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.

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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