As soon as I sent my message I realized that my conjecture was mostly wrong. As a reminder, I conjectured that
> Conjecture: Suppose that in an election with 3 or more candidate and > Ordinacy Approval Voting the CW is elected with a majority, and at > least one other candidate is also approved by a majority. If voters > cast identical ballots with MCA, except that some voters indicate > distinctions among those whom they approved, AND ALL VOTERS WHO > APPROVED BOTH THE CW AND THE OTHER CANDIDATE(S) APPROVED BY THE > MAJORITY DISTINGUISH AMONG THEM WITH THE PREFERRED AND ACCEPTABLE > RATINGS then the CW will still win. This is wrong. However, we can modify it: Theorem: Suppose that in an Ordinary Approval election the CW is elected with the approval of the majority, and that the set M of candidates approved by the majority includes at least one other (losing) candidate. Suppose also that we change election methods and use MCA, and all voters rate as Preferred or Acceptable those whom they approved previously. If all voters who approved at least one candidate in the set M now rate as Preferred no more than one candidate in the set M then the CW will still be elected. Proof: The only candidates who might be Preferred by a majority are those in the set M. If each voter Prefers no more than one of them, then only one of them can be Preferred by a majority, and that candidate must be the CW, since only a minority of the electorate will Prefer a given candidate in M over the CW. If no candidate is Preferred by a majority of the electorate then the Acceptable votes will also be counted, and the result will be the same as in the election held with Ordinary Approval. The one question remaining to me is what happens if in general some voters are indifferent between candidates in M and rate as Preferred either their favorite candidate in M or those tied for favorite (and, of course, also those candidates not in M whom they rank higher than those in M). In the case of exact clones (whom all voters agree are identical) nothing changes. Otherwise I'm not so sure. Input from others? Alex ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
