Talk:Landau set

example
Could anybody explain the Landau set by means of an example? Also, I'd like to know what it is important for. Martinwilke1980 (talk) 16:43, 22 October 2008 (UTC)

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name
I searched through every single source listed here and on electowiki, and found only one use of the term "Landau set" (https://www.google.com/books/edition/Applications_of_Combinatorics_and_Graph/7HPSBwAAQBAJ?hl=en&gbpv=1&bsq=landau). Here is a breakdown of how each source used in the Wikipedia article uses each term:

Since only one source uses the term "Landau set" and nearly every source uses the term "uncovered set" (including the source that uses the term "Landau set"), I propose renaming this article to "uncovered set". Any objections?

Solomon Ucko (talk) 18:59, 27 October 2023 (UTC)


 * Here are the parts (except examples, most proofs, and diagrams) that are most relevant to this article, of :
 * (Formatting modified.)
 * "Landau set: If the top cycle set is too large, we could try to restrict it by including only those alternatives which beat all other alternatives either directly or indirectly by a chain of length two (i beats j, j beats k). This set is called the Landau set after . See for a clear and entertaining exposition. [...] Theorem 1. The Landau set cannot contain a Pareto inferior alternative."
 * "The Landau set has a number of other good properties. For example, it is non empty [...] Moreover, Miller showed it can be characterized game theoretically. Suppose [...] that two politicians P and Q must each choose one of the alternatives [...] as his 'platform.' P will beat Q and get a payoff +1 precisely if a majority of voters prefers P's platform to Q's platform. [...] Theorem 2. The Landau set is exactly the set of undominated platforms. [...] Miller called the relation of domination covering, and the Landau set the uncovered set. With all these nice properties, it is unfortunately true that the Landau set can still be large. In fact, also proved that in a random tournament of size n, the probability that every alternative is in the Landau set, goes to one as n gets large."
 * "Landau set (Uncovered set): In the spatial context, it is convenient to use Miller's alternative formulation of the Landau set. Definition. For a point x in Rk, DEF(x) is the set of all points which beat x under majority rule. Definition. For points x and y, x covers y if x beats y, and DEF(x) ⊆ DEF(y). See Figure 9. Definition. The uncovered set (Landau set) is the set of all points which are not covered by any other point."
 * "Landau proved that any Copeland winner must be in the Landau set (which is why the Landau set must be non-empty)."
 * "If the top cycle set is all of Rk, and in the discrete case the Landau set is not much smaller, one might be pessimistic about the size of the uncovered set in a spatial voting situation. However, it is easy to show that the Landau set cannot be all of Rk. First, in a spatial voting situation the set of non-Pareto-inferior alternatives ('Pareto optimal alternatives') is exactly the convex hull C of the voter ideal points. The argument of Theorem 1 goes through to show that the uncovered set cannot contain any Pareto inferior alternatives, hence must be contained in C. In fact, recent results show that the uncovered set is often considerably smaller than C."
 * "Definition. The yolk of a spatial voting system in R2 is the smallest disk B(x, r) which intersects all median lines. Here x is the center of the disk, and r is its radius. See Figure 10. Theorem 5. The uncovered set is contained in B(x, 4r) [...] One consequence of this theorem is that if a voting situation almost has a Condorcet winner (so the yolk is small), the uncovered set will be small. Hence, the uncovered set looks like an attractive solution concept in spatial voting, although its precise geometry is not yet well understood. The importance of the uncovered set is increased by a beautiful result of which characterizes it as the set of alternatives which can be reached from any other point by a chain of majority votes when the voters vote sophisticatedly (taking into account what other voters will do and adjusting their vote accordingly) rather than sincerely."
 * "A version of Landau ' s original argument goes through to show that the Copeland winner is in the uncovered set."
 * Solomon Ucko (talk) 21:38, 31 October 2023 (UTC) Solomon Ucko (talk) 21:38, 31 October 2023 (UTC)


 * It's my impression as well that uncovered set is the main term used in the literature. (Thanks for doing the research to back that up.) I have no objection to the move. - CRGreathouse (t | c) 01:39, 3 November 2023 (UTC)
 * Perfect! Since you created the page and contributed most of the content and you say you have no objection, no one else has raised any objection either and it's been 6 days since I proposed the move, and disagreement seems unlikely, I'll go ahead and move the page. If anyone objects, they can initiate a discussion process following Requested moves/Controversial, and it can be moved back if there is consensus to do so. Solomon Ucko (talk) 01:54, 3 November 2023 (UTC)