Impossibility and Possibility Theorems; Single-peaked Preferences

In this chapter we show Arrow’s (Social Choice and Individual Values, 2nd edn. Yale University Press, New Haven, 1951/1963) characterization of the dictatorial ranking function: In case of three or more alternatives, any social ranking function that is transitive, Pareto-optimal and Independent of Irrelevant Alternatives (IIA) is dictatorial. In other words: there is no social ranking function that is transitive, Pareto optimal, IIA and not dictatorial. This result is called Arrow’s Impossibility theorem (1951). We prove this result first for the case of two individuals—we call them Romeo and Julia—because this case shows perfectly the role of the different conditions in making one of the persons in fact a dictator. Next we present a proof for the general case of two or more voters (and n, n ≥ 3 $$n \geq 3$$ , alternatives). Gibbard (Econometrica 41:587–601, 1973) and Satterthwaite (J. Econ. Theory 10:187–217, 1975) proved that in the case of three or more alternatives every social choice function that is Pareto optimal and strategy-proof is dictatorial. In other words, in the case of three or more alternatives there exists no social choice function that is Pareto optimal, strategy-proof and non dictatorial. We shall prove this theorem by reducing it to Arrow’s impossiblity theorem. But, because of its transparency, we also give a proof of this result for the special case of only two individuals/voters and three alternatives. We also present three theorems due to Balinski and Laraki (Majority Judgment. MIT Press, Cambridge, 2010) (Chapter 4) which show that several desirable properties concerning electing and ranking are incompatible. Arrow’s and Gibbard-Satterthwaite’s impossibility results assume that the domain of the social ranking function, resp. social choice function, is unrestricted, i.e., that the social ranking function, resp. social choice function, should assign a social order, resp. a social choice, to all logically possible individual preference profiles. But in practice the individual preference profiles frequently satisfy certain restrictions. We shall show Black’s (Theory of Commitees and Elections. Cambridge University Press, Cambridge, 1958) result that under some of these restrictions, in particular in the case of single-peaked preferences, nice (i.e., satisfying a number of desired properties), non dictatorial social choice functions do exist.