Here's Levin's & Nalebuff's definition of Simpson-Kramer, as quoted by Markus:
in that paper (Jonathan Levin, Barry Nalebuff, "An Introduction to Vote-Counting Schemes", Journal of Economic Perspectives, vol. 9, no. 1, pp. 3--26, Winter 1995) the Simpson-Kramer method is described as follows:
> For our purposes, we assume that voters rank all the > candidates on their ballots, and do not score candidates > as ties. (...) The Simpson-Kramer min-max rule adheres to > the principles offered by Condorcet in that it emphasizes > large majorities over small majorities. A candidate's > "max" score is the largest number of votes against that > candidate across all head-to-head matchups. The rule > selects the candidate with the minimum max score. > A Condorcet winner will always be a min-max winner. > When there is a cycle, we can think of the min-max > winner as being the "least-objectionable" candidate.
Markus says:
Thus, this paper supports my claims (1) that Levin and Nalebuff explicitly presume that each voter casts a complete ranking of all candidates and (2) that the Simpson-Kramer method _is_ the MinMax method.
Then Marklus continues:
Well, when Levin and Nalebuff write that they "assume that
voters rank all the candidates on their ballots, and do not score candidates as ties" then this doesn't mean that this assumption is a part of the definition of the Simpson-Kramer method; it simply means that Levin and Nalebuff don't discuss partial individual rankings.
I reply:
As you yourself said, Levin & Nalebuff explicitly say that, for the purpose of defining Simpson-Kramer, they assume that no one truncates. That means they're defining Simpson-Kramer only for complete rankings. That means that they're saying that their definition of Simpson-Kramer doesn't apply unless no one truncates. As I said, that makes Simpson-Kramer very different from PC.
Which part of that don't you understand?
But, even aside from that, if they hadn't stated that assumption, Simpson-Kramer still wouldn't be PC, because, if it weren't stated that the definition is only for complete rankings, and so truncation were allowed, Simpson-Kramer wouldn't distinguish between pairwise defeats and pairwise victories. With lots of truncation, a candidate could have his greatest pairwise vote against him in one of his pairwise victories. So, even if Levin & Nalebuff hadn't made it quite clear that, unlike PC, Simpson-Kramer is defined only for complete rankings, Simpson-Kramer would still be very different from PC.
Mike Ossipoff
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