# Thoughts on Vote Unitary

I want to preface this by stating that Keith has been an extremely valuable member of the Equal Vote Coalition Research Committee and has worked harder then anyone else to test various methods in a variety of ways.

Over my past 2 years on the Equal Vote Coalition Research Committee, I have given many criticisms about his vote unitary (latest revision here) concept. @Keith_Edmonds I know this is going to sound be harsh, but to me it seems like a pseudo-concept rather then something based in rigorous mathematics.

Keith talks about it as if it is a mathematically rigorous criterion for which he knows what methods pass or fail it. Can anybody write a computer program that takes in some election result (as well as the order the candidates were elected along with the weights of each voter throughout the process and the single winner selection method those weights effect if necessary even though to any mathematical criterion this information should be irrelevant and all that should matter is strictly the vote inputs and winner outputs) and determines whether that election violates vote unitary?

Another thing that bothers me about it is that itâs definition seems very inconsistant. When he describes the concept in his electowki page, he describes it as being about vote weight being able to be transferred to candidates but not being able to be created nor destroyed. If he applied this definition found at the top consistently, then all allocated systems like allocated score and sequential Monroe would also pass this criterion. However he doesnât apply this definition throughout the page as latter when he gets to the compliance of multiwinner systems, he adds additional requirements to the criterion:

On the other side, a voter who has not been fully satisfied should still have some level of influence. This means that systems which allocate votes such as Single Transferable Vote and Sequential Monroe violate Vote Unitarity if they allocate the whole vote weight to a candidate the voter did not express maximal endorsement for.

This is an additional requirement concept that has nothing to do with vote weights being conserved that could be better described as itâs own criteria. But it gets worse:

In sequential multi-member systems this concept become especially relevant due to the different rounds of tabulation. Specifically, a voter whose favorite has been elected should not have influence over subsequent rounds.

This would imply that SSS fails vote unitary because even under SSS, voters whose favorite candidate is elected donât always loose all their influence. When too much weight is exhausted this way, SSS has a check to make sure that it no more then a quota of weight is exhausted, at which point SSS reduces how much weight each vote would lose proportionally so that only a quota of weight is exhausted. In itâs current form, no voting method can satisfy this criteria without violating proportionality which is why even SSS (the voting method based off of it) fails it.

Finally, Keith makes the criterion means something completely different altogether for single winner elections:

In single member systems this property is defined by the Equal Vote Criterion.

It would be one thing if the criterion naturally reduced to this when applied to single winner elections, but you canât just define one criterion to behave completely differently based on how many winners you have.

What concerns me the most about it is that Keith keeps describing it as the concept of one person one vote when I have yet to see any substantive argument for that perspective (which he states, not as a perspective but as a fact). The criterion has now ended up in various Equal Vote Coalition Research Committee reports.

Thats not harsh. In fact it is totally fair. I would say it is more like an extension of Proportional Representation. Similar to the Thiele reweighting concept. I would say more that I discovered it and am finding out more and more about the concept as I learn more. A good analogy that was taught to me on this forum is that of the relation to the party list case. It is fairly obvious that SPAV and RRV are related to the Highest average system. I however did not see that Vote unitarity was related to the Largest Remainder system.

Similarly again to PR it is intuitively pretty obvious but when you dig into the details it gets tricky. Vote Unitarity was invented after Sequentially Spent Score. I invented it in order to name the mechanism or that system and explain why I did not like RRV. I liked allocation better but felt it needed refinement. The idea was that we should be allocating on the score level not the ballot level. And in this way it is very related to the KP-Transform. However, the KP transform does lose all connection to the original ballot. Either way, they both move to a finer gain than ballot allocation. The justification for it being better than ballot allocation is that finer grain would make better splitting decision when things like rounding errors come up.

No, it is not like winner set stability or something. It is about the process of the computation. It is the procedure which is or is not Unitary. If your gain in utility/representation is directly proportional to your loss in influence then the mechanism is Unitary.

Maybe the wording is bad there. Let me explain. In an allocation system you could have given somebody a score of 3 and had your whole ballot allocated. This means that you are 3/5 satisfied but have lost all your influence. The remaining 2/5 was destroyed without explanation. This is a violation of vote unitarity.

Maybe I am being too arbitrary with the definition. For example, when you do surplus handling in SSS you get more than you spend but it is justifiable because of the others. Does this violate vote Unitarity? I have defined it not to but I am open to being told that is bullshit. This is definitely something I canât verbalize well but it âfeels rightâ.

I am pretty sure that the scaling variant of SSS is a violation of vote unitarity. This is why I do not like it.

That was not me who wrote that. I had that space filled with " In single member systems this property is trivially satisfied due to the simplicity of such systems." There is a IP address but no user assigned to that edit. I think I know who it was because I recall a thread about this a while ago. Around thanksgiving. Anyway, if nobody is elected then nobody should have influence reduced so it is always satified. Or at least that is how I view it. Its like asking if score is a largest remainder or highest averages method. It is a bit of a meaningless question. Anyway, I do not recall what the âequal vote criterionâ is. Is it the same as âtest of balanceâ because I do not really like that criteria. There is more on that here but I have never edited this page.

That is the essence of what I was trying to do. âone person one voteâ can mean a few different things in different context. Everybody should be given one vote at the beginning. That is not vote unitarity. The concept I am trying to promote is that if winners are chosen in steps then the fractional allocation of the ballot is done in proportion to the support. I know you know what I mean here.

There are likely some good argument why preserving this relation is not wanted. Maybe the assumption of the linear relation between score and utility is unfounded. Anyway, I am not the arbiter of what is and is not a Unitary transformation of vote. If you can formalize this system better than me please do so. I know there is a kernel of usefulness in this concept. This is how ideas evolve. I would not expect to have come up with the idea perfectly formed all on my own.

No, this is good. This is exactly what the forum is for.

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Not on the electowiki page. On that page, this is your definition:

Vote Unitarity ensures that each person should have one vote and that vote should not change in power during the rounds of tabulation in any system.

This includes no mention of when voters can or cannot loose influence.

Though to give you the benefit of the doubt, you do provide a secondary definition:

More mathematically, it is the condition that the time evolution of the vote according to the tabulation procedure is mathematically represented only by Unitary Transformations

The language used here is from a specialized field of mathematics that Iâm not well versed in and would like to learn more if itâs relevant to electoral reform, but to be honest, I have no idea what you mean by unitary transformations. The wiki page that unitary transformation links to is very deep. But perhaps thatâs just on my end and this one statement is enough to fully describe all of the requirements you add to youâre vote unitary definition later in the page.

This is one of the things that separates mathematics and science from many other fields. âfeels rightâ is not allowed. Human intuition often leads us astray.

Thatâs why I put this in parenthesizes:

as well as the order the candidates were elected along with the weights of each voter throughout the process and the single winner selection method those weights effect if necessary even though to any mathematical criterion this information should be irrelevant and all that should matter is strictly the vote inputs and winner outputs

If for a voting method, I give you a set of votes, tell you the single winner voting method the algorithm uses to select the winner, and walk you through the process by which the winners are chosen, providing you with the weight of each vote at each round in the process, can you tell me if my method is violating vote unitary? Iâd imagine it would be extremely difficult to write a program that reads in the definition of the voting method in English or whatever programing language you want and determine if that method passes vote unitary so this is the next best thing.

Reworded in computer speak:

Can you write a function that takes in a 2d array of votes and for each vote: that voteâs scores, an array of electionState structs (one for each round) which contain:

1. a boolean array of which candidates were currently elected during that round
2. a 1d array containing the various weights each voter currently have

as well as a function pointer to a single winner selection method that:

1. takes the 2d array of votes as well as one of the 1d weight arrays from one of the electionState structs
2. outputs the winner for under the weights for that given round (to validate that the weights are actually being used correctly and that whatever voting method you are testing isnât just giving you garbage weights to pass vote unitary)

and outputs a boolean telling me whether my method is violating vote unitary?

Is that enough information to work with or would your function need more? The only information that youâre function cannot have is the whole internal logic behind why weights are reduced the way they are because that level of information can go on for ever. Is there any amount of information that will allow a generic sequential voting method to be plugged in and be determined to fail vote unitary? If itâs impossible to write a function that determines whether a voting is violating vote unitary, then your concept can not be treated as a rigorous pass/fail criteria.

This is important because without this you are the arbiter of what methods pass/fail vote unitary, and if you are the arbiter how can I even test the boundaries of the criteria to determine itâs worth when I myself canât rigorously determine what methods pass it?

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I also do not have the best understanding of the mathematics used by unitary transformations. However, I do know that the identity function preserves inner products along with everything else, and therefore would be a unitary transformation. This would seem to imply that all bloc methods meet that definition, so I donât really see a connection between it and the concept that @Keith_Edmonds is trying to capture.

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OK this works. This is something that can be tested in the function I mentioned. Though it still leaves out extra requirements you have included on the electowiki page. It says nothing about votes that donât give a full score to their preferred candidate not being able to be fully de-weighted during a round or votes that do give a candidate a minimum score being required to loose all influence over a given round (which as we previously discussed, SSS canât pass this unless some sort of exception for which this is OK is added at which point your criteria is kind of just describing your own methods). So perhaps you might want to separate those into separate criteria?

I still think that a rigorous criterion should only be defined in terms of inputs and outputs and internal parameters like weights should be none of the criterionâs business, but this is still much easier to work with then what you had before.

Finally, you canât keep calling it one-person-one-vote. This is one opinion of what one-person-one-vote translates to in multi-winner methods, not the end all be all. You stating that vote unitary is the concept of one person one vote would be like FairVote stating that LNH is the concept of fair voting methods. It de-legitimizes all voting methods that donât pass it as somehow violating one person one vote. You canât just redefine a term that is synonymous with democracy. Sure you can argue why you believe methods that pass vote unitary preserve vote equality more then methods that donât (and I would disagree with you on that point), but please donât state it as fact on what is supposed to be an objective informative resource like eletowiki.

Yes and no. I come from theoretical physics. There is a concept we refer to a âphysics intuitionâ. Einstein made nearly all of his breakthroughs in this manner not through proofs. Of course he later fleshed out the idea and defined it mathematically. The point being is that if you understand the system you get a feel for what is right without doing the math.

This might help. I am thinking more in terms of optics or quantum mechanics. There is something called the S-mtraix which is a unitary operation. It takes the initial state like a beam of light and scatters it into sub components. What I am envisioning is that we are taking the original score matrix and transforming it with a similar process to this scattering. When a winner is elected there would then be two resultant score matrices; one for the score attributed to the winner and one for the score remaining. The splitting transform must be unitary. The unitary transform I am thinking of is not simple addition because each row representing each voter has a maximum power. I think what it needs to preserve is the sum() across all of the matrices of the max() in each matrix for each row. In a 5 winner race you end up with 1 matrix for each winner and a remainder matrix showing what was not spent. This was all explained at the group level but there is an individualistic component at the individual voter level too. That is that the sum of the elements have to add up too but not such that they are equal. You want it such that the sum of the output does not exceed the input. It can be less because of the total being spent. This is I think where the problem where my non-mathematical and mathematical definitions clash in surplus handing. In the mathematical one it is fine that the full score is not given to the winner in such cases.

I think the tricky part is that there are some parts which I might have put into the definition which are not part of vote unitarity. I think much of that is the stuff which is required to be PR. For example a system which elected people but did not assign them any score ever would obey vote unitary because everything is conserved unitaritly.

So yes, I suppose Bloc methods do hold to this. However, when I defined it I also rolled in the concept of Monroe type PR. So maybe what I am trying to say is that Vote Unitarity is an enhancement on PR and should not be considered outside of that realm.

It is the preservation of one person one vote. It is likely best to take a step back and consider some other systems for context.

I hope it is pretty clear how SSS passes this (or at least with some hand waving around surplus handling). It should also be clear that allocation fails this since a score of 2 could result in a total of MAX influence being allocated.

Things get more tricky with SPAV. It actually has a concept similar to Vote unitarity but I would argue is a different concept. It might be equally valid, though. I did not understand it myself until a few months back. In SPAV the vote power is spread out between the elected candidates and the potentially next candidates. So if you have 1 candidate then half goes to them and half goes into electing the future candidate. So the half going to the next candidate is your ballot weight. If you have two candidates then a third goes to each of them and the final third goes into you ballot weight. This is a different way to conceptualize one person one vote.

This is more something which makes sense for an optimal system. Since an optimal system does all the selections at the same time there is no time evolution and hence Unitarity does not really apply. There is however a similar concept of preserving the vote. I think it should be named and have said so in the past. Thieleâs method is what we tend to use.

When moving to score voting one can transfer this concept by going from SPAV to Single Distributed Vote. The name is intended to reflect the Unitary like property that there is one vote which is just being spread out. I am not sure what the optimal version of Single Distributed Vote is. It might be Harmonic Voting.

Anyway, the âThiele one person one voteâ is different than the âUnitary one person one voteâ because in the Thiele method the amount of vote power given to each winner changes over time. In the Unitary method once the winners matrix is made it never changes. In an optimal method the Thiele concept seems better to me. Sequential Thiele systems seem better with Webster reweighting but Jefferson is more natural since it treats future and past winners symmetrically. However, I am not sure that putting more weight on already elected candidates is not reasonable. This is why in sequential system the unitary concept makes more sense to me.

I hope that clears up some stuff. I know the electowiki page could be better but I would like to point out that there are no pages for the Thiele or PhragmĂŠn mechanisms. I would think that these are the âAlternativesâ to Unitary. I do not feel like any of them are super rigorously defined.

Lets take a moment to talk about single winner systems since there are sequential elimination methods like IRV, STAR and IRNR. Since these eliminate candidates there should be some adjusting of vote power, right. I feel like IRV fails this concept because it does not keep track of the amount of endorsement. On the other hand all your vote is on one person at a time with full power so maybe it is fair. I would think IRNR is more in line with this since the vote power is clearly conserved through normalization. STAR and STLR have different interpretations for what power preservation means. I get into this in the other post a lot but I think they both pass. It seems like all single winner systems pass with the exception of maybe IRV.

You could also do something similar with RRV: Each round a voterâs vote weight is evenly split between each previous candidate (proportional to the score they gave that candidate) as well as to themselves for giving to future candidates (so treat the voter as itâs own candidate in this process for which they rate themselves a 5). To me, this way of understanding RRV is no more meaningful then the raw mathematical definition because to me any system where votes are cast anonymously (thus the voting method canât distinguish which vote was cast by myself and decide to give me twice as much weight as any other voter with the preferences would normally get throughout the process, and I would contend that this is the right way to think of one-person-one-vote because itâs political implications align the most with the political implications of violating the version of one-person-one-vote that the average voter is used to. Ex. the dictatorship method clearly violates one-person-one-vote). But to you this alternative way of looking at RRV might be of some value.

Thatâs why I donât refer to them as rigorously defined criteria. I have never said you canât do that without violating Thiele or to preserve Thiele in this instance you must do this. When I want to point out how one method is deviating away from one of these philosophies in an undesirable way objectively, then I find or create an objective criteria that one of the concepts passes and the method in question doesnât (ex. independence of irreverent ballots). You have defined and used youâre vote unitary concept as a criterion.

This is good and insightful but it is the first I am hearing of it. I think a page justifying Thiele systems with explanations like this is really needed. Nearly all people view RRV as a formula with no justification from a philosophical point of view. Like I said, I was unaware of it until this year. I do however think the RRV explanation is less compelling than the SDV explanation but I am glad there is at least a way to word it.

I guess that is fair but I would argue that it would be better for Thiele type systems to start talking about even distribution of votes. SSS does not do this just like how RRV does not use the unitary mechanism. I think that I likely over stated vote unitarity because I did not know that there was such a concept for Theile systems. I find the concept of Vote Unitarity very compelling but others may not. And then there is Monroe and Phragmen systemsâŚ

There was a thread a while back where we were trying to define each philosophy in a short phrase. Some of that made it onto this page. The lines between Phragmen and Thiele are a little blurred and Unitarity is sort of an extension of Monroe.

I do not see how such a philosophy really differs from criteria when you get right down to it. Especially the fake criteria like LNH. Of these 4 PR philosophies there should be one that you like the best. It is sort of like the choice between Majoritarianism and Utilitarianism. If somebody invents a system and I do not prefer it because it is majoritarian and not Utilitarian it is sort of like saying I do not like it because it is not Unitary. I could even take it a step further. Participation and Quota criteria are basically incompatable. When I say I want a system to pass hare quota I am saying I care more about the philosophical implications of that then participation. Same as when I say I like Unitary systems over Thiele systems. There is just dissagreeemnt about which criteria even matter. People cant even rally behind the important ones like CloneÂ­proof, MonoÂ­tone and ConsisÂ­tency

I think in the end I will take your point. I have been doing a bit of advocacy in a place where I should not. I would sell SSS by showing why I think vote unitarity is important. If you want to come at me with SPAV as an alternative then me saying it is not Unitary is meaningless if you think that the Theile philosophy is better. If we ever get a citizens assembly up here in Canada I would want to pitch both philosophies and see which one the masses find appealing. I would stress though that it is up to the advocates of their preferred system to come up with compelling justifications.

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This RRV âexplanationâ I gave is literally just something I came up with on the spot to show how it can fit with a lot of peopleâs (I would say misguided) STV-like intuition about votes being things that need to be distributed between candidates. I definitely donât regard it as the definitive explanation for why RRV works the way it does. For me the justification lies in the math behind the optimal systems it is related to.

It would matter in an advocacy campaign.

OK then if thatâs what you mean then hereâs a more rigorous definition.

First, suppose that the method we are examining only needs the proportion of votes that were cast in each possible way to determine the winner, not the raw number (i.e. if every vote were duplicated k times for some k, then the outcome would not change). I donât think we ever considered any methods that didnât pass this. (Also assume that the method can use a continuous scale).

Suppose candidate c is elected in a round of a sequential election with a surplus, and that changing the score for c on any proportion of the vote less than or equal to Î´* (for some Î´* with 0<Î´*<1) would not change this or the outcome of any previous round.

Let Îľ>0. Then there exists a Î´ such that 0<Î´<Î´* and

If

• V is a proportion of the total votes equal to Î´ that all vote in the same way
• M_V(s) is what the average maximum score that a vote in V could give to any candidate in the subsequent round would be if they all gave c a score of s (i.e. the total maximum number of points that the votes in V would be able to contribute to a candidate in the subsequent round divided by Î´),

Then there exist real numbers Îą, Î˛ such that |M_V(s) - (Îąs+Î˛)|<Îľ for all s in (min_score, max_score).

More simply, if we change the score for an elected candidate c on a vote that comprises a very small proportion of total vote (particularly small enough that we donât have to worry about it affecting the outcome), and there is a surplus, then that ballotâs influence in the next round should be approximately linear with respect to the size of the change.

The reason that the order of election needs to be maintained is obvious. The reason that the change in influence has to be approximately linear (and that the proportion has to be small) is that changing the score will change the size of the surplus. In the first round of an SSS election, if a voteâs score for winner c is raised by k points (and was originally S_v,c), and order of election is maintained, then the vote will have its cap reduced by (S_v,c+k)*(Quota)/(OriginalSurplus+k+Quota), which is only approximately linear with respect to k. (Note however that after the first round capping messes with the linearity, since M_V will be piecewise).

This criterion is effective at screening out RRV. If the range is 0<=s<=1, then in the first round M_V(s)=1/(1+s) always, which is clearly nonlinear.

Yes, to a rounding error this is true.

This is A pretty good characterization of the surplus handling in SSS. I am not sure that it really covers Vote Unitarity in general. I would hope that there are other systems than SSS that would be considered Unitary.