[EM] Mutual majority set methods

[Kristofer Munsterhjelm]

Leon Smith wrote:
> How does the mutual majority set differ from either the Smith Set or
> the Schwartz Set?  Can't you just sum the "a beats b" half-matrices
> and then run a strongly connected component algorithm on the result?
> Kosaraju's algorithm produces the SCCs in a topological order,  so
> you don't even need serious processing on the output,   just take the
> first or last SCC depending on how it's implemented.
> 
> But I'm guessing you already know this,  so I'm guessing there is
> something I'm missing.

The Smith set differs from the mutual majority set in a manner similar 
to how the Condorcet winner differs from a majority winner. For 
instance, if you have something like:

49: A > B > C > D
  2: D > B > C > A
49: C > B > D > A

then the Smith set consists of B (since B is the CW), but the mutual 
majority set is {C, B, D}. Every time there's a CW that isn't the 
majority favorite, the sets differ.

I think the Smith set is a subset of the mutual majority set, though, so 
there is *some* similarity. Each candidate in the MM set will beat each 
candidate outside it by a majority, and thus the Smith set can be no 
larger than the mutual majority set. Unless I'm missing something, 
myself, that should mean that there is an n so that the union of the n 
first nested Smith sets is the mutual majority set.

(The nested Smith set would consist of the Smith set, then the Smith set 
were the first Smith set excluded, then the Smith set were the first two 
sets excluded, and so on.)