Computational application of the mathematical theory of democracy to Arrow’s Impossibility Theorem (how dictatorial are Arrow’s dictators?) Original Article First Online: 14 January 2010 Received: 22 May 2009 Accepted: 14 November 2009 DOI :
10.1007/s00355-009-0433-1

Cite this article as: Tangian, A. Soc Choice Welf (2010) 35: 129. doi:10.1007/s00355-009-0433-1
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Abstract The article is based on three findings. The first one is the interrelation between Arrow’s (Social choice and individual values, Wiley, New York, 1951) social choice model and the mathematical theory of democracy discussed by Tangian (Aggregation and representation of preferences, Springer, Berlin, 1991; Soc Choice Welf 11(1):1–82, 1994), with the conclusion that Arrow’s dictators are less harmful than commonly supposed. The second finding is Quesada’s (Public Choice 130:395–400, 2007) estimate of their power as that of two voters, implying that Arrow’s dictators are not more powerful than a chairperson with an additional vote. The third is the model of Athenian democracy (Tangian, Soc Choice Welf 31:537–572, 2008), where indicators of popularity and universality are applied to representatives and representative bodies. In this article, these indicators are used to computationally evaluate the representativeness/non-representativeness of Arrow’s dictators. In particular, it is shown that there always exist Arrow’s dictators who on the average share opinions of a majority, being rather representatives. The same holds for dictators selected by lot, which conforms to the practice of selecting magistrates and presidents by lot in ancient democracies and medieval Italian republics. Computational formulas are derived for finding the optimal “dictator–representatives”.

In some former publications, the author’s name has been spelled as Andranick Tanguiane.

References Abramowitz M, Stegun I (1972) Handbook of mathematical functions. Dover, New York

Google Scholar Aristotle (1981) The politics. Penguin, Harmondsworth

Google Scholar Armstrong TE (1980) Arrow’s theorem with restricted coalition algebras. J Math Econ 7(1): 55–75

CrossRef Google Scholar Armstrong TE (1985) Precisely dictatorial social welfare functions. Erratum and addendum to ‘Arrow’s theorem with restricted coalition algebras’. J Math Econ 14(1):57–59

Google Scholar Arrow KJ (1951) Social choice and individual values. Wiley, New York

Google Scholar Aumann RJ (1964) Markets with a continuum of traders. Econometrica 32: 39–50

CrossRef Google Scholar Aumann RJ (1966) Existence of competitive equilibria in markets with a continuum of traders. Econometrica 34: 1–17

CrossRef Google Scholar Aumann RJ, Shapley LS (1974) Values of non-atomic games. Princeton University Press, Princeton

Google Scholar Dowding KM (1997) Why democracy needs dictators. Année Sociologique 47(2): 39–53

Google Scholar Fishburn PC (1970) Arrow’s impossibility theorem: concise proof and infinite voters. J Econ Theory 2(1): 103–106

CrossRef Google Scholar Geanakoplos J (2005) Three brief proofs of Arrow’s impossibility theorem. Econ Theory 26:211–215.

http://cowles.econ.yale.edu/~gean/art/p1116.pdf
Google Scholar Gibbard A (1973) Manipulation of voting schemes: a general result. Econometrica 41: 587–601

CrossRef Google Scholar Grofman B, Feld S (1988) Rousseau’s general will: a condorcetian perspective. Am Political Sci Rev 82: 567–578

CrossRef Google Scholar Kelly JS (1978) Arrow impossibility theorems. Academic Press, New York

Google Scholar Kirman A, Sondermann D (1972) Arrow’s theorem, many agents, and invisible dictators. J Econ Theory 5(2): 267–277

CrossRef Google Scholar Korn GA, Korn ThM (1968) Mathematical handbook for scientists and engineers. McGraw-Hill, New York

Google Scholar Monjardet B (1978) An axiomatic theory of tournament aggregation. Math Oper Res 3: 334–351

CrossRef Google Scholar Monjardet B (1979) Duality in the theory of social choice. In: Laffont JJ (ed) Proceedings of the summer econometric society european seminar on preference aggregation and preference revelation, Paris, June 1977. Amsterdam, North-Holland

Mueller DC (1989) Public choice II. Cambridge University Press, Cambridge

Google Scholar Quesada A (2007) 1 dictator = 2 voters. Public Choice 130: 395–400

CrossRef Google Scholar Regenwetter M, Grofman B, Marley AAJ, Tsetlin I (2006) Behavioral social choice: probabilistic models, statistical inference, and applications. Cambridge University Press, Cambridge

Google Scholar Reny PJ (2001) Arrow’s theorem and the Gibbard–Satterthwaite theorem: a unified approach. Econ Lett 20:99–105.

http://home.uchicago.edu/~preny/papers/arrow-gibbard-satterthwaite.pdf
Google Scholar Rousseau J-J (1968) The social contract. Penguin, Harmondsworth

Google Scholar Russell B (1945) A history of western philosophy, and its connection with political and social circumstances from the earliest times to the present day. Simon and Schuster, New York

Google Scholar Schmitz N (1977) A further note on Arrow’ impossibility theorem. J Math Econ 4(3): 189–196

CrossRef Google Scholar Tangian AS (1980) Hierarchical model of group choice. Ekonomika i matematicheskiye metody 16(3):519–534 (in Russian)

Tangian (Tanguiane) AS (1991) Aggregation and representation of preferences. Springer, Berlin

Google Scholar Tangian (Tanguiane) AS (1994) Arrow’s paradox and mathematical theory of democracy. Soc Choice Welf 11(1): 1–82

Google Scholar Tangian AS (2003) Combinatorial and probabilistic investigation of Arrow’s dictator. FernUniversität Hagen, Discussion Paper 336, presented at the 7th international meeting of the society for social choice and welfare, Osaka, July 21–25, 2004

Tangian AS (2008) A mathematical model of Athenian democracy. Soc Choice Welf 31: 537–572

CrossRef Google Scholar Tutubalin VN (1972) Theory of probabilities. Moscow State University Press, Moscow (in Russian)

Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton

Google Scholar Authors and Affiliations 1. WSI in der Hans Böckler Stiftung Düsseldorf Germany