[EM] Generalized symmetric ballot completion (was Hybrid/generalized ranked/approval ballots)

Sat May 28 22:53:28 PDT 2011

Dear all,

I apologise for some less fortunate attempts to resolve the problem of the
generalized incomplete ballot.
Now I think I have finally arrived at a good unified treatment the problem,
generalized symmetric ballot completion, which respects the power of the
blank vote to block elections and allows for equal preferences.

This generalized symmetric ballot completion completes the ballot and adds
"blank" candidates (dummy-candidates if you whish).
It gives same election results for margins, winning votes and quotas.
In an STV election there is no need to introduce static or dynamic quotas,
as the ballots are complete.

Generalized symmetric ballot completion is defined by the following process:
Say "s" marks the number of seats to be filled in an election.
Each ballot is processed as follows:
1. New candidates called X(1),..., X(s) are added to the ballot and ranked
lower than all other ranked candidates. These new candidates are ranked as
follows X(1)>X(2)...>X(2)
2. If the ballot is incomplete then the missing candidates are ranked lower
than all new candidates X(1),...,X(s) above. The missing candidates are
given equal ranking.
3.  In IRV: Ballots with k equally ranked first preferences are split into k
ballots with a weight 1/k the old weight of the ballot. Each of these
ballots have a different candidate of the k candidates as first preference
and the remaining k-1 candidates are  all second preferences. In a Condorcet
election, candidates with equal ranking 0.5 are given votes each in a
pairwise comparison.
4. If any seat is filled with X(1),..., X(s) then this seat is deemed vacant
and nominations may be re-opened for new elections.

The procedure should work for any IRV or Condorcet single or IRV-STV
multiple winner election.
For Condorcet-STV it I haven't though thought it through, but the treatment
should be analogous to single-winner Condorcet and IRV-STV.

The procedure is equivalent with the requirement that the voter to ranks all
candidates approved of (either using ">" or "=") and that she/he disapproves
all the other candidates. Here the approval of a candidate means that the
voter considers it is better that the candidate is elected than that nobody
is elected. A disapproved candidate is a candidate, where the voter
considers it better to have no-one elected than that candidate.

Example election 1: Kevin Venzke's example:
35 A>B
25 B
40 C

The ballots are thus completed as follows:
35:A>B>X>C
25:B>X>A=C
40:C>X>A=B
In a Condorcet election B wins (as Kevin Venzke argued).
If candidate X wouldn't have been included, then A would win.
(see http://www1.cse.wustl.edu/~legrand/rbvote/calc.html)

In an IRV election, the ballots are further processed as follows:
35:   A>B>X>C
12.5:B>X>A>C
12.5:B>X>C>A
20:  C>X>A>B
20:  C>X>B>A
C wins.
(see http://www1.cse.wustl.edu/~legrand/rbvote/calc.html with 10x the number
of ballots to eliminate the fractional votes)

Example 2:
41:A
45:B
14:Blank votes

The ballots are processed as follows for a Condorcet election:
41:A>X>B
45:B>X>A
14:X>A=B
X wins.

For an IRV election we get:
41:A>X>B
45:B>X>A
7:X>B>A
7:X>A>B

The idea of the generalized symmetric ballot completion was inspired
by: http://www.mvcwestsussex.org.uk/glossary.html#Re-open
nominations

The treatment above is different from the one previously proposed in my
email below, as it allows the voters to elect a candidate, if he/she get
more than 50% explicit preferences on the ballot in a single-winner
election.

In a multiple winner IRV-STV election, the seat is left vacant, if candidate
X gets a Droop quota of votes.
For STV-Condorcet I am not sure.

Woodall's plurality criterion is not violated as X is not an elected
candidate.
Thus I retract all I previously wrote about the invalidity of Woodall's
plurality criterion :o)

Political motivation why blank votes are appropriate:
A blank vote basically says, that "I'd rather have no candidate elected than
any of these".

Imagine you have two-seat election with a ballot with "Kim Jong Il" and "Pol
Pot" as the only candidates.
Electing any one of them to any seat would be worse than not electing any
one of them (in fact it might mean the end of democracy and the beginning of
atrocities).
If no-one is elected, nominations might be re-opened and new candidates
might appear which are acceptable.

Now say we have a two-seat election with "Kim Jong Il", "Pol Pot" and
"Mahatma Gandhi".
Suppose we like Gandhi to be seated but neither of the two other ones.
Thus we bullet-vote "Mahatma Gandhi" and hope that the other seat will
remain vacant, so that new elections can be held and nominations re-opened
in order to avoid genocides and other nasty stuff.

When I run the following Condorcet single winner election without the blank
vote option:
1:Pol Pot
2:Kim Jong Il
9997:Blank,
then Kim Jong Il wins (see http://condorcet.ericgorr.net/ - replace "Blank"
with nothing, and http://www1.cse.wustl.edu/~legrand/rbvote/calc.html -
replace "Blank" with A=B).
Thus Condorcet elections in their current form do not support blank votes
and can lead to very unfavourable election results.

Thus I think the blank vote is quite an important and useful institution,
which should be implemented for Condorcet elections.
Equal preferences are good for building coalitions.

In Kevin's example above, B won, because more than 50% of the voters ranked
B above X, i.e. thought that it is better that B is elected than nobody,
which is an argument, that my email below lacked (Fri, May 27, 2011 at 8:49
PM).
I.e. in a single candidate election between B and X, B would win.
The solution above is different from the more strict solution in my email
below, which required that a candidate needs >50% in a pairwise comparison
to score a win and this without any "symmetrical completion" for unranked
candidates.
That solution could lead to no candidate being elected, even if the majority
would think that it is better to elect some candidate than none.

I think, that for now, I would prefer generalized symmetric completion as
proposed in this email.
Maybe this solution has already been proposed for Condorcet elections, I
don't know.
In any case I think it could be useful.
Comments are welcome.

Best regards
Peter Zborník

On Fri, May 27, 2011 at 8:49 PM, Peter Zbornik <pzbornik at gmail.com> wrote:

> Hi Kevin,
>
> I am sending you a small hopefully clarifying P.S. to my email below.
> 1] My appologies for some unfinished sentences, please disregard them
>
> 2] In my email below I state three things:
>
> a] Giving explicitly equally ranked candidates 0.5 votes each in their
> pairwise comparison, but not for unranked candidates does not violate
> Woodalls's plurality criterion in a condorcet election where winning
> votes are used. Woodall's plurality criterion is only violated if symetrical
> completion is used and the previously unranked candidates are given 0.5
> votes each.
>
> b] In order to preserve the power of the blank vote to prevent a candidate
> from being elected in a Condorcet election, it is necessary to introduce a
> blank vote criterion (static quota or absolute majority criterion),
> which entails the following rule: in a Condorcet single winner election a
> candidate wins a pairwise comparison only if he/she gets a majority of the
> total votes cast (including blank votes).
>
> c] combinations of the rules
> (i) Rule b] can be combined with rule a] (winning votes) without violating
> Woodall's plurality criterion.
> (ii) Rule b] can be applied for winning rules: margins and quotas similarly
> as for case (i) above
> (iii) With losing votes I am not sure (haven't studied this criterion), but
> I guess there should be a natural extension along the principles in (i)
> above.
>
> I hope this makes the email below somewhat clearer.
>
> Best regards
> Peter Zbornik
>
> On Fri, May 27, 2011 at 7:35 PM, Peter Zbornik <pzbornik at gmail.com> wrote:
>
>> Hi Kevin,
>>
>> I think an additional rule, "the absolute majority rule" is needed in
>> Condorcet elections in order to preserve the power of a blank vote to block
>> the election of a candidate and force new elections.
>>
>> This rule might be used "on top" of the winning votes rule and would
>> require a candidate to get that more than 50% of the votes cast in order to
>> get a win in a pairwise comparison in a Condorcet election.
>>
>> This extra rule, if combined with winning votes, would
>> not violate Woodal's plurality criterion furthermore it would obey what I
>> call the
>>
>> I think that the discussion so far was confounded by requiring equal
>> treatment to two different "phenonema": (i) incomplete ballots (or partially
>> blank votes as I would like to call them) and (ii) equally ranked
>> candidates, where the ranking is explicitly made on the ballots.
>>
>> This leads to equal treatment between unranked candidates and explicitly
>> ranked candidates with equal ranking.
>>
>> I propose different rules for:
>> (i) unranked candidates on partially blank votes (incomplete ballots).
>> (ii) equally ranked candidates, where the ranking is explicitly made on
>> the ballot
>>
>> Rule (i) above applies for candidates not given any ranking on the ballot.
>>
>> Rule (ii), gives two explicitly equally ranked candidates 0.5 points each
>> in a pairwise comparison.
>> Rule (i) however a new winning rule (or maybe it has been proposed before)
>> in order to preserve the majority criterion in condorcet elections with
>> partially blank votes (incomplete ballots).
>>
>>  If we have the election
>> 30 A>B
>> 40 B>A
>> 30 Blank,
>> then in a condorcet election (Schulze<http://en.wikipedia.org/wiki/Schulze_method#Ballot>) B
>> is elected, while in a majority election requiring 50 percent of the votes
>> cast, no candidate is elected.
>>
>> Thus Schulze elections
>>
>> We can compare this situation with voting, where you can vote "yes", "no"
>> and "abstain", in order to get the vote passed 50% of the votes cast are
>> required to be "yes" votes.
>> It might thus be appropriate to retain this blocking property of the
>> abstention (or blank) vote for Condorcet elections.
>>
>> I think that the so far proposed winning criteria do not allow for
>> abstention voting in condorcet elections.
>>
>> I guess that the only way to retain the expressive power of the blank
>> vote, is through adding an additional rule for when a pairwise comparison
>> to qualifies as a win.
>>
>> This rule would state that a pairwise comparison results in a win only if
>> the candidate gets more than 50% of all votes cast in the election.
>>
>> Thus in the election
>>  40 A>B
>> 30 B>A
>> 40 Blank
>> A vs B would end 30% vs 40%.
>> No candiate would win.
>> New elections would be held.
>> Maybe this rule could be called the "absolute majority" rule for instance
>> (or whatever).
>>
>>  I.e. winning votes, losing votes, ratio and margins do not respect the
>> something we might call the "blank vote criterion" or the "static quota
>> criterion", which for single winner elections states that: a candidate can
>> win a two-candidate election only if he/she is preferred by a majority of
>> the voters".
>>
>> The general case of the "blank vote criterion" or the "static quota
>> criterion" would read: a candidate can win a multiple member election only
>> if he/she is preferred by a static quota number of the voters" (the quota
>> used can be Droop, Hare, etc.).
>>
>> However, in order not to penalize explicit equal rankings on the ballot by
>> giving both equally ranked candidates 0 wins thus making it more difficult
>> for these candidates to meet the "static quota criterion", separate
>> treatment is needed for explicit equal rankings and for candidates left out
>> of the ballot, in the same way as I propose these two cases to be separately
>> treated in an IRV-STV election.
>>
>> Thus we need to add two new rules to a Condorcet election.
>>
>> *The generalized symmetric completion rule for condorcet elections:*
>> *Equal rankings explicitly made on the ballot are counted as 0.5 win for
>> each candidate.*
>>
>>  Any candidate left out from the ballot is counted as ranked lower than
>> all candidates explicitly ranked on the ballot. This rule is currently
>> implemented for Schulze, so I just state it for completenes.
>>
>> *Absolute majority rule: a pairwise comparison between two
>> candidates results in a win only if more than 50% of the total votes cast
>> are in favour of any candidate.*
>> **
>> The absolute majority rule might thus lead to the case where there is no
>> winner of the election.
>>
>> In that case a new election might be held, or the voters can go home.
>>
>> It seems most natural to combine the absolute majority rule with winning
>> votes, but in theory it might maybe be combined with any other rule
>> (margins, ratios, losing votes). I have no firm oppinion on this.
>>
>> Turning to your example to apply these new rules:
>>  35 A>B
>> 25 B
>> 40 C
>>
>> Let us first count the votes cast.
>>  Total votes cast are 100 with the following matrix:
>> X   A    B   C
>> A   X    35  35
>> B   25    X  60
>> C   40  40  X
>> We only count as a win >50% of the votes casts.
>> Thus the election results in no candidate being elected as no candidate
>> scores a win against both the other candidates.
>>
>> To see the similarity with the blank vote cast, let us imagine, that a
>> second round of the election is held.
>> C has no chance of winning, as B beats C with more than 50% of the votes.
>> Thus let's assume that only A and B go through to the second round
>> and that the voters keep their preferences from the first election intact.
>>
>> Then we get the following result.
>> 35 A>B
>> 25 B, which is completed to B>A using the current rules of Schulze.
>> 40 Blank votes (these voted for C before)
>>
>> Thus we get A vs B: 35% vs 25%.
>> No candidate is elected, as no one got more than 50% of the votes.
>> Thus the blank vote criterion is not violated.
>>
>> This procedure allows the voters to find a candidate, who has better
>> support in the electorate.
>> Of course it also allows for "sabotaging" elections.
>> In the example above C's voters can prevent any candidate from being
>> elected.
>> However, that is exactly how elections are done in our party today, and
>> the blank votes are thus respected.
>>
>> I guess that the extention of the approach above to Condorcet-STV is a
>> rather trivial excersise (static quotas used), but I haven't looked at that
>> case.
>>
>>  Maybe the "blank vote" criterion above somehow "crashes" the
>> Condorcet method, I don't know, even though I hope it doesn't.
>>
>> Woodalls plurality criterion (a retraction):
>> The criterion reads: *If the number of ballots ranking A as the first
>> preference is greater than the number of ballots on which another candidate
>> B is given any preference, then A's probability of winning must be no less
>> than B's.*
>>  http://en.wikipedia.org/wiki/Plurality_criterion
>>
>> If the the method described above (the generalized symmetric
>> completion rule for condorcet elections and the absolute majority rule) is
>> used together with the winning votes rule, then Woodal's plurality criterion
>> is not violated.
>>
>> Thus I have to retract my statement "that Plurality is "a rather arbitary
>> property that surely mustn't hold in any real election", which I wrote in my
>> email to Kristofer today (Fri, May 27, 2011 at 12:28 PM).
>> That bold statement did not last a day even.
>>
>> A more correct statement is that "Plurality is a property that might not
>> lead to proportional representation in multiple-winner elections"
>>
>>  A short disambiguation:
>> With "biggest win", I meant "winning votes".
>> I think my calculation of the method "winning votes" using symmetrical
>> completion with 0.5 wins to each candidates in case of equal ranking was
>> correct, as I controlled it with the calculations on
>> http://www1.cse.wustl.edu/~legrand/rbvote/calc.html
>> I entered:
>>  35:A>B>C
>> 25:B
>> 40:C
>> and pressed the button "Schulze".
>>
>>  To sum up my argument so far:
>> 1] symmetrical completion is not a good way to process *incomplete
>> ballots (or partially blank votes)*, as it removes the possibility to
>> "protest" in the election.
>>
>> 2] Generalized symmetrical completion is a good way to process *equally
>> and explicitly ranked candidates* in an IRV-STV election, if the
>> algorithm is modified to "dissolve" only one equal sign at a time (i.e.
>> A=B=C is broken up to three ballots A>B=C, B>A=C, C>A=B).
>>
>> 3] Generalized symmetrical completion for Condorcet elections would give
>> each candidate 0.5 points in a Condorcet election only if both candidates
>> explicitly were equally ranked on the ballot.
>>
>> 4] My preferred way to handle incomplete ballots in IRV-STV for now, is
>> through using static quotas and no ballot completion as it retains the power
>> of the blank vote to block elections.
>>
>> 5] Absolute majority is proposed as an additional winning rule for
>> Condorcet elections which retains the power of the blank vote to block
>> elections. The rule requires the candidate to get more than 50% of the votes
>> cast in order to get a win in a pairwise comparison. Thus a plurality of the
>> votes is not enough to qualify for a win. This rule does not violate
>> something I call the "blank vote criterion", i.e. partially blank votes have
>> the power to block the election of a candidate.
>>
>> 6] The extention of the absolute majority rule to Condorcet-STV elections
>> seems to be trivial if static quotas are used.
>>
>> Best regards
>> Peter Zborník
>>
>> On Fri, May 27, 2011 at 4:59 PM, Kevin Venzke <stepjak at yahoo.fr> wrote:
>>
>>>   Hi Peter,
>>>
>>> Let me say first of all that proportional representation isn't my area of
>>> interest, so you
>>> shouldn't take anything I say to apply also to a PR situation.
>>>
>>> And although STV has a single-winner case, my thoughts on equal ranking
>>> don't apply
>>>
>>> there either.
>>>
>>>
>>>
>>>
>>>
>>> --- En date de : *Ven 27.5.11, Peter Zbornik <pzbornik at gmail.com>* a
>>> écrit :
>>>
>>>
>>> [end quote]
>>>
>>> I think you forgot Schulze as it is usually done: Weakest biggest loss.
>>>
>>>
>>> With "weakest biggest loss", do you mean losing votes (
>>> http://m-schulze.webhop.net/, page 7)?
>>>
>>>
>>>
>>> No I mean "winning votes" on that page. Is that what you meant by
>>> "biggest win"?
>>> I can't really see how those could be the same thing.
>>>
>>>
>>>
>>>
>>> Experimentally, in simulations: When you treat equal-ranking as split
>>> votes, voters will have to compromise more often, instead of just
>>> compressing the top ranks. This suggests weaker, non-frontrunner
>>> candidates are more likely to be best advised to drop out of the race,
>>> because their presence is more likely to harm the voters that support
>>> them.
>>>
>>>
>>> Could you please send me a link to these simulations?
>>>
>>>
>>>
>>> There is no complete set of simulations currently/yet. If you want to get
>>> a sense of
>>> what I was doing, you can go to the archives:
>>> http://lists.electorama.com/pipermail/election-methods-electorama.com/
>>> and read my March 2011 posts in particular. My simulations involve
>>> voters who do not
>>> initially know anything about the method except the valid ballot
>>> types, but try to
>>> determine their ideal vote in a given situation via repeated and
>>> hypothetical polling.
>>>
>>> I have explained (probably five years ago) why we should expect margins
>>> to have more
>>> favorite betrayal incentive than WV though. Suppose that you want to vote
>>> A>B, but
>>> so doing causes C to win instead of B, because A defeats B pairwise. In
>>> WV both
>>> reversing the order to be B>A or compressing the top to be A=B have the
>>> same effect
>>> in reducing the magnitude of B's loss to A. But in margins reversal is
>>> twice as effective
>>> as compression.
>>>
>>> Kevin Venzke
>>>
>>>
>>>
>>
>
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