# [EM] Majority Enhanced Approval generalized

Forest Simmons fsimmons at pcc.edu
Fri May 9 16:57:20 PDT 2014

```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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