Subcycles and the Rich Party Problem

[Steve Eppley]

In recent messages, the "rich party problem" (which led to the
development of my Independence from Twins criterion and a
near-identical criterion defined by Mike O.) was described. 
Several methods we've been considering, such as Regular-Champion,
fail these criteria.

In other recent messages, Mike O. described a modification called
the "subcycle rule" which can be made to pairwise methods.  This
modification affects their properties.  I'll try to restate it from
memory (let me know if I screw it up):  

  The subcycle rule(?):
  A cycle is any set of two or more candidates such that none of
  them was defeated pairwise by all the others in the set.  A subcycle
  is a cycle which is also a subset of a larger cycle.  Before deter-
  mining the winner of any cycle, every subcycle (if there are any) of
  that cycle must be replaced by the winner of that subcycle (elimin-
  ating all the pairings which involve the losers of that subcycle).  
  Note: This rule is recursive.

My question: Can the subcycle rule be used to "fix" methods which 
would otherwise fail the rich party criteria?

In the example I provided to demonstrate Regular-Champion's violation
of Independence from Twins, the Blue twins are a two-candidate
subcycle and the Red twins are a two-candidate subcycle.  When these
subcycles are each replaced by just one Blue and one Red, the extra
pairdefeats of all the other candidates to the eliminated twins get
discounted.  So that example ends with a three-way Copeland tie
broken finally by Plurality (most first-ranks). 

---Steve     (Steve Eppley    seppley at alumni.caltech.edu)