Score Distribution Voting Method

The voting method I am proposing is as follows, depending on a single real variable parameter B in the interval [0,1] and a positive integer N:

1. Voters score candidates on an integer scale from 0 to N.

2. Each candidate is assigned the following metric:

SUM[from k=0 to N-1] p(k)B^k*(1-B^{N-k})

where p(k) is the fraction of voters who scored that candidate as a k.

1. The candidate with the smallest metric wins. If there are ties, choose the candidate with the smallest score variance. If there are still ties, choose the candidate with the highest score mean. If there are still ties, choose a winner by some arbitrary method.

I explained my reasoning for this metric in a previous post:

In my opinion, N=5 is reasonable and B~0.63 or so seems reasonable as well. These parameters donâ€™t matter that much as voters will adapt strategically to whatever system you give them, but I think the more pleased people are with the initial results, the less likely they will be to strategize excessively.

I think such a method is subject to the same problems as most score methods, but I believe it will also tend to yield superior results and reduce majoritarianism whenever possible.

Iâ€™m curious if anyone would care to examine this method or its variants.

This is what I understand:

Find a matrix like this, with:

• first line with candidates.
• first column with the possible ratings, in range [0,5].
• the % indicates how many voters have assigned a certain rating to a certain candidate.
.
A B C D E F constant
0 30% 5% ... ... ... ... 0.9
1 10% 25% 0.53
2 10% 15% 0.29
3 5% 20% 0.15
4 5% 25% 0.05
5 40% 10% 0 (?)

At this point, you multiply the values of the rows by a certain constant (depending on the rating), and then do a sum by columns, to find the metrics of each candidate. If the constant is the rating, Score Voting is obtained.
I donâ€™t understand what the constant of the rating N is, since in the sum k goes from 0 to N-1.

If you want to get an â€śexponential scaleâ€ť wouldnâ€™t it be easier only to change the rating values?
That is, in the ballot these ratings are shown:
{0,1,2,3,4,5}
but the actual value associated with them is:
{0.9,0.5,0.3,0.15,0.05,0}
and then you add up the points where the one with the highest sum wins (Score Voting).
Basically, your system seems to me to be a normalization of the range before applying Score Voting (if I understand correctly).

Well, itâ€™s actually the candidate with the smallest metric who wins. So in that way it is mathematically different from score voting. You can think of the metric as measuring a sort of â€śdistanceâ€ť from the actual distribution to the ideal distribution where everybody scores a 5. The metric is 0 if and only if everybody scores a 5, it is maximal if and only if everybody scores a 0, and it strictly decreases if lower scores are replaced with higher scores.

Mathematically speaking, I believe it is not a formal metric. There might be a way to generalize/fix it and turn it into one so that it satisfies the triangle inequality. I donâ€™t know, I might look into that.

Anyway you are right about assigning the scores values, as long as â€śno indicationâ€ť gives a default score of 0. And you are correct that the coefficient on p(5) is 0.

If you do this conversion:
{0,1,2,3,4,5} -> {0.9, 0.5, 0.3, 0.15, 0.05, 0}
and then you apply Score Voting, it seems to me that you get a mathematically equivalent method to yours, but easier to explain.

You can possibly do this conversion:
{0,1,2,3,4,5} -> {-0.9, -0.5, -0.3, -0.15, -0.05, 0}
and say that the candidate with the lowest sum (metric) wins.

In any case, even using an â€śexponential scaleâ€ť for the values, the method remains subject to the min-maxing tactic (which for me is the worst of all).

That conversion is not mathematically equivalent to the system Iâ€™m proposing, which is what Iâ€™m trying to explain. It is a minimization of a â€śweighted averageâ€ť rather than a maximization.

You are right that it is susceptible to min-maxing, but otherwise it is fairly utilitarian. A candidate with many 0 scores and a bit more 5 scores will have a much higher metric than a candidate with few 0 scores and some medium scores, which means that â€śhonestlyâ€ť evaluated candidates can more easily beat out min-maxed candidates.

The reasoning for the form of the metric is so that the evaluation process is â€ścloseâ€ť to a system where if two score distributions have the same values of p(k) for k from 0 up to X, then the score of X+1 becomes as close as possible to something like the â€śnew 0â€ť in the comparison of those two distributions. Iâ€™m not sure how to explain that effectively, I hope it makes sense.

Having that property is demonstrably impossible for a system with a finite number of score bins, so what I did is take an information projection of an infinite system satisfying that property onto a finite system, hoping to preserve the property to the best degree possible.

Assume that the votes are 100, and 30% of the votes assign a 0 rating to candidate A.
30% of the votes should be multiplied by â€śB^k*(1-B^{N-k})â€ť which equals about 0.9 so you would have:
0.3 * 0.9 = 0.27
which is like saying:
(30 votes) * 0.9 = 27
For rating 1 you will have metric â€śB^k*(1-B^{N-k})â€ť approximately equal to 0.5 and so on.

If itâ€™s a metric to be applied to purely honest votes, then the system is not equivalent, the problem is that once the system is understood, the voter will see the values â€‹â€‹{0.9, 0.5, 0.3, 0.15, 0.05, 0} instead of the values {0,1,2,3,4,5}.

Itâ€™s as if the voter saw ratings with exponential distribution instead of the classical linear [0,5].

Example, my honest vote like this, in Score Voting:
A[5] B[4] C[3] D[2] E[1] F[0]
Iâ€™d write it like this, in your voting system:
A[5] B[2] C[1] D[0] E[0] F[0]
If I wanted to use the min-maxing tactic, then it would become like this
A[5] B[5] C[0] D[0] E[0] F[0]
or so
A[5] B[0] C[0] D[0] E[0] F[0]

Yes certainly min-maxing is still a serious problem with this kind of system. I just think that it may be even more risky to do that kind of thing in this system than in a standard score system, especially if there are many candidates.

I will have to try to run some simulations. I think you may be correct that, like standard scoring, the Nash Equilibrium is approval.

Also, frankly I donâ€™t take that much issue with approval voting. I understand that media influence may distort perceptions of voters, but I feel like that isnâ€™t something that can be avoided. I also tend to feel like itâ€™s OK for voters to strategize based on other votersâ€™ expected ballots, that seems like it could actually encourage compromises.

These kinds of strategy problems are why I think itâ€™s important to have voters enter a state of artificial empathy with the other members of the electorate and to have them really face the risks involved with strategizing directly. The hope is that facing those risks in a transparent way will discourage manipulation tactics. I tried writing about that here under â€śThe Veil of Ignorance: Combating Manipulationâ€ť yada yada.

Also, I basically just realized that the metric actually simplifies considerably to be

SUM[k=0 to N] p(k)*B^k

I.e. minimization of this metric is equivalent to minimization of the previous metric (just distribute the B^k, use the fact that the p(k)'s sum to 1, and ignore constant terms).

I understand that media influence may distort perceptions of voters, but I feel like that isnâ€™t something that can be avoided. I also tend to feel like itâ€™s OK for voters to strategize based on other votersâ€™ expected ballots, that seems like it could actually encourage compromises.

It cannot be avoided, but it can be reduced.
Specifically, I noticed that the main thing to avoid is maximization, while minimization is acceptable.

In practice, if a voter knows the 2 frontrunners A and B (A>B), then his only interest must be to minimize the worst frontrunners B. Maximizing A must in no way increase Aâ€™s chance of winning over B.

Systems with ranges that give a partial solution to this problem are STAR, CB, and DV which, in the last step, if the 2 frontrunners remain, itâ€™s only necessary that the worst of the 2 has been minimized in the initial vote.
However, this is a partial solution because in intermediate steps it doesnâ€™t apply.
Eg. STAR using the sum in the first round, becomes subject to the min-maxing strategy (even if less than Score Voting).
DV resists a little longer because it isnâ€™t possible to maximize more than one candidate.

There are other â€śsolutionsâ€ť in range systems, but I donâ€™t want to change the subject.

Also, I basically just realized that the metric actually simplifies considerably to be:
SUM[k=0 to N] p(k)*B^k

In this case the conversion is about:
{0,1,2,3,4,5} --> {1, 0.6, 0.4, 0.25, 0.15, 0.1}
with B = 0.63

Thatâ€™s definitely a fair critique. I feel like this method of scoring could be used to modify something like STAR to make it less majoritarian. Rather than the highest two score sums, it could be the lowest two distribution metrics, for example, and then a runoff in the same way.

I donâ€™t think talking about what people have considered to eliminate those problems with score is off topic. This is a score method after all.

I still think that all the benefits you talk about end when the voters notice that the ratings are converted like this, before being added up:
{0,1,2,3,4,5} --> {-1, -0.6, -0.4, -0.25, -0.15, -0.1}
Note, that by converting them to negative metrics, the greater sum wins (equivalent to saying that, with positive metrics, the smaller sum wins).

I donâ€™t want to talk about other voting systems, just about the idea used to avoid maximization.

FAIR-Max
STAR uses the sum of the points in the first round (utilitarian) to get only 2 candidates, on which it then makes an automatic-runoff (majoritarian). The first round introduces the problem of min-maxing while the second round introduces the â€śproblemâ€ť of the majority.
FAIR-Max treats the votes as rankings as it eliminates the candidates (very majoritarian) and when only 2 candidates remain, it adds up the points (utilitarian). Overall itâ€™s very utilitarian but also resistant to maximization.
The method itself also has an actual mathematical property that effectively nullifies maximization as a tactic, leaving only minimization.

Extended DV
In Score Voting, to maximize a candidate itâ€™s sufficient to give him 5 points, while to minimize it is given 0 points.
By voting in one of the following two ways:
A [5] B [5] C [0] D [0]
A [5] B [0] C [0] D [0]
A is always maximized in the same way, that is, it always has 5 points of difference (maximum difference) from all those minimized.
This ensures that even minimized candidates are always minimized in the same way.

In DV, effective maximization and minimization is possible only in one case:
A [5] B [0] C [0] D [0] which then (with 100 points distributed) becomes
A [100] B [0] C [0] D [0]
If you vote like this instead:
A [5] B [5] C [0] D [0] which then (with 100 points distributed) becomes
A [50] B [50] C [0] D [0]
A is no longer maximized because it no longer has 100 points of difference with all other candidates. Likewise, all other candidates are no longer minimized to the maximum.
All this prompts voters to accumulate 100 points on their preferred candidate (similar to bullet voting, in single-winner case).

Extended DV (or Reversed DV), instead of assigning 100 â€śpositiveâ€ť points to approved candidates, assigns -100 â€śnegativeâ€ť points to disapproved candidates.
This means that, in Reversed DV, the only way to maximize-minimize is by voting like this:
A [5] B [5] C [5] D [5] E [5] F [0] invert the vote
A [0] B [0] C [0] D [0] E [0] F [-5] and distribute the -100 points
A [0] B [0] C [0] D [0] E [0] F [-100]
This means that the more candidates, the harder it becomes to maximize.
A vote like this (bullet vote):
A [5] B [0] C [0] D [0] E [0] F [0] inverted in
A [0] B [-5] C [-5] D [-5] E [-5] F [-5] and distribute the -100 points
A [0] B [-20] C [-20] D [-20] E [-20] F [-20]
as you see, this converted vote is much less maximized-minimized than this one (the previous one):
A [0] B [0] C [0] D [0] E [0] F [-100]
In practice, while DV favors bullet voting, Reversed DV favors the opposite of bullet voting (which is much less harmful).

Again, these are just examples to expose anti-maximization ideas in ranged votes.

2 Likes

I just want to point out that score sum maximization is â€śutilitarianâ€ť in a very particular sense that is not necessarily distributionally just, which is why I developed this system in the first place. I donâ€™t believe that kind of utilitarianism is necessarily idealâ€”it does not account for the actual distribution of scores/utility or properties of such distributions that would actually be preferable for a group or society. A distribution with a high expected score value relative to other distributions can still be divisive.

For example, a distribution with 60% 5s and 40% 0s has a higher mean than a distribution with 30% 4s, 30% 3s, and 40% 2s, yet I feel that the second distribution would be preferableâ€”most people are reasonably pleased, and nobody is horribly upset. The metric I have in place agrees with my own intuition about this for most values of B, but maybe thatâ€™s just how I feel. Nonetheless I imagine what we want is a distribution that represents a compromise between involved parties that are in disagreement. In that sense I donâ€™t feel like the utilitarian method you are pointing to is actually much of a contrast to what you are deeming majoritarian (not to be read aggressively).

The system I am considering basically evaluates an area beneath a particular cumulative distribution function of the scores given to each candidate. Doing this causes divisive distributions automatically to have a relatively high lower bound for their metric. To have a low metric, it is necessary to have both a decently high expected value and a decently low variance, and the metric is also relatively easy to calculate.

I do think that the extended DV is interesting. I feel it might even be sufficient to just select the candidate with the highest sum in that caseâ€”that would totally eliminate non-monotonicity, obviously the redistribution is a part of the whole point, but otherwise I could easily be missing something important!

A distribution with a high expected score value relative to other distributions can still be divisive.

In your example the difference is small so whether A or B wins is quite irrelevant.
A more striking example would be needed to support what you are saying.

The metric I have in place agrees with my own intuition about this for most values â€‹â€‹of B, but maybe thatâ€™s just how I feel.

If a voter uses these ratings {0,1,2,3,4,5} and then applies the metric, I think you are right, but if the voter reasons by applying the metric himself before voting, then I think the system fails (i.e., it becomes similar to Score Voting but with exponential ratings instead of linear).

I imagine what we want is a distribution that represents a compromise between involved parties that are in disagreement. In that sense I donâ€™t feel like the utilitarian method you are pointing to is actually much of a contrast to what you are deeming majoritarian.

When I say Iâ€™m against the majoritarian, I mean Iâ€™m against this:
55%: A [5] B [4] C [0]
45%: A [0] B [5] C [0]
With majority methods, A would win, even if obviously B should win.
The problem in this example is much more obvious than in your example.

To have a low metric, it is necessary to have both a decently high expected value and a decently low variance, and the metric is also relatively easy to calculate.

I had thought of this idea too but in the context of multiple winners. I had thought of doing this:
For each candidate we calculate:

• average.
• the simple average deviation (I prefer it over variance because itâ€™s easier to explain to an average voter).

then create two ranks:

1. sorts candidates from the one with the highest average.
2. sorts the candidates from the one with the lowest average deviation.

The candidate with the highest position in the two rankings is chosen as the winner.

Example (yours):
30: A [5] B [4] C [0]
30: A [5] B [2] C [0]
40: A [0] B [2] C [5]
Ranks:
Avg[ A(3), B(2.9), C(2) ]
AvgDev[ B(0.84), A(2.4), C(2.4) ]
The difference between A and B values â€‹â€‹in Avg is much smaller than the difference of the values â€‹â€‹in AvgDev, so the Avg rank is worthless and B wins.

The general rule is:

• sum the values between Avg and -AvgDev and the greater sum wins.
• before adding them, however, I have to multiply the Avg values by a certain% (eg 0.8) and those of AvgDev by the remaining% (eg 0.2).

What I miss is figuring out the best way to assign those %.
Eventually (online) you can create a system where you can change the weight of Avg and AvgDev at will, to see how the result changes.

I think that is circular reasoning. It is only irrelevant if you are using the mean as a standard for evaluating distributions in the first place, but that is the very premise I am challenging. You can make it 70% to 30% in the first distribution and the logic is still the same, I would try to avoid having 30% of the electorate totally upset if at all possible.

This is something I was working on regarding means and variances. I started here and thought my way to the system on this post, not that that means much:

I am also not sure how effective reasoning with the metric would be anyway. Because itâ€™s discrete and exponential, voters will have to be sure about which candidates they want to minimize if they are unsure about how others intend to vote. At the same time, the differences between higher ratings become smaller, which punishes voters less for voting their actually preferred candidate as highly as possible, and their backup as second-highest or so. Even voting a candidate just one or two slots lower is a pretty significant defense against them beating out your more preferred candidate, so maybe voters will be more inclined to indicate their preferences somewhat honestly.

On a theoretical level I agree that:

• better fewer points, more evenly distributed among the voters.
• rather than more points, but accumulated on a few voters.

But, on a practical level, with a really utilitarian system, I think the problem solves itself.
In a utilitarian system, after the first election, the parties will immediately begin to understand (more or less) which is the best â€śhalf wayâ€ť, that is, the candidate who would have a good score from the most voters, and in subsequent elections it is likely that such candidate appears and wins (if there are no min-maxing strategies, especially maxing ones).
For this I try to eliminate maximization from range methods.

P.S.
The main problem of giving weight to the variance (rather than the mean) is that a candidate with all 0s, gets the best variance, so I think the main problem comes down to understanding what is the best weight balance between variance and mean, but this what it seems to me that does not have an â€śobjectiveâ€ť answer.
There may be contexts where a smaller variance is better and others where a larger mean is better.

Regarding the idea of favoring candidates with a low variance (but with a sufficiently high average), I suggest you also take a look at the PRO-V system, which is extremely simple:
in Score Voting you add the ratings, while in PRO-V the product is made, thatâ€™s all (obviously, the range must be [1,MAX] without values <= 0).

Example (range [1,9]):
A[1] B[4] C[8] D[9]
A[9] B[4] C[2] D[1]
Sum: A[10] B[8] C[10] D[10]
Prod: A[9] B[16] C[16] D[9]

In Score voting candidate C would be on a tie with A and D.
According to your idea, instead, candidates A and D should be clearly worse than C (which has more points equally distributed).

In PRO-V the difference between C and A/D becomes clear, furthermore, even candidate B (although having a smaller sum than C) is equal to C, this too agrees with your idea.

Yes, in fact the principle of PRO-V is almost identical to the system I am suggesting. You can instead think of PRO-V as a score system where the scores are associated with their logarithms.

When you say â€śas score systemâ€ť do you mean that you could somehow add up the votes (instead of making the products) to get a system equivalent to PRO-V?

Given these rating {1,2,3,4,5} (range [1,5]) which logarithmic values should I associate with it to obtain a â€śScore Votingâ€ť (sum of the votes) equivalent to PRO-V?

Yes sorry by â€śscore systemâ€ť I do mean the simple score voting where the scores are summed. To turn PRO-V into a weighted score system you would use the association

{1,2,3,4,5}â€”>{log(1),log(2),log(3),log(4),log(5)}

This is because log(xy)=log(x)+log(y), and logarithms are monotonic, meaning that maximizing the logarithm is the same as maximizing its argument.

So for example,

xyz>abc

is equivalent to

log(xyz)>log(abc)

which is equivalent to

log(x)+log(y)+log(z)>log(a)+log(b)+log(c.)

So basically what I am suggesting is a (sort of) generalization of PRO-V, except instead of maximizing an increasing logarithmic sum I would prefer to minimize a decreasing exponential sum, which is a somewhat different concept.

Youâ€™re right, I didnâ€™t think about it.

Better this way because to an average voter I can say that the product is made instead of the sum (which is extremely simple to understand), while mathematically I can use logs to simplify the calculation.
I think any other metric would be more complex to explain than this one.