Multiwinner elections and Bayesian Regret

[Toby Pereira]

Bayesian Regret for multiwinner systems has been discussed on here before, although it seems that calculations have never been done. http://www.rangevoting.org/BRmulti.html However, I was wondering whether Bayesian Regret is really the best measure here. Well, it could be if you're just looking at multiwinner methods per se, but if you're specifically looking at proportional systems, then it might make sense to simply look at which methods give the best proportionality. It's an open question (I think) whether proportional representation gives the lowest Bayesian Regret, so if you're looking at the best proportional system, it makes sense to look at proportionality as your measure.

 

If it then turns out that proportionality matches Bayesian Regret figures then that's good, and it's possible that it's simpler to find the most proportional method than to find the method that best minimises BR, so you can kill two birds with one stone.

 

If it turns out that proportionality doesn't match BR figures, then you could end up with a situation where the "best" proportional method isn't very proportional. It might obey some proportionality theorem in theory but in practice not do so well. Would we then decide that this was the best PR method, or would it be better just to ignore PR and look at BR? Personally, I'd argue that it would be best to have one or the other and not end up with a system that just happens to obey a PR theorem in theory and so isn't best for PR or BR.

 

I also found this interesting discussion https://groups.google.com/forum/#!topic/electionscience/0EfwmBL9TzM but I haven't read all of it.

 

So how would you find the method that gives best proportionality? I think it can be done, but I think it might have to involve having an agreed-upon theoretical most proportional method that can be used on voters' open-ended honest utilities for candidates (as well as being usable for scores out of a set number), just like normal score voting is for single winners.

 

You run elections (using honest/strategic voters as required) and look at the results for several different PR methods. You then take the winning sets of candidates for each method and look at their "satisfaction score" (or whatever the measure is) under the theoretical best system under honest utilities. You can then compare these results with the winning set using this method. This last bit would involve a comparison of all possible winning sets, but isn't an entirely necessary part because you can still compare the methods with each other even if not with the magic best system.

 

[Warren D Smith]

Well, what does "the best proportionality" mean?
As far as I can tell, this can only be given (or anyhow, so far this
is all that has been done) a meaning in some postulated oversimplified
unrealistic model. (For example, several "colors" of people, and a
Pink person has absolutely zero Blueness...)

Now one can in various such unrealistic models actually prove theorems
that some voting method is actually perfect, or anyhow as perfect as
can possibly be done, in terms of proportionality, but I would contend
those theorems do not actually mean all that much
in the real world. First, they would not distinguish among voting
methods both of which enjoyed such theorems. Second, just look at the
real countries that have PR and ask, e.g. how many women are in their
parliaments vs in their populations.
Obviously, in the real world PR is not delivering proportional governments.

I'm in favor of methods that enjoy such theorems, but those theorems
are a minimal standard, not the overall goal.

So. I return to my claim that what we actually want, is the best happiness
for all of society, i.e. minimize Bayesian Regret BR. And this
should be under much more realistic conditions than those silly
models. Which computer simulation should be able to handle.
Now BR has a lot to do with PR, in the sense of this

PR Optimality Theorem: if the parliament has the same composition as
(e.g. large enough random sample of) the population, then it will, on
any binary (yes/no) decision, make the same choice the whole
population would have made (using simple
majority votes).

However, actually, we want for the parliament, not a random sample,
but preferably something "better," e.g. smarter, more knowledgeable,
more honest, etc. people,
if possible. That would yield something better than "optimal."

The last time a pure random sample was tried, I think, was ancient Athens.
It worked pretty well for a while, but made some seriously stupid decisions.
Athenian democracy was 508BC to 322BC, albeit after 404BC Athens was often
under the control of others, or part of a league, not independent.

Also, even if you believe in a pure random sample, you have to realize
it is impossible
in the sense that the very act of picking you randomly to be an MP,
distorts your
circumstances so that you no longer are a random sample. Oops.





--
Warren D. Smith
http://RangeVoting.org <-- add your endorsement (by clicking
"endorse" as 1st step)

[Toby Pereira]

I certainly agree with a lot of what you say, Warren, but looking at, for example, whether a system would in practice deliver women in governments might be beyond a computer simulation that could be realistically produced. And when I think in terms of proportional representation I normally think in terms of what people have voted for rather than what characteristics people have themselves. Proportional representation doesn't necessarily mean being represented by people who share all your characteristics (e.g. male, brown hair), but have the characteristics that you vote for. So that means that you won't necessarily get a 50/50 male/female split unless voters specifically want to only be represented by someone of their own sex. So I would argue against the random sample as the best from of proportionality, as it seems you do anyway. Also not everyone wants to be in parliament or would trust themselves to be competent so wouldn't vote for "clones" of themselves.

 

If we set certain criteria that a "perfect" PR system should conform to, then as you say there might be more than one method that achieves this. But if this is the case, then it's likely that they will yield very similar results anyway and you could in any case compare other systems with both/all of these systems. But there's certainly more to it than different colours of people that have nothing in common, and more criteria can be defined. I discussed a few things in this thread https://groups.google.com/forum/#!topic/electionscience/c_YRZ5ocvVg that could be turned into more rigorous criteria.

 

And if we have a system (or more than one) that conforms to our criteria, we can measure, using realistic simulated voting behaviour, how close results would be to the "ideal" under these criteria. Yes, there are real-life considerations that might mean it ends up not being as great as we predict, but that's also the case for single-winner BR figures. Proportional systems are obviously more complex than single-winner systems so there's probably more that can go wrong in the simulations, but that's the case whether you're measuring for proportionality or directly trying to measure Bayesian Regret.

[Warren D Smith]

Well, let's see. First of all, even the Athenian random sample
method, only sampled rich men. No women. No slaves. No children.
So there has never been a random sample parliament yet.

And as far as I can tell, no PR country, ever, has allowed children as MPs.
Unless Toby wants to do that, then he, himself, has admitted that he
actually does not
want proportionality -- he instead wants to deny proportionality in a
huge way. Therefore, the very quality measure he, himself proposes
(in an extremely vague manner -- I saw no mathematical definition of
whatever he wanted) is, he admits, not correct.

So it is not so easy.

And if you leave out (say) children, then clearly you have sacrificed the
PR Optimality Theorem. Every PR country thus-sacrifices it. And
children are pretty important, I think few would dispute.

So then we have to ask -- just what are the PRians trying to accomplish? And
the answer is, they, themselves, by their own obvious admission, are
not striving for proportionality. They are striving for something
else. And hopefully it is more
like "total happiness." Can this be addressed by computer? Partially. It can
be addressed a good deal better by computer (flawed though that will
necessarily be) than basically just proposing a design, saying it
obeys a "perfect" PR theorem (or even worse, not), and then declaring
done. Which is the approach that has previously been used.

So I'm convinced we can do better, it is just that nobody has done it.
They just talk about it uselessly, without ever doing the computer
study.

[Toby Pereira]

You're right that I didn't define proportionality. It's quite tricky to define, but it's something we can work towards by adding in extra criteria step by step. The basic start is obviously that if people are completely separated into factions that each have their own candidates and there's no cross-voting, then these factions would elect a number representatives proportional to their size (subject to rounding).

 

But anyway, it seems that we've got (at least) two distinct sorts of proportionality that we're talking about. There's proportionality of what we vote for and proportionality of what we are. It's the former that I'm looking at. This does not require every human characteristic to be represented proportionally. If x% of people have brown hair, it doesn't mean that x% of representatives have to have brown hair unless x% of people are specifically voting purely on that basis. It's only proportionality of what we are (every characteristic we have) that would require that. And you can define proportionality in those terms if you want, but I'm not going to for my purposes, and I don't think most people would. I see it as a red herring.

 

And as for whether children can stand etc., that's also a red herring. I'm looking at proportionality within a voting system. That means looking at the voters and the candidates that are participating in a particular election. I'm looking for a system that gives good proportionality given the voters and candidates that are allowed to stand. Who can stand and who can vote is a separate issue to that. It's not a property of the mathematical voting system. And this would apply to whether you look at proportionality or Bayesian Regret. If you run your Bayesian Regret analysis, it's not going to come out with "And the winner is - allowing people over the age of 12 to stand and vote in an election". It's going to come up with a multiwinner voting system.

 

But maybe I was too hasty to dismiss a Bayesian Regret analysis. I think it would be interesting, although obviously the assumptions would be more stretched than in a single-winner case, and people would probably dispute them more. However, it doesn't have to be one or the other. A proportionality analysis would be done as well (subject to more rigorous definitions), and it may be simpler to do that a Bayesian Regret analysis.

[Warren D Smith]

On 2/23/14, Toby Pereira <tdp...@yahoo.co.uk> wrote:
> You're right that I didn't define proportionality. It's quite tricky to
> define, but it's something we can work towards by adding in extra criteria
> step by step. The basic start is obviously that if people are completely
> separated into factions that each have their own candidates and there's no
> cross-voting, then these factions would elect a number representatives
> proportional to their size (subject to rounding).

--this is basically the axioms underlying various PR theorems. However,
it is unrealistic and also many voting systems are "perfect" under
these criteria, even though undoubtably some are better than others.
So you need other criteria.

> But anyway, it seems that we've got (at least) two distinct sorts of
> proportionality that we're talking about. There's proportionality of what
> we vote for and proportionality of what we are. It's the former that I'm
> looking at. This does not require every human characteristic to be
> represented proportionally. If x% of people have brown hair, it doesn't
> mean that x% of representatives have to have brown hair unless x% of people

--the trouble is (a) what parliaments vote for is unpredictable. For
example, they could
hold a vote "YES if you have brown hair, NO otherwise." Random samples
work for any vote including ones we cannot predict.
(b) you do not know what people vote for & care about. Nor how much
they care. Nor
how it interacts with other stuff they also care about.

> And as for whether children can stand etc., that's also a red herring.

--I disagree. Look, children have specific issues & concerns. Ditto,
say, women.
The problem with children is not the lack of validity or importance of
children's issues, it is that they are too ignorant and easily
manipulated to be good MPs.
However, the same is also true of some adults...

> But maybe I was too hasty to dismiss a Bayesian Regret analysis. I think it
> would be interesting, although obviously the assumptions would be more
> stretched than in a single-winner case, and people would probably dispute
> them more. However, it doesn't have to be one or the other. A
> proportionality analysis would be done as well (subject to more rigorous
> definitions), and it may be simpler to do that a Bayesian Regret analysis.

--it is definitely easier to do a proportionality analysis, indeed you
can construct voting systems essentially perfect. But this will not
satisfy you. Is RRV or STV better? Both are "perfect" under some PR
theorem. So are some other systems.

With BR and a simulator you will be able to explore a lot more
territory and get more realistic. You will also discover new
surprising phenomena you would not have been able to think of yourself
without the computer pointing it out to you.

[Toby Pereira]

On Sunday, 23 February 2014 17:21:16 UTC, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

On 2/23/14, Toby Pereira <tdp...@yahoo.co.uk> wrote:
> You're right that I didn't define proportionality. It's quite tricky to
> define, but it's something we can work towards by adding in extra criteria
> step by step. The basic start is obviously that if people are completely
> separated into factions that each have their own candidates and there's no
> cross-voting, then these factions would elect a number representatives
> proportional to their size (subject to rounding).

--this is basically the axioms underlying various PR theorems.  However,
it is unrealistic and also many voting systems are "perfect" under
these criteria, even though undoubtably some are better than others.
So you need other criteria.

I understand and I basically said as much.

 

 

> But anyway, it seems that we've got (at least) two distinct sorts of
> proportionality that we're talking about. There's proportionality of what
> we vote for and proportionality of what we are. It's the former that I'm
> looking at. This does not require every human characteristic to be
> represented proportionally. If x% of people have brown hair, it doesn't
> mean that x% of representatives have to have brown hair unless x% of people

--the trouble is (a) what parliaments vote for is unpredictable.  For
example, they could
hold a vote "YES if you have brown hair, NO otherwise." Random samples
work for any vote including ones we cannot predict.
(b) you do not know what people vote for & care about.  Nor how much
they care.  Nor
how it interacts with other stuff they also care about.

But when measuring different voting systems we have to go by assumptions. I'm not sure exactly how your single-winner BR calculations were done, but I'm presuming that when voters voted for candidates based on their estimate of utility, that was in some sense borne out in the BR measurements. In other words your model wouldn't have had candidates with hidden hair-colour agendas that worked against the people who voted for them. It also doesn't need to matter that I don't know what people vote for or care about. The model would just require that there are candidates standing on non-defined issues that voters each care about to a certain degree.

 

> And as for whether children can stand etc., that's also a red herring.

--I disagree.  Look, children have specific issues & concerns.  Ditto,
say, women.
The problem with children is not the lack of validity or importance of
children's issues, it is that they are too ignorant and easily
manipulated to be good MPs.
However, the same is also true of some adults...

Yes, children have specific issues and concerns. But that's not against what I was saying. I was just saying look at PR voting methods and see how well they do for the voters and candidates that happen to be standing. The issues of people not allowed to vote are not unimportant, but they just won't be modelled by this. I don't see how they could be. Did your single-winner BR figures take into account the utility of non-voters? I don't see how a computer model is going to come up with figures that say how well a voting method is going to do for such specific issues. It's going to come up with a list of numbers for utility for the voters, not statements like "This system will improve the education of children" or "This system will increase the number of women in parliament". If it does, it's likely to be a result of egregious and unwarranted assumptions.

 

> But maybe I was too hasty to dismiss a Bayesian Regret analysis. I think it
> would be interesting, although obviously the assumptions would be more
> stretched than in a single-winner case, and people would probably dispute
> them more. However, it doesn't have to be one or the other. A
> proportionality analysis would be done as well (subject to more rigorous
> definitions), and it may be simpler to do that a Bayesian Regret analysis.

--it is definitely easier to do a proportionality analysis, indeed you
can construct voting systems essentially perfect.  But this will not
satisfy you.  Is RRV or STV better? Both are "perfect" under some PR
theorem.  So are some other systems.

This is potentially a problem, I will grant you. I intend to come up with a (not necessarily exhaustive) list of certain criteria that I think a "perfect" system should have, but obviously not everyone has to agree with these criteria. But a discussion can be had and there may be consensus on some criteria. But also it could be shown even now that with strategic voting some systems are better than others under their own criteria - i.e. the strategic results are closer to the results from honest voting than with some other methods.

 

With BR and a simulator you will be able to explore a lot more
territory and get more realistic.   You will also discover new
surprising phenomena you would not have been able to think of yourself
without the computer pointing it out to you.

I hope it does happen.

 

Toby

[Toby Pereira]

I was thinking about some of the criteria that the "magic best" system might have for finding the "most proportional" set of candidates from all of those standing. This is not exhaustive and any one of them is open to debate and/or further explanation.

 

1. Independence of irrelevant alternatives. This is pretty obvious. If a slate A is better than a set B, then it will remain so regardless of candidates not in either slate being added to or removed from the election. The rules out any ranked-ballot method for being guaranteed to find the best set (assuming a few basic background assumptions like universal domain etc.)

 

2. Partly based on the above, the system should work on open-ended utility ratings. That's not to say I am advocating a voting system that allows voters to score candidates arbitrarily high, but just that a simulation to find the most proportional candidate set would not have a score limit. The same system could still be used for elections but with a score limit.

 

3. Basic proportionality. If a group of voters rate a set of candidates each at a certain utility that is at least as high as any other utility rating of any voter, and have zero utility for all other candidates, then that group of voters should, subject to seat rounding, be able to elect the proportion of candidates that equals their proportion of the electorate.

 

4. Monotonicity. This should be obvious, but specifically if candidates in set A have the same scores from the voters as set B except that at least one of the candidates in A has strictly higher scores from one or more voters than the replaced candidate(s) and none lower, then set A is better than set B. I've brought this up because there are some systems where this would reach a limit. For example:

 

Approval voting (for simplicity), 2 to elect

 

1 voter: A, B, C

1 voter: A, B, D

 

There are some systems I've seen that would give a maximum score to C, D. However, an optimal system would award the election to the set A, B over and above anything else.

 

5. A voter's candidate utility is additive. For a voter where A=10, B=5, C=5, D=0, then A, D is equal to B, C. Obviously you could have a situation where someone's preferences for a candidate is based on who else is elected making it not so simple. However, taking this to its logical conclusions, each voter would end up with a utility score for every possible slate of candidates, and the simulation would simply be running a utility analysis on these treating it as a single-winner slate system, and this would be Warren's BR project anyway. The idea of proportional representation is distinct from this and I would argue that it is based on individual utilities for candidates.

 

6. Independence of ratings multiplication. If every voter multiplied their ratings for every candidate by a constant c, then the result would remain the same. I think a system should conform to this anyway, but it's essential if we want to handle open-ended utilities in a consistent and non-arbitrary way. RRV fails this.

 

7. Independence of universally rated candidates. If a group of voters all rate a particular candidate at the same level, then if this candidate is elected, it is ignored when working out the rest of the proportionality between these voters. To give an example:

 

Approval voting, proportional representation, elect 6

 

20 voters: A, B, C, D, E, F

10 voters: A, B, C, G, H, I

 

A, B, C, D, E, G would be elected (or at least some combination of two from D, E, F and one from G, H, I). A, B and C are universally rated so you ignore them and the ratio of elected candidates should be 2:1 between the factions. RRV fails this.

[Warren D Smith]

On 2/24/14, Toby Pereira <tdp...@yahoo.co.uk> wrote:
> I was thinking about some of the criteria that the "magic best" system
> might have for finding the "most proportional" set of candidates from all
> of those standing. This is not exhaustive and any one of them is open to
> debate and/or further explanation.
>
> 1. Independence of irrelevant alternatives. This is pretty obvious. If a
> slate A is better than a set B, then it will remain so regardless of
> candidates not in either slate being added to or removed from the election.
>
> The rules out any ranked-ballot method for being guaranteed to find the
> best set (assuming a few basic background assumptions like universal domain
> etc.)

--in other words, Toby is saying Arrow's theorem rules out ranked ballot methods
given his Ind-Irrel-Alt demand.

> 2. Partly based on the above, the system should work on open-ended utility
> ratings. That's not to say I am advocating a voting system that allows
> voters to score candidates arbitrarily high, but just that a simulation to
> find the most proportional candidate set would not have a score limit. The
> same system could still be used for elections but with a score limit.

> 3. Basic proportionality. If a group of voters rate a set of candidates
> each at a certain utility that is at least as high as any other utility
> rating of any voter, and have zero utility for all other candidates, then
> that group of voters should, subject to seat rounding, be able to elect
> the proportion of candidates that equals their proportion of the
> electorate.

> 4. Monotonicity. This should be obvious, but specifically if candidates in
> set A have the same scores from the voters as set B except that at least
> one of the candidates in A has strictly higher scores from one or
> more voters than the replaced candidate(s) and none lower, then set A is
> better than set B. I've brought this up because there are some systems
> where this would reach a limit.

--that was is a very specific kind of monotonicity demand. One could consider
other kinds of monotonicity demands which Toby is not demanding.

> For example: Approval voting (for simplicity), 2 to elect
> 1 voter: A, B, C
> 1 voter: A, B, D
>
> There are some systems I've seen that would give a maximum score to C, D.
> However, an optimal system would award the election to the set A, B over
> and above anything else.

> 5. A voter's candidate utility is additive. For a voter where A=10, B=5,
> C=5, D=0, then A, D is equal to B, C. Obviously you could have a situation
> where someone's preferences for a candidate is based on who else is elected
> making it not so simple. However, taking this to its logical conclusions,
> each voter would end up with a utility score for every possible slate of
> candidates, and the simulation would simply be running a utility analysis
> on these treating it as a single-winner slate system, and this would be
> Warren's BR project anyway. The idea of proportional representation is
> distinct from this and I would argue that it is based on individual
> utilities for candidates.

> 6. Independence of ratings multiplication. If every voter multiplied their
> ratings for every candidate by a constant c, then the result would remain
> the same. I think a system should conform to this anyway, but it's
> essential if we want to handle open-ended utilities in a consistent and
> non-arbitrary way. RRV fails this.

--RRV would pass it if "MAX" were the maximum score actually used
by any voter, as opposed to the max score allowed.
In realistic large elections, the difference between these two kinds
of RRV rules
would not be detectable since the two definitions of MAX would in
practice yield the
same number.
A third RRV rules variation which actually would be detectably different:
MAX would be the max score used by the particular voter who authored
this particular ballot.

> 7. Independence of universally rated candidates. If a group of voters all
> rate a particular candidate at the same level, then if this candidate is
> elected, it is ignored when working out the rest of the proportionality
> between these voters. To give an example:
>
> Approval voting, proportional representation, elect 6
>
> 20 voters: A, B, C, D, E, F
> 10 voters: A, B, C, G, H, I
>
> A, B, C, D, E, G would be elected (or at least some combination of two from
> D, E, F and one from G, H, I). A, B and C are universally rated so you
> ignore them and the ratio of elected candidates should be 2:1 between the
> factions. RRV fails this.

--RRV fails this? It seems to me, RRV would elect A,B,C, at each step
downweighting
all voters equally. We then would reach a situation
20*w: D,E,F
10*w: G,H,I
where w is some common weight. Now, elect D. Then
20/2: E,F
10: G,H,I
and then it is a tossup who gets the next seat, but whoever does,
the next seat goes to the other faction.

So I'm not seeing RRV failing in Toby's example; it does what he wanted.

Also, more generally, I object to voting methods criteria which only apply
in situations which in practice will essentially never happen. Toby
criterion 7
sucks because in practice nobody will ever be truly 100% universally rated.
His criterion 4 also is predicated on assumptions which in practice
will essentially never happen... and his criterion 3 also is
attackable in same sort of sense, for example if you really had
unrestricted score ranges as he demands in (2), then his criterion 3
would essentially never happen since some loontog voter would give
somebody a score of 96594769985697436599999999999999999999999999.
His demand (2) is anyhow also stupid since it essentially amounts to ballot
renormalization to fixed score-range, so we might as well make a fixed
range 0 to 9999 say
to begin with. The advantages or disadvantages of that are surely not
so huge as to be worth demanding as a foundational axiom.

So I think Toby has some good ideas spiritually speaking, but the
details of how he here tried to codify those ideas, were poor.

Next, about "additive utility" (5), be careful what you wish for.
See, the utility of a committee is NOT the sum of individual members.
In fact that totally contradicts the demand Toby also made, for proportionality.
With 51% Democrat voters, the highest "additive" utility is got by
electing 100% Democrats, highly disproportionally.
Oops.

Also, if I (a single voter) tend to like Democrats best, I still might consider
a government which was NOT entirely Democrat, to be better because I
prefer somebody else keep an eye on them to stop unbridled corruption.
So even from my point of view alone with no other voters involved,
utility is not
additive.

So really, utility is not about single candidates, it is about the
decisions they collectively make once elected, which is why I proposed
"two stage BR" outlined in
http://rangevoting.org/BRmulti.html

[Toby Pereira]

On Monday, 24 February 2014 22:42:44 UTC, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

On 2/24/14, Toby Pereira <tdp...@yahoo.co.uk> wrote:
> I was thinking about some of the criteria that the "magic best" system
> might have for finding the "most proportional" set of candidates from all
> of those standing. This is not exhaustive and any one of them is open to
> debate and/or further explanation.
>
> 1. Independence of irrelevant alternatives. This is pretty obvious. If a
> slate A is better than a set B, then it will remain so regardless of
> candidates not in either slate being added to or removed from the election.
>
> The rules out any ranked-ballot method for being guaranteed to find the
> best set (assuming a few basic background assumptions like universal domain
> etc.)

--in other words, Toby is saying Arrow's theorem rules out ranked ballot methods
given his Ind-Irrel-Alt demand.

 

Yes, but just to be clear, I didn't mean it to be just a case of stating the obvious - I was just pointing out that a ranked-ballot method wouldn't be guaranteed to produce the "best" winner, even if you could get inside people's heads to find out their exact true preferences. Earlier you said that there are several PR systems that obey perfect proportionality according to some theorem, including STV methods. So my point is that if there is some form of true perfect proportionality, we couldn't use a ranked ballot to guarantee finding the best slate of candidates. Also this isn't the same as ruling out ranked-ballot methods for use in real elections, but just as the comparison point in a computer simulation.

 

 

> 2. Partly based on the above, the system should work on open-ended utility
> ratings. That's not to say I am advocating a voting system that allows
> voters to score candidates arbitrarily high, but just that a simulation to
> find the most proportional candidate set would not have a score limit. The
> same system could still be used for elections but with a score limit.

> 3. Basic proportionality. If a group of voters rate a set of candidates
> each at a certain utility that is at least as high as any other utility
> rating of any voter, and have zero utility for all other candidates, then
> that group of voters should, subject to seat rounding, be able to elect
> the proportion of candidates that equals their proportion of the
> electorate.

> 4. Monotonicity. This should be obvious, but specifically if candidates in
> set A have the same scores from the voters as set B except that at least
> one of the candidates in A has strictly higher scores from one or
> more voters than the replaced candidate(s) and none lower, then set A is
> better than set B. I've brought this up because there are some systems
> where this would reach a limit.

--that was is a very specific kind of monotonicity demand.  One could consider
other kinds of monotonicity demands which Toby is not demanding.

 

Certainly, and actually I would demand more general monotonicity for an optimal system. The post was a basic sketch of some ideas, and it was just that I had in mind a couple of systems that would fail this. For example, this quite nice method from Ross Hyman http://lists.electorama.com/pipermail/election-methods-electorama.com/2014-February/032560.html

If you took "MAX" to be the maximum score actually used, then I'm pretty sure it would fail IIA. I think your third variant would pass IIA and independence of ratings multiplication, however.

 

> 7. Independence of universally rated candidates. If a group of voters all
> rate a particular candidate at the same level, then if this candidate is
> elected, it is ignored when working out the rest of the proportionality
> between these voters. To give an example:
>
> Approval voting, proportional representation, elect 6
>
> 20 voters: A, B, C, D, E, F
> 10 voters: A, B, C, G, H, I
>
> A, B, C, D, E, G would be elected (or at least some combination of two from
> D, E, F and one from G, H, I). A, B and C are universally rated so you
> ignore them and the ratio of elected candidates should be 2:1 between the
> factions. RRV fails this.

--RRV fails this?  It seems to me, RRV would elect A,B,C, at each step
downweighting
all voters equally.  We then would reach a situation
20*w:  D,E,F
10*w:  G,H,I
where w is some common weight.  Now, elect D.  Then
20/2: E,F
10: G,H,I
and then it is a tossup who gets the next seat, but whoever does,
the next seat goes to the other faction.

So I'm not seeing RRV failing in Toby's example; it does what he wanted.

 

I don't think that's right. Because each faction has had three candidates elected with A, B, C, the weightings would be 20*w and 10*w as you say, but the 20*w wouldn't halve after D is elected as it is the fourth not first candidate elected for this faction. Using the normal divisors, the weighting of each ballot would change from 1/4 to 1/5, so it would be more like:

 

16: E, F

10: G, H, I

 

Elect E (1/5 to 1/6), which leaves:

 

13.3: F

10: G, H, I

 

 Also, more generally, I object to voting methods criteria which only apply
in situations which in practice will essentially never happen.   Toby
criterion 7
sucks because in practice nobody will ever be truly 100% universally rated.
His criterion 4 also is predicated on assumptions which in practice
will essentially never happen... and his criterion 3 also is
attackable in same sort of sense, for example if you really had
unrestricted score ranges as he demands in (2), then his criterion 3
would essentially never happen since some loontog voter would give
somebody a score of 96594769985697436599999999999999999999999999.
His demand (2) is anyhow also stupid since it essentially amounts to ballot
renormalization to fixed score-range, so we might as well make a fixed
range 0 to 9999 say
to begin with.  The advantages or disadvantages of that are surely not
so huge as to be worth demanding as a foundational axiom.

So I think Toby has some good ideas spiritually speaking, but the
details of how he here tried to codify those ideas, were poor.

 

I don't see criterion 7 as completely insane, but it is probably one of the less important ones. While a candidate won't be universally rated, I see a parallel with cloneproof voting systems. You're unlikely to get exact clones, but systems that are cloneproof (and not just in a really forced way) should not be adversely affected by several similar candidates. Similarly here, I would want a system that wouldn't penalise a smaller factions with their other choices for supporting a generally popular candidate.

 

On the unrestricted scores in criterion 2, remember this is only for computer simulations in which not everyone would have the same utility score for their favourite candidate. Similarly for single-winner BR calculations, the best winner could result from one person's utility rating of 4579824752043 for one candidate. But as long as a simulation was realistic, one voter would not swing it in such a way - either in the single-winner case or the PR case. But just to clarify again, I would not advocate unrestricted scores in actual elections.

 

And on these criteria generally, I'm not saying that any PR method used in a real election has to satisfy them all to be any good. Just that if there is a "magic best" system, then I think it would. Obviously I also think that these criteria are generally positive features for real election methods (criterion 2 is clearly not required though).

 

Next, about "additive utility" (5), be careful what you wish for.
See, the utility of a committee is NOT the sum of individual members.
In fact that totally contradicts the demand Toby also made, for proportionality.
With 51% Democrat voters, the highest "additive" utility is got by
electing 100% Democrats, highly disproportionally.
Oops.

 

The additive utility is just for each individual concerning which slate of candidates they prefer. I wouldn't then add them together across all voters to find the winning slate.

 

Also, if I (a single voter) tend to like Democrats best, I still might consider
a government which was NOT entirely Democrat, to be better because I
prefer somebody else keep an eye on them to stop unbridled corruption.
So even from my point of view alone with no other voters involved,
utility is not
additive.

So really, utility is not about single candidates, it is about the
decisions they collectively make once elected, which is why I proposed
"two stage BR" outlined in
   http://rangevoting.org/BRmulti.html

 

That's fine, and I basically said as much in my previous post. I said that voters' utility ratings for candidates may depend on who else gets elected, but then if you take it to its logical conclusions you just get a separate utility for each slate of candidates. That might be in some sense better, but it's not what I'd call PR. Well, it might happen to end up with PR, but not by definition alone. So for my PR purposes, it makes sense to consider each candidate separately. I don't think there's another sensible way of doing it that isn't simply your BR analysis on whole slates. You'd end up with some weird compromise between the two.

 

I think this is a worthwhile discussion to have. I hope you do too. I get the feeling you get annoyed, but I think it's all good.

 

Toby

[Warren D Smith]

>> --RRV would pass it if "MAX" were the maximum score actually used
>> by any voter, as opposed to the max score allowed.
>> In realistic large elections, the difference between these two kinds
>> of RRV rules
>> would not be detectable since the two definitions of MAX would in
>> practice yield the
>> same number.
>> A third RRV rules variation which actually would be detectably different:
>> MAX would be the max score used by the particular voter who authored
>> this particular ballot.
>>
>
> If you took "MAX" to be the maximum score actually used, then I'm pretty
> sure it would fail IIA.

--again this "failure" would in practice be immaterial since the first
2 rules variants in practice would be identical. Good criteria
should have some reasonably large
amount of applicability to the real world.

> I think your third variant would pass IIA and
> independence of ratings multiplication, however.

--I think it would fail IIA?


>> --RRV fails this? ...

>> So I'm not seeing RRV failing in Toby's example; it does what he wanted.
>>
>
> I don't think that's right. Because each faction has had three
> candidates elected with A, B, C, the weightings would be 20*w and 10*w as
> you say, but the 20*w wouldn't halve after D is elected as it is the fourth
> not first candidate elected for this faction. Using the normal divisors,
> the weighting of each ballot would change from 1/4 to 1/5, so it would be
> more like:
> 16: E, F
> 10: G, H, I
>
> Elect E (1/5 to 1/6), which leaves:
> 13.3: F
> 10: G, H, I

--aha, now I see your point. Think you are right. And further, this seems to
be a good criticism of RRV? Not sure. One might argue than in
the situation
20: A,B,C,D,E,F approved
10: A,B,C,G,H,I approved
elect: A,B,C,D,E,F
then the 10 get represented by A,B,C, and the 20 get represented by
A,B,C,D,E, and/or F.
So it is "perfectly proportional."

Whereas, if we elected
A,B,C,D,E,G
this would violate perfect proportionality?
Or would it be a "better" kind of proportionality?

What do you think?

Now I think about it, I think I prefer ABCDEG
on the basis of (probably) better Bayesian Regret.

> I don't see criterion 7 as completely insane, but it is probably one of the
> less important ones. While a candidate won't be universally rated, I see a
> parallel with cloneproof voting systems. You're unlikely to get exact
> clones,

--indeed, and that is a problem with the cloneproofness criteria.

> but systems that are cloneproof (and not just in a really forced
> way) should not be adversely affected by several similar candidates.
> Similarly here, I would want a system that wouldn't penalise a smaller
> factions with their other choices for supporting a generally popular
> candidate.
>
> On the unrestricted scores in criterion 2, remember this is only for
> computer simulations in which not everyone would have the same utility
> score for their favourite candidate. Similarly for single-winner BR
> calculations, the best winner could result from one person's utility rating
>
> of 4579824752043 for one candidate. But as long as a simulation was
> realistic, one voter would not swing it in such a way - either in the
> single-winner case or the PR case. But just to clarify again, I would not
> advocate unrestricted scores in actual elections.

--??

[Toby Pereira]

On Tuesday, 25 February 2014 00:57:31 UTC, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

> I think your third variant would pass IIA and
> independence of ratings multiplication, however.

--I think it would fail IIA?

 

I think it would actually.

 

--aha, now I see your point. Think you are right.  And further, this seems to
be a good criticism of RRV?  Not sure.   One might argue than in
the situation
20: A,B,C,D,E,F approved
10: A,B,C,G,H,I approved
elect: A,B,C,D,E,F
then the 10 get represented by A,B,C, and the 20 get represented by
A,B,C,D,E, and/or F.
So it is "perfectly proportional."

Whereas, if we elected
A,B,C,D,E,G
this would violate perfect proportionality?
Or would it be a "better" kind of proportionality?

What do you think?

Now I think about it, I think I prefer ABCDEG
on the basis of (probably) better Bayesian Regret.

 

I see it as a "better" kind of proportionality personally, and I have thought about it a lot before. It's partly because I don't see the 10 voters and the 20 voters as two completely separate factions so the 2:1 ratio doesn't necessarily apply. They agree on certain things and not on others so they can be seen as between one and two factions. ABCDEG also does just seem intuitively more proportional to me, not that that has to be taken seriously. I do think that it's a criterion that a lot of people might disagree with, however.

 

> but systems that are cloneproof (and not just in a really forced
> way) should not be adversely affected by several similar candidates.
> Similarly here, I would want a system that wouldn't penalise a smaller
> factions with their other choices for supporting a generally popular
> candidate.
>
> On the unrestricted scores in criterion 2, remember this is only for
> computer simulations in which not everyone would have the same utility
> score for their favourite candidate. Similarly for single-winner BR
> calculations, the best winner could result from one person's utility rating
>
> of 4579824752043 for one candidate. But as long as a simulation was
> realistic, one voter would not swing it in such a way - either in the
> single-winner case or the PR case. But just to clarify again, I would not
> advocate unrestricted scores in actual elections.

--??

 

Which bit was that referring to in particular? But as far as I understood, your BR figures for single-winner elections worked on the basis that each voter had a utility rating for each candidate and that these utility ratings were open-ended. So one person's favourite candidate might give them a lower utility than another person' favourite would give them. That being the case, it would be logically possible for one voter's utility ratings to effectively make them the dictator over the "best winner". But only *logically* possible. Obviously the simulation wouldn't use utility ratings that had such big differences due to considerations of realism so the simulation wouldn't be affected by it. So my point is that the same would apply with the open-ended utility ratings for a PR election. No "loontog" voter could ruin the simulation in that manner.

[Warren D Smith]

>>Now I think about it, I think I prefer ABCDEG
>>on the basis of (probably) better Bayesian Regret.

>I see it as a "better" kind of proportionality personally, and I have thought about it a lot before. It's partly because I don't see the 10 voters and the 20 voters as two completely separate factions so the 2:1 ratio doesn't necessarily apply. They agree on certain things and not on others so they can be seen as between one and two factions. ABCDEG also does just seem intuitively more proportional to me, not that that has to be taken seriously. I do think that it's a criterion that a lot of people might disagree with, however.

--In that example, if not all, but only say 99% of voters approved A,B,C
then it is less clear what to do.

[Toby Pereira]

It would be less clear, but I would want a voting system that handled that criterion "automatically" rather than as an artificial add-on. For example, you could make a voting system cloneproof by observing if any pairs/triplets of candidates were scored identically on everyone's ballot or ranked in order with no candidates in between, and having a separate rule for what to do in that case. So while it might be cloneproof in some technical sense, it wouldn't solve any of the problems that non-cloneproof systems have. So essentially I'd like a method that just happened to obey independence of universally rated candidates, and then trust it on cases where you can't immediately intuitively see what the best result would be.

[Jameson Quinn]

Which rewards free-riding more- a system which gives ABCDEF or one which gives ABCDEG? The ABCGHI voters can force GH to win by free riding, so I guess that the answer is the ABCDEF is farther from that, and thus encourages free riding more. Do you agree? If so, that would be another argument for ABCDEG as the "right" answer.

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[Warren D Smith]

On 2/25/14, Jameson Quinn <jameso...@gmail.com> wrote:
> Which rewards free-riding more- a system which gives ABCDEF or one which
> gives ABCDEG? The ABCGHI voters can force GH to win by free riding, so I
> guess that the answer is the ABCDEF is farther from that, and thus
> encourages free riding more. Do you agree? If so, that would be another
> argument for ABCDEG as the "right" answer.

--I do not know. I will say that:
1. Thoughts of this nature can be playing a dangerous
and speculative game...
2. I find it really a lot harder to think about multiwinner system
strategy than single winner
(and even that can be quite tricky) in particular JQ here already was
making various unstated debatable assumptions such as that certain
groups of voters act/think in a coordinated fashion.
3. I'll also say that notably missing from Toby P's proposed list of
multiwinner systems criteria, were any about strategy (e.g. free
riding, favorite betrayal, burial...)
and a "participation" criterion was also absent.

(I sent TP a longer email off-list which he might want to think about...)

[Toby Pereira]

A participation criterion would also be essential. I never intended my list to be exhaustive. The list was more to head off specific concerns I had in my head at the time, and to demonstrate why certain methods wouldn't be "perfect" on the basis that they fail them. As for strategy, because I was mainly talking about a "perfect" system that could work straight off voters' honest utilities, I wasn't really concerned about them. A real-life system would of course have to consider these.

 

But as it is, I'm not sure which would reward free-riding more. I don't want to speculate too much, as I'm not the best qualified person.

[Warren D Smith]

> But as it is, I'm not sure which would reward free-riding more. I don't
> want to speculate too much, as I'm not the best qualified person.

--another good reason to have a simulator is:
it gets hard to think about this with the unaided brain...

[Toby Pereira]

Having said all that, I think the general assumption is that if it is easier for one group to force a result in a particular direction, then a strategy-resistant method would work to make that the result anyway. For example, if two factions are at a 3:1 size ratio, and there are two seats, then logically I'd see a tie between 2-0 and 1-1, but many methods would award both seats to the larger faction because they could split into two groups to force this result.

 

But if the "ideal" result is A and the strategically forced result is B, then by using a method that isn't resistant to strategy, and some people strategise anyway, you're likely to get a result somewhere between A and B I would have thought. Which is still "better" than by using a strategy-resistant method. Obviously an over-simplification, but that's why I'm sometimes wary of methods that are more resistant to strategy. They might in some cases do so by producing a worse result by default.

 

Having said that, I'm not going to complain if a strategy-resistant method builds in a bias towards ABCDEG, because I think it's the better result anyway.

[Gabriel Bodeen]

On Saturday, February 22, 2014 6:27:32 PM UTC-6, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

Well, let's see.  First of all, even the Athenian random sample
method, only sampled rich men.  No women. No slaves. No children.
So there has never been a random sample parliament yet.

Your comment here reminded me of the recent article on "Lottocracy" by Alexander Guerrero.  http://aeon.co/magazine/living-together/forget-elections-lets-pick-reps-by-lottery/  He makes an interesting case which to me can be summed up by his sentence, "Because individuals are chosen at random from the jurisdiction, they are much more likely to be an ideologically, demographically, and socio-economically representative sample of the people in the jurisdiction than those individuals who are capable of successfully running for office."  Then with a large enough legislature size and the use by the legislature of something like range voting over societal options, we'd have a cheap, fairly reliably accurate substitute for direct democracy, which might be worthwhile.

[Toby Pereira]

I've been thinking more about the possible implications of failing my "independence of commonly rated candidates" criterion. ("Commonly" might be better than "universally" because I don't want it to imply that every voter has to rate them equally because it should apply for subfactions within factions etc).

 

To recap, the example I've been using is:

 

6 to elect, approval voting (or score voting where voters just top rate or bottom rate)

 

20 voters: A, B, C, D, E, F
10 voters: A, B, C, G, H, I

 

There may be some debate whether to elect A, B, C, D, E, F or A, B, C, D, E, G.

 

But what if we add two further factions that vote independently of anyone else?

 

Faction 1a (20 voters): U1, U2, U3, U4, U5, U6, A1, A2, A3.....

Faction 1b (10 voters): U1, U2, U3, U4, U5, U6, B1, B2, A3.....

Faction 2a (20 voters): C1, C2, C3.....

Faction 2b (10 voters): D1, D2, D3.....

 

I've named them 2a and 2b because even though they have no candidates in common, I'm going to assume that the candidates they vote for all happen to be from the same party. This is important. Also, I've left it open-ended how many candidates each faction votes for, so we can fill any number of seats.

 

Let's say there are 48 seats (sorry for the high number but I struggled to get this to work). What's the best result? I would argue that factions 1a and 1b should get 24 candidates between them, as should 2a and 2b because that gives party proportionality for one thing. For me, the ideal result would be:

 

U1-U6, A1-A12, B1-B6, C1-C16, D1-D8

 

This means that in terms of number of seats, we have:

 

Faction 1a: 18

Faction 1b: 12

Faction 2a: 16

Faction 2b: 8

 

Perhaps, if you argue against my criterion, you might have:

 

U1-U6, A1-A14, B1-B4, C1-C16, D1-D8

 

This gives:

 

Faction 1a: 20

Faction 1b: 10

Faction 2a: 16

Faction 2b: 8

 

However, according to my calculations, RRV (and I think probably several other methods) would give:

 

U1-U6, A1-A12, B1-B3, C1-C18, D1-D9

 

This gives:

 

Faction 1a: 18

Faction 1b: 9

Faction 2a: 18

Faction 2b: 9

 

This means that the party that 1a/1b have in common have 21 candidates elected, and 2a/2b have 27 candidates elected. This is arguably not just an undesirable result but a proper violation of proportionality! It might superficially look more proportional, but I don't see it like that, and factions 2a/2b have gained candidates from the co-operation of 1a/1b. Somebody get on the phone to the Academy of Motion Picture Arts and Sciences - we need them to cancel the Oscars!

 

I'd be interested to know what other methods do. Jameson, Ted, do you know how AT-TV and GATV would handle this?

 

I might have a look at STV methods to see if there is an equivalent form of violation.

 

So while my criterion might be seen as a bonus although not essential, or even undesirable in the original case, I think this case shows that non-compliance of this criterion can lead to violations of proportionality. Warren thinks it might be impossible to have this criterion, however.

[Toby Pereira]

A slightly simpler example is:

 

6 to elect

 

2 voters: U1, U2, A1
1 voter: U1, U2
3 voters: C1, C2, C3, C4

 

RRV seems to go for U1, U2, C1, C2, C3, C4, regardless of whether you use D'Hondt or Sainte-Laguë divisors.

 

I think the result should be U1, U2, A1, C1, C2, C3

 

I put this into an STV calculator (assuming preferences are left to right), and it gave what I consider to be the correct result. http://paul-lockett.co.uk/stv.html

 

Philosophically, I prefer score/approval ballots for multiwinner elections to a greater degree than I do for single-winner elections. But I do think that this is an area of research that needs more work.

[Jameson Quinn]

Hmm...

OK, first, I think that BTV (Bucklin Transferrable Voting) has supplanted AT-TV; it has the same principal advantages, while being simpler. 

These systems (roughly: "algorithms which sequentially assign droop quotas of voters to winning candidates, greedily attempting to maximize the minimum rating of a voter for their assigned candidate") certainly don't have the resolution to get a precise answer for factions with identical votes over 20 candidates. But scale the problem down, and I think they'd get the "right" ABCDEG answer here.

Jameson

--

[Toby Pereira]

I could change that example, to make it clear there are two parties.

 

6 to elect

 

2 voters: A1, A2, A3
1 voter: A1, A2, A4
3 voters: B1, B2, B3, B4

 

RRV elects A1, A2, B1, B2, B3, B4.

STV elects A1, A2, A3, B1, B2, B3.

[Toby Pereira]

I suppose here, my "independence of commonly rated candidates" has evolved into a "exploitation of partial agreement" or something along those lines. The B voters would normally expect to have three candidates elected given the size of the faction, but because there is partial agreement among the A voters, the voting system doesn't elect as many A candidates, which wouldn't happen if there was no agreement or full agreement. e.g.

 

2 voters: A1, A2, A3

1 voter: A4, A5, A6

3 voters: B1, B2, B3, B4

 

There is no agreement here and you'd get A1, A2, A4, B1, B2, B3.

 

2 voters: A1, A2, A3

1 voter: A1, A2, A3

3 voters: B1, B2, B3, B4

 

There is full agreement and you'd get A1, A2, A3, B1, B2, B3. Only with partial agreement do B get 4 candidates elected.

 

Sorry for the serial posting. I should probably save up my thoughts.

[Toby Pereira]

STV gives the ABCDEG result, so it might be something to do with quota-based systems. That's not to say that non-quota systems *necessarily* fail, however.

[Warren D Smith]

It might be that the sort of multiplicative reweighting employed by
BTV and STV,
is helpful for achieving "ABCDEG result" while the kind of weighting RRV
uses is unhelpful for that. On the other hand it might be that RRV's kind of
weighting procedure, is helpful for electing winners who have a lot of support
(e.g. Condorcet winners) in situations where STV would refuse to elect
them in a foolish attempt to get more representation for fringe
groups, like this:
http://rangevoting.org/PRcond.html

But, as I said in another post, BTV's algorithm could be redefined if desired
to use a weighting method like RRV uses. (BTV appears to have stolen
its reweighting process from STV, but I think a case could be made
that both procedures would be "natural.") In which case, you'd want to
ask which BTV-algorithm-variant performed better, and whether some
class of hybrid-mixed algorithms might be better still.

[Toby Pereira]

On Wednesday, 5 March 2014 12:08:50 UTC, Toby Pereira wrote:

STV gives the ABCDEG result, so it might be something to do with quota-based systems. That's not to say that non-quota systems *necessarily* fail, however.

 

 

 

Of course, you could turn STV into an approval system anyway. I might have seen this described somewhere before, but if every candidate you approve has 1/c votes from you (if you approve c of the remaining candidates), and then it proceeds as an STV election, then it might work better than STV because the elimination phase wouldn't be quite as bad. No candidate would be eliminated because they have lots of second places and not enough firsts. They are just approved or not.

[Toby Pereira]

On the ABCDEG / ABCDEF result, I think you can measure proportionality on the basis of agreement with the other voters. So:

 

20 voters: A, B, C, D, E, F

10 voters: A, B, C, G, H, I

 

You look at the proportion of the electorate that agrees with each voter for each candidate and total them for the number of candidates they "should" have. So for one of the majority of 20, you'd have:

 

(1+1+1+2/3+2/3+2/3)/6 = 5

 

For one of the minority of 10, you'd have:

 

(1+1+1+1/3+1/3+1/3)/6 = 4

 

That's a start, but it doesn't say what you'd do in general to achieve this result. For a start, these numbers are purely for measuring the proportionality of a particular set of candidates being looked at. In more realistic elections, changing the set of candidates being looked at would change the "agreement level" for voters, so it doesn't tell you actually how many candidates each voter should have elected. But it does tell you whether a particular result is proportional relative to the voting behaviour for that set of candidates.

 

I think there are different ways to proceed from here. You could change the relative weights of each voter, but a general method for doing this isn't obvious. In this case, if the faction of 20 have their votes weighted at 5/6 and the faction of 10 have their votes weighted at 4/3, then it would get the right result.

 

One possibility is this: For each set of candidates you want to find the "score" for, consider its "negative" set. So for ABCDEG, you also have A' B' C' D' E' G'. A' is an imaginary candidate where every voter has the opposite voting pattern A. If someone votes for A they haven't voted for A' and vice versa. Find the winning set of candidates among all the real candidates and their negatives where only one of a candidate and it's negative can be elected. The "optimum" result might actually be for half a candidate to be a elected and half its negative or some other fraction, so we'd allow this (or multiply candidates and seats up accordingly to keep integers).

 

Using normal proportional approval voting with Sainte-Lague or D'Hondt divisors, ABCDEG' (which is in this case the same as ABCDEF) would beat ABCDEG and be the winning result. But once we've found our winning result, we then weight the voters accordingly. The weight for a voter is their agreement level above divided by the number of seats they win in the election with negative candidates.

 

These weightings are then used for the actual candidate set being measured. This should then give the ABCDEG result.

[Toby Pereira]

I think I have now come up with a system that works for approval voting.

For each voter, their score is the number of elected candidates that they "possess". If v voters have voted for a particular candidate, then each voter possesses 1/v of that candidate. If there are c candidates and v voters, then as long as each elected candidate has received at least one approval, then the average possession score for a voter is c/v. Full proportionality is achieved if every voter has a score of c/v. The measure of a set of candidates is the average squared deviation from c/v for the voters' scores (lower deviation being better). For example:

4 to elect

3 voters: A, B, C, D

1 voter: E, F, G, H

The obvious proportional result is A, B, C, E. And you can see that the 3 ABCD voters would each possess 1/3 of each of A, B, C, so would have a score of 1. The single EFGH voter would fully possess E, so would also have a score of 1. c/v in this case is 1 so this is trivially a proportional result.

But this also works where there are commonly rated candidates. So to use my favourite example:

6 to elect

20 voters: A, B, C, D, E, F

10 voters: A, B, C, G, H, I

RRV would elect A, B, C, D, E, F. But in this case the 20 ABCDEF voters would all have a score of 3 * 1/30 + 3 * 1/20 = 1/4. The 10 ABCGHI voters would score 3 * 1/30 = 1/10.

Whereas with A, B, C, D, E, G, the 20 ABCDEF voters would score 3 * 1/30 + 2 * 1/20 = 1/5. The 10 ABCGHI voters would score 3 * 1/30 + 1 * 1/10 = 1/5. The equal scores imply a proportional result. To check, c/v = 6/30 = 1/5 so it all adds up.

This also works for other examples that I've tried. Where there isn't an exact proportional result available, the squared deviation seems to work as the best measure. For example:

4 to elect

5 voters: A, B, C, D

3 voters: E, F, G, H

1 voter: I, J, K, L

I won't do the maths here, but this would give an exact 3-way tie between ABCE, ABEF and ABEI. This is the same as the result that Sainte-Laguë would give (which I would argue is objectively more proportional than D'Hondt).

It's not obvious how to score a set of candidates where at least one candidate has no approvals, but this should never cause a problem. As long as there are as many approved candidates as seats, there is no need to consider unapproved candidates. If there aren't, you would always elect all the approved candidates anyway.

For a similar reason, it's not obvious what to do with score voting. For example, if a single voter has given a candidate a score of 1 out of 10, then does this voter possess 1/10 of this candidate or the whole candidate? And if 10 voters have given a score of 1/10 then do they each possess 1/100 or 1/10? If it's 1/100, then candidates won't be "fully possessed" and we end up with a no obvious way of working out the best set of candidates because the average c/v would not be the same for each set of candidates and we'd be looking at more than just the squared deviation. We'd have to consider both average and deviation, which would make it messy. But otherwise we end up with a voter fully possessing a candidate that they've given a low score to, and this would count against that voter. So at the moment, this is just a system for approval voting.

I think this is superior to the approval version of RRV because it is independent of commonly approved candidates, and so it also avoids the proportionality problems that this can cause, which I mentioned in other posts above.

It is also superior to STV methods because in these methods a voter is considered to just have a single vote and if that vote is fully assigned to a candidate then it doesn't matter what else happens. For example:

4 to elect

10 voters: A, B, C, D

10 voters: A, B, C, D, E, F

The best result here is A, B, C, D, but STV is indifferent between this and, say, A, B, E, F. The 10 ABCD voters can have their votes assigned to A and B (5 to each) and the 10 ABCDEF voters can have their votes assigned to E and F. This would heavily favour the ABCDEF voters. In practice, most STV methods wouldn't do this because of the sequential way they operate. But STV methods that consider sets of candidates non-sequentially would be indifferent to these results. So avoiding this result would be more accidental than ideological.

However, my method is also indifferent between results that you might expect a method not to be. For example:

2 to elect

10 voters: A, B, C

10 voters: A, B, D

You might think that A, B is the obvious result. But my system would say this is equal to C, D. So in this sense it could be said to fail monotonicity. However, I do think it makes sense in the way that proportional systems work. If A and B are elected, then they are both "shared" among all 20 voters. Where if it's C and D then although each voter only has one candidate, they share the candidate with half as many people, and their possession score would be the same under either result.

If this all stands up, I think this is better than the other systems of proportional representation based on approval voting that I've seen.

[Toby Pereira]

I was thinking about a score voting version of this, and I think I've got it to work. I was struggling because the system only works with exact approvals rather than fractional scores. But instead of fractional scores, a score of e.g. 9 out of 10 can be translated into 0.9 people approving the candidate and 0.1 people not approving them. If I score a 9 and a 7 to two candidates, then this would be 0.9 * 0.7 approving both, 0.9 * 0.3 approving just the former, 0.1 * 0.7 approving the latter and 0.1 * 0.3 approving neither.

If I give a candidate 1 out of 10, and no-one else gives them anything, then if that candidate gets elected then the 0.1 voter "possesses" 1/number of voters, so 10 of the candidate. This might seem a weird anomaly, but it doesn't actually break the system. It's just what happens so that every candidate provides the same total amount of representation for the voters.

This system should also work with arbitrarily high scores. For example, if someone was allowed to give a score of 20 then it would simply be equivalent to two people giving an approval. It also doesn't matter what score is considered to be the approval score. In the above example, I could work on the assumption that 5 was the "maximum", so the scores of 9 and 7 would be 9/5 and 9/7 of an approval instead of 9/10 and 7/10. The system seems to be independent of multiplication factors, so can be used with arbitrary utilities rather than scores out of a set number, which is something I wanted to achieve as I discussed earlier in this thread.

Interestingly, this could also work as a patch for reweighted range voting, which is not independent of multiplication factors. If everyone gave scores of 10/10 under RRV, then you can get a different result from a case where everyone gives the same candidates 9/10 instead. By having 9/10 of the voters giving full approvals instead, this would cancel out the anomaly.

This system should always elect the score winner where there is a single winner. For example with scores out of 100:

Elect 1

1 voter: A=100, B=49

1 voter: A=0, B=49

Elected candidates/voters = 1/2 = 0.5, so that's the target score that we work out the deviation from.

Normal score voting would obviously elect A. If A is elected, then 1 voter has a score of 1 and the other has a score of 0. So the total squared deviation from 0.5 (candidates/voters) is 0.5^2 + 0.5^2 = 0.5

If B is elected, 0.98 voters have a score of 1/0.98 = 1.02. 1.02 voters have a score of 0. The squared deviation from 0.5 is 0.98*0.52^2 + 1.02*0.5^2 = 0.52. This is a higher deviation so A is elected.

[Jameson Quinn]

I think this is impressive. Until I saw this, I was not enthusiastic about any approval- or score-based proportional voting method, and I had begun to think that the common problems with such systems were unavoidable. With this system, I'm not saying that it is entirely immune to free-riding or other common problems, but it seems to me to be less subject to them than most non-asset PR systems I know of.

The questions I have are:

1. Run time. For common situations, I think "greedy"-type algorithms would be optimal; but the more I think about it, the more I suspect that getting a provably-optimal winner set could in pathological situations be NP-complete. Is there some "good enough" algorithm that would use these ideas, but always be easy to calculate?

 2. Is there any Bucklin-like extension to this idea? In other words, can it be helpful to consider different cutoff points, where the common cutoff is used to convert all ballots to approval-style ballots? Or does that give no advantage over the "score" version you mentioned? 

3. What happens when you minimize other norms besides the squared deviation? Certainly minimizing the L0 norm (number of voters with the wrong representation) or the L∞ norm (mimimax deviation) both seem worse than the L2, so I suspect that squared deviation is best; but it would be interesting to at least look at a few examples under other norms.

Anyway... good work, and I'm interested to hear more.

Jameson

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[Jameson Quinn]

Wait a minute... I think this system does not pass mono-raise. I don't think failure would be common, but it's certainly hard to justify if you can ever show that it did happen. 

[Jameson Quinn]

Hmm... thinking further about this, I think that a greedy algorithm based on these ideas might pass mono-raise; which is another argument, besides computability, for specifying a greedy algorithm.

[Toby Pereira]

I'm glad you think the method shows promise, Jameson. The way I imagined it, if it were ever used in an election, it would probably be done sequentially. So the highest scoring candidate would be elected. Then the best set of two that includes the first candidate and so on. Is that what you mean by a greedy algorithm? I'm not sure if you can guarantee finding the best set without testing all possible winning sets of candidates, which might be too much computationally. But I'm not really an expert in such matters.

Not sure about a Bucklin version really. There's no quota with this system. Does there not need to be a quota for a Bucklin method? If no candidate reaches the quota then the B grades are converted to A grades etc.

Minimising the squared deviation was the obvious one to look at for me and it gave the results I expected from the system. Looking at absolute deviation is likely to lead to ugly results, although I'm not sure what the best examples would be to try. I have to say, I find the method quite clean as it is, and picking other measures of deviation would give it a more arbitrary feel. They'd have to be demonstrably better, and that would probably take more than just picking out one or two examples.

As for mono-raise, it might fail it, but I think only in that it would be a feature of the system rather than a bug. According to this system, a voter is better represented by a candidate that fewer other people have voted for than one that a lot of other people have. This is an example I gave earlier:

2 to elect

10 voters: A, B, C

10 voters: A, B, D

The method would be indifferent between AB and CD, despite A and B being universally approved. I suppose mono-raise specifically requires finding an example where a candidate would be elected under one set of ballots, but then wouldn't if it was raised on some ballots, which isn't demonstrated here. So I'm not sure, but people might be just as put off by this result. But as you say, by electing candidates sequentially, this sort of thing is less likely to happen. A or B would be elected first followed by the other one of A/B.

On the ideology of the method, I was never really happy with the existing approval/score systems. There were always properties I felt they should have but none seemed to (as I have discussed in other posts above), and I thought that there must be a system that can do those things - it was just a matter of finding it (and hopefully I have). With proportional approval/score systems generally, you say you weren't that enthusiastic about them, but I always thought that in particular it was important to have a good approval method. A lot of committees are elected by just voting for any number of candidates up to the number to be elected with no scores or ranks. Normally they're just done on totals, but a move to proportional approval voting would be a fairly small and positive step from there. So I think having a good proportional approval method is important. Whether or not this is that method, I suppose it's for others to judge it.

[Gabriel Bodeen]

On Monday, June 2, 2014 3:31:47 PM UTC-5, Toby Pereira wrote:

I'm glad you think the method shows promise, Jameson. The way I imagined it, if it were ever used in an election, it would probably be done sequentially. So the highest scoring candidate would be elected. Then the best set of two that includes the first candidate and so on. Is that what you mean by a greedy algorithm? I'm not sure if you can guarantee finding the best set without testing all possible winning sets of candidates, which might be too much computationally. But I'm not really an expert in such matters.

There are some cases where the sequential method doesn't work, of course.

Elect 2:
1 voter: A, B
1 voter: A, C

If I understood correctly, your method chooses the more proportional winning set BC, skipping the more popular candidate A.  For elections where the number of candidates and winners is enough to be computationally difficult, though, the sequential method probably does very, very well relative to the ideal case.

[Toby Pereira]

Yes, you're right. BC is the optimum result under this method. I think any proportional system that can elect candidates either all at once or sequentially will have cases where they will give different results. So I don't see it as a massive problem. 

[Toby Pereira]

On the score voting version of this, I don't think the way I suggested is actually the best way to go about it. I suggested that if, for example, someone gives scores of 9/10 and 7/10 to A and B respectively, then this would count as 0.9 * 0.7 people approving both, 0.9 * 0.3 approving just A, 0.1 * 0.7 approving just B, and 0.1 * 0.3 approving neither. This seemed to me at the time to be the most balanced way of doing it. However, this causes the maximum score to have relevance causing a possible failure of independence of multiplication factors.

For example, someone gives scores of 10/10 and 5/10 to A and B respectively. This would result in 0.5 people approving both and 0.5 approving just A.

But consider scores of 10/20 and 5/20 to A and B respectively. This would mean that 0.125 would approve both, 0.375 would approve just A, 0.125 would approve just B, and 0.375 would approve neither. This gives completely different ratios, and so could potentially lead to different result if everyone gave the same scores but out of a different maximum.

So it seems that a better way would be to clump them together as much as possible and do away with the mixing and matching. So for example, scores of 10/10, 8/10 and 6/10 to A, B and C would mean 0.6 of a voter would approve ABC, 0.2 would approve AB and 0.2 would approve A. For the same scores out of 20, it would be the same but scaled down - 0.3, 0.1 and 0.1. This method also means far less "splitting" of the voter. Instead of splitting into up to 2^c parts where c candidates are awarded a non-zero score, voters would only ever be split into a maximum of c parts.

While I initially thought this way if doing it was asymmetrical and lopsided, I now see it as simpler and less arbitrary, and it's trivial to see that it passes independence of multiplication factors. There is still the question - is it The Right Way, but I now think it is. I initially saw it as asymmetrical because I pictured each voter like a square with approvals going down from left to right. Using the example above (scores of 10, 8, 6), it would be 3333332211 where the left 6 tenths approve all 3 of ABC, the next 2 tenths approve of AB and the right two tenths approve of just A. But it could equally be seen like this: 1233333321, which is nice and symmetrical. Because of this, I now view it in a more positive light and it assuages my doubts on whether it's The Right Way. But that's just an insight into my weird mind I suppose.

One final thing. On this "paradox":

2 to elect

10 voters: A, B, C

10 voters: A, B, D

where AB is no better than CD, it would only ever really come up where there is an exact tie. In all other cases, such as:

2 to elect

99 voters: A, B, C

99 voters: A, B, D

1 voter: C

1 voter: D

then CD is justified because it is objectively more proportional. If you pick AB because of something like greater overall satisfaction, you have to come up with some potentially arbitrary trade-off between satisfaction and proportionality, as well as a definition of satisfaction. The sequential version would pick the most popular overall candidate first anyway, and subsequent candidates would be picked to get the best proportionality, so I would see it as "RRV done right". On the non-sequential version, there could be some sort of tie-break system based on something like total votes where there is an exact tie, which could then elect AB over CD. But that would be an add-on rather than an intrinsic part of the system.

I do have quite a lot of confidence in this system now, to the extent that I would back this system as the one to use as the "objective arbiter" to measure other methods against that I started this whole thread with.

[Toby Pereira]

On Tuesday, 25 February 2014 14:59:08 UTC, Warren D. Smith (CRV cofounder, http://RangeVoting.org) wrote:

>>Now I think about it, I think I prefer ABCDEG
>>on the basis of (probably) better Bayesian Regret.

>I see it as a "better" kind of proportionality personally, and I have thought about it a lot before. It's partly because I don't see the 10 voters and the 20 voters as two completely separate factions so the 2:1 ratio doesn't necessarily apply. They agree on certain things and not on others so they can be seen as between one and two factions. ABCDEG also does just seem intuitively more proportional to me, not that that has to be taken seriously. I do think that it's a criterion that a lot of people might disagree with, however.

--In that example, if not all, but only say 99% of voters approved A,B,C
then it is less clear what to do.

I can look at this again with your modification now, Warren.

6 to elect

198: ABCDEF

99: ABCGHI

2: DEF

1: GHI

In this example, exactly 99% approve ABC. Is this what you had in mind? I have to say, pre-working it out, I'm fairly confident that ABCDEG would win under my system.

If ABCDEF wins, then 198 people would each have representation of 1/297 + 1/297 + 1/297 + 1/200 + 1/200 + 1/200 = 0.025101

99 people would have representation of 1/297 + 1/297 + 1/297 = 0.010101

2 people would have representation of 1/200 + 1/200 + 1/200 = 0.015

1 person would have representation of 0.

The proportional amount of representation per person is 6/300 = 0.02

The total of the squared deviation from 0.02 is 0.0153.

If ABCDEG wins, then 198 people would have representation of 1/297 + 1/297 + 1/297 + 1/200 + 1/200 = 0.0201

99 people would have representation of 1/297 + 1/297 + 1/297 + 1/100 = 0.0201

2 people would have representation of 1/200 + 1/200 = 0.01

1 person would have representation of 1/100 = 0.01

The total squared deviation from 0.02 is 0.000303. ABCDEG wins fairly comfortably with a much smaller deviation. Only the 2 DEF voters are nearer their proportional level of representation under ABCDEF.

[Gabriel Bodeen]

On Saturday, May 31, 2014 9:20:47 AM UTC-5, Toby Pereira wrote:

 The measure of a set of candidates is the average squared deviation from c/v for the voters' scores (lower deviation being better).

Do you want to give a name to the measure, Toby?  I'm inclined to try it out in some simulations as a multiwinner analogue to Bayesian Regret, to see how well it rates different PR election methods.

[Toby Pereira]

I'm not sure what I'd call it actually other than "distance from full proportionality". Maybe "Distance of result from objective proportionality" or DORFOP for a catchy acronym! So in your simulations, would each voter have a utility score for each candidate? The system should work with unbounded utility scores. But anyway, I'd definitely look forward to seeing the results from this.

[Rob Wilson]

Recently I've been considering what a Condorcet version of PR would look like. I thought up of a system and I wonder if it would allow for enough diversity in opinion or if every winner would be pretty moderate. This is how it would work:

1) Start off with all votes equaling 1 vote.

2) Tabulate rankings and get the Condorcet win (doesn't matter which method).

3) Reweigh each persons vote for the next seat. This would involve multiplying the current weight of each voter's vote by a scale. The value of this scale would be determined by what position they ranked the winner of the last seat.

4) Repeat to step two if there are more seats to fill.

To calculate the scale:

1) Find the average scale. This would be (number of seats-1)/(number of seats)

2) calculate the average ranking value of winner of last winning candidate( can be a decimal)

3) calculate constant C = (100- Average Scale)/ (number of candidates - average ranking of winner)

4) set the function of each voter scale where R = individuals ranking of winning candidate. f(R)=[ 100 - (Number of candidates-R)*C] / 100

[Rob Wilson]

I just realized that using 100 instead of 1 was completely unnecessary.  It should have been written:

1) Find the average scale. This would be (number of seats-1)/(number of seats)

2) calculate the average ranking value of the last winning candidate( can be a decimal)
3) calculate constant C = (1- Average Scale )/ (number of candidates - average ranking of winner)
4) set the function of each voter scale where R = individuals ranking of winning candidate. f(R)= 1 - (Number of candidates-R)*C

[Gabriel Bodeen]

Ha, catchy.  I ran into a snag, though, in that the mono-raise failures were frequent and annoyed me too much to let them be.  My sense is that they occur because a set of candidates can be proportional to the voters independently of how well it satisfies their preferences, and that consequently DORFOP is great for measuring proportionality, but not so good for choosing well-liked candidates.  So what I've done for now is this hybrid:

0. I've chosen to use relative utilities, and so scores in the range [0-1].  Using scores outside that range works fine, though.

1a. A voter's 'possession' of a candidate equals the score the voter gave the candidate divided by the sum of all scores given to the candidate.

1b. A voter's 'possession' of a set of candidates is the sum of the voter's possessions of those candidates.

2a. Proportionality Score (PpS) of a set of candidates equals 1-2*MAD(voters' possessions of the set of candidates).  It ranges from 1 at perfect proportionality to 0 at perfect unproportionality.

2b. Preference Score (PfS) of a set of candidates equals the sum of scores given to the candidates divided by the maximum sum of scores the candidates could have been given (assuming range [0-1]).  It ranges from 1 for unanimous complete support to 0 for unanimous complete opposition.

3. Multiwinner Quality (MQ) equals SQRT(PpS * PfS).

Is that broken anywhere versus your proposal?  It chooses the expected set of winners in the examples you've given above, and it looks like it behaves as expected for variations in the scores.

[Jameson Quinn]

What's MAD({x...})?

--

[Gabriel Bodeen]

Mean absolute deviation.  L1 norm in this case.  I wanted more linear behavior.  If that broke it, well, it's easy to undo.

Oh, and a copying error.  I copied a special case on line 2a.  The general case is 1-MAD(the voters' possessions)/(the max possible MAD, which occurs when all the candidates in the set are possessed by only one voter).

[Toby Pereira]

Thank you for your response, Gabriel. I suppose my main worry is that by having a hybrid of proportionality and well-likedness, you can end up having to make arbitrary decisions with the weighting system. If there's a clean way of doing it, then all the better obviously. Looking at your specific points:

1a. My problem with doing it that way is that if a voter gave a score of 1/10 to a candidate and everyone else gave 0/10, then that voter would have full possession of that candidate. That's why I chose the method of splitting the voters. So 1/10 of a voter gives a full approval and 9/10 of a voter gives nothing. However, because you have a separate overall preference score, this would still be taken into account at a different stage.

2a. I'm not convinced by mean absolute deviation. I tried it with my three-way tie example reproduced below (using fully my method) and it didn't give a tie. I think generally squared deviations give more sensible results (with the mean of a set of data being the point that minimises squared deviation whereas the median minimises absolute deviation).

4 to elect (approval)

5 voters: A, B, C, D

3 voters: E, F, G, H

1 voter: I, J, K, L

The 3-way tie between ABCE, ABEF and ABEI breaks down with absolute deviation.

Also, the zero level of proportionality varies with number of voters. One person getting all the representation when there are 10 voters is very different from when there are 100. I'm not sure how well it works as a true zero point.

2b. I don't think the preference score works very well. Take this example:

2 to elect (approval)

3 voters: AB

1 voter: CD

My system would award an exact tie between AB and AC (or any combination where each faction gets 1 candidate). AB would score 3/4 of the possible vote whereas  AC would only score 1/2. The problem is that whenever there is a tie between two results either side of proportionality it would always be pushed in the direction of the larger faction when your preference score is added, regardless of what the initial method is. I think a better measure to look at is something like voter agreement level. So for example:

2 to elect (approval)

10 voters: A, B, C

10 voters: A, B, D

You might want to award it to AB over CD not because AB has a higher proportion of the available score, but because under this result there is 100% agreement between voters. Under the CD result you'd probably measure it at 50% or 0.5, because you'd count a voter agreeing with themselves. So you would just count up the agreements. I think I've considered this before and encountered other problems but I've have to come back to you on what. But naively I might consider taking (agreement level)/(mean squared deviation). This would mean division by zero when there is full proportionality though so you couldn't distinguish between these results without further specifying that you ignore the deviation when it's 0 and there's a tie.

[Toby Pereira]

Without having looked in detail at your method, there are already some methods that would elect a Condorcet winner in the single-winner case, such as Schulze STV and CPO-STV. http://en.wikipedia.org/wiki/Schulze_STV http://en.wikipedia.org/wiki/CPO-STV

[Gabriel Bodeen]

On Friday, June 13, 2014 2:40:00 PM UTC-5, Toby Pereira wrote:

I suppose my main worry is that by having a hybrid of proportionality and well-likedness, you can end up having to make arbitrary decisions with the weighting system.

Quite so; proportionality and preference are distinct concepts and I thought that should be kept clear.  For example:

Elect 1

2 voters: 1.0 for A, 0.5 for B, 0.0 for C

Regardless of which candidate wins, the result is equally proportional.  But there are clear differences of preference.

 

1a. My problem with doing it that way is that if a voter gave a score of 1/10 to a candidate and everyone else gave 0/10, then that voter would have full possession of that candidate.

Pardon, I'm not clear on why that's a problem.  Can you elaborate?

 

2a. I'm not convinced by mean absolute deviation. ... I think generally squared deviations give more sensible results (with the mean of a set of data being the point that minimises squared deviation whereas the median minimises absolute deviation).

It's only one character difference to switch the code from L1 to L2, so I'm happy to make that change.  (I'm indifferent between use of the mean or the median, but I like to keep the code linear unless it needs to not be.)

 

Also, the zero level of proportionality varies with number of voters. One person getting all the representation when there are 10 voters is very different from when there are 100. I'm not sure how well it works as a true zero point.

Right.  Having more people opens up a possibility for more severe inequality.  It seems to me to be just a fact of life.

 

The problem is that whenever there is a tie between two results either side of proportionality it would always be pushed in the direction of the larger faction when your preference score is added, regardless of what the initial method is. I think a better measure to look at is something like voter agreement level. So for example:

2 to elect (approval)

10 voters: A, B, C

10 voters: A, B, D

You might want to award it to AB over CD not because AB has a higher proportion of the available score, but because under this result there is 100% agreement between voters. Under the CD result you'd probably measure it at 50% or 0.5, because you'd count a voter agreeing with themselves. So you would just count up the agreements.

What's the difference, exactly?  For this example, agreement on AB is 100% and on CD is probably 50%, and the preference scores are the same (40/40=100% for AB, 20/40=50% for CD).

[Toby Pereira]

On Friday, 13 June 2014 21:58:00 UTC+1, Gabriel Bodeen wrote:

On Friday, June 13, 2014 2:40:00 PM UTC-5, Toby Pereira wrote:

I suppose my main worry is that by having a hybrid of proportionality and well-likedness, you can end up having to make arbitrary decisions with the weighting system.

Quite so; proportionality and preference are distinct concepts and I thought that should be kept clear.  For example:

Elect 1

2 voters: 1.0 for A, 0.5 for B, 0.0 for C

Regardless of which candidate wins, the result is equally proportional.  But there are clear differences of preference.

 

1a. My problem with doing it that way is that if a voter gave a score of 1/10 to a candidate and everyone else gave 0/10, then that voter would have full possession of that candidate.

Pardon, I'm not clear on why that's a problem.  Can you elaborate?

My problem is as in your case there would be no difference between A and B. Also in this example:

2 to elect

1 voter: A=10/10, B=1/10, C=0/10

1 voter: A=0/10, B=0/10, C=10/10

In this case there would be no difference between AC and BC proportionally-wise. You might argue that there isn't so it's not a problem, but I'm not comfortable with that, and also it doesn't reduce to normal score voting in the single-winner case. However, having a separate preference score can counteract that.

 

 

2a. I'm not convinced by mean absolute deviation. ... I think generally squared deviations give more sensible results (with the mean of a set of data being the point that minimises squared deviation whereas the median minimises absolute deviation).

It's only one character difference to switch the code from L1 to L2, so I'm happy to make that change.  (I'm indifferent between use of the mean or the median, but I like to keep the code linear unless it needs to not be.)

 

Also, the zero level of proportionality varies with number of voters. One person getting all the representation when there are 10 voters is very different from when there are 100. I'm not sure how well it works as a true zero point.

Right.  Having more people opens up a possibility for more severe inequality.  It seems to me to be just a fact of life.

Yes, but your method wouldn't measure it as more severe inequality. If there was an election and A and B were elected, I would want the same inequality score if all the voters were cloned, so there were double the voters and exactly the same result. Your system would say there was less inequality because it's further away from the extreme case where one person out of double the voters has all the representation, whereas the mean squared deviation would be the same in both cases.

 

 

The problem is that whenever there is a tie between two results either side of proportionality it would always be pushed in the direction of the larger faction when your preference score is added, regardless of what the initial method is. I think a better measure to look at is something like voter agreement level. So for example:

2 to elect (approval)

10 voters: A, B, C

10 voters: A, B, D

You might want to award it to AB over CD not because AB has a higher proportion of the available score, but because under this result there is 100% agreement between voters. Under the CD result you'd probably measure it at 50% or 0.5, because you'd count a voter agreeing with themselves. So you would just count up the agreements.

What's the difference, exactly?  For this example, agreement on AB is 100% and on CD is probably 50%, and the preference scores are the same (40/40=100% for AB, 20/40=50% for CD).

The numbers are the same in that case. I only used that example for how you would calculate the numbers, because it was simple. But in the other example I gave it would change the result:

2 to elect (approval)

3 voters: AB

1 voter: CD

My system (like Sainte-Laguë) would award a tie between AB and AC, but your preference score system would automatically nudge it in favour of the larger faction and award it to AB. It would be the same with any system. If we had a tie under a D'Hondt-style system, adding in your preference scores would favour the larger faction as well.

[Gabriel Bodeen]

Ah, I see.  You're arguing in favor of a complete, usable system.  OK.  I was only interested it as a way to measure the proportionality of different PR methods.  That's why, on further examination, I needed to draw a separation between proportionality and preference.

[Gabriel Bodeen]

(version 0.2 - it probably has errors still)

(0.) Use scores in the range [0-1].  An honest score given by a voter to a candidate is here defined to be, not a utility, but the degree to which that candidate represents the voter.

(1a.) A voter's degree of possession (DP) of a candidate equals the score the voter gave the candidate divided by the sum of all scores given to the candidate.

(1b.) A voter's DP of a set of candidates is the sum of the voter's DPs of the individual members of the set.

Given a vector V containing the voters' DPs of a set of candidates:

(2a.) The Proportionality Score (PpS) of the set of candidates equals MEAN(V)^2/MEAN(V^2).  For example, PpS is 100% if all voters have equal DP, 50% if half the voters have equal positive DP and the rest have zero DP.  PpS approaches 0% in the limit as the number of voters rise and only one voter has nonzero DP.

(2b.) The Preference Score (PfS) of a set of candidates equals the sum of scores given to the candidates divided by the maximum sum of scores the candidates could have been given (nVoters*nCandidatesInSet, assuming range [0-1]).  It ranges from 100% for unanimous complete support to 0% for unanimous complete opposition.

(3.) The Representativeness (REP) of a set of candidates versus the voters is defined to be the product PpS*PfS.

-

In the single-winner case, this always selects the same winner (i.e. most representative set) as normal Score Voting.

For C candidates of which E will be elected, the idealized method would check all NCHOOSEK(C,E) sets of E candidates.  It's definitely intractable for large C and E, but REP can readily be calculated for the winning set of more practical algorithms.

PfS is a rescaling of Bayesian Regret or Voter Satisfaction Efficiency in the single-winner case and is closely related to them in multiwinner cases.  It can be used to make the same comparisons as BR and VSE.  PpS measures only how equally the voters' votes contribute to a set of candidates.  It can be used to compare how proportional different voting systems are.

When voters give honest score votes, I think REP has a reasonable claim of describing the degree to which a set of candidates represents the voters.

[Toby Pereira]

On Friday, 13 June 2014 23:41:16 UTC+1, Gabriel Bodeen wrote:

Ah, I see.  You're arguing in favor of a complete, usable system.  OK.  I was only interested it as a way to measure the proportionality of different PR methods.  That's why, on further examination, I needed to draw a separation between proportionality and preference.

Well yes, but if you've got a measure you've basically got a system. The system is just to find the set of candidates that is best under that measure. 

[Toby Pereira]

On Saturday, 14 June 2014 01:11:45 UTC+1, Gabriel Bodeen wrote:

(version 0.2 - it probably has errors still)

(0.) Use scores in the range [0-1].  An honest score given by a voter to a candidate is here defined to be, not a utility, but the degree to which that candidate represents the voter.

(1a.) A voter's degree of possession (DP) of a candidate equals the score the voter gave the candidate divided by the sum of all scores given to the candidate.

(1b.) A voter's DP of a set of candidates is the sum of the voter's DPs of the individual members of the set.

Given a vector V containing the voters' DPs of a set of candidates:

(2a.) The Proportionality Score (PpS) of the set of candidates equals MEAN(V)^2/MEAN(V^2).  For example, PpS is 100% if all voters have equal DP, 50% if half the voters have equal positive DP and the rest have zero DP.  PpS approaches 0% in the limit as the number of voters rise and only one voter has nonzero DP.

(2b.) The Preference Score (PfS) of a set of candidates equals the sum of scores given to the candidates divided by the maximum sum of scores the candidates could have been given (nVoters*nCandidatesInSet, assuming range [0-1]).  It ranges from 100% for unanimous complete support to 0% for unanimous complete opposition.

(3.) The Representativeness (REP) of a set of candidates versus the voters is defined to be the product PpS*PfS.

-

In the single-winner case, this always selects the same winner (i.e. most representative set) as normal Score Voting.

For C candidates of which E will be elected, the idealized method would check all NCHOOSEK(C,E) sets of E candidates.  It's definitely intractable for large C and E, but REP can readily be calculated for the winning set of more practical algorithms.

PfS is a rescaling of Bayesian Regret or Voter Satisfaction Efficiency in the single-winner case and is closely related to them in multiwinner cases.  It can be used to make the same comparisons as BR and VSE.  PpS measures only how equally the voters' votes contribute to a set of candidates.  It can be used to compare how proportional different voting systems are.

When voters give honest score votes, I think REP has a reasonable claim of describing the degree to which a set of candidates represents the voters.

I don't think this does always elect the normal score winner in the single-winner case. For example:

1 voter: A=1, B=0.1

8 voters: A=0, B=0.1

A wins (1 to 0.9) under normal score voting.

A has a PpS of (1/81) / (1/9) = 1/9

A has a PfS of 1/9

REP = 1/9 * 1/9 = 1/81

B has a PpS of 1 (everyone equally represented)

B has a PfS of 1/10

REP = 1 * 1/10 = 1/10.

So B would win in this case.

I would also still prefer a "voter agreement" measure rather than your PfS. This way, the measure would remain constant when factions remain constant. If there are two parties and everyone votes for all the candidates of just one party, then under any result, the voter agreement would remain the same. So the result would only be affected by proportionality rather than being skewed towards larger factions. Whereas if there were some candidates that had support from both factions, then their inclusion in the winning set would increase the agreement level, so would produce a better overall score than an equally proportional result where there is less agreement. So it would just be the proportion of votes that agree with each other. For example:

1 voter: A=1, B=0.6

1 voter: A=1, B=0.4

A voter counts as agreeing with themselves. If A and B are elected, the voters agree 100% with each other on one candidate, and 80% on the other (1 minus the difference between the two), so 90%. This would make a total agreement level of 95% or 0.95.

I think voter agreement works better because where two results are equally proportional, it would favour the result with the higher total score anyway, and in other cases, there would be no systematic large-faction bias.

[Toby Pereira]

On Saturday, 14 June 2014 01:11:45 UTC+1, Gabriel Bodeen wrote:

(2a.) The Proportionality Score (PpS) of the set of candidates equals MEAN(V)^2/MEAN(V^2).

Is this a standard statistical method? Does it have a name outside PpS? Will a lower mean squared deviation from proportionality always have a higher PpS, making it just a normalised 0 to 1 score for standard deviation?

[Jameson Quinn]

2014-06-14 12:23 GMT-04:00 'Toby Pereira' via The Center for Election Science <electio...@googlegroups.com>:

On Saturday, 14 June 2014 01:11:45 UTC+1, Gabriel Bodeen wrote:

(2a.) The Proportionality Score (PpS) of the set of candidates equals MEAN(V)^2/MEAN(V^2).

Is this a standard statistical method?

Yes; it's the variance of the DP.

 

Does it have a name outside PpS? Will a lower mean squared deviation from proportionality always have a higher PpS, making it just a normalised 0 to 1 score for standard deviation?

--

[Gabriel Bodeen]

On Saturday, June 14, 2014 8:49:12 AM UTC-5, Toby Pereira wrote:

I don't think this does always elect the normal score winner in the single-winner case. For example:

Yeah, I realized that couldn't be true immediately after hitting "Post".  Funny how that works.  I'd forgotten that MATLAB code treats matrices' singleton dimensions differently, so the columns showing 100% for "PpS" under a range of circumstances were just a foolish bug.

 

A voter counts as agreeing with themselves. If A and B are elected, the voters agree 100% with each other on one candidate, and 80% on the other (1 minus the difference between the two), so 90%. This would make a total agreement level of 95% or 0.95.

Ah, thanks! I wasn't clear on how you were defining it.

On Saturday, June 14, 2014 11:23:43 AM UTC-5, Toby Pereira wrote:

On Saturday, 14 June 2014 01:11:45 UTC+1, Gabriel Bodeen wrote:

(2a.) The Proportionality Score (PpS) of the set of candidates equals MEAN(V)^2/MEAN(V^2).

Is this a standard statistical method? Does it have a name outside PpS? Will a lower mean squared deviation from proportionality always have a higher PpS, making it just a normalised 0 to 1 score for standard deviation?

Probably; I can't recall what and my search failed.  It's not the variance.  ( VAR(X)=E(X^2)-E(X)^2 .  E(X)^2/E(X^2) = ??? )

[Toby Pereira]

On Saturday, 14 June 2014 21:00:18 UTC+1, Gabriel Bodeen wrote:

On Saturday, June 14, 2014 8:49:12 AM UTC-5, Toby Pereira wrote:

I don't think this does always elect the normal score winner in the single-winner case. For example:

Yeah, I realized that couldn't be true immediately after hitting "Post".  Funny how that works.  I'd forgotten that MATLAB code treats matrices' singleton dimensions differently, so the columns showing 100% for "PpS" under a range of circumstances were just a foolish bug.

 

A voter counts as agreeing with themselves. If A and B are elected, the voters agree 100% with each other on one candidate, and 80% on the other (1 minus the difference between the two), so 90%. This would make a total agreement level of 95% or 0.95.

Ah, thanks! I wasn't clear on how you were defining it.

If you used voter agreement as the criterion here though, then the way you measure proportionality would cause problems. If someone gives a candidate a score of 0.1 and no-one else gives that candidate anything, then that voter would fully possess that candidate (as we've discussed). Using your preference score would counteract that, but agreement level wouldn't. That's why I'd recommend using my way of calculating proportionality where for a score of e.g. 0.1 out of 1, the voter would "split" into 0.1 of a voter that gives a score of 1 and 0.9 of a voter that gives a score of 1. This also has the advantage of reducing to normal score voting in the single-winner case. Then you can combine proportionality with voter agreement.

 

On Saturday, June 14, 2014 11:23:43 AM UTC-5, Toby Pereira wrote:

On Saturday, 14 June 2014 01:11:45 UTC+1, Gabriel Bodeen wrote:

(2a.) The Proportionality Score (PpS) of the set of candidates equals MEAN(V)^2/MEAN(V^2).

Is this a standard statistical method? Does it have a name outside PpS? Will a lower mean squared deviation from proportionality always have a higher PpS, making it just a normalised 0 to 1 score for standard deviation?

Probably; I can't recall what and my search failed.  It's not the variance.  ( VAR(X)=E(X^2)-E(X)^2 .  E(X)^2/E(X^2) = ??? )

What made you pick that measure? It seems to give a nice 0 to 1 scale, but I'm not really sure where it comes from. Do you know whether higher PpS always means lower variance? If it doesn't, then I'm probably less convinced by it.

[Gabriel Bodeen]

On Sunday, June 15, 2014 11:19:17 AM UTC-5, Toby Pereira wrote:

A voter counts as agreeing with themselves. If A and B are elected, the voters agree 100% with each other on one candidate, and 80% on the other (1 minus the difference between the two), so 90%. This would make a total agreement level of 95% or 0.95.

Hm. If we add another voter to your example calculation:

1 voter: A=1, B=0.6

1 voter: A=1, B=0.4

1 voter: A=0, B=0.3

...it's not clear what to do.  1-2*MAD(vector of scores given to one candidate) looked at first like the most obvious generalization among several plausible choices.  But in reading up on measures of rater agreement, no method stood out as particularly good.

 

 What made you pick that measure? It seems to give a nice 0 to 1 scale, but I'm not really sure where it comes from. Do you know whether higher PpS always means lower variance? If it doesn't, then I'm probably less convinced by it.

 I chose it because it has the desired behavior.  Using a vector of voter's possessions for a set of candidates (DPs) as I previously defined it without voter splitting...

1. Cloning voters doesn't change the measure:  e.g. PpS([0 1])=0.5, PpS([0 0 1 1])=0.5, PpS([0 0 0 1 1 1])=0.5, ...

2. If a candidate set was chosen equally by a subset of the voters, the measure is the percent of voters in that subset: e.g. PpS([1 1])=1, PpS([0 0 1])=0.333, PpS([0 1 1])=0.667, PpS([0 0 0 0 1])=0.2, PpS([0 0 0 0 0 1 1 1])=0.375

3. Scaling the DP by a constant factor (i.e. electing a different number of candidates) doesn't change the measure:  e.g. PpS([0 10])=0.5, PpS([0 0.5 0.5])=0.667, PpS([0 0 0 0 0 0.667 0.667 0.667])=0.375

4. If the voters have unequal influence in choosing a candidate set, the measure responds in the right direction: e.g. for 0<x<1 and 0<e<<1,  PpS([0 x-e 1]) < PpS([0 x 1]) < PpS([0 x+e 1])

5. The measure is more sensitive at large differences in influence than at small ones: e.g. PpS([0.10 1])-PpS([0.09 1]) > PpS([0.91 1])-PpS([0.90 1])

No, higher PpS does not always mean lower variance.  Variance has the wrong behavior for points 2 through 5.

Given point 2 and the consistency provided by the other points, PpS can be interpreted very simply as the equivalent percent of equally influential voters, or as the equivalent percent of voters whom a candidate-set represents equally (regardless of how well it does so).

Elect 3 of A B C D

2 voters: 1 1 0 0

2 voters: 0 0 1 1

For the candidate set CD, the voters have DP [0 0 1 1] and PpS=50%.  That matches the interpretation in the ordinary way.

Elect 3 of A B C D

1 voter: 1 0 0 0

2 voters: 0 1 0 0

1 voter: 0 0 1 1

For the candidate set BCD, the voters have DP [0 0.5 0.5 2] and PpS=50%.  Three quarters of the voters would have at least a little influence in choosing this candidate set, but one of the voters would have 2/3rds of that influence, so the equivalent percent of effective voters is less than 3/4ths.  In this case, we can say the distribution of influence is as if the candidates had been chosen equally by 50% of the voters, or as if the candidates represented 50% of the voters equally.

If you used voter agreement as the criterion here though, then the way you measure proportionality would cause problems. If someone gives a candidate a score of 0.1 and no-one else gives that candidate anything, then that voter would fully possess that candidate (as we've discussed). Using your preference score would counteract that, but agreement level wouldn't. That's why I'd recommend using my way of calculating proportionality where for a score of e.g. 0.1 out of 1, the voter would "split" into 0.1 of a voter that gives a score of 1 and 0.9 of a voter that gives a score of 1. This also has the advantage of reducing to normal score voting in the single-winner case. Then you can combine proportionality with voter agreement.

 I'm actually fine with it not producing the same winner as Score Voting, because Score Voting is based on choosing the highest-relative-utility candidate as winner.  To select multiple winners with that basis, we'd just choose the N candidates with highest total scores, with no regard for proportionality.  A voting method that aims at proportionality requires an additional conceptual basis.  I suspect something like "representativeness" could be a good hybrid since it's the percent of voters whom a candidate-set represents ("PpS") multiplied by how well it represents them ("PfS").  REP approaches zero for candidate-sets that the voters dislike and also for candidate-sets that only represent very small groups.  For a constant sum, PpS+PfS, the max REP occurs at PpS=PfS.  So it's similar to the Chiastic method recently discussed.

Consequently the highest-PfS candidate is the same as the Score winner given the same ballots, but the highest REP candidate is likely to be a bit more centrist.

 

[Toby Pereira]

On Sunday, 15 June 2014 21:15:02 UTC+1, Gabriel Bodeen wrote:

On Sunday, June 15, 2014 11:19:17 AM UTC-5, Toby Pereira wrote:

A voter counts as agreeing with themselves. If A and B are elected, the voters agree 100% with each other on one candidate, and 80% on the other (1 minus the difference between the two), so 90%. This would make a total agreement level of 95% or 0.95.

Hm. If we add another voter to your example calculation:

1 voter: A=1, B=0.6

1 voter: A=1, B=0.4

1 voter: A=0, B=0.3

...it's not clear what to do.  1-2*MAD(vector of scores given to one candidate) looked at first like the most obvious generalization among several plausible choices.  But in reading up on measures of rater agreement, no method stood out as particularly good.

I would have said voter 1 has agreement with voter 2 of 0.9 and with voter 3 of 0.35. Voter 2 has agreement with voter 3 of 0.45. They all fully agree with themselves. So (1 + 1 + 1 + 0.9 + 0.35 + 0.45) / 6 = 0.783. But this is just the method I came up with for it. I don't have any evidence that it's the "best" or that it does what I think it should do, other than that it works in some very simple cases. It seems intuitively sensible to me though.

But I still think something like this is necessary. If there are two separate factions with no voting overlap, then I think the only important measure here is proportionality. Introducing a separate measure skews it towards larger factions. Whereas with my example from before:

Elect 2

10 voters: A, B, C

10 voters: A, B, D

then AB and CD have the same proportionality, so we have to use a different measure from proportionality to separate them. We can use PfS, but it only makes sense to do that in a case like this when their proportionality is identical. Otherwise it doesn't work.

If we have:

Elect 2

20 voters: A, B, E

10 voters: C, D, E

we could elect AB, AC, or CD, and although these would all give different levels of proportionality, there is intuitively a scale in which they are the same. Whereas if E is awarded a seat along with any of the others, then regardless of proportionality, then in this scale this result is different from the other results. And it's this scale in addition to proportionality that I think we should look at. PfS is not that scale. It needs to be orthogonal to PpS, which PfS isn't.

 

 

 What made you pick that measure? It seems to give a nice 0 to 1 scale, but I'm not really sure where it comes from. Do you know whether higher PpS always means lower variance? If it doesn't, then I'm probably less convinced by it.

 I chose it because it has the desired behavior.  Using a vector of voter's possessions for a set of candidates (DPs) as I previously defined it without voter splitting...

1. Cloning voters doesn't change the measure:  e.g. PpS([0 1])=0.5, PpS([0 0 1 1])=0.5, PpS([0 0 0 1 1 1])=0.5, ...

2. If a candidate set was chosen equally by a subset of the voters, the measure is the percent of voters in that subset: e.g. PpS([1 1])=1, PpS([0 0 1])=0.333, PpS([0 1 1])=0.667, PpS([0 0 0 0 1])=0.2, PpS([0 0 0 0 0 1 1 1])=0.375

3. Scaling the DP by a constant factor (i.e. electing a different number of candidates) doesn't change the measure:  e.g. PpS([0 10])=0.5, PpS([0 0.5 0.5])=0.667, PpS([0 0 0 0 0 0.667 0.667 0.667])=0.375

4. If the voters have unequal influence in choosing a candidate set, the measure responds in the right direction: e.g. for 0<x<1 and 0<e<<1,  PpS([0 x-e 1]) < PpS([0 x 1]) < PpS([0 x+e 1])

5. The measure is more sensitive at large differences in influence than at small ones: e.g. PpS([0.10 1])-PpS([0.09 1]) > PpS([0.91 1])-PpS([0.90 1])

No, higher PpS does not always mean lower variance.  Variance has the wrong behavior for points 2 through 5.

OK. I actually meant for a given election, so that the two methods wouldn't ever give a different result. And because (MEAN V)^2 would always be the same, whenever (MEAN V)^2 / MEAN (V^2) goes up, MEAN (V^2) - (MEAN V)^2 would always go down. So that's fine. They'd agree.

 

Given point 2 and the consistency provided by the other points, PpS can be interpreted very simply as the equivalent percent of equally influential voters, or as the equivalent percent of voters whom a candidate-set represents equally (regardless of how well it does so).

Elect 3 of A B C D

2 voters: 1 1 0 0

2 voters: 0 0 1 1

For the candidate set CD, the voters have DP [0 0 1 1] and PpS=50%.  That matches the interpretation in the ordinary way.

Elect 3 of A B C D

1 voter: 1 0 0 0

2 voters: 0 1 0 0

1 voter: 0 0 1 1

For the candidate set BCD, the voters have DP [0 0.5 0.5 2] and PpS=50%.  Three quarters of the voters would have at least a little influence in choosing this candidate set, but one of the voters would have 2/3rds of that influence, so the equivalent percent of effective voters is less than 3/4ths.  In this case, we can say the distribution of influence is as if the candidates had been chosen equally by 50% of the voters, or as if the candidates represented 50% of the voters equally.

If you used voter agreement as the criterion here though, then the way you measure proportionality would cause problems. If someone gives a candidate a score of 0.1 and no-one else gives that candidate anything, then that voter would fully possess that candidate (as we've discussed). Using your preference score would counteract that, but agreement level wouldn't. That's why I'd recommend using my way of calculating proportionality where for a score of e.g. 0.1 out of 1, the voter would "split" into 0.1 of a voter that gives a score of 1 and 0.9 of a voter that gives a score of 1. This also has the advantage of reducing to normal score voting in the single-winner case. Then you can combine proportionality with voter agreement.

 I'm actually fine with it not producing the same winner as Score Voting, because Score Voting is based on choosing the highest-relative-utility candidate as winner.  To select multiple winners with that basis, we'd just choose the N candidates with highest total scores, with no regard for proportionality.  A voting method that aims at proportionality requires an additional conceptual basis.  I suspect something like "representativeness" could be a good hybrid since it's the percent of voters whom a candidate-set represents ("PpS") multiplied by how well it represents them ("PfS").  REP approaches zero for candidate-sets that the voters dislike and also for candidate-sets that only represent very small groups.  For a constant sum, PpS+PfS, the max REP occurs at PpS=PfS.  So it's similar to the Chiastic method recently discussed.

Consequently the highest-PfS candidate is the same as the Score winner given the same ballots, but the highest REP candidate is likely to be a bit more centrist.

I've often wondered whether in a PR system something like two votes of 0.4 should count for more than a single vote of 1, but when my system as I came up with it resulted in the normal score winner winning, I was quite happy to accept that as the result. I didn't particularly go out of my way to force it to happen. The voter splitting was to avoid someone fully possessing a candidate they'd given a low score to rather than to fix the single-winner result.

So I suppose in conclusion, it seems we disagree on both PpS (I want to "split" voters") and on PfS (I want to look at voter agreement instead). But I think this is a useful discussion!

[Toby Pereira]

I might have to reconsider how I would calculate agreement levels to ensure that it works with unbounded cardinal scores. For example, currently if one voter gives a score to a candidate of 6/10 and another gives a score of 4/10, then this would give an agreement of 0.8. The only disagreement is between 4/10 and 6/10. But this assumes a cut-off maximum score and so agreement between 6/10 and 10/10. But if the agreement levels use the same voter splitting as the proportionality scores, this problem disappears but it gives different results.

In this case we end up with:

0.6 voters: 1

0.4 voters: 0

0.4 voters: 1

0.6 voters: 0

So:

1 voter: 1

1 voter: 0

This gives an agreement level of 0.5. This might seem a bit weird at first because it would also give an agreement level of 0.5 if both voters gave a score of 5/10 in a case where there is no disagreement! But all this means is that two voters giving 5/10 would still equal one voter scoring 10/10. And I think this is fine. The total "amount of vote" is the same in both cases, so PfS would be the same, and intuitively it doesn't seem a bad result to me.

So in conclusion, if I was going to use voter agreement/integration in my method, I would split the voters the same way as I would with score voting. So forget the other method.

[Toby Pereira]

So to summarise, I think between us we've come up with a fairly decent method. We have my original proportional score/approval method, and then Gabriel turned the variance into a 0 to 1 score where intuitively twice as proportional means twice the score on the 0 to 1 scale. We also have my method of voter agreement which also works on an intuitive 0 to 1 scale, where for a given level of proportionality, twice the score give to the candidates from the voters means twice the score on the 0 to 1 scale (I'm pretty sure of that). So by multiplying the two together, we have a method that combines proportionality and overall voter preference in a non-arbitrary way.

[Toby Pereira]

On Monday, 16 June 2014 20:42:05 UTC+1, Toby Pereira wrote:

So to summarise, I think between us we've come up with a fairly decent method. We have my original proportional score/approval method, and then Gabriel turned the variance into a 0 to 1 score where intuitively twice as proportional means twice the score on the 0 to 1 scale. We also have my method of voter agreement which also works on an intuitive 0 to 1 scale, where for a given level of proportionality, twice the score give to the candidates from the voters means twice the score on the 0 to 1 scale (I'm pretty sure of that). So by multiplying the two together, we have a method that combines proportionality and overall voter preference in a non-arbitrary way.

It seems I was wrong about how the agreement level would relate to the preference score. If you have one faction that has all the elected candidates, then as the size of that faction goes below half the electorate, then the agreement level actually increases as the faction size decreases because the unrepresented voters all agree with each other. So if we have three columns of faction size (=proportionality score), agreement level and product of the first two, we have:

1.0 * 1.00 = 1.000

0.9 * 0.82 = 0.738

0.8 * 0.68 = 0.544

0.7 * 0.58 = 0.406

0.6 * 0.52 = 0.312

0.5 * 0.50 = 0.250

0.4 * 0.52 = 0.208

0.3 * 0.58 = 0.174

0.2 * 0.68 = 0.136

0.1 * 0.82 = 0.082

0.0 * 1.00 = 0.000

So even though the agreement score does increase as proportionality goes down below 0.5, the product of the two is strictly decreasing here. The agreement scores aren't as intuitive as the proportionality scores in how they behave, but I'm struggling at the moment to think of a better way of doing it.

[Gabriel Bodeen]

On Sunday, June 15, 2014 6:04:37 PM UTC-5, Toby Pereira wrote:

 Introducing a separate measure skews it towards larger factions.

PpS*PfS is definitely broken this way.  A single score of 0 with a single score of 1 together have the same weight as two scores of only 0.25.  Doubling a faction's size (and keeping the total of the scores constant) has the same effect as doubling the scores.

One thing I tried is (PpS^(E-1))*PfS, where E is the number of candidates to be elected.  It's quite arbitrary, but it patches the large-faction favoritism.  It is just Score Voting for single-winner elections since the first term goes to 1.  As the number of candidates to be elected increases, proportionality comes to dominate over preferences, without entirely eliminating the latter's tie-breaking use.

On Monday, June 16, 2014 5:29:32 PM UTC-5, Toby Pereira wrote:

The agreement scores aren't as intuitive as the proportionality scores in how they behave, but I'm struggling at the moment to think of a better way of doing it.

Since your method splits voters, can you omit the unrepresented voter-parts?

[Toby Pereira]

On Tuesday, 17 June 2014 00:03:04 UTC+1, Gabriel Bodeen wrote:

On Sunday, June 15, 2014 6:04:37 PM UTC-5, Toby Pereira wrote:

 Introducing a separate measure skews it towards larger factions.

PpS*PfS is definitely broken this way.  A single score of 0 with a single score of 1 together have the same weight as two scores of only 0.25.  Doubling a faction's size (and keeping the total of the scores constant) has the same effect as doubling the scores.

One thing I tried is (PpS^(E-1))*PfS, where E is the number of candidates to be elected.  It's quite arbitrary, but it patches the large-faction favoritism.  It is just Score Voting for single-winner elections since the first term goes to 1.  As the number of candidates to be elected increases, proportionality comes to dominate over preferences, without entirely eliminating the latter's tie-breaking use.

Is this definitely right? For two equally proportional results but where one has a higher PfS, won't your new measure still favour the one with higher PfS? For example:

2 to elect

30 voters: A, B

10 voters: C, D

Proportionally AB = AC. But AB has higher PfS, so PpS * PpS favours AB. Whereas with your new measure, (PpS^(E-1)) would be the same for AB and AC because PpS and E are the same for both. So then we just multiply by PfS again and it favours AB.

 

On Monday, June 16, 2014 5:29:32 PM UTC-5, Toby Pereira wrote:

The agreement scores aren't as intuitive as the proportionality scores in how they behave, but I'm struggling at the moment to think of a better way of doing it.

Since your method splits voters, can you omit the unrepresented voter-parts?

It's seems better to do that. Just count positive agreement as long as you take into account the number of people agreeing, so one person with all the representation doesn't count as full agreement. Since I turn everything into approvals, it would be a case of counting matched approvals. But I'm not sure how to do that without also favouring larger factions.

However, I've just seen that my measure doesn't work anyway either way. I did some calculations where there were three factions, and the numbers didn't add up right.

It might be that there isn't a way to sensibly resolve this. I don't know though.

[Toby Pereira]

I just want to follow up on the Preference Scores (PfS) that Gabriel was talking about. On further reflection, I don't think there is a way to combine proportionality with overall preference in a workable way. If we have two to elect and approval voting with these votes:

1 voter: A, B, C

1 voter: A, B, D

AB and CD would come out as equally proportional under my system. We could fix it in this case by saying that AB has 100% approval, whereas CD has just 50% approval, so we could multiply proportionality by approval in this case, as Gabriel suggested. But such a method fails here:

2 to elect

3 voters: A, B

1 voter: C

My system gives an exact tie between AB and AC. But AB has a higher approval percentage, so multiplying proportionality by approval would fail us here because this isn't analogous in any way to the other case where we have equal proportionality but where one result is clearly more popular. The point is we don't want to fix the result here. AB does equal AC in the relevant way.

One possible idea from here is not to multiply by the approval level, but to multiply by the approval level of the "equivalent proportional result" (EPR). So we look at the result and the sizes of the factions, and level of agreement between voters, and use that to fabricate a result that would be proportional under these conditions. So in this case both AB and AC would revert to 1.5 of A and B being elected and 0.5 of C being elected. This way, the proportionality and EPR for the AB and AC results would be the same as each other, so it's still a draw. But in the first example, it would still elect AB over CD because the EPR of AB is double that of CD. So far so good.

However, it fails when there are more than two factions:

2 to elect

3 voters: A, B

1 voter: C

1 voter: D

Proportionally AB is equal to AC and AD. But let's look at the result AB. What's the EPR? Well, it seems obvious that 1.2 of AB should be elected, 0.4 of C and 0.4 of D and we'd calculate the approval percentage of that. But this doesn't work. The C voters and the D voters have no candidate elected, so under this result they are not in any way separate. A result has to stand on its own, not relative to unelected candidates. The point is that there is no faction data in the result for voters who have no candidates elected, so whether we count them as one faction, two or 77 is arbitrary. And this matters because we need to know the faction size to calculate the EPR. But the proportionality under any sensible system has to be the same for AB here as it is for AB with these ballots:

3 voters: A, B

2 voters: C, D

So basically EPR can't exist.

It seems that the best way to avoid monotonicity violations is to elect candidates sequentially.

On Tuesday, 17 June 2014 00:03:04 UTC+1, Gabriel Bodeen wrote: