[EM] Variable Inferred Approval Sorted Margins Elimination

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[EM] Variable Inferred Approval Sorted Margins Elimination

C.Benham

Ted,

I don't see the two methods (VIASME and Smith/IBIFA ) as being in competition with each other because they use
two very different types of ballot and and VIASME is probably much harder to explain and sell.

Smith/<something IBI> at least has the benefit of satisfying later-no-help ...

I'm afraid not.  IBIFA fails Later-No-Help because adding a lower (or "later") preference (i.e. rating another candidate X
above Bottom) can trigger another (say a second) round that is won by a candidate (not X) you prefer to the one (also not X)
who would have otherwise won (say in the first round).

I thought of a possible kludge to try and fix that but it makes the method much more complicated and and less Condorcet
efficient.

*(Say we are using 3-slot IBIFA.) We consider the IBIFA winner A to be provisional. Then we truncate all the ballots below A
and if A is still the IBIFA winner we elect A.

But if instead there is a new IBIFA winner B, we un-truncate the ballots below A and truncate below B and if B is still the
IBIFA winner then we elect B.

But if instead there is a new IBIFA winner C then repeat the process. If we run out of candidates or a previous provisional
winner appears, then we simply elect the most approved candidate.*

A very ugly answer to a question no-one was asking, and I'm not even completely sure it works. Median Ratings methods
(such as Bucklin and MJ) do meet Later-no-Help.  Arguably it is desirable that Later-no-Help and Later-no-Harm should
either both be met (like IRV) or both failed  (like IBIFA and Condorcet methods). Otherwise you either get a random-fill
incentive (yuck) or a very strong truncation (or only use the top and bottom rating slots) incentive.

And complying with Later-no-Help is one of the properties that Woodall has proved is incompatible with Condorcet, so
"Smith/ anything" can't meet it.  The other criterion compliances in the same boat are Later-no-Harm, Particpation,
Mono-raise-random, Mono-raise-delete, Mono-sub-plump, Mono-sub-top.

http://groups.yahoo.com/group/election-methods-list/files/wood1996.pdf

Election 6:  
bca   3    
bac   2             
cab   3    
cba   2
abc   3    
acb   2

Theorem 2 says that if an election rule satisfies Condorcet's principle, then it cannot possess any of the seven properties that are crossed in the column headed 2 in Table 1.
This is a lot to prove. Fortunately most of it can be proved by considering variants of Election 6 above. The only bit that cannot is the incompatibility of Condorcet with
participation; this is proved by Moulin2, and I shall not attempt to reproduce his proof here. The following proof of the rest of Theorem 2 invokes the axioms of symmetry
and discrimination, for a precise statement of which see Woodall4.

So suppose we have an election rule that satisfies Condorcet. By symmetry, the result of this rule applied to Election 6 above must be a 3-way tie. But by the axiom of
discrimination, there must be a profile P very close to the one in Election 6 (in terms of the proportions of ballots of each type) that does not yield a tie. So our election rule,
applied to profile P, elects one candidate unambiguously; and there is no loss of generality in supposing that this candidate is a. However, there are ways of modifying the
profile P so that c becomes the Condorcet winner, so that our election rule must then elect c instead of a. This happens, for example, if all the bac ballots are replaced by a;
and the fact that this causes c to be elected instead of a means that our election rule does not satisfy mono-raise-random, mono-raise-delete, mono-sub-top or mono-sub-plump.
It also happens if all the abc ballots are replaced by a, and this shows that our election rule does not satisfy later-no-help.

To prove that our election rule does not satisfy later-no-harm, it is necessary to consider a slight modification of the profile in Election 6, in which the second and third choices
are deleted from all the abc, bca and cab ballots. Again, our election rule, applied to this profile, must result in a 3-way tie. But again, there must be a profile P' very close to this
(in terms of the proportions of ballots of each type) that does not give rise to a tie, and we may suppose that our election rule elects a when applied to profile P'. But if we replace
the a ballots in P' by abc, then b becomes the Condorcet winner, and so must be elected by Condorcet's principle; and this shows that our election rule does not satisfy later-no-harm.
Together with the result of Moulin2 already mentioned, this completes the proof of Theorem 2, that an election rule that satisfies Condorcet cannot satisfy any of the seven properties
crossed in the column headed 2 in Table 1.


Chris Benham


On 20/06/2019 5:20 am, Ted Stern wrote:
Just as I'm warming up to Smith/Relevant-Ratings (or Smith/IBIFA), you introduce another method. :-)

This seems to be in the same vein as MinLV(erw)SME.

I like the general idea, but would prefer to avoid doing multiple tabulations as that makes the method not precinct summable.

Smith/<something IBI> at least has the benefit of satisfying later-no-help and mono-raise without requiring multiple passes through the ballots. 

On Wed, Jun 19, 2019 at 10:59 AM C.Benham <[hidden email]> wrote:

This is my favourite Condorcet method that uses high-intensity Score ballots (say 0-100):

*Voters fill out high-intensity Score ballots (say 0-100) with many more available distinct scores
(or rating slots) than there are candidates. Default score is zero.

1. Inferring ranking from scores, if there is a pairwise beats-all candidate that candidate wins.

2. Otherwise infer approval from score by interpreting each ballot as showing approval for the
candidates it scores above the average (mean) of the scores it gives.
Then use Approval Sorted Margins to order the candidates and eliminate the lowest-ordered
candidate.

3. Among remaining candidates, ignoring eliminated candidates, repeat steps 1 and 2 until
there is a winner.*

To save time we can start by eliminating all the non-members of the Smith set and stop when
we have ordered the last 3 candidates and then elect the highest-ordered one.

https://electowiki.org/wiki/Approval_Sorted_Margins

In simple 3-candidate case this is the same as Approval Sorted Margins where the voters signal
their approval cut-offs  just by having a large gap in the scores they give.

That method fulfils Forest's recent 3-candidate, 3-groups of voters scenarios requirements, resists Burial
relatively well and meets mono-raise. The motivation behind this version is to minimise any disadvantage
held by naive (and/or uninformed) sincere voters.

Chris Benham

Forest Simmons [hidden email]
Thu May 30

In the example profiles below 100 = P+Q+R, and  50>P>Q>R>0. 

I am interested in simple methods that always ...

(1) elect candidate A given the following profile:
P: A
Q: B>>C
R: C,

and
(2) elect candidate C given
P: A
Q: B>C>>
R: C,

and
(3) elect candidate B given
P: A
Q: B>>C  (or B>C)
R: C>>B. (or C>B)





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