[EM] Immunization against Second Place Complaints

[Simmons, Forest]

MinMax Pairwise Opposition (MMPO) is an example of a method that satisfies the FBC, but fails Smith.  If we could immunize it against 2nd place complaints, then the resulting immunized method I(MMPO) would satisfy Smith, and therefore would not be as clone dependent as MMPO.

Here's an attempt at this:

1. For each alternative (i.e. candidate) X let X' denote the alternative that would (by this method applied recursively) win the election if alternative X were withdrawn from consideration.

2. For each alternative X, if X is beaten pairwise by X', disqualify X.

3. If step two would disqualify all alternatives, then disqualify none.
 
4. Use MMPO to choose an alternative from among the undisqualified alternatives.

This immunizatin attempt will be successful if and only if step three turns out to be unnecessary.
 
In other words, the method I(MMPO) will be immune from second place complaints as long as there is always at least one disqualified alternaive to choose from.  But if there is a case in which all alternatives are disqualified, then any winner would be beaten by the second place alternative.
 
So the big question: is there always some X that is not beaten by X' ?
 
Perhaps this could be proven by induction.
 
Note that there are two ways to interpret step four.   One (in the spirit of irrelevant alternatives) is to completely remove the disqualified alternatives and only consider pairwise opposition from the rest of the alternatives.  But I prefer considering their pairwise opposition as well, since this information helps reveal the relative defensive strength of the remaining alternatives.  If a disqualified alternative scores highly against some qualified alternative X, then a weakness in X has been revealed that might have been hidden otherwise.
 
If this attempt works for MMPO, perhaps there might be some other methods worth immunizing as well.
 
Forest
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