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

[Forest Simmons]

> From: Jobst Heitzig <heitzig-j at web.de>
> Subject: [EM] Re: Definite Majority Choice, AWP, AM
>
> 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.


This is a very nice proof, and another interesting and valuable 
characterization of DMC:

When defeat strength is measured by the approval of the defeating 
candidate, there is only one possible immune method, namely DMC.

All of the main competing Condorcet methods collapse into simple little 
old DMC by the device of measuring defeat strength by approval.

And measuring defeat strength by approval in no way decreases any of the 
strategy resistance or other nice properties of the winning votes versions 
of those methods.

Also, it's very nice to have the great variety of other descriptions and 
characterizations of this method that we have seen lately.

Watch out IRV !

The only thing that worries me is this: what if DMC gets adopted all over 
the place, and it turns out that Donald gets the credit because he proves 
that he came up with an equivalent version before we did?

I don't really waste any time worrying about such things, but wouldn't 
that be irony in the extreme?

[I don't care. Let him have the credit. It would be worth it!]

Forest