[EM] Generalized symmetric ballot completion

[Juho Laatu]

On 31.5.2011, at 19.52, Peter Zbornik wrote:

> Juho,
>  
> a correction:
> I wrote: "If a candidate is ranked on >50% of the ballots, then this method will always produce a winner"
>  
> That is not correct. Say we have 4 A>B, 3 B>A, 3 blank.
> Noone wins that election in the modified IRV election I proposed, neither candidate has 50% of the total vote in the first and final round, but both candidates are explicitly ranked on >50% of the ballots.
>  
> The example above illustrates the difference between the two rules I proposed for blank voting in Condorcet elections in my previous emails. I restate the rules again:
>  
> Rule 1 (proposed above):
> In a pairwise comparison of a Condorcet election (or in the final round of an IRV election), a winning candidate needs to get >50% of the total votes (including blank votes). This rule amounts to: "A majority of all voters support A before B". Technically this is a three part election with A vs B+X and B vs A+X. Thus we can have elections with no winner.
>  
> Rule 2 (from the generalized ballot):
> A winning candidate needs to be explicitly ranked on >50% of the ballots. This rule amounts to "A majority of all voters support the election of A rather than having no election winner". Technically this rule means that any candidate A has to win the election of A vs X in order to have a chance of winning the election.

Yes, this kind of rules are quite possible for elections where one wants to add a criterion that one must be "approved" by many enough voters, or when one need not make the decision if the result is not clear enough. As you can already guess I like the explicit "additional preference indicators" more than the implicit one (because of the possible loss of information). Additional markings put however a small burden on the voters, so it is good if all voters can use them efficiently (or leaving them out does not harm them). People should know what the impact of their marking is. Plain "negative vote" or "burial" style thinking is not usually the best. A better approach (naming) to (explicit and also implicit) cutoffs would be something like "I prefer keeping the old elected official" or "I prefer second election round of to electing anyone below this line" or "don't elect anyone this time" or "let these guys rule only until we can arrange a new election in 6 months" or "I want to see these guys as candidates in the second and final round". My point thus is that I prefer sincere rankings of candidates and additional markings to strategic voting (like trying to harm the strongest competitors of one's favourite).

(I thus think there are many different kind of elections. In some additional rules should be used while in some basic rankings are enough.)

>  
> If the proposed method (call it Static-IRV) fails to produce a winner (i.e. all IRV winners, who don't satisfy Rule 1 above are deleted), then the IRV election would be repeated only for candidates having least 50% of explicit ballot rankings (applying rule 2 instead of rule 1). If no candidate has at least 50% of explicit ballot rankings, then the IRV winner would be elected.
>  
> The same heuristic could be applied for Condorcet elections.
>  
> Heuristics are frowned upon, I know, but even Schulze uses heuristics, and a lot of them.
>  
> The benefits of the proposed Static-IRV election method is
> 1] to keep the LNH property as long as possible and

I guess the key idea of LNH is to allow people to rank all the candidates without any fear that ranking all of them would work against themselves. This is the same reason why I'm afraid that implicit cutoffs might have a bad impact on the voting behaviour. On the other hand even if a method fails LNH that does not mean that it would be automatically beneficial to the voter to truncate. I think Condorcet methods should in general be considered to "almost meet the LNH" in the sense that it is a good and sincere advice to the voters to (as a main rule, maybe to be followed every time) rank all relevant candidates and not to truncate.

> 2] respect the blank vote and get a winner with 50% of all votes in the last round (the run-off), if possible

People should be aware of this too. They should know what the difference between staying at home and casting an empty ballot is.

> 3] to generate candidates with strong support for the runoff

Here we might come a bit to the IRV territory. IRV proponents sometimes refer to the Condorcet winner as a potentially weak candidate. They tend to emphasize the role of first preferences as opposed to being acceptable to all but not the first preference to many. If there are such first preference requirements they should be made explicit. People should know and agree. Also the "additional markings" could be used to indicate such "strong or first preference" support. If one of the candidates of the election (A) is my good and very competent friend that is still unknown to most voters, ann candidate B is the main candidate of my favourite party, then I might vote A>B>"strong support">C>D. I want to say that also candidate B should be considered as my "first" choice although A is even better. IRV could be bad heuristics in the sense that if my party has many excellent new candidates (like A) it might happen that IRV eliminates B at the first round although B is the main candidate of the largest party. Ok, probably not, but the point is that additional markings could be used (at least in theory) to measure strong support better than the heuristics of IRV do.

(Term "strong support" should be well understood in order (again, to make sincere voting possible). It could mean something like "I will support these guys by actions or verbally or just inside my head throughout their period in power", i.e. "I will not be in opposition against these guys".)

>  
> IRV can be seen as a heuristic to generate two good candidates for a head-to-head election.

See above :-). IRV is a practical option but one should remember that its serial elimination process gives somewhat "randomish" results.

>  
> If the blank vote is not respected and the winner is not required to have 50% of the vote, then we have a plurality voting system.
>  
> In the Czech senatorial elections, it is not possible to vote blank in the second round and some senators are elected with less than 50% support of the voters, counting "invalid" votes and abstentions.
> If blank votes were allowed in the second round of run-off elections, then double-voting could be allowed too (A=B, half a vote to each) and possible allowing for ranking of the candidates (A>B) in order to allow the voters to compensate the blank votes.
>  
> In the Czech parliament, >50% of the votes (including abstentions) is required for a decision.
> A good argument for the blank vote and for the 50% requirement in elections is to refer to the voting in parliament.
>  
> Do you know of any nice paper or post on this list, which discusses possible significant modifications/improvement of the general mechanics of Condorcet elections (apart from the debate on ranked-pairs, maximin, minimax, Schulze, Beatpath, Copeland etc.)?

There have been too many discussions to remember them all. I don't have any clear favourites in my mind. Maybe a search in the archives could give some indication on what kind of various discussions there have been. One interesting approach that pops in my head is the Debian project and its use of "None Of The Above".
http://www.debian.org/vote/2007/vote_001

Juho




>  
> Best regards
> Peter Zborník
>  
>  
> On Tue, May 31, 2011 at 11:58 AM, Peter Zbornik <pzbornik at gmail.com> wrote:
> Juho,
>  
> comments in the text below.
> Mostly details.
>  
> Below I propose a new election method using IRV, which is closer to Condorcet than regular IRV and would have elected Montroll in Burlington.
>  
> Method:
> If the IRV winner doesn't get >50% of the votes (including blank ballots or "write-in candidates") then he/she is deleted and the IRV election is re-run on the same ballots without the candidate.
> Repeat until we have a winner with >50%.
> If no candidate is ranked on >50% of the ballots, then a new election is called.
> If a candidate is ranked on >50% of the ballots, then this method will always produce a winner
>  
> That would be, I think the smallest improvement on IRV, which could make a positive change in real life and would support centrist candidates.
>  
> The generalized ballot completion procedure will not work in an IRV-STV election, I think, but adding null-candidates at the end of the empty ballot will work, if the null-candidate cannot be deleted. However static quotas is easier to understand in IRV-STV, than null candidates, I think. I cannot see how to integrate negative rankings in STV elections.
>  
> The rest in the text below
>  
> Best regards
> Peter Zborník
> 
> On Tue, May 31, 2011 at 12:19 AM, Juho Laatu <juho4880 at yahoo.co.uk> wrote:
> On 30.5.2011, at 18.41, Peter Zbornik wrote:
> 
>> Juho,
>>  
>> summarize my argument concerning generalized ballot and generalized ballot completion and in the end of this email I suggest a new single-member Condorcet election system.
>>  
>> Nomenclature: I think that "null-candidate" (marked "X") is a fitting name for voting for not filling a seat. The other names given do not have that chique mathematical sound: "White", "None of the Above", "Re-open nominations", "Ficus (the plant)", etc.
>>  
>> In the discussion, I think I showed the following
>> If blank voting ("null candidates") is not allowed, then truncated/incomplete ballots give different election results for winning votes and for margins.
>> Compare Kevin Venzke's example:
>> 35:A>B
>> 25:B
>> 40:C
>> If we complete this election (Woodall's original proposal) to
>> 35:A>B>C
>> 25:B>A=C
>> 40:C>A=B,
>> then the election gives different results whether the candidates in the ties are resolved as 0.5 vs 0.5 (margins - A winner) or 0 vs 0 (winning votes - B winner)
>> (compare the results of the election at  http://condorcet.ericgorr.net/ and http://www1.cse.wustl.edu/~legrand/rbvote/calc.html)
>>  
>> For margins, Woodall's plurality criterion is violated.
>>  
>> If the same election is completed to allow for blank voting:
>> 35:A>B>X>C
>> 25:B>X>A=C
>> 40:C>X>A=B,
>> then the election gives same result (B - winner) both for margins and for winning votes and the parwise comparison matrix will be identical for both methods if a an equality  awarded 0.5 votes for both candidates.
> 
> To summarize my thoughts...
> - I think explicit cutoffs work fine when the cutoff carries some agreed message (e.g. approved vs. not approved)
> - Using explicit cutoff just as an extra candidate that voters can use as a strategic tool to generate big defeats to some candidates is more problematic (you can try to bury someone under X without any risk of electing X)
> You can try to bury someone under all other candidates anyway. Introducing a null-candidate as a "cuttoff" does not change that.
>  
> - Implicit cutoff is problematic since it may encourage truncation
> - Woodall's plurality criterion assumes an implicit cutoff (i.e. voters are expected to vote so that unlisted candidates are considered "bad" and listed candidates "good"; unlisted candidates are thus not just purely "tied last")
> - In elections where unlisted candidates should be considered purely "tied last" Woodall's criterion is not relevant (i.e. when one wants "B" to mean "B>A=C" and nothing more than that)
> Well, I guess the relevance of any criterion depends on what the method is supposed to achieve.
>  
> - There are many alternative rules for cutoffs (one could e.g. not use the cutoff as a regular candidate that can win and lose to others but require that n% of the votes must approve the winner)
> Yes, I think the rule in the parentesis is the same as having a null-candidate, if approval is defined as explicitly ranking the cadidate on the ballot. As I wrote below I cannot show it though.
>  
> 
>>  
>> Thus, truncated/incomplete ballots can be completed using the following generalized symmetric ballot completion algorithm, in order to give same election results for margins and winning votes and to not violate Woodall's plurality criterion for margins:
>> 1.  add s "null candidates" under the ranked candidates, where s is the number of seats
>> 2.  rank the unranked candidates equally and under the "null candidate".
>> 3.  equalities are resolved by giving each candidate 0.5 votes in the pairwise comparison.
>>  
>> If margins are used in Condorcet elections with generalized symmetric ballot completion, then Woodall's plurality criterion is not violated, since the "blank votes" are actually represented and the ballot is complete.
>>  
>> Maybe the entry in Wikipedia could be updated, where we read "Only methods employing winning votes satisfy Woodall's plurality criterion."
>> http://en.wikipedia.org/wiki/Condorcet_method#Defeat_strength
>>  
>> I think an equality on the ballot between two candidates A=B should intuitively mean nothing else than giving half a vote to A>B and B>A, i.e. the pairwise comparison matrix should not change and Woodall's plurality criterion should be kept at the same time. This is only possible if the generalized symmetric ballot completion algorithm is used.
> 
> I think the original margins style of simply completing the ballots as "tied last" without any implicit cutoff is ok and from that point of view it is not a problem that it does not meet Woodall's plurality criterion (since no implicit cutoff (meaning "approval" of the candidates) was intended). So maybe the new method should not be considered an improved margins method but as one of the approaches that have an implicit cutoff and that also meet Woodall's plurality criterion.
> 
>>  
>> The rule of requiring the candidate to score more than 50% in a pairwise comparison which I proposed in a previous email is enforced if generalized symmetric completion is used.
>>  
>> Furthermore, the Wikipedia entry could also mention the inclusion of "null-candidates" as the natural way to enable blank voting and avoid elections of candidates, where the voters would rather like to see an empty seat.
> 
> Note that Wikipedia does not want to have original research. So the correct approach would be to first publish the new approach somewhere and only then refer to it. (Note that the electorama web site contains many new proposed methods, so it can also serve as a storage place for new methods. Not a wikipedia though.)
> 
>> I.e., A wins the following election with current Condorcet implementations (disregarding if we use margins or winning votes):
>> 45:A
>> 40:B
>> 15:Blank
>>  
>> If we use generalized ballot completion, then the null-candidate wins in a Condorcet election (but not in an IRV election):
>> 45:A>X>B
>> 40:B>X>A
>> 15:X>A=B
>> Woodall's plurality criterion is not violated because X is not a candidate to win a seat.
>>  
>> Introducing a cutoff, like saying that "a winning candidate needs to be explicitly ranked on 50% of the ballots" maybe is equivalent to the generalized ballot completion algorithm (I don't know). However such a cutoff doesn't allow for ranking between disfavoured alternatives, which the generalized ballot does.
>>  
>> I aggree that it is better to require the voter to rank all candidates, as an incomplete ballot is completed in any case and the voter might not know the ballot completion algorithm.
> 
> Having complete rankings is good but it may be ok to accept also ballots that have accidentally failed to rank some of the candidates. This depends also on the number of candidates (ranking 100 of them could be too much for most voters).
>  
> Well, a truncated ballot is a shorthand for a specific type of ballot. Of course it could be used, but the voter should know the algoritm to translate the shorthand to a complete ballot. But this is essentially only technical details. In essence, I aggree.
>  
> 
>>  
>> I don't think that introducing a null candidate in a Condorcet election has any impact on its violation of Later-no-harm, i.e..the incentive of the voter to bullet-vote to maximize the success of "His" candidate. Even if the equalities and null candidates would be disallowed on the ballot, later-no-harm would still not hold for Condorcet elections and burying would still be an efficient strategy (slightly OT: the claim that Condorcet methods elect centrist canidates is questionable, since the centrist candidate will be the prime target for burying attempts, since he/she has the highest chance of winning, thus losing his "centricity" even before it is measurable in a election).
> 
> My approach to the various criteria is that one should take into account also how much some method violates some criterion. No proper method meets them all. Condorcet methods are very good from this point of view in the sense that although they fail Later-no-harm there is "usually and by default" no harm ranking also "later" candidates. Same with burial. They are vulnerable to burial but "usually and by default" one need not worry about burial (=not a practical strategy in typical large public elections with independent voters).
>  
> OK for public elections, but for a political party, where voting strategy is the name of the game?
>  
> This "usually and by default" rule applies also e.g. to risk of one party naming multiple candidates and minmax not being clone proof.
> 
> If people start using burial in Condorcet, I believe in most cases their strategy is not a good one since using burial efficiently is so difficult. Typically (I guess "usually and by default") burial attempts will just cause more harm than good to the strategists.
>  
> Do you have any references for your statements concerning "usually and by defaults"?
>  
> 
> I noted already above that having a "candidate" that can not win but that can be used for burial (="X") may make burial easier and more tempting than what it would be with "normal" candidates only.
>  
> Well, burial applies for complete ballots too and I think it is just as easy and tempting than with an added null candidate.
>  
> 
>>  
>> Thus, I think that the voter by default should be able to give a partially blank vote, by completely ranking the candidates and the "null candidates" using ">" and "=".
>>  
>> Definition of a generalized ballot:
>> Maybe the discussion could focus more on constraints that can be put on the generalized ballot, than on ballot completion algorithms.
>>  
>> A generalized ballot is defined as:
>> i a partiall ordering (i.e. using only "=", ">") of the set C, where C contains 
>> ii. s enumerated instances of the h candidates in the election for s seats: A11,..,A1s,...,Ah1,...,Ahs  and
>> iii. s enumerated instances of the "null candidates" X1,...,Xs.
> 
> (I just note that there are many possible ballot formats. For example one where all candidates are listed and next to them there are possible ratings from 1 to 20 (to be ticked) and a clear cutoff borderline between numbers 10 and 11 (=approval cutoff).)
>  
> I agree.
>  
> 
>>  
>> Some constraints on the candidate set:
>> 1. Normally we put the constraint in the election that there may only be one instance of each candidate in C, i.e. C={A1,...,Ah, X1,...,Xs - each elected candidate has only one seat and one vote, except for the Null-candidate.
>> 2. We might restrict H in the previous point to only contain candidates , i.e. C={A1,...,Ah} and no null-hopefuls, disallowing the blank vote and thus requiring a complete ranking of the candidate list.
> 
> (You didn't define and discuss basic uses of multiple null candidates and multi-winner elections very much.)
>  
> Basically in a multiwinner elections you have as many null-candidates as seats, which I think is covered by the definition above and by constraint 1 above, as the number of instances of the null candidate equals the number of seats (s) in the election.
>  
> 
>>  
>> Some ideas:
>> An other interesting issue, is if election systems with several election election rounds can improve results in Condorcet elections, for instance, an STV Condorcet election could be held with three seats.
>>  
>> Those who get one of the seat go through to the second round (which maybe can be automatical), where one of the candidates is elected in a Condorcet election, where a Condorcet winner is guaranteed.
>>  
>> Maybe an election type could be devised which makes a bottom-up proportional ranking. At the start of the election, as many seats as there are candidates are elected, then in each subsequent round one candidate is dropped util we have a Condorcet winner. 
>>  
>> Example: start with six candidates and elect five of them in a five-seat Condorcet-STV election, check if we have a Condorcet winner, if not, out of these five, elect four of them in a four-seat election and check if we have a Condorcet winner if not elect three of them in a three-seat election. Amon the three elected there is always a Condorcet winner.
>>  
>> Well, it's a new method at least.Could be worth trying out, maybe it will help resist burying or have some other nice properties.
>>  
>> Do you or anyone else around on this list have a reference to where the debate between IRV and Condorcet stands today (pros and cons of the methods respectively)?
>>  
>> Personally I am not yet convinced that Condorcet is a "better method" than IRV when it comes to resisting tactical voting.
> 
> They are quite different methods with respect to strategic voting. To me the promise of Condorcet methods is that in typical political elections they may avoid (rational) strategic voting even completely.
>  
> If you mean public elections, then maybe. If you by "typical political elections" mean elections in a political party, then I do certainly not aggree.
>  
> If there is a top level cycle, then people may afterwards think "I should have voted that way", but it is not easy to know what to do (except to vote sincerely) before the election.
>  
> I don't aggree. There is polling and the voter normally knows who is the biggest competitor to the "favored" candidate. The competitor is buried. The voters for the competitor bury your favorite candidate, and the winner is a "nobody" that no-one cared enough about to out-maneuver and noone supports, but also noone dislike. In a polarized environment that is not an unlikely scenario.
>  
> I do not personally like the idea of keeping the voter "uninformed" of the workings of an election system and their different strategies.
> That is a path I do not want to walk.
>  
> In IRV one may end up sooner in situations where e.g. some voter group knows that it should compromise (and thereby improve the result of the election). This may happen e.g. when a Condorcet winner is about to be eliminated at the first round and as a result "the other side" is likely to win. This example is not really on "resisting tactical voting" but on "requiring tactical voting". Maybe this describes my first thoughts on this topic well enough. I will not try to prove these claims here (that would require too many lines of text :-). IRV had some problems at least in Burlington in 2009 (the Condorcet winner was eliminated).
>  
> Well I think that IRV might be a good approach to find the two or three candidates to meet in the second round.
>  
> When I look at the Burlington result, then what first comes to mind is that the winner (Bob Kiss) didn't get 50% of the votes, but only 4313 out of 8980 votes (48%), since there were 606 "Exhausted votes" in the final round, i.e. IRV used dynamic droop quotas.
>  
> Thus IRV didn't respect the partially blank vote and this might be a reason why there is so much controversy around this election.
> A second option would have been to require complete ballots without the possibility to blank vote, which however might have triggered a new candidate "None of the Above" OR "Mr. Blank" in the election :o).
>  
> So let us assume Bob Kiss wasn't elected, since he didn't get >50% of the votes in the end, what would have happened?
>  
> Well, one approach mighet have been to hold a second round election, would be held, which is how presidents and such are elected most over Europe.
>  
> In the second round either two or three candidates could meet depending on the favoured result (IRV or Condorcet).
>  
> If there was only one round for the election, then I would have favoured to eliminate Bob Kiss (he got his chance, but didn't make 50%), and re-run the election.
>  
> With Bob Kiss eliminated, Andy Montroll would have won and everyone would have been happy. I did a quick and dirty run on the reduced election data with Kiss, Wright and Montroll (http://rangevoting.org/JLburl09.txt, at the end) and Bob Kiss eliminated. Andy Montroll got more than the 4490 votes needed (4968 votes)
>  
> Maybe a new IRV method could be considered: IRV with static quotas.
> If the IRV winner doesn't make >50%, then the IRV winner deleted and the IRV election is re-run.
> The generalized ballot can also be used for IRV-STV, but then we would have to add the rule that the null candidate(s) cannot be deleted.
>  
> Ballot files used for Burlington (X are the blank ballots):
>  
> With Kiss (K)
> 1332 M>K>W 
> 767 M>W>K 
> 455 M 
> 2043 K>M>W 
> 371 K>W>M 
> 568 K 
> 1513 W>M>K 
> 495 W>K>M 
> 1289 W 
> 147:X
>  
> Without Kiss (K):
> 1332:M>W
> 767:M>W
> 455:M
> 2043:M>W
> 371:M>W
> 568:X
> 1513:W>M
> 495:W>M
> 1289:W
> 147:X
> 
> To summarize my thoughts also after reading the mail...
> - I like explicit cutoff marks when they carry a clear agreed message that voters can easily and sincerely (not to implement a strategy) rank (e..g. between acceptable and non-acceptabe candidates)
> - Ranked ballots can thus be efficiently used for collecting also additional information in addition to basic ranking data
> - In elections where there is no clear cutoff information to be collected, basic rankings will work fine (i.e. no need for fixes in the basic case, it works fine as it is)
> - There are many possible rules on how to take the cutoffs into account in the vote counting process (check impact on strategic voting)
>  
> Yes here I am OK with you.
>  
> 
> Juho
> 
> 
> 
>>  
>> Best regards
>> Peter Zborník
>>  
>>  
>> On Sun, May 29, 2011 at 4:29 PM, Juho Laatu <juho4880 at yahoo.co.uk> wrote:
>>  
>> On 29.5.2011, at 16.06, Peter Zbornik wrote:
>>  
>> > On the other hand I might rather prefer "My Political Opponent" to be elected than "Pol Pot".
>> > Thus a ballot on the form A>X>My Political Opponent>Pol Pot, might be a good idea to allow.
>>  
>>  
>> I like this kind of explicit cutoffs more than implicit ones (at the end of the ranked candidates) since implicit cutoff easily encourages truncation. If people like to truncate their strongest opponents we might end up having bullet votes only. That would mean that we would be back in plurality, and all useful information of the ranked votes would be gone.
>>  
>> The explicit cutoff works well in elections where it is possible not to elect anyone (maybe keep the old elected alternative, or maybe arrange a new election after a while). One could also have elections where there are many possible outcomes, e.g. a seat for 6 months or a seat for 2 years (A>2y>B>C>6m>D). In these cases it is possible to measure quite reliably which candidates fall into which categories (e.g. "approvable enough"). The detailed rules on how to interpret e.g. a pairwise defeat to a cutoff entity have to be agreed.
>>  
>> Using the cutoff to give "negative votes" to candidates below the cutoff line (in the sense that such "negative votes" would really decrease their chance of winning candidates above the cutoff line) may be problematic since people could start giving negative votes to their worst competitors as a default strategy.
>>  
>> There have been also various proposals allowing strength of preference to be expressed (e.g. A>B>>>C>D>>E).
>>  
>> Juho
>>  
>>  
>>  
>>  
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