[EM] Plurality criterion and the "sincere CW"

[Juho Laatu]

> On 01 Jul 2019, at 23:32, C.Benham <cbenham at adam.com.au> wrote:
> 
> Juho,
> 
>> If vote A>B=C is not the same as vote A in a three candidate election, then there is an implicit cutoff at the truncation point. (Or maybe someone wants to put the special "given any preference" point after A in the first vote.)
> To be clear, I interpret "vote" as meaning vote above at least one other candidate and not just any mark or number next to a candidate's name on the ballot paper.

Ok, "preferred over at least one candidate" then, which is not the same as truncation in the case that the voter lists all the candidates in the ballot paper.

>> Ranked methods work well only if voters give us their preferences (of all the relevant candidates)
> As long as the truncation incentive isn't stronger than the voters like and generally the incentives for voters to misrepresent their true 
> preferences are as low as possible then if voters choose not reveal their weaker preferences that should be no problem.
> 
>> But when reading the definition, a natural thought is that if you do not rank some of the candidates that you don't like at all, that might improve the chances of your favourite candidates to win.
> Yes, it might or it might not. No-one suggests that the Plurality criterion should be posted in the polling booth. 

I guess the behaviour of voters (sincere vs strategic) depends in this case on how strategic (and paranoid) the voters and the party officials and media are expected to be.

> 
> 
>> One quite possible (real life?) risk is that competing factions start truncating the candidates of the other factions. That might lead to bad results.
> Yes, I suppose it could lead to slightly "bad" results. But I think it is more likely to lead to relatively good results. Generally speaking I would guess that
> pairwise preferences among candidates the voter "doesn't like at all" would be weaker, probably less well-informed, if the method meets Later-no-Harm
> the preferences could be light-minded (almost arbitrary). That could result in a winner with relatively low Social Utility and less legitimate-looking.
> 
> And of course since all Condorcet methods have some Burial incentive, preferences among candidates the voter doesn't like are less likely to be 
> sincere in any case.  So arguably we should err on the side of trying to avoid a "garbage in, garbage out" scenario rather than fret about electing
> the candidate who would be the CW if only the voters could be induced to reveal their sincere full rankings.

The most probable problem might be one where there are two factions, and the bigger faction has two candidates, one centrist and one non-centrist, and the non-centrist one is more popular within that faction. If the other faction truncates enough, the CW will not be elected. (The worst case of truncation problem that I can imagine now would be E winning with 50: A>B>D, 50: C>D>E, 1: E>A with winning votes.)

> 
>>> 46: A
>>> 44: B>C
>>> 10: C
>> 
>> To people who are used to methods where the first positions are the key thing, and that's everything that is important in the election, these votes seem to say that A is the strongest of the candidates.
> Here you are somewhat missing the point again. It's not just "first" positions, it's any (above bottom) positions.  And it's not that A is "the strongest 
> of the candidates". It is that A is so much stronger than B that electing B can't be justified.  The Plurality criterion says nothing about C.

Ok, maybe Approval users my read these votes as "all marked candidates have been approved". Maybe that could be also called one type of "first position" or "black and white" thinking.

> 
>> Since information given in the votes above is very limited, we can imagine various reasons why all A and C supporters truncated their vote.
> It would be much more to the point if you instead try to "imagine various reasons" why all the B supporters didn't.

Their thinking is at least one step clearer since they gave more information (don't know if they are sincere or strategic though). A and C supporters might be sincere tie voters, but maybe not. They didn't rank all potential winners.

>> One way to see those votes is to ask what would happen if A, B or C wins. There would be an interest to change B to A, but just a small interest.
> That is "one way" which I regard as very stupid and completely reject. Instead of scratching your head wondering why the A and C supporters
> deprived us of so much "information", why don't you seriously consider the possibility that the B supporters' ranking of  C is completely insincere?!

When thinking about performance with sincere votes, I think about if the elected candidate is the best for the job. After the election the working relations and strength of opposition are important and obvious topics to discuss. With strategic voting and Condorcet elections my first question usually is, would the electorate really try something, and would they be able to successfully implement one of the strategies. I tend to think that in many cases (typically large public elections in places that are not used to strategic voting, and would not like people that would try to plot against others) Condorcet methods may well be strategy proof enough.

> 
> With no explicit approval (or any other sort of ratings) information the best winner is arguably C.  Then the 46 A voters have no strong complaint: C
> pairwise beats A, C is voted above bottom on more ballots than is A.  The 44 B>C voters have no complaint: if they hadn't ranked C then A would have
> won, if they really prefer A to C then it serves them right for telling lies.

I guess the B voters would complain about the 44-10 opinion in favour of B.

Strategic thoughts are always a mystery. I hope Condorcet methods encourage mostly sincere votes. If not, then maybe we need to change the method to something else since results would not be nearly as good if we assume that half of the electorate doesn't tell us their sincere opinions.

> 
> But electing B and then telling the A supporters "If only two of you later change your A>B preference to B>A then everything will be ok, and maybe
> the B supporters were all sincere in ranking C and it was just their good luck that doing so caused their favourite to be elected instead of yours" to
> me just doesn't wash.

I really hope Condorcet elections would not be a fight of different strategies, with all voters changing their plans depending of how they expect the other voters to change their plans. I hope most elections will need no such concerns, and I believe most elections will not need to worry about this.

> 
>> I'd like my favourite method to meet all the sensible sounding criteria. Unfortunately that is not possible. My philosophy is that the best method might violate all of the mutually incompatible useful criteria a bit, but only so little that those violations do not cause any problems in practice.
> 
> So have I been wrong in assuming that you insist on compliance with the Condorcet criterion?

I definitely do not insist compliance with Condorcet criterion (nor with most others), but I think Condorcet methods (and the CW philosophy) would be a good choice in very many cases.

Juho


> 
> Chris Benham
> 
> 
> On 1/07/2019 5:38 pm, Juho Laatu wrote:
>> I guess the key point that I was referring to is that if you read the definition verbally, it has words "votes in total". Or the wikipedia version of the definition (https://en.wikipedia.org/wiki/Plurality_criterion <https://en.wikipedia.org/wiki/Plurality_criterion>) has words "given any preference". Usually people talk about "truncation" of the vote. That seems to indicate that the point of truncation has some special meaning (in addition to just indicating that the unlisted candidates should be seen to be in the "shared last" position in the pure rankings).
>>> So what is the point of the Plurality criterion?  To my mind it is simply about not offending obvious fairness and common-sense.
>>> 
>> 
>> I believe that is the case. I just don't like the idea of giving the truncation point any special meaning. If vote A>B=C is not the same as vote A in a three candidate election, then there is an implicit cutoff at the truncation         point. (Or maybe someone wants to put the special "given any preference" point after A in the first vote.) Different methods may meet this criterion in different ways (some trivially). But when reading the definition, a natural thought is that if you do not rank some of the candidates that you don't like at all, that might improve the chances of your favourite candidates to win. (Also if there is a fourth totally hopeless candidate D, those two votes should be in practice similar, in all typical methods in which the existence of D has no effect on the outcome, or otherwise they seem to have a meaningful implicit cutoff.)
>> 
>> My philosophy is thus that pure ranked votes are pure ranked votes (usually completed so that unlisted candidates are seen as "shared last"). If methods derive some (cutoff like) additional information from the ballots, then I typically prefer methods where that cutoff is explicit (not implicit at the truncation point). And the reason is that I want to see complete rankings (of all the relevant candidates) instead of truncated (= lost) preferences. Ranked methods work well only if voters give us their preferences (of all the relevant candidates). One quite possible (real life?) risk is that competing factions start truncating the candidates of the other factions. That might lead to bad results. For these reasons the idea of rewarding truncation in some cases is not a very good idea. Sometimes it may be acceptable though, just like violations of most criteria sometimes are, if there are no better ways available around the problem in question.
>>> 46: A
>>> 44: B>C
>>> 10: C
>> 
>> To people who are used to methods where the first positions are the key thing, and that's everything that is important in the election, these votes seem to say that A is the strongest of the candidates. Condorcet methods (that I guess we are mainly discussing here) however can be said to aim at electing the best compromise candidate. That candidate might not have any first preference supporters, and still be a Condorcet winner. Since information given in the votes above is very limited, we can imagine various reasons why all A and C supporters truncated their vote. One way to see those votes is to ask what would happen if A, B or C wins. There would be an interest to change B to A, but just a small interest. One may consider the interest to change the others to some other candidate to be stronger.
>> 
>> If there is a Condorcet winner, as in votes 49: A>>>B>C, 03: B>A>>>C, 48: C>>>B>A, one could ask the voters, would they prefer to change the winner to B, if A or C would win. B may not be very popular, but maybe still a better choice than electing one of the more "extreme" alternatives. If people want the first preferences to have a strong influence, they might prefer methods like IRV (where candidates with small amount of first preferences support may often be eliminated quickly.). Having that kind of a "cutoff" would be another interesting discussion.
>> 
>>> If voters decline to (or don't bother to) express some or all of their very weak (possibly light-minded) pairwise preferences by truncating, then I don't classify that as "insincere" voting
>> 
>> I would not call those votes "insincere" either. But they would be "incomplete", and possibly "lazy" in some cases. It is not important to give opinions on the "irrelevant" candidates, but it is important to give opinions on the "relevant" candidates (unless one really thinks they are tied).
>> 
>>> Since therefore there could be several (or even many) alternative "sincere voting" profiles it follows that there could be more than one "sincere CW".
>> 
>> 
>> I wouldn't say so. There would be only (max) one sincere CW, based on what we know about the opinions of the voters. Information that is not there is not information to the method. Next day voters might vote differently, but that would be another day, and possibly another CW.
>>> Later-no-Help criterion ...
>>> 
>> 
>> I'd like my favourite method to meet all the sensible sounding criteria. Unfortunately that is not possible. My philosophy is that the best method might violate all of the mutually incompatible useful criteria a bit, but only so little that those violations do not cause any problems in practice. It is however possible that a method like that would be quite complex. I put some considerable weight also on simplicity and understandability, so I might prefer some simpler method instead of the "theoretically optimal" one. I also often tend to emphasise performance with sincere votes in cases, since in many elections strategic voting may not emerge even if there are some small theoretical possibilities of some strategy possibly being successful sometimes. It is important to elect the best winner (= performance with sincere votes), and not tweak the method to do something else because of some far fetched strategy concerns. Often the situation is thus such that there is no need to defend against strategies that are not likely to emerge and succeed anyway. Different elections have different needs. A repeated competitive poll among few EM strategy experts is different from a public election with millions of voters, clear frontrunners, and a wide mixture of continuously changing opinions.
>> 
>> Juho
>> 
>> 
>>> On 30 Jun 2019, at 19:00, C.Benham <cbenham at adam.com.au <mailto:cbenham at adam.com.au>> wrote:
>>> 
>>> Juho (and interested others),
>>> 
>>> The Plurality criterion was coined in 1994 by Douglas Woodall. Quoting him exactly from then:
>>>> The following rather weak property was formulated with single-seat elections in mind, but it makes sense also for multi-seat elections and, again, it clearly holds for STV .
>>>> 
>>>> Plurality. If some candidate a has strictly fewer votes in total than some other candidate b has first-preference votes, then a should not have greater probability than b of being elected.
>>>> 
>>> No mention of any "implicit approval cutoff".  I know that at the time Woodall was only thinking about strict rankings from the top with truncation allowed.  
>>> If equal-first ranking is allowed, then for the purpose of this criterion we should be using the fractional (summing to 1) interpretation of the number of 
>>> "first-preference votes".
>>> 
>>> Juho seems to think that the Plurality criterion is a "feature" or strategy device  that somehow encourages truncation.  It isn't and doesn't. 
>>> 
>>> If the method uses one of the traditional Condorcet algorithms that are almost the same as each other (Smith//MinMax, Schulze, River, Ranked Pairs)
>>> and uses Winning Votes as the measure of pairwise defeat strength, then the method meets Plurality and also has, at least in the zero-info case, a weak
>>> random-fill incentive.
>>> 
>>> IRV, and IRV modified to meet Smith by before each elimination checking to see if there is pairwise-beats-all candidate among those remaining, both meet
>>> the Plurality criterion.  In those methods do the voters have any have any incentive "not to rank the candidates of the competing groupings" ?  No they 
>>> don't.
>>> 
>>> So what is the point of the Plurality criterion?  To my mind it is simply about not offending obvious fairness and common-sense.
>>> 
>>> Juho, try to imagine that you have no interest in or knowledge about voting algorithms, you've never thought about the split-vote problem. You are accustomed
>>> to voting in plurality elections (or even perhaps Approval elections) and you've never been interested in doing anything other than voting for your sincere
>>> favourite, who regularly wins.  You are content with the current voting method and can't see any point in changing it.
>>> 
>>> Now imagine some voting-reform movement succeeds and the new method is, say, MinMax(Margins).  You hear that voters can now rank more
>>> than one candidate and you simply seek assurance that you will be allowed to go on voting as before and you assume that the government must
>>> more-or-less know what it's doing and assume the method won't in any way be less fair than before.
>>> In this election your favourite is A.
>>> 46: A
>>> 44: B>C
>>> 10: C
>>> 
>>> It is announced that the winner is B.  At first you think "A got more first-preference votes than B, it must have something to do with some voters'
>>> second preference votes", but then you notice that B got the same number of second-preference votes as A (zero), and then you ask "How on earth
>>> did this crazy new method elect B over my favourite A, who very clearly got more "votes" (marks next to his name on the paper ballots) all of which
>>> were first-preference votes!"
>>> 
>>> On hearing the reply "Oh, that's because B was the fewest votes shy of being the Condorcet winner" do you (a) say "Oh how silly of me, obviously
>>> that's fair!" or  (b) say .. something much less understanding and accepting ?
>>> This scenario also works if the old method was IRV.  You might also notice that this first MMM election scenario is also a massive egregious failure
>>> of the Later-no-Help criterion (because if the B voters had truncated then B wouldn't have won).  Do you like that criterion?
>>> 
>>> If the old method had been Approval, you would then presumably be understanding and resigned if it is announced that C won. 
>>> In fact electing A is a failure of the Minimal Defense criterion. Do you like that one?   So methods that meet both MD and Plurality (such as Winning
>>> Votes and Smith//implicitA) must elect C.
>>> 
>>>> ... methods might not elect the best winner (sincere Condorcet winner).
>>> If voters decline to (or don't bother to) express some or all of their very weak (possibly light-minded) pairwise preferences by truncating, then I don't
>>> classify that as "insincere" voting.  Since therefore there could be several (or even many) alternative "sincere voting" profiles it follows that there
>>> could be more than one "sincere CW".  It seems obvious to me that the one of of those that is based on only the relatively strong pairwise preferences
>>> will have a higher "social utility" than one based on all pairwise preferences which include a lot of very weak ones.
>>> 
>>> 49: A>>>B>C
>>> 03: B>A>>>C
>>> 48: C>>>B>A
>>> 
>>> Say these are the sincere preferences. If the voters care to express all their pairwise preferences then the "sincere CW" is B, but if they choose
>>> what I consider to be an alternative way of sincere voting and truncate where that will only "conceal" some weak pairwise preferences then
>>> an alternative "sincere CW" is (the apparently higher Social Utility candidate) A.
>>> 
>>> In fact if the method used was the tweaked IRV  method with an explicit approval cutoff that I recently suggested and the cast votes were
>>> 49: A>>B
>>> 03: B>A>>
>>> 48: C>>B
>>> 
>>> then only C would be disqualified (because A both pairwise beats C and is more approved than C) and then B is eliminated and A wins.
>>> I doubt that there would much blood flowing in the streets caused by the failure to elect the voted CW (B).
>>> As consolation for not meeting the Condorcet criterion we would have a method much more resistant to Burial strategy than any Condorcet
>>> method (and maybe more appealing to people who like IRV).
>>> 
>>> 
>>> Juho Laatu juho.laatu at gmail.com  <mailto:election-methods%40lists.electorama.com?Subject=Re%3A%20%5BEM%5D%20What%20are%20some%20simple%20methods%20that%20accomplish%20the%20following%0A%20conditions%3F&In-Reply-To=%3C06DA7FE6-AAF5-4684-B448-7FD93DCA0E35%40gmail.com%3E>
>>> Sat Jun 29 07:43:29 PDT 2019 
>>>> P.S. I don't like the plurality criterion. It actually sets an implicit approval cutoff at the end of the listed candidates. The worst part of that idea is that it encourages voters not to rank the candidates of the competing groupings. That (potentially huge amount of missing information) is not good for ranked methods. If voters learn to use that feature, methods might not elect the best winner (sincere Condorcet winner).
>>> 
>>> 
>>> 
>>> The following rather weak property was formulated with single-seat elections in mind, but it makes sense also for multi-seat elections and, again, it clearly holds for STV .
>>> Plurality. If some candidate a has strictly fewer votes in total than some other candidate b has first-preference votes, then a should not have greater probability than b of being elected.
>>> 
>>> 
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>> 
>> 
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