Sh:E37

• Shelah, S. What majority decisions are possible. Now 816. Preprint. arXiv: math/0303323
  [Sh:816]
• Abstract:
  The main result is the following:

  Let X be a finite set and {\mathfrak D} be a non empty family of choice functions for {X}\choose{2} closed under permutation of X. Then the following conditions are equivalent:

  (A) for any choice function c on {X}\choose{2} we can find a finite set J and c_j\in {\mathfrak D} for j\in J such that for any x\neq y\in X: c\{x,y\}=y\ \Leftrightarrow\ |J|/2<|\{j\in J:c_j\{x,y\}= y\}| (so equality never occurs)

  (B) for some c\in {\mathfrak D} and x\in X we have |\{y:c\{x,y\}= y\}| \neq (|X|-1)/2.

  We then describe what is the closure of a set of choice functions by majority; in fact, there are just two possibilities (in §3). In §4 we discuss a generalization.

• Version 2004-03-01_10 (29p)

@unpublished{Sh:E37,
 author = {Shelah, Saharon},
 title = {{What majority decisions are possible. Now 816}},
 note = {\href{https://arxiv.org/abs/math/0303323}{arXiv: math/0303323}  [Sh:816]},
 arxiv_number = {math/0303323},
 refers_to_entry = { [Sh:816]}
}