[EM] Something Different

[Forest Simmons]

Our Friendly Method experiments have led in this direction:

Remember Bottom  Count Dsapproval?

The Bottom Count Disapproval of X ... BotCountD(X) is the number of ballots
on which X outranks no candidate.

The Bottom Count Anti-Favorite Lottery probabilities f'(X) are obtained
from the Bottom Count Disapproval scores D(X) the same way the Martin
Harper lottery probabilities f(X) are ontained from the Equal Top Approval
scores A(X):

f'(X) is the percentage of ballots B on which the disapproval D(X) is the
largest disapproval of any candidate disapproved on ballot X ... meaning
not ranked above any candidate on ballot B ... i.e. contributing to the
Bottom Count Disapproval of X.

For example ...
48 C
28 A>B
24 C

D(A)=72
D(B)=48
D(C)=52

So f'(A)=72, f'(B)=0, and f'(C)=28.

Our first anti-favorite lottery method is to elect argmax Q(X), where Q(X)
is the quotient Sum{f'(Y|X defeats Y}/D(X)

In this case the sums reduce to one term each because no candidate X
defeats more than one candidate Y.

Q(A)=f'(B)/D(A)=0/72
Q(B)=f'(C)/D(B)=28/48
Q(C)=f'(A)/D(A)=72/72

The winner is argmax Q(X) = C.

For the less lottery minded, here's a different version:

Let Q(X)=min{f'(Y)| X defeats Y}/D(X)
Elect argmaxQ(X).

The result is the same ... because min S and sum S are the same when the
set S has only one member.

-Forest
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