[EM] Generalized Median

[Forest Simmons]

For our purposes a generalized median voting method is any method that
elects the candidate that minimizes the total distance from the ballots or
voters to the candidate to be elected.

If the "candidates" are proposed locations for a community center , the
distances to the voters are easy to visualize.

Here's a rather general way of specifying the distance from a voter ballot
B to a candidate X:

d(B, X)=Sum {P(Y)| Y both defeats X pairwise AND outranks X on B}, where
P(Y) is the percentage of ballots on which Y is the lowest ranked candidate
that covers every candidate ranked above it.

If a candidate covers every candidate ranked above it, then it will either
be the top ranked candidate or it will cover the top ranked candidate as
well as any other candidates ranked between them.

For each candidate X let T(X) be the total

Sum over B of d(B,X).

Then elect argminT(X), the candidate X that minimizes T(X), the total
distance from all of the ballots to X.

Compare this to Kemeny-Young. K-Y minimizes the sum of Kendall-tau
distances from the ballots to all possible finish orders of the candidates,
instead of to the nominated individual candidates themselves ... a lot of
un needed computation if all you need is one winner.

-Forest
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