[EM] Generalized symmetric ballot completion

[Peter Zbornik]

On Tue, May 31, 2011 at 6:06 PM, Kristofer Munsterhjelm <
km_elmet at lavabit.com> wrote:

> 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.
>>  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 <
>> https://mail.google.com/wiki/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.
>>  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.
>>
>
> How would you deal with Condorcet cycles? Is the rule "must score more than
> 50% in at least one comparison"? If not, there could be problems when no
> beats-all winner exists.


See later posts. Especially the recent email to Juho.


>
>
> 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.
>>
>
> It seems you've rediscovered the concept of an approval cutoff, but in
> another context. There are two sorts of approval cutoff: implied and
> explicit. In the former case, truncation means that you don't want those you
> didn't rank to win. In the latter, there's an explicit mark - your X - after
> which the same holds true.
>
> Implied cutoff methods are less expressive but they can guard against
> burial. Kevin Venzke shows how this works in Condorcet//Approval: you can't
> effectively bury a candidate because ranking him last means you give him
> approval, which strengthens his position. I imagine that like with explicit
> cutoff, you could bury more powerfully by saying something like A>B>D>X>C
> instead of a honest A>B>C>D (or strategic A>B>D>C).


Maybe I misunderstand something. If I truncate A>B>D, then this is completed
to A>B>D>C with C disapproved and buried.
X is just the cutoff, so I really don't see how burying would be more
powerful.


>
> As for whether X wins whenever the winner is not ranked above him on a
> majority of the ballots, that would be a majority version of the Pareto
> criterion. "If a majority ranks A above B, then B must not win". But we know
> that it's possible for a majority to rank A above B, another majority rank B
> above C, and a third majority rank C above A. Therefore I don't think it's
> equivalent, although I can't find an example.
> (In other words, a winning candidate could be not-X even when X is ranked
> above him by a majority, but then this candidate must be in a cycle with X.)
>


Yes, there are two rules, see later posts.


>
>
> 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.
>>
>
> On the other hand, forcing voters to rank all candidates can lead them to
> taking the easy way out - in the short term by donkey-voting and in the long
> term by encouraging (or at least not resisting) a change of the ballots to
> include above-the-line voting. Both of these things have happened in
> Australia, where the type of IRV they use means that voters have to rank
> every candidate, and the consequence is that the parties wield greater
> power.


Yes, several other shorthand ranking formulas/algorithms are thinkable so
that the voter would save time and ink from writing less on the ballot, but
that is really concerning only the execution of the elections. I would
rather not focus on this issue in this discussion.


>
>
> 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).
>>
>
> I think I found out earlier that to immunize completely against burial,
> you'd need both LNHarm and LNHelp, because there are two ways burial could
> work, and one is blocked by LNHarm whereas the other is blocked by LNHelp.
> The Condorcet criterion is incompatible with both, in any event.


OK, noted.


>
> Your OT would probably hold no matter what tendency Condorcet has. Say that
> Condorcet tends to elect winners of type K. Then type-K candidates are the
> most visible targets, and thus could be buried if the electorate is
> strategic enough.
>
>
> 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.
>>
>
> I have a hunch (as you put it :-) that you can't both have Droop
> proportionality and monotonicity. My conjecture is based on that the
> apportionment/party list version of these methods (largest remainder Droop)
> fails population-pair monotonicity, i.e. transferring votes from one party
> to another can in some cases make the former party gain seats. I have not
> been able to prove this, but if I'm right, that means that runoff methods
> based on STV (even Condorcet STV, if it passes the Droop proportionality
> criterion) would be nonmonotone.
>
> Further, such methods wouldn't be cloneproof. To slightly alter an example
> of Woodall's, consider:
>
> 700: x
> 300: y
>  1: z
>
> and two winners should elect x and y.


Not if we use static quotas as I propose.
 Droop quota is >1001/3=333.67, so y doesn't pass.
If we append the Xs as I propose then the ballots are (it's not really
needed)
700: x>NULL
300: y>NULL
 1: z>NULL
x is elected and the other seat is left vacant.
x's voters are stupid not to add a second candidate (the clone below).


> But now clone x:
>
> 350: x>x'
> 350: x'>x
> 300: y
>  1: z
>
> Then the DPC forces the two winners to be x and x', so cloning can push
> other candidates out of the runoff. Perhaps a better runoff algorithm would
> be something akin to LeGrand's minmax approval. Minmax approval picks the
> set of candidates for which the candidate who disagrees the most about the
> composition of the council, as measured by Hamming distance, disagrees the
> least. I'm unsure how to generalize it, however.


>
> 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.
>>
>
> I don't have a reference. You're probably right that IRV in general resists
> strategy better than does the ordinary Condorcet methods discussed on this
> list (like Schulze, Minmax, or Tideman); but note that James
> Green-Armytage's paper shows Smith-limited IRV to be as good as, if not
> better than, ordinary IRV with respect to strategy resistance, while always
> choosing winners from the Smith set (and thus always electing the CW where
> it exists).
>
> His working paper is here: http://www.econ.ucsb.edu/~armytage/hybrids.pdf
>
> (One does pay for this resistance by having a method that fails
> monotonicity, though.)
>

I will check it out.
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