[EM] Swap Cost Approval Agenda

[Forest Simmons]

The Kendall-tau cost of reversing a permutation of n candidates is n(n-1)/2
transposition ... independent of the original permutation.

But our Swap Cost metric is more sensitive ... it does depend on the
original order ... which gives an objective standard for the approval of a
permutation of the candidates:

The approval of a candidate ranking R is the difference of the swap cost of
converting R to its reversed order R' to the swap cost of converting it
back.

Let's compare the cost of converting AB to BA, with the cost of converting
it back:

 cost(AB to BA)/cost(BA to AB) is ...

ab'/(ba') = (a/a')/(b/b'), ...

which tells us that when (a/a')>(b/b') it's more costly to move A down and
B up than vice-versa.

So our Swap Cost Approval Agenda Order is according to the size of
f(X)/f'(X), where f and f' are the probability density functions of the
random ballot favorite  and  anti-favorite lotteries, respectively.

Example:

48 C>A=B
28 A>B>C
24 B>A=C

Counting Bottom fractionally or using symmetric completion ... we get

a/a'=28/40
b/b'=24/24
c>c'=48/36

So from least favorable to most favorable the agenda order is A<B<C.

Our baby-step agenda method says to elect the most favorable agenda item
that defeats the least favorable agenda item:

Since A is least favorable, and C beats A, we elect C.

Example2

a ABC
b BCA
c CAB

Assume c<b<a  so that  c/c' =c/a, etc..

Then the candidate that beats C is the winner ... namely B.

In sum ... our simple method is to elect    the candidate with the largest
first to last ratio f(X)/f'(X), that defeats the candidate with the
smallest such ratio (when no undefeated candidate exists).

It seems to me that this method is so simple and effective, that there is
really no good excuse for imposing a complicated, flawed method like IRV
onto the public.

-Forest
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