Vitaliy Dolgorukov's talk "Dictators and Ultrafilters: Arrow's and Fishburn's Theorems as Corollaries of the Kirman–Sondermann Result"

The 109th session of the "Formal Philosophy" seminar was held on January 28.

Abstract

The famous Arrow's theorem (1951, 1963) asserts the impossibility of a social welfare function that simultaneously satisfies the principles of unanimity, independence from irrelevant alternatives, and is not dictatorial. Fishburn (1970) introduces a significant amendment: Arrow's theorem cannot be extended to an infinite number of voters. Kirman and Sondermann (1972) observed that the key concept for Arrow's theorem, the "set of decisive coalitions," possesses the properties of an ultrafilter, which allows for deriving both Arrow's theorem and Fishburn's theorem as corollaries. This report will present a modified proof of the result by Kirman and Sondermann. We will also discuss the utility of applying logical methods in social choice theory and issues surrounding the interpretation of infinitary constructions in the social sciences and formal philosophy. The report is based on joint research with E.L. Popova.

References

– Arrow K.J. Social Choice and Individual Values. New York: Wiley, 1951 (1963). 

– Fishburn P. C. Arrow’s impossibility theorem: Concise proof and infinite voters // Journal of Economic Theory. 1970. Vol. 2, № 1. P. 103–106.

– Kirman A.P., Sondermann D. Arrow’s theorem, many agents, and invisible dictators // Journal of Economic Theory. 1972. Vol. 5, № 2. P. 267–277.