Does anyone know whether the theorem that all elimination methods suffer failure of monotonicity applies to one-at-a-time elimination only, or would include any kind of elimination? I am, in particular, interested in knowing whether bifurcation can be used to find candidates' rankings. I have proposed median elimination as one method, which would eliminate all candidates whose median rank is not in the top half (this could alternatively be seen as grouping candidates, if a complete ranking was desired -- then each grouping could be further median-cut, until a complete ranking was determined). A similar method could be used for evaluating ranked pairs: It seems clear that the top contenders in RP are going to be the ones with victories against the most opponents, so a grouping could be done based on that: all candidates who beat three candidates are in one group, those who beat two are in a separate, lower-ranked group, etc. Then, within each group, the contests that include candidates not in the group are discarded, and the candidates are re-grouped based on victories against each other. In this situation, it seems clear to me that lowering a candidate's position, if it changes anything, can only result in his moving into a lower group, or that that elevated candidate moves into a higher group, both of which would "eliminate" the lowered candidate. Thus, no failure of monotonicity.
