Social Choice and Individual Values

Kenneth Arrow's monograph Social Choice and Individual Values (1951, 2nd ed., 1963, 3rd ed., 2012) and a theorem within it created modern social choice theory, a rigorous melding of social ethics and voting theory with an economic flavor. Somewhat formally, the "social choice" in the title refers to Arrow's representation of how social values from the set of individual orderings would be implemented under the constitution. Less formally, each social choice corresponds to the feasible set of laws passed by a "vote" (the set of orderings) under the constitution even if not every individual voted in favor of all the laws. The work culminated in what Arrow called the "General Possibility Theorem," better known thereafter as Arrow's (impossibility) theorem. The theorem states that, absent restrictions on either individual preferences or neutrality of the constitution to feasible alternatives, there exists no social choice rule that satisfies a set of plausible requirements. The result generalizes the voting paradox, which shows that majority voting may fail to yield a stable outcome.

The Introduction contrasts voting and markets with dictatorship and social convention (such as those in a religious code). Both exemplify social decisions. Voting and markets facilitate social choice in a sense, whereas dictatorship and convention limit it. The former amalgamate possibly differing tastes to make a social choice. The concern is with formal aspects of generalizing such choices. In this respect it is comparable to analysis of the voting paradox from use of majority rule as a value. In the simplest case of the voting paradox, there are 3 candidates, A, B, and C, and 3 voters with preferences listed in decreasing order as follows. Voter 1: A B C
Voter 2: B C A
Voter 3: C A B
By majority rule for 2 candidate votes, A beats B, B beats C, but C beats A. Majority rule works for an individual selecting consistently among the 3 candidates but not necessarily for the "social choice" in any general sense. Arrow asks whether other methods of taste aggregation (whether by voting or markets), using other values, remedy the problem or are satisfactory in other ways. Here logical consistency is one check on acceptability of all the values. To answer the questions, Arrow proposes removing the distinction between voting and markets in favor of a more general category of collective social choice. The analysis uses ordinal rankings of individual choice to represent behavioral patterns. Cardinal measures of individual utility and, a fortiori, interpersonal comparisons of utility are avoided on grounds that such measures are unnecessary to represent behavior and depend on mutually incompatible value judgments (p. 9). Following Abram Bergson, whose formulation of a social welfare function launched ordinalist welfare economics, Arrow avoids locating a social good as independent of individual values. Rather, social values inhere in actions from social decision rules (hypostatized as constitutional conditions) using individual values as input. Then 'social values' means "nothing more than social choices" (p. 106). Topics implicated along the way include game theory, the compensation principle in welfare economics, extended sympathy, Leibniz's principle of the identity of indiscernibles, logrolling, and similarity of social judgments through single peaked preferences, Kant's categorical imperative, or the decision process.

The proof is in two parts (Arrow, 1963, pp. 97–100). The first part considers the hypothetical case of some one voter's ordering that prevails ('is decisive') as to the social choice for some pair of social states no matter what that voter's preference for the pair, despite all other voters opposing. It is shown that, for a constitution satisfying Unrestricted Domain, Pareto and Independence, that voter's ordering would prevail for every pair of social states, no matter what the orderings of others. So, the voter would be a Dictator. Thus, Nondictatorship requires postulating that no one would so prevail for even one pair of social states. The second part considers more generally a set of voters that would prevail for some pair of social states, despite all other voters (if any) preferring otherwise. Pareto and Unrestricted Domain for a constitution imply that such a set would at least include the entire set of voters. By Nondictatorship, the set must have at least 2 voters. Among all such sets, postulate a set such that no other set is smaller. Such a set can be constructed with Unrestricted Domain and an adaptation of the voting paradox to imply a still smaller set. This contradicts the postulate and so proves the theorem.

The book proposes some apparently reasonable conditions for a "voting" rule, in particular, a 'constitution', to make consistent, feasible social choices in a welfarist context. But then any constitution that allows dictatorship requires it, and any constitution that requires nondictatorship contradicts one of the other conditions. Hence, the paradox of social choice. The set of conditions across different possible votes refined welfare economics and differentiated Arrow's constitution from the pre Arrow social welfare function. In so doing, it also ruled out any one consistent social ordering to which an agent or official might appeal in trying to implement social welfare through the votes of others under the constitution. The result generalizes and deepens the voting paradox to any voting rule satisfying the conditions, however complex or comprehensive. The 1963 edition includes an additional chapter with a simpler proof of Arrow's Theorem and corrects an earlier point noted by Blau. It also elaborates on advantages of the conditions and cites studies of Riker and Dahl that as an empirical matter intransitivity of the voting mechanism may produce unsatisfactory inaction or majority opposition. These support Arrow's characterization of a constitution across possible votes (that is, collective rationality) as "an important attribute of a genuinely democratic system capable of full adaptation to varying environments" (p. 120). The theorem might seem to have unravelled a skein of behavior based social ethical theory from Adam Smith and Bentham on. But Arrow himself expresses hope at the end of his Nobel prize lecture that, though the philosophical and distributive implications of the paradox of social choice were "still not clear," others would "take this paradox as a challenge rather than as a discouraging barrier." The large subsequent literature has included reformulation to extend, weaken, or replace the conditions and derive implications. In this respect Arrow's framework has been an instrument for generalizing voting theory and critically evaluating and broadening economic policy and social choice theory.