[EM] EM Metrics

[Paul Kislanko]

Seeing "RRV" used in a post reminded me of some earlier discussions.
Analysis of team computer rankings in sports that do NOT have round-robin
schedules use "Retrodictive Ranking Violations" to characterize computer
ratings with A>B after B has won a match against A. (This is not an "error",
since the rating that has A>B may have noticed A has 10 wins over {C}s, each
of which has beaten B.)

But this leads to a thought.

Suppose an Election Method results in an ordered list of alternatives {1st
2nd 3rd....} (trivial for Plurality, and well-defined for any method if we
accept {2nd, 2nd, 5th, 5th, 5th,...} for ranked methods that result in
"ties")

For each ranked ballot (for plurality we assume a "ranked ballot" that looks
like {1st, last, last, last, ...}, for approval we assume a "ranked ballot"
that looks like {1st, 1st, 1st, .... last, last, last} ) we can find
RRV(ballot) = SUM over pairs(x,y) (Altx > Alty in EM but Alty > Altx in
ballot)

This is just the Kendall tau rank correlation "distance" = # of swaps
required by a bubble sort to turn {ballot} into {EM result}

Something like that could be exploited to avoid the nebulous notion of
utility and define (voter(s) who cast {ballot})'s (lack of) "satisfaction
index" unambiguously and entirely with respect to the EM used to form the
election results list.

Paul Kislanko

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