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In the study of electoral systems, the uncovered set (also called the Landau set or the Fishburn set) is a set of candidates that generalizes the notion of a Condorcet winner whenever there is a Condorcet paradox.^[1] The Landau set can be thought of as the Pareto frontier for a set of candidates, when the frontier is determined by pairwise victories.^[2]

The Landau set is a nonempty subset of the Smith set. It was first discovered by Nicholas Miller.^[2]

Definition

The Landau set consists of all undominated or uncovered candidates. One candidate (the Fishburn winner) covers another (the Fishburn loser) if they would win any matchup the Fishburn loser would win. Thus, the Fishburn winner has all the pairwise victories of the Fishburn loser, as well as at least one other pairwise victory. In set-theoretic notation, $x$ is a candidate such that for every other candidate $z$ , there is some candidate $y$ (possibly the same as $x$ or $z$ ) such that $y$ is not preferred to $x$ and $z$ is not preferred to $y$ .

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, "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, "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.

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^ Miller, Nicholas R. (1980-02). "A New Solution Set for Tournaments and Majority Voting: Further Graph- Theoretical Approaches to the Theory of Voting". American Journal of Political Science. 24 (1): 68. doi:10.2307/2110925. {{cite journal}}: Check date values in: |date= (help)
^ ^a ^b Miller, Nicholas R. (1977-11). "Graph-Theoretical Approaches to the Theory of Voting". American Journal of Political Science. 21 (4): 769. doi:10.2307/2110736. {{cite journal}}: Check date values in: |date= (help)

[1] Miller, Nicholas R. (1980-02). "A New Solution Set for Tournaments and Majority Voting: Further Graph- Theoretical Approaches to the Theory of Voting". American Journal of Political Science. 24 (1): 68. doi:10.2307/2110925. {{cite journal}}: Check date values in: |date= (help)

[:0-2] Miller, Nicholas R. (1977-11). "Graph-Theoretical Approaches to the Theory of Voting". American Journal of Political Science. 21 (4): 769. doi:10.2307/2110736. {{cite journal}}: Check date values in: |date= (help)

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+In the study of [[voting system|electoral system]]s, the '''uncovered set''' (also called the '''Landau set''' or the '''[[Peter Fishburn|Fishburn]] set''') is a set of candidates that generalizes the notion of a [[Condorcet winner criterion|Condorcet winner]] whenever there is a [[Condorcet paradox]].<ref>{{Cite journal |last=Miller |first=Nicholas R. |date=1980-02 |title=A New Solution Set for Tournaments and Majority Voting: Further Graph- Theoretical Approaches to the Theory of Voting |url=https://www.jstor.org/stable/2110925?origin=crossref |journal=American Journal of Political Science |volume=24 |issue=1 |pages=68 |doi=10.2307/2110925}}</ref> The Landau set can be thought of as the [[Pareto front|Pareto frontier]] for a set of candidates, when the frontier is determined by pairwise victories.<ref name=":0">{{Cite journal |last=Miller |first=Nicholas R. |date=1977-11 |title=Graph-Theoretical Approaches to the Theory of Voting |url=https://www.jstor.org/stable/2110736?origin=crossref |journal=American Journal of Political Science |volume=21 |issue=4 |pages=769 |doi=10.2307/2110736}}</ref>
-In [[voting system]]s, the '''uncovered set''' (or '''Landau 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.
+The Landau set is a nonempty subset of the [[Smith set]]. It was first discovered by Nicholas Miller.<ref name=":0" />
+== Definition ==
+The Landau set consists of all ''undominated'' or ''uncovered'' candidates''.'' One candidate (the ''Fishburn winner'') covers another (the ''Fishburn loser'') if they would win any matchup the Fishburn loser would win. Thus, the Fishburn winner has all the pairwise victories of the Fishburn loser, as well as at least one other pairwise victory. In set-theoretic notation, <math>x</math> is a candidate 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>.
 ==References==