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

On The Probability of a Rational Outcome for Generalized Social Welfare Functions on Three Alternatives