Electoral Rules, Properties and Impossibilities

In this chapter we will discuss electoral functions and rules which take the individual preference orders over the alternatives by the voters as input. In particular, we shall discuss Plurality Rule (PR), Majority Rule (MR) and the Borda Rule (BR). For each of these we shall establish whether they satisfy a number of (at first sight) desirable properties: unrestricted domain, anonymity, neutrality, monotonicity, completeness, Independence of Irrelevant Alternatives (IIA) and transitivity. We shall see that in the case of only two alternatives the only electoral function satisfying the first five conditions is Majority Rule (May, Econometrica 20:680–684, 1952) and that in the case of three or more alternatives there cannot be an electoral function satisfying all seven conditions (Balinski and Laraki, Majority Judgment vs Majority Rule. Social Choice and Welfare, 2019. https://doi.org/10.1007/s00355-019-01200-x ). In particular, none of the electoral rules PR, MR and BR satisfies all these properties. We shall also see that the different electoral functions, applied to the same profile, may give very different outcomes. In other words, the result of an election does not depend so much on the preferences of the voters, but rather on the electoral function used (Saari, Decisions and Elections. Cambridge University Press, Cambridge, 2001). We discuss Saari’s geometrical representation of voting profiles and explain how paradoxical results in (minimal) liberalism, judgment aggregation and in the allocation of tenders all boil down to violations of IIA and can be reduced to the Condorcet paradox.