[EM] carrying Warren's approval equilibrium idea to its logical conclusion
Simmons, Forest
simmonfo at up.edu
Tue Dec 26 12:59:24 PST 2006
Warren's idea of using range ballots to find a kind of equilibrium approval candidate has led me to the following idea based on range ballots:
For each candidate X let MPO(X) be the maximum pairwise opposition received by that candidate, i.e. the maximum number of ballots on which any given competitor Y is rated above X.
The essence of Warren's idea is that if X is to be an approval equilibrium winner, then X must have more approval than MPO(X), since candidates (like Y) that are rated above the approval equilibrium winner on a ballot would surely be approved on that ballot (whether or not X would be approved).
So let R(X) be the highest possible range value that could be used as a common approval cutoff in order for X to get more approval than MPO(X).
The higher the value of R(X) the more likely that X would be an approval equilibrium candidate.
Let M = max over all candidates X of R(X).
Among all candidates with R(X) = M, the winner is the candidate X with the greatest surfeit of approval
Approval(X) - MPO(X),
when the approval cutoff is set at M .
To efficinetly calculate this winner we can use two summable matrices.
The first matrix is the ordinary pairwise matrix P, whose (x,y) entry is the number of ballots on which candidate x is rated above candidate y.
The (x, r) entry of the second matrix, Q, is the approval that candidate x would get if the approval cutoff were set at range level r on each ballot.
To find the winner, find MPO(X) for each candidate X by picking out the highest number in column X of the matrix P.
Then find R(X) as the maximum value of r such that the difference D(X) = Q(X, r)-MPO(X) is positive.
Find the max value M of R(X) as X varies over the candidates.
Among those candidates X such that R(X)=M, elect the one with the greatest value of D(X).
It seems to me that this candidate X has the best claim on the title "Approval Equilbrium Winner."
Forest
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