[EM] Majority Enhanced Approval generalized

[C.Benham]

> "ME(et-eb) is chicken proof,..."

35 A>B
25 B
40 C

B>C>A>B    et-eb scores:  B-15 > C-20 > A-30

Forest's suggested ME(et-eb) method elects B, but the Chicken Dilemma 
criterion says that the winner must not be B.

http://wiki.electorama.com/wiki/Chicken_Dilemma_Criterion

Also this method can fail to elect a candidate that is voted below no 
other on more than half the ballots.

46 A>C
10 B>A
10 B>C
34 B=C

C>B>A>C   et-eb scores:  C24 > B8 > A2     100 ballots, e-t scores  B54 
 > A46 > C34

ME(et-eb) elects  C.

Also it fails Unburiable Mutual Dominant Third.

34 A>B
17 C>A
16 B>C
31 B
02 B>C (sincere is B or B>A)

B>C>A>B    et-eb scores: B32 > A-15 > C-48

A is the sincere MDT winner, but ME(et-eb)  easily elects the buriers' 
favourite B.


Chris Benham



On 5/10/2014 9:27 AM, Forest Simmons wrote:
> Here's how majority enhanced approval works:  It elects the approval 
> winner unless she is covered by some other candidate.  In that case 
> from among those that cover her it elects the one with the most 
> approval.  Unless she also is covered, in which case from among those 
> that cover her, it elects the one with the most approval, etc.
>
> Another fancier way to articulate this goes like this: First 
> initialize a list with the name of the approval winner. Then while at 
> least one candidate covers every candidate named on the list, from 
> among such candidates add to the list the one with the greatest 
> approval.  Elect the candidate whose name was added last.
>
> Obviously, the MEA winner is uncovered.  This means that to every 
> other candidate she has a short beat path, i.e. if she doesn't beat 
> him, she beats someone who does.  Since she has a beatpath to every 
> other candidate she is a member of Smith.
>
> We can majority enhance other kinds of methods that generate a social 
> order.  For example, we could list the candidates in order of max 
> pairwise opposition, initialize the list with the name of the 
> candidate with the best score, etc. While some candidate covers all 
> candidates listed, from among those covering candidates add to the 
> list the one with the best score, etc.
>
> Currently the score that I like best because of simplicity and other 
> properties is what I call et-eb, Equal Top minus Equal Bottom.
>
> A candidate's et-eb score is the difference in the number of ballots 
> on which she is ranked below no other candidate and the number of 
> ballots on which she is ranked above no other candidate.
>
> ME(et-eb) is chicken proof, monotone, clone proof, and elects an 
> uncovered candidate from Smith.  It satisfies Independence from Pareto 
> Dominated Alternatives and the Plurality criterion. It does all of 
> these things seamlessly from the et-eb order and the pairwise defeat 
> graph, which are easily assembled from a summable matrix..
>
> Here's how it works on Kevin's famous chicken example:
>
> 49 C
> 27 A>B
> 24 B
>
> The et-eb scores are C(49-51)>B(24-49)>A(27-73)
>
> Candidate C is elected because she has the best score and is uncovered 
> (because she has a short beatpath to each of the other candidates).
>
> Notice that when there are only three candidates in Smith, this method 
> always gives the same result as Smith//(et-eb), but is more seamless. 
> .  Furthermore (in the case of three candidates) the et-eb scores 
> yield the same order as the Borda scores, so in the case of three 
> candidates this method is equivalent to Black (provided Black allows 
> equal ranking and truncation)..
>
> With any number of candidates you can think of there being three 
> levels: equal bottom, equal top, and in between. The in between ranks 
> do not affect the score, but they do contribute to the pairwise 
> matrix, and thereby help determine the covering relation.
>
> Note that (by definition) candidate X covers candidate Y iff for each 
> candidate Z, whenever Y defeats Z, then so does X.
>
> So if Y is not covered by X, there is some Z that if beaten by Y but 
> not by X, which gives a short beatpath from Y to X, namely Y>Z>X .
>
> This short beatpath idea allows for an alternative definition of covering:
>
> Candidate X covers candidate Y iff there is no short beatpath from Y to X.
>
>
>
>
> ----
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