[I've been working with random elections to examine how well various voting mechanisms conform to monotonicity.]
"Hybrid theory" violates monotonicity if we consider the default option as a candidate: Using the "hybrid theory" proposal, the j wins the election where a and b require 3:1 majority and j is the default option, and the votes are: ba bc ca jb If the "ca" vote is changed to "ac", c wins. If we drop a and b before calculating the schwartz set (since in both cases neither satisfy supermajority), c wins both elections. I've not been able to prove, to my satisfaction, that "drop options which don't satisfy supermajority" satisfies monotonicity, but after simulating over a million elections I have not been able to find any cases where it fails to satisfy monotonicity. [As an aside: the ballots I'm testing with are quite a bit more complicated than this example -- problems tend to show up more often on complicated ballots. I just went with a simple example for presentation purposes.] FYI, -- Raul
