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

[John Wong]

That sorta answers my question about Landau, but what about the Schwartz criterion? Is it important? 

Date: Sat, 22 Sep 2007 21:50:16 +0930
From: chrisjbenham at optusnet.com.au
To: johnwong00 at hotmail.com
CC: election-methods at electorama.com
Subject: Re: [Election-Methods] How important is the Schwartz criterion? Also, what is the Landau set, and how is different from the

  

John Wong wrote:
...what is the Landau set, and how is differentfrom 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  
  Sun, 9 Apr 2000 09:58:20 -0600  
      Previous message: [EM] New Smith,Schwartz Algorithms         Next message: [EM] YES versus Head toHead Tiebreakers         Messages sorted by: [ date ]      [ thread ] [ subject ] [author ]       Here is another message from Markus answering some of my questions about'Landau Winners'.  This issue arose because Markus included the algorithmfor Landau along with his Schwartz algorithm, and I had some questions aboutit. Again, I thought it might be something of interest to EM generally, so I amforwarding 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 somereferences.******A Landau winner is a candidate, who defeats every other candidate with apath of length 1 or 2.Candidate A is a Landau winner iff for every other candidate B at least oneof 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) thevoters are sophisticated and (3) the used election method meets themajority criterion, then the winner must always be a Landau winner.Therefore, many scholars consider the Landau winners to be the naturalgeneralization of the Condorcet winner.[a] Nicholas R. Miller, 'Graph-Theoretical Approaches to the Theory ofVoting,' American Journal of Political Science, vol. 21, p. 769-803, 1977,[b] Nicholas R. Miller, 'A New Solution Set for Tournaments and MajorityVoting: Further Graph-Theoretic Approaches to Majority Voting,' AmericanJournal 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 SocialSciences, edited by Fred S. Roberts, New York-Berlin, Springer, 1989,page 315-335.******I mentioned the Fishburn set only because the calculation of the Fishburnset is almost identical to the calculation of the Smith set and becausesomebody 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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