The Value of Social Choice Theory for Normative Political Theorists

Social choice theory is the study of collective decision-making, focusing on the procedures and strategies for aggregating individual preferences. It generally takes a formal approach, relying on axiomatic and mathematical analyses. It often produces results that have troubling implications for democracy. This essay shows how political theorists can benefit from social choice theory. By looking at democracy from a social choice perspective, new normative issues emerge that might otherwise remain dormant and the consideration of certain perennial problems and concerns is reinvigorated. As an example, I discuss the ramifications of intransitive social orderings for the notion of group rationality, the concept of majority rule, and democratic decision-making institutions. My primary aim is to show that an analysis of intransitive social orderings clearly demonstrates the importance of critically examining institutions from a normative point of view. In the course of this essay, I will occasionally raise questions that I think political theorists may find interesting. Hopefully, the subject matter itself will also inspire useful insights and ideas.

Intransitive Social Orderings

Kenneth Arrow's General Possibility Theorem (1963) proves that, whenever there are at least three choosers and three alternatives, no voting procedure can simultaneously satisfy a set of ostensibly reasonable postulates. The original proof consists of two rationality axioms (connectedness and transitivity) and five moral conditions (universal admissibility, monotonicity, independence of irrelevant alternatives, non-imposition, and non-dictatorship). ¹ There is no question about the logical validity of Arrow's theorem. All voting procedures must sacrifice one of its axioms or conditions. This result, however, would not be troubling if a voting procedure violated an axiom or condition that was thought to be trivial or misconceived. The most basic philosophical issue, therefore, is whether or not these postulates are really all that reasonable. Almost all philosophical critiques of Arrow's theorem focus on the transitivity axiom and the independence of irrelevant alternatives condition. The other five postulates are widely considered reasonable democratic requirements and, as a consequence, rarely attract scrutiny. In this essay, I take a closer look at the implications of violating the transitivity axiom.

According to the transitivity axiom, collective choices must be represented by complete and transitive rankings of all alternatives. When social orderings are transitive, there is always one alternative that is preferred in pairwise comparisons to all other alternatives in the choice set. This alternative is commonly called a Condorcet winner. ² Collective choices, however, may end up being intransitive any time three or more persons individually rank three or more alternatives. ³ Take, for example, the simple case of three choosers (1, 2, and 3) and three alternatives (A, B, and C). Suppose that Chooser 1 prefers A to B to C, Chooser 2 prefers B to C to A, and Chooser 3 prefers C to A to B. The pairwise aggregation of transitive individual ordinal rankings produces the following intransitive relationship: A is preferred to B, B is preferred to C, and C is preferred to A. Every alternative can be defeated by at least one of the competing alternatives in pairwise comparisons. ⁴

Consider a less formal description of intransitive social orderings using the same individual preference rankings. Assume that the status quo is A. Because a majority prefers C to A, simple majority rule dictates that C should replace A as the status quo. There appears to be nothing at all controversial about this action. In fact, from the perspective of majority rule, it would seem to be warranted. Having established C as the status quo, a majority now claims that it would instead prefer B. Once again, it seems reasonable to allow the majority to rule. The majority would rather have B as opposed to C; therefore, B should supersede C. With B as the status quo, it just so happens that a majority prefers A. If majority rule is respected (and why should it not be?), then A is installed as the status quo and we are...