[EM] Richard Moore's idea continued (was Martin Harper's "Majority Potential" Idea Applied to Social Orderings)
Forest Simmons
fsimmons at pcc.edu
Sat Apr 26 12:35:06 PDT 2003
Here's a concise summary of one way of applying Richard's idea.
(1) Convert all marked ballots, as well as all possible permutations
(rankings) of the candidates, to vectors whose components have mean zero
and variance one.
(2) The winning social ordering (i.e. ranking of the candidates) is the
one represented by the vector which is most likely to project on the same
side of a random vector as the majority of vectors representing the marked
ballots.
Of course the set of random vectors considered are supposed to be
uniformly distributed among the vectors of the appropriate type. By the
appropriate type we mean having the same number of components, the same
component mean, and the same component variance as the social ordering and
ballot vectors.
The side of projection is determined by the sign of the dot product. For
example, if most of the ballots have a negative dot product with the
random vector, and the ranking in question also has a negative dot
product, then it is on the same side as the majority.
In practice when the probability calculations are too long or complicated,
the winning social ordering can be determined by Monte Carlo simulation,
using many randomly chosen test vectors until the desired level of
certainty about which ordering is most likely to agree with the majority
in the long run is established, according to established principles of
probability and statistics.
Forest
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