### On the Arrow property

by Shelah. [Sh:782]

Advances in Applied Math, 2005

Let X be a finite set of alternatives. A choice function c
is a mapping which assigns to nonempty subsets S of X an element
c(S) of S . A rational choice function is one for which
there is a linear ordering on the alternatives such that c(S) is
the maximal element of S according to that ordering. Arrow's
impossibility theorem asserts that under certain natural conditions,
if there are at least three alternatives then every non-dictatorial
social choice gives rise to a non-rational choice function.
Gil Kalai asked if Arrow's theorem can be extended to the case when
the individual choices are not rational but rather belong to an
arbitrary non-trivial symmetric class of choice functions. The main
theorem of this paper gives an affirmative answer in a very general
setting.

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