Landau set

In voting systems, the Landau set (or uncovered set, or Fishburn set) is the set of candidates ${\displaystyle x}$ such that for every other candidate ${\displaystyle z}$, there is some candidate ${\displaystyle y}$ (possibly the same as ${\displaystyle x}$ or ${\displaystyle z}$) such that ${\displaystyle y}$ is not preferred to ${\displaystyle x}$ and ${\displaystyle z}$ is not preferred to ${\displaystyle y}$. In notation, ${\displaystyle x}$ is in the Landau set if ${\displaystyle \forall \,z}$, ${\displaystyle \exists \,y}$, ${\displaystyle x\geq y\geq z}$.

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, "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.