[Election-Methods] How important is the Schwartz criterion? Also, what is the Landau set, and how is different from the

[Chris Benham]

John Wong wrote:

> ...what is the Landau set, and how is different from the Smith and the 
> Schwartz set?
>
http://lists.electorama.com/mmsearch.cgi/election-methods-electorama.com

http://lists.electorama.com/htdig.cgi/election-methods-electorama.com/2000-April/003908.html

>
>   [EM] Landau Winners/Fishburn Set
>
> *Norman Petry *election-methods-list at eskimo.com 
> <mailto:election-methods-list%40eskimo.com>
> /Sun, 9 Apr 2000 09:58:20 -0600/
>
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> ------------------------------------------------------------------------
>
>Here is another message from Markus answering some of my questions about
>"Landau Winners".  This issue arose because Markus included the algorithm
>for Landau along with his Schwartz algorithm, and I had some questions about
>it.
>
> Again, I thought it might be something of interest to EM generally, so I am
>forwarding it to the list for further discussion.
>
>N.
>
>**********
>
>Dear Norman,
>
>you wrote (8 Apr 2000):
>>/ You mentioned the Landau set in your message, but I do not recall that
>/>/ Landau has ever been discussed on the EM list.  Does it have any merits or
>/>/ uses we should consider?  I did a quick search on the Internet, but turned
>/>/ up nothing useful, so if you have any references to Landau I would
>/>/ appreciate it.
>/
>I should have said that the set of Landau winners is called "uncovered set"
>or "Fishburn set." If you search for these words, then you will find some
>references.
>
>******
>
>A Landau winner is a candidate, who defeats every other candidate with a
>path of length 1 or 2.
>
>Candidate A is a Landau winner iff for every other candidate B at least one
>of the following two statements is correct:
>(1) A >= B.
>(2) There is a candidate C such that A >= C >= B.
>
>******
>
>There must always be at least one Landau winner.
>
>******
>
>Miller demonstrated that if (1) the electorate is 2-dimensional, (2) the
>voters are sophisticated and (3) the used election method meets the
>majority criterion, then the winner must always be a Landau winner.
>Therefore, many scholars consider the Landau winners to be the natural
>generalization of the Condorcet winner.
>
>[a] Nicholas R. Miller, "Graph-Theoretical Approaches to the Theory of
>Voting," American Journal of Political Science, vol. 21, p. 769-803, 1977,
>
>[b] Nicholas R. Miller, "A New Solution Set for Tournaments and Majority
>Voting: Further Graph-Theoretic Approaches to Majority Voting," American
>Journal of Political Science, vol. 24, page 68-96, 1980,
>
>[c] Norman J. Schofield, "Social Choice and Democracy," Berlin,
>Springer-Verlag, 1985,
>
>[d] Philip D. Straffin, "Spatial Models of Power and Voting Outcomes,"
>Applications of Combinatorics and Graph Theory to the Biological and Social
>Sciences, edited by Fred S. Roberts, New York-Berlin, Springer, 1989,
>page 315-335.
>
>******
>
>I mentioned the Fishburn set only because the calculation of the Fishburn
>set is almost identical to the calculation of the Smith set and because
>somebody might ask in the future how to calculate the Fishburn set.
>
>******
>
>You wrote (8 Apr 2000):
>>/ Also, I think your message would be a valuable contribution to the EM list
>/>/ archives, for anyone trying to implement Smith, Schwartz, etc.  May I have
>/>/ your permission to forward the message to the list?
>/
>Of course, you may.
>
>Markus Schulze
>  
>



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