[EM] Immunity from Second Place Complaints implies Smith

[Simmons, Forest]

This result is probably known, but since I plan on using it in another post to the EM list, I put it here for reference.
 
If a method is immune to second place complaints (I2PC), then it elects winners from Smith.
 
Proof sketch (by induction on the numbert of candidates, and ignoring the possibility of pairwise ties):
 
1. If there are fewer than two candidates there is nothing to prove.
 
2. If there are just two candidates, then I2PC means that the winner beats the only other candidate, and so is in the Smith set.
 
N.  If there are N candidates, then (by induction) withdrawal of W gives the win to a member W' of the Smith set of the reduced election, and so this W' has beatpaths to all other candidates (except possibly W).   But by I2PC, W' beats W, so W has a beatpath to every candidate (except self), and so W is a member of Smith.
 
So  I2PC  is stronger than the Smith Criterion.
 
However, we have to realize that this is "Ballot Smith" that we are talking about.  In particular, Approval satisfies I2PC, in the sense that the second place winner has less approval than the winner.  The above argument only implies that Approval satisfies Ballot Smith.  In fact (except in the case of ties) Approval always elects the ballot CW.
 
Note that Approval satisfies the FBC and this I2PC property, as well.  Is there a way to "immunize" other FBC methods like MMPO to give them the I2PC property, and hence Ballot Smith?
 
FWS
 
 
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