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Uncovered set: Difference between revisions
Copied most of current w:Landau set article (this version: https://en.wikipedia.org/w/index.php?title=Landau_set&oldid=1020697182 ). I plan to incorporate the text a bit better later
(Cleaned up definitions and added a formal definition section about the covering relation.) |
(Copied most of current w:Landau set article (this version: https://en.wikipedia.org/w/index.php?title=Landau_set&oldid=1020697182 ). I plan to incorporate the text a bit better later) |
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{{Wikipedia|Landau set}}
In [[voting system]]s, the '''Landau set''' (or '''uncovered set''', or '''[[Peter Fishburn|Fishburn]] set''') is the set of candidates <math>x</math> such that for every other candidate <math>z</math>, there is some candidate <math>y</math> (possibly the same as <math>x</math> or <math>z</math>) such that <math>y</math> is not preferred to <math>x</math> and <math>z</math> is not preferred to <math>y</math>. In notation, <math>x</math> is in the Landau set if
<math>\forall \,z</math>, <math>\exists \,y</math>, <math>x \ge y \ge z</math>.
The Landau set is a nonempty subset of the [[Smith set]]. It was discovered by Nicholas Miller.
==References==
*Nicholas R. Miller, "Graph-theoretical approaches to the theory of voting", ''American Journal of Political Science'', Vol. 21 (1977), pp. 769–803. {{doi|10.2307/2110736}}. {{JSTOR|2110736}}.
*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 (1980), pp. 68–96. {{doi|10.2307/2110925}}. {{JSTOR|2110925}}.
*Norman J. Schofield, "Social Choice and Democracy", Springer-Verlag: Berlin, 1985.
*Philip D. Straffin, "Spatial models of power and voting outcomes", in ''Applications of Combinatorics and Graph Theory to the Biological and Social Sciences'', Springer: New York-Berlin, 1989, pp. 315–335.
*Elizabeth Maggie Penn, "[https://web.archive.org/web/20060913022520/http://www.people.fas.harvard.edu/~epenn/covering.pdf Alternate definitions of the uncovered set and their implications]", 2004.
*Nicholas R. Miller, "In search of the uncovered set", ''Political Analysis'', '''15''':1 (2007), pp. 21–45. {{doi|10.1093/pan/mpl007}}. {{JSTOR|25791876}}.
*William T. Bianco, Ivan Jeliazkov, and Itai Sened, "[https://web.archive.org/web/20181220033859/https://pdfs.semanticscholar.org/b15e/72fc147a0421f710b349bf346deeb30aef8b.pdf The uncovered set and the limits of legislative action]", ''Political Analysis'', Vol. 12, No. 3 (2004), pp. 256–276. {{doi|10.1093/pan/mph018}}. {{JSTOR|25791775}}.
The '''uncovered set''' is defined for a set of [[preferential voting|rank-order]] preferences. Usually, the uncovered set is defined only for situations without pairwise ties. When there are no pairwise ties, then the uncovered set is identical to the set called '''Fishburn winners''':
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