[EM] Generalized symmetric ballot completion

[Juho Laatu]

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)
- 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)
- 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)

>  
> 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).

>  
> 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). 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.

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.

>  
> 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).)

>  
> 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.)

>  
> 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 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. 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).

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)

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
>  
>  
>  
>  
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info
>  
>  
> ----
> Election-Methods mailing list - see http://electorama.com/em for list info

-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20110531/3c5e5a95/attachment-0004.htm>