[EM] carrying Warren's approval equilibrium idea to its logical conclusion

[Simmons, Forest]

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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