### What majority decisions are possible

by Shelah. [Sh:E37]

The main result is the following:
Let X be a finite set and 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 D for j in J such that for
any x not= 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 D and x in X we have
| {y:c {x,y}= y}| not= (|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 section
3). In section 4
we discuss a generalization.

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