[EM] two new hybrid approval-ranking methods by Brams & Sanver

[Warren Smith]

On Brams' web page
http://www.nyu.edu/gsas/dept/politics/faculty/brams/brams_home.html
 is found
Steven J. Brams & M. Remzi Sanver:
Voting Systems That Combine Approval and Preference,
Dec 2005.

This paper introduces two new hybrid approval-preference voting systems.

I. Preference-Approval-Voting:
0. The votes are full strict rank orderings
(no ballot truncation or equality-rankings allowed).
Also voters "approve" or "disapprove" or each candidate, in a manner that
has to be compatible with the rank order.
It is not illegal to approve everybody.
It is not illegal to disapprove everybody.

1. If no candidate, or exactly one candidate, 
receives a majority of approval votes, 
then the PAV winner is just the AV winner (i.e most-approved candidate).

2.  If two or more candidates receive a majority of approval votes, then  
(i)  If one of these candidates is preferred by a majority to every other majority-approved
     candidate, then he is the PAV winner (even if not the AV or Condorcet winner).   
(ii) If there is a cycle among the majority-approved candidates, then the
     AV winner among them is the PAV winner (even if not the AV or Condorcet winner).   


II. Fallback-Voting:
0. Votes are strict rankings of a subset of the candidates - ranked candidates 
are automatically "approved".

1. If a majority of voters top-rank a candidate, he wins.
We call this candidate a level-1 winner.      

2.  If there is no level 1 winner, the next-highest ranked candidate of all voters is considered.
If a majority of voters rank a candidate as either their top or their second-top,
then he wins.  (Call this a level-2 winner.)

3. If there is no level-2 winner, the voters descend one level at a time,
until, for the first time, one or more candidates are approved of by a
majority of voters, or no more candidates are ranked.  
If  exactly one candidate receives majority approval, this candidate is the FV winner.  
If more than one candidate receives majority approval, then the candidate with the largest  
majority is the FV winner.  If the descent reaches the lowest rank of all voters and no 
candidate is approved of by a majority of voters, the candidate with the most approval is
the FV winner. 


WDS comments:
It appears that "fallback voting" is a
rediscovery of "Bucklin voting" which was used to elect governors in several US states,
with ballot-truncation being allowed.
See  http://wiki.electorama.com/wiki/ER-Bucklin 
     http://wiki.electorama.com/wiki/Bucklin_voting
Oddly, Brams & Sanver were aware of Bucklin but seem to think their method is different.
I don't really see why.  Also strangely, B&S do not consider ER-Bucklin, which is
quite an interesting system since it avoids "favorite betrayal."  

P.A.V. disobeys favorite-betrayal criterion and add-top
(since if lots of candidates are approved,
then it just is a Condorcet method).

wds