Computational application of the mathematical theory of democracy to Arrow’s Impossibility Theorem (how dictatorial are Arrow’s dictators?)

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”.