  In [G. Kalai, A Fourier-theoretic Perspective on the Condorcet Paradox and
Arrow's Theorem, Adv. in Appl. Math. 29(3) (2002), pp. 412--426], Kalai
investigated the probability of a rational outcome for a generalized social
welfare function (GSWF) on three alternatives, when the individual preferences
are uniform and independent. In this paper we generalize Kalai's results to a
broader class of distributions of the individual preferences, and obtain new
lower bounds on the probability of a rational outcome in several classes of
GSWFs. In particular, we show that if the GSWF is monotone and balanced and the
distribution of the preferences is uniform, then the probability of a rational
outcome is at least 3/4, proving a conjecture raised by Kalai. The tools used
in the paper are analytic: the Fourier-Walsh expansion of Boolean functions on
the discrete cube, properties of the Bonamie-Beckner noise operator, and the
FKG inequality.
