The Informational Basis of Social Choice

Sen received the Sveriges Riksbank Prize in particular for his contributions to welfare eco­nomics and the theory of social choice. The latter area has its modern origin in Arrow’s famous monograph Social Choice and Individual Values from 1951 (2nd edition 1963) where the author proved that under certain rather intuitive conditions, a so-called social welfare function does not exist.

Sen’s own reaction to Arrow’s findings culminated in his monograph from 1970 Collective Choice and Social Welfare, though in the second half of the 1960s already, Sen had published a couple of fundamental articles on this topic in economics journals that emphasize the formal or mathematical approach, such as Econometrica, Journal of Economic Theory and the Review of Economic Studies. Sen’s book, which turned out to be the gateway into the field of collective decisions for many who, in the course of time, became distinguished social choice theorists, highlighted at least three fundamental aspects within the area: the informational parsimony within the Arrovian approach and various escape roads therefrom, the role of individual rights and liberties within col­lective decisions, and the effectiveness of the majority method and related rules under various domain conditions of voters’ preferences.

The informational basis in Arrow’s set-up is a set of individual ordinal preferences to be mapped into a unique social ordering. Preference intensities not only make no sense in such a framework, they cannot even be expressed meaningfully in an ordinal frame­work. Arrow also excluded the interpersonal comparability of preference rankings. Once ordinal preferences are combined with the possibility to compare levels of utility across persons so that it is, for example, possible to say that under a certain policy x, let’s say, person i is better off than person j, the maximin or leximin rule (that is, the lexicographic extension of maximin) a la Rawls (1971) focusing on the worst-off in society is a non­contradictory and, perhaps, attractive aggregation rule.

respect to a status-quo point. If utilities are not only cardinal but also comparable across persons, classical utilitarianism and its modern version based on the Bayesian concept of rationality, as proposed by Harsanyi (1953, 1955, 1977), do not run into the Arrovian impossibility. In other words, the informational aspect, the degree of available utility information, is a powerful tool to distinguish among different approaches, and it was Sen in particular who made us aware of this “taxonomy” very convincingly. As is argued above, Arrow’s approach is parsimonious and, in a certain sense, unsuccessful, ending in a cul de sac.

Sen (1970a, 1970b) was the first to combine the mechanism of aggregating preferences with the idea that individuals or citizens within a community should be able to exercise certain personal rights. While Arrow showed the existence of a global dictator, Sen asked whether it would be possible to permit individuals to be “local dictators” over purely private matters. The latter was called a libertarian right. To the surprise of many, Sen came up with another impossibility result, the “impossibility of a Paretian libertarian”. Under an unrestricted domain of individual preferences, the weak Pareto principle and the right to be individually decisive over a minimal personal sphere are incompatible. The number of papers that tried to circumvent this negative finding runs into hundreds. Individuals’ rights can be restricted, though Sen’s requirement is already minimal, the application of the weak Pareto rule can be limited or the domain of individual prefer­ences can be restricted. However, are these proposals satisfactory? Sen’s negative result has an analogy in non-cooperative game theory where it is well known that Nash equi­libria can be Pareto inefficient.

That the majority rule can lead to cyclicity if preferences are not restricted has been known for several centuries. The Marquis de Condorcet had already suggested ways out of this dilemma. It was only around 1950 that Arrow (1951) and Black (1948) independ­ently came forward with a domain condition on individual preferences that is not only easily interpretable but can also be witnessed in certain real-life situations, the property of single-peaked preferences. Given a certain number of alternatives arranged along the real line, individuals have a most preferred object somewhere along the line, and to the left and to the right of this object, preferences decline. This can be taken literally. One of the exam­ples that depict this structure comes from the left-to-right structure of political parties. The mirror-image of single-peakedness is single-cavedness. Sen (1966) proposed his con­dition of value restriction which encompasses both properties and a third of “not being in the middle” between the other two alternatives, given any triple of options, and showed that this condition is sufficient for the existence of a majority or Condorcet winner under the simple majority rule. If the number of voters who are not indifferent among all alter­natives is odd, the simple majority rule yields a social ordering. In Arrow’s terminology, under value restriction plus oddness, the method of simple majority decision is a social welfare function. Various other sufficient conditions, for example, limited agreement and dichotomous preferences, such that the simple majority rule becomes an Arrow social welfare function were formulated, but space does not allow us to go into greater detail.