[EM] Re: Definite Majority Choice, AWP, AM

[Jobst Heitzig]

Dear Ted!

You wrote:
> Chris & Jobst: Please take careful note -- the DMC defeat strength
> assertion has not been proved rigorously, to my knowledge!  It is
> worth a very careful look before basing any other assumptions on it.

The following proves that the only immune candidate is the least
approved not strongly defeated candidate, assuming no pairwise defeat or
approval ties:

Let A be that candidate, with approval a.

To prove that A is immune, assume that B1 defeats A, with approval b1.
We show that there is a beatpath A>...>B1 with all defeats at least as
strong as B1>A, that is, with all intermediate candidates having
approval at least b1. Because of a>b1, and since B1 does not defeat all
more approved ones, there is B2 with approval b2>b1 and B2>B1. If a>b2,
also B2 does not defeat all more approved ones, hence there is B3 with
approval b3>b2 and B3>B2, and so on until we find some Bk with approval
bk>=a and Bk>...>B1. Now either Bk=A or A>Bk, QED.

Now assume that B is a candidate other than A, with approval b. We show
that B is not immune. If b>a then the defeat A>B has strenght a but all
defeats against A have strength below a, hence all beatpaths B>...>A
have strength below A, so B is not immune. If, on the other hand, b<a,
then B is beaten by some C with approval c>b, but any defeat B>... has
strength b<c, hence any beatpath B>...>C has strength below c, so again
B is not immune. QED.

This proves that all immune methods, especially RP, River, Beatpath, are
equivalent to DMC when defeat strength := approval of defeating
candidate, and when no pairwise ties exist.

Yours, Jobst