Social Choice and Individual Values


's monograph Social Choice and Individual Values 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" 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 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.

Introduction

The Introduction contrasts voting and markets with dictatorship and social convention. 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.
Arrow asks whether other methods of taste aggregation, 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.
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 using individual values as input. Then 'social values' means "nothing more than social choices".
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.

Terminology

The book defines a few terms and logical symbols used thereafter and their applied empirical interpretation. Key among these is the "vote" of the society composed of individuals in the following form:
The ordering of each voter ranks social states, including the distribution of commodities, not merely direct consumption by that voter. So, the ordering is an "individual value," not merely, as in earlier analysis, a purely private "taste." Arrow notes that the distinction is not sharp. Resource allocation is specified in the production of each social state in the ordering.
The comprehensive nature of commodities, the set of social states, and the set of orderings was noted by early reviewers.
The two properties that define any ordering of the set of objects in question are:
The earlier definition of an ordering implies that any given ordering entails one of three responses on the "ballot" as between any pair of social states : better than, as good as, or worse than.
An ordering of a voter is denoted by R. That ordering of voter i is denoted with a subscript as '.
If voter i changes orderings, primes distinguish the first and second, say ' compared to ' . The same notation can apply for two different hypothetical orderings of the same voter.
The interest of the book is in amalgamating sets of orderings. This is accomplished through a 'constitution'.

A social ordering of a constitution is denoted R.
For any two social states x and y of a given social ordering R:
x P y is "social preference" of x over y.
x I y is "social indifference" between x and y.
x R y is either "social preference" of x over y or "social indifference" between x and y.
A social ordering applies to each ordering in the set of orderings. This is so regardless of similarity between the social ordering and any or all the orderings in the set. But Arrow places the constitution in the context of ordinalist welfare economics, which attempts to aggregate different tastes in a coherent, plausible way.
Arrow shows how to go from the social ordering R for a given set of orderings to a particular 'social choice' by specifying:
The social ordering R then selects the top-ranked social state from the subset as the social choice set.
Less informally, the social choice function is the function mapping each environment S of available social states for any given set of orderings to the social choice set, the set of social states each element of which is top-ranked for that environment and that set of orderings.
The social choice function is denoted C. Consider an environment that has just two social states, x and y: C = C. Suppose x is the only top-ranked social state. Then C =, the social choice set. If x and y are instead tied, C =. Formally, C is the set of all x in S such that, for all y in S, x R y.
The next section invokes the following. Let R and R' stand for social orderings of the constitution corresponding to any 2 sets of orderings. If R and R' for the same environment S map to the same social choice, the relation of the identical social choices for R and R' is represented as: C = C'.

Conditions and theorem

A constitution might seem to be a promising alternative to dictatorship and vote-immune social convention or external control. Arrow describes the connectedness of a social ordering as requiring only that some social choice be made from any environment of available social states. Since some social state will prevail, this is hard to deny. The transitivity of a social ordering has an advantage over requiring unanimity to change between social states if there is a maladapted status quo. Absent deadlock, transitivity crowds out any reference to the status quo as a privileged default blocking the path to a social choice.
Arrow proposes the following "apparently reasonable" conditions to constrain the social ordering of the constitution.
Each voter is permitted by the constitution to rank the set of social states in any order, though with only one ordering per voter for a given set of orderings.
Arrow describes this condition as an extension of ordinalism with its emphasis on prospectively observable behavior. He ascribes practical advantage to the condition from "every known electoral system" satisfying it.
The conditions, particularly the second and third, may seem minimal, but jointly they are harsh, as may be represented in either of two ways.
  • Arrow's Theorem : The 3 conditions of the constitution imply a dictator who prevails as to the social choice whatever that individual's preference and those of all else.
An alternate statement of the theorem adds the following condition to the above:
# Pareto is stronger than necessary in the proof of the theorem that follows above. But it is invoked in Arrow for a simpler proof than in Arrow. In the latter, Arrow uses 2 other conditions, that with above imply Pareto :
Arrow describes social welfare here as at least not negatively related to individual preferences.
Under imposition, for every set of orderings in the domain, the social ranking for at least one x and y is only x R y''. The vote makes no difference to the outcome.

Proof

The proof is in two parts. The first part considers the hypothetical case of some one voter's ordering that prevails 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 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.

Summary, interpretation, and aftereffects

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 as "an important attribute of a genuinely democratic system capable of full adaptation to varying environments".
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.