Variety of Voting Systems

If DV is not chaotic I can definitely get behind it! Also, I guess there is a “limiting case” argument for proportional redistribution. For example, using the string voting as a crude model (supposing it is a reasonable representation of interests) you could assume that as the allowed number of characters gets arbitrarily large, voters will tend to approach a stable distribution among the candidates, i.e. far out in the string, the probability that a random character represents a particular candidate should depend only on the candidate. I think that’s a reasonable assumption, and in that case the long string method will, at first, conform to a discrete analogue of proportional redistribution in terms of costs to voters. And if the marginal costs are distributed more-or-less proportionally, then so should be any marginal gains.

I agree that the string method is more theoretical than practical. It allows voters to control their marginal utility functions for a small number of discrete votes, but for proper redistribution you would want to be able to analyze the marginal utility functions for an arbitrarily large number of discrete votes. Pushing the characters to the left achieves some level of redistribution, but the problem is illustrated in the example I gave, where Voter 1 at some point in the algorithm ends up with a string like AAAAAAAXXXX, and hence one of their available votes isn’t counted at all. If they had submitted a longer string, basically “loading up” the marginal utility function beyond the characters that are counted in the algorithm, they would have a better chance at having their votes fully redistributed according to their marginal utility.

So I was thinking, maybe the theoretical voters could submit significantly longer strings, and then have the algorithm only look at the beginning portions, and allow the characters beyond that to slide into place when candidates are eliminated. But I think that opens the floodgate to tactical voting of the kind you suggested. When the strings are totally truncated, I’m not sure that tactical voting in that sense would actually be a good idea, because the tactical voter runs the risk of their alternative choices being eliminated early, and then having no fall-back if their first choice gets eliminated. So it’s risky. Basically I think honest voters would have more consistent control over the entire election, while tactical voters would have disproportionate control over the beginning stages, but then their influence would die out quickly.

The other method similar to DV I was thinking of is analogous to the string method, where voters incur costs in voting capital proportional to their investments in protected candidates. So again, rather than eliminated the least-grossing candidate, candidates are protected in turn, costs are incurred for protection, and then the final remaining vulnerable candidate is eliminated and remaining capital is redistributed. I think that would tamp down on strategic voting, but again I am not sure how it fairs in practice, and I know that there are problems with it. For example, if you have two voters with distributions

Voter 1: [1,0,0,0,0]
Voter 2: [0,1/4,1/4,1/4,1/4]

I think we can agree that the most rational approach to deciding a victor is to uniform-randomly select one of the four candidates that Voter 2 supports, and then select uniform-randomly between that candidate and the candidate that Voter 1 supports. But with incurred costs and redistributions, I think Voter 1 comes out on top, basically because Voter 1 incurs costs to protect his candidate once at the beginning of the first round, but then Voter 2 incurs costs three times for the remainder of the round before one of their candidates is eliminated and the points are redistributed. So vote-splitting exists. I don’t know how it works with many voters though. Vote splitting is avoided in DV because there are no incurred costs, but then there is the (minor?) risk of tactical voting.

If tactical voting is a very minor problem in DV, I think it turns out to be superior to a protection-cost/redistribution system. But if there were a way to incur costs without vote splitting, I think that would be a great theoretical development. I was having costs incurred basically like an auction against the average of the remaining candidates, but there might be a better way. For example, maybe voters pay the cost of the difference between the highest-grossing candidate and the second-highest, so it’s kind of a grass-trimming situation. I don’t know, just spit-balling here.

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When the strings are totally truncated, I’m not sure that tactical voting in that sense would actually be a good idea

Yes, truncation should avoid those tactical votes.

Vote splitting is avoided in DV because there are no incurred costs, but then there is the (minor?) Risk of tactical voting.

In practice I don’t know, because I have yet to test it, but I have already created a site that allows you to create polls with DV (maybe you want to try it):

Regarding the “chaos”, DV for single winner is a subcategory of the IRNR method, and at this site you can compare IRV with IRNR to see the difference (select the 2 voting systems at the top right):
I also did tests on the actual DV and the results were like that.

Are you familiar with Yee Diagrams? Basically, each pixel represents an election where the average voter is located at that pixel, and voters prefer closer candidates. The voters’ X & Y coordinates are normally distributed, with sigma=75 in my code for each axis. The pixel is given the color of the winning candidate (the candidates are drawn on the diagram after the simulations; they’re the circles.) Ideally, each pixel should be the color of the nearest candidate. Here’s one I made for distributed voting.

Areas that appear “blurry” with many differently colored pixels near each other are ones where there is a lot of chaos, since it means that the winner is unclear and changes from simulation to simulation.

Here’s a picture indicating the win probability of the blue candidate (5 trials per pixel; white indicates that blue won all 5 trials; black indicates that blue lost all 5.) Gray areas are high chaos, and white areas are low chaos. Black areas may be high or low chaos.


For comparison, here’s a Yee Picture for score voting

For thoroughness, here’s the probability picture for score voting


If you are making such diagrams can you do STAR voting and STLR voting?

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No I had not seen Yee Diagrams before, I’ve gotten into this subject more recently, and I’ve been learning more about the abstract properties of systems. Specifically I’ve been reading through a textbook called Economics and Computation, and doing my own additional research online. Wikipedia, Electowiki and organizations like the Center for Election Science have been very helpful. I’m really happy to have found this forum too, and I’m kind of sad to hear it will be closing on the 30th. Hopefully it will just be moving elsewhere, because I think this is a very important and very interesting discussion to be having.

Lately I’ve been more interested in learning about the practical aspects of voting systems as in through simulations or comparisons with survey data. Yee Diagrams seem like a very interesting method of analysis–basically if you draw random lines between two points of the same color, you don’t want to find intersections with other colors. Using that logic, by generating line segments randomly you could probably give a numerical measure of convexity for higher-dimensional Yee Diagrams too, although it probably wouldn’t matter much now that I consider it, since adding extra dimensions shouldn’t really change anything… So according to the Yee Diagram you shared @Marylander, it does seem that Distributed Voting is quite chaotic, and Range Voting is not. I’m curious about what you have to say about this @Essenzia. Thank you for sharing!

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We’re on that. :slight_smile: We’ll have a new forum up soon. Equal Vote is taking charge, we’ve got a little organization/committee formed to own it, and I’ve volunteered to get it up and running. It will technically be a new forum (using NodeBB as the discussion forum software), but all the old content will be statically archived, etc.

If you want to get involved in it, please let us know.


That sounds like a great project, I’d love to get involved! Although I wonder what I would be able to offer in terms of helping with the actual setup of the forum. I studied math but not computer programming, and I just recently graduated lol. Either way I’ll definitely try to keep up with the migration and keep the discussion going. Let me know if there is anything particular I can do to help or where I can get started.

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I’ve already had to deal with wrong DV codes, so could you share the function with which you do DV counting?
If you use java, I can send you mine.

I would be interested in seeing your code and the Yee Diagram associated with the correction. I also still think for these Yee Diagrams it would be nice to have some numerical measures. You could establish a variable line thickness, and then generate random line segments between points of the same color, and determine how many lines intersect with points of a different color when many random line segments are generated. That would give some indication of how convex the voting system is, because sometimes the relative fuzziness at the borders is hard to evaluate visually.

Additionally I definitely hope that DV is not chaotic lol. If it is then I think the system concepts I’ve tried to develop would also be chaotic.

Now I show you a classic case where you see the failure of monotony, with various methods (sigma 40) starting from the one with the best result. Remember, however, that these simulations use totally honest votes so they are theoretical. Real simulations should also deal with resistance to various types of tactical votes.

Score Voting




Now three diagrams with 7 randomly positioned candidates, with DV system (sigma 40):

DV fails the monotony and this creates chaos, but it seems to me something acceptable in most contexts.
The @Marylander example occurs when candidates are arranged in a “circular” way (very rare case) and when sigma is high. The sigma value can be understood as chaos.
To clarify, this is DV with sigma 40 and sigma 0:

If your system tries to eliminate the worst from time to time (by changing the votes), then it will probably fail the monotony and therefore introduce more or less chaos.
In DV I wanted to minimize this problem.

I want to clarify that the DV count pushes the voters to give 0 points to all the more or less disapproved candidates, and in this way the chaos is reduced.
In the diagrams I posted above, only one candidate receives 0 points (the worst) although in reality more than 1 typically receives 0 points (reducing chaos).
In the DV I tried to minimize the tactical votes (the tactics actually usable) so the merits of this method compared to the others should be seen more in tactical contexts. However, I still have to do these simulations so for now you will have to settle for the Yee diagrams.

Yes for sure, that makes sense. I think it’s reasonable to assume that actual candidates would be spread throughout some elongated oval shape in this kind of space, as in your first and second examples with 7 randomly positioned candidates, and that makes circular symmetry very unlikely. It seems that DV is “mostly” monotonic and in almost all cases is not chaotic. I’m interested in some of its formal properties and also in its performance regarding those properties.

These are the properties I know of that I think are the most important, besides obvious ones like non-dictatorship, determinism, citizen’s sovereignty and polynomial time:

  • Monotonicity
  • Independence of Clones
  • Favorite Betrayal Criterion
  • Strategy Hardness
  • Independence of Irrelevant Alternatives
  • Participation
  • Reversal Symmetry

I’m not really concerned with Condorcet/Smith criteria, Majoritarianism, Separability or Homogeneity. I’m not really sure what to make of the Later No Help/Harm criteria. I guess I wonder how DV fairs with these criteria, and if you think there are any other criteria that I might be reasonably concerned with.

I thought this was a really informative page about other voting simulations as well:

It seems like STAR and 3-2-1 do exceptionally well. I wonder where DV would fall in comparison.

I’m interested in some of its formal properties and also in its performance regarding those properties.

I know, I’m still working on it.

Monotonicity, Independence of Clones, Favorite Betrayal, Criterion, Strategy Hardness, Independence of Irrelevant Alternatives, Participation, Reversal Symmetry

The problem with these properties is that when they are not satisfied, they are not asked how much they are not satisfied.
DV fails monotony but IRV fails more.

I think the only real solution is to do various simulations in many contexts, and understand which method gives the best results. The problem is that even simulations can have arbitrary rules that are not necessarily correct or accepted by everyone.

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The fuzziness has a lot to do with the number of voters as well as the sigma. In all my pictures there were 2000 voters per election.

The way I’m converting distances to cumulative ballots is by setting utility to 1/√(12000+Distance²) (which is what Warren Smith’s IEVS used). Then I subtract the lowest utility from all the utilities and distribute the 100 points by applying Sainte-Laguë to the offset utilities. How are you converting distances to ballots?

Also: why are you not getting the Voronoi diagram with sigma=0? In such a case the nearest candidate should be preferred unanimously.

How are you converting distances to ballots?

I’m going to do different simulations and explaining them without code is complicated. However, I don’t think I have to prove that there are arbitrary choices, it’s obvious.

why are you not getting the Voronoi diagram with sigma=0? In such a case the nearest candidate should be preferred unanimously.

I noticed that in code used by me it’s sigma = 1 which returns Voronoi diagram.

The original page

has some very well chosen points. Do you think you could copy them and put them somewhere? It would be really good to have a growing set of good examples and have them for as many methods as possible. Electowiki seems a good place for such documentation. @robla would this be too detailed? It would evolve over time so I am not sure there is a place better than a wiki that multiple people could add plots to.

@Marylander told me that with sigma = 0 I should get Voronoi diagram, but in the code that I use, the Voronoi diagram I get them with sigma = 1 so during the modification of the code I could have made some little mistake.
Maybe it’s an error that doesn’t affect the graphs when sigma is greater than 1, but I am not sure of it so I don’t feel like proposing such graphs.

@Keith_Edmonds, @Marylander and @cfrank i was wrong 2 times :sweat_smile:
The diagram with sigma = 0 was not Voronoi because I used range [0,9] in ballots. Using instead range [0,1000] results Voronoi so the graphs are right.

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After a test with the scenario [Yellow: (101, 143), Blue: (73, 126), Red: (156, 155), Green: (17, 54)], it seems that the utility function I used for the original example was particularly bad for DV.



If we rerun the octogonal scenario with Utility=1/(12000+Distance²), the nonmonotonic region shrinks a great deal.

So it seems that DV’s performance in the Yee diagram scenario depends a lot on the choice of utility function.

Edit: nevermind; I forgot to change part of the code and so the effect was essentially to eliminate the part of the code that subtracted the lowest utility rather than change the utility formula. With the proper code it actually doesn’t seem to matter as much: