Why a Popular Will Can Exist: Generalizing Majority Rule’s Five Arguments

• In Event: Normative Formal Theory

Abstract

Majority rule is considered so elementary, it is typically accepted uncritically in democratic theory (within certain parameters). A few, like Robert Dahl, Duncan Black, and Christian List, have compiled normative arguments for majority rule on two alternatives. The 5 most common arguments are: contractual (Dahl), social choice (Dahl, List), epistemic (Black, Dahl, List), statistical [central tendency] (Black), and utilitarian (Dahl, List).
It is well established that when majority rule is interpreted as majority preference, and there are more than two alternatives, majority preference is intransitive. Dahl and List agreed, this intransitivity made generalizing the normative arguments for majority rule to multiple alternatives difficult. William H. Riker went so far as to contend that this intransitivity and Arrow’s theorem make meaningful aggregation of individual votes into a popular will impossible; thus elections should be interpreted only as a means for the populace to throw out unresponsive elites.
I argue that Riker is incorrect. Rather, if we revive an interpretation of majority rule commonly used by Locke, Rousseau, and Rawls, we can generalize the normative arguments for majority rule to multiple alternatives. That interpretation is consent of the majority. When generalized to multiple alternatives, it is approval voting. Specifically, we show that for each of the five arguments, an Arrovian version of approval voting better generalizes majority rule to multiple alternatives than any other Arrovian social welfare function (SWF).
(1) Contractual: The argument contends that majority rule selects the alternative more voters would voluntarily choose. We provide a formal axiomatization of consent with respect to utility theory, which shows that Arrovian approval voting (AAV) chooses an alternative which maximizes the number of voters who consent to the winning alternative.
(2) Social Choice: May’s theorem showed that majority rule on two alternatives is uniquely characterized by four normative conditions: decisiveness, anonymity, neutrality, and positive responsiveness. We demonstrate that AAV is the SWF with the least restrictive domain which satisfies May’s four conditions and independence of irrelevant alternatives.
(3) Epistemic: The argument is Condorcet’s jury theorem. Define the majority aggregation condition as requiring that if more voters prefer x over y than prefer y over x, then x must be socially ranked above y. We show that AAV is the unique SWF with the least restrictive domain which generalizes the jury theorem to multiple alternatives while satisfying decisiveness, neutrality, and the majority aggregation condition.
(4) Statistical: The median voter theorem (MVT) assumes candidates are on a single dimension, and voters have single peaked preferences. The theorem demonstrates that the top preference candidate of the median voter beats all other candidates via majority rule. The theorem has many failings. First, there is no a priori necessity that candidates are on a single dimension; the McKelvey-Schofield theorem shows that when there is more than one dimension, there is no guarantee of majority transitivity. Second, there is no a priori necessity to voters having single peaked preferences. Third, MVT’s SWF violates neutrality. AAV provides a better means of measuring the central tendency of the voters. Imagine that each voter is a set, and if a voter V consents to an alternative x, then x is a member of V; otherwise, x is not a member of V. If this is true of all voters and alternatives, the AAV winner will be the alternative with the largest intersection of sets. This measure of central tendency is true regardless of the number of dimensions and the utility functions of the voters. Additionally it is neutral and gives voters access to basically twice as many ballots as they have access to with MVT’s SWF.
(5) Utilitarian: The Rae-Taylor theorem states that when there are two alternatives and certain background conditions, majority rule maximizes the utility of voters. What we show is that the Rae-Taylor theorem is a two-alternatives special case of a multiple alternatives generalization of the Rae-Taylor theorem. Additionally, we provide two more utilitarian arguments for AAV. First, we show that if we use Herbert Simon’s concept of satisficing, AAV can maximize the number of satisficed voters. Second, if each voter behaves as a von Neumann-Morgenstern utility (vNM) function, and we want to aggregate the preferences of the voters with an SWF which is decisive, anonymous, neutral, monotonic, and Pareto satisfying, and we want the group of voters, as a group, to behave as if it is behaving in accordance with a vNM function, then AAV is a SWF on the least restrictive domain which does this.
All of this suggests that if we wish to generalize the arguments for majority rule to multiple alternatives, we should interpret majority rule as consent of the majority, not majority preference.

Author

• Mahendra Prasad, University of California, Berkeley