes back to A to balance. Example 2: A:35 B:35 C:30, add 12 votes to C to make C win with 42, but you cannot revert it to an A-B tie with 12 additional votes.)

One could argue using waugh’s argument that you only need Weak Balance to guarantee democracy. But both sides of that appear shaky.

One could also consider a generalization of Frohnmayer Balance: a set of ballots that would cancel each other out if they were the only ballots cast in the election, should also cancel each other out if they are all added to an election with other ballots. Condorcet, STAR, and probably a slew of other majoritarian methods fail this, but not Approval and Score.

Example modeled off of https://rangevoting.org/TobyCondParadox.html:

3 A:5>B:4>C:0

2 B:5>A:4>C:0

2 B:5>C:4>A:0

2 C:5>A:4>B:0

Scores are A 31, B 32, C 18, with A pairwise beating B and thus being the STAR winner. Removing 6 votes that constitute a cycle and a kind of pairwise tie and a definite scored tie for A, B, and C (2 A:5>B:4>C:0, 2 B:5>C:4>A:0, 2 C:5>A:4>B:0 votes, which give a total of 9 points to A, B, and C, and create a Condorcet cycle between the three where A>B, B>C, and C>A are all matchups of 4 to 2) yields:

1 A>B>C

2 B>A>C

Without even looking at the scores, B must win here, since A and B are unanimously preferred as the top 2 candidates and a majority prefers B>A.

Given a starting point, if I push the needle in a certain direction and you can bring it back to the starting point, then the equality holds

This I accept.

Given a starting point, if I push the needle in a certain direction and you cannot bring it back to the starting point, then the equality is not worth it

You haven’t shown this and I don’t accept it.
In the DV you cannot bring the needle back to the starting point simply because you cannot pull it back (remove points) but only push it forward (add points) and if everyone pushes it forward with the same intensity, the equality still applies.

More precise demonstration:
By “equality” I mean “one person, one vote (100 points)”.

  • In the DV 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, you can make sure that all voters continue to have 100 points each, always distributed according to their interests, therefore equality is satisfied.
  • the result is one of the steps in which equality continues to apply.

End.
There is no passage in the DV where equality doesn’t apply, and at the same time DV doesn’t satisfy “Balance Criterion”.

The Balance Criterion in RV only applies in theory, while in practice it can fails.

V1 loves the right, hates the left.
V2 loves the left, hates the right.
V1 and V2 are opposite.
A,B,C are left candidates (from the most extreme to the least extreme).
D,E,F are right candidates (from the most extreme to the least extreme).

V1 and V2 vote considering the category of candidates:
V1: ABC[0] DEF[10]
V2: ABC[10] DEF[0]
Balance Criterion satisfied.

V1 and V2 vote considering the order of their preferences:
V1: A[0] B[2] C[3] D[7] E[8] F[10]
V2: A[10] B[8] C[7] D[3] E[2] F[0]
Balance Criterion satisfied.

In practice, however, V1 and V2 could reason differently (ambiguity) and vote honestly like this:
V1: A[0] B[0] C[0] D[10] E[10] F[10]
V2: A[10] B[8] C[7] D[3] E[2] F[0]
The right wins, even if V1 and V2 had opposite interests.
The ambiguity in RV means that the Balance Criterion may not be satisfied in practice.
Satisfying the Balance Criterion in theory is useless if then in practice it can fail.

The AV instead satisfies the Balance Criterion both in theory and in practice.

In RV and AV you also only add points, but the points you add can perfectly cancel one other person’s vote, unlike in DV.

But @waugh can you please construct a concrete set of elections to show how the balance criterion fails?

1 Like

If you use a range [-5,+5] you can see that the concept of “subtracting points” exists in RV. Normally this is not seen because it’s masked with ranges like [0,10].