Forest Simmons wrote (12 May 2010):
Here's another proposal. Let M be the matrix whose (i,j) element is the number of ballots on which
candidate i is ranked ahead of candidate j. I think that this is what you mean by the "normal gross
pairwise matrix" that you mention below.
For each candidate i, let d(i) be the difference of the maximum number in column i and the minimum
number in row i. In other words d(i) is the difference is the maximum number of points scored against
candidate i in a pairwise contest and the minimum number of points that candidate i scored in a pairwise
contest.
Generally speaking, the smaller d(i), the stronger candidate i.
So list the candidates in increasing order of d(i) instead of the order of decreasing approval, and apply
the enhancement as before:
Let D1 be the candidate i with the smallest difference d(i). Elect D1 if uncovered, else let D2 be the
smallest d(i) candidate among those that cover D1, etc.
This method wastes the diagonal slots of matrix M just like all of the other standard Condorcet
methods. But I would be interested if you would run it by your standard test cases.
Forest,
Your suggested method fails both the Minimal Defense and Plurality criteria.
49: A
24: B
27: C>B
"Forest scores"
A: 51-49 = 2, C: 49-27 = 22, B: 49-24 = 25.
A has the lowest score and is uncovered and so wins, violating Minimal
Defense (which says that A can't win because on more than
half the ballots A is ranked below B and not above equal bottom).
7: A>B
5: B
8: C
"Forest scores"
A: 8-7 = 1, B: 8-5 = 3, C: 12-8 = 4.
A has the lowest score and is uncovered and so wins, violating the
Plurality criterion (which says that A can't win because C has more
top-preference votes than A has above-bottom votes).
Chris Benham
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