[EM] Two Theorems: Please criticize or point out errors

[Alex Small]

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


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