[EM] Generalized symmetric ballot completion

[Peter Zbornik]

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.

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.
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.

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).

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.

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.

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.

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
>
>
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