[EM] Least Resentment Voting (LRV) in the context of Approval

[fsimmons at pcc.edu]

If I remember correctly Kevin Venzke's first post to this list was a geometric argument that 
the MMPO winner was apt to be closer to the voter median position in Approval than the 
Approval winner.  The scenario he had in mind was something like this

Scenario One:

26 A
24 A=C
24 B=C
26 B

The geometry was something like this:

AAAAAAAAAAAAAAAAAAAAAAAAABBBBBBBBBBBBBBBBBBBBBBBBB
_________________CCCCCCCCCCCCCCCCCCCCCCCC________________

It weemed pretty obvious that the MMPO winner C was more likely the true majority choice 
than than either of the tied approval winners A.or B.

Not long after that Kevin came up with his MMPO bad example:

Senario Two:

48 A
2 A=C
2 B=C
48 B.

The MMPO winner C lacked too much approval to be the real winner. Kevin took back his 
proposal, and went on the bigger and better things like trying to convince Juho that MinMax
(margins) was not an acceptable public proposal. either.

But as I mentioned in a recent posting MMPO with symmetric completion is the same as 
MinMax(margins), and by exempting the top rank or slot from the symmetric completion, we 
get a really nice compromise between MMPO and MinMax(margins), and what's more this 
version gives the "right" answer in both scenarios above.  Kevin was ever so close to 
proposing what I now call LRV "Least Resentment Voting" or more technically Bottom-
Symmetric-Completion MMPO.

In the first scenario above both A and B have max opposition of fifty, while the max opposition 
of C is 26 + 13 = 39, so C wins.

In the second scenario, A and B still have max opposition of 50 each, but now C's max 
opposition is  48 + 24 = 72, so C loses, to a tie betwen A and B.

Where is the cutoff where all three are tied?

Scenario Three:

34 A
17 A=C
17 B=C
34 B

In this scenario the max opposition (with the bottom symmetric completion rule in effect) is 
34+17 for A, 34+17 for B, and 34 +17 for C.

The second largest opposition to either A or B is 17+17=34, while the second largest 
opposition to C is still 34+17, so C loses the tied situation , and the winner is a toss up 
between A and B.