Kenneth Arrow's monograph Social Choice and Individual Values (1951; revised in 1963 and 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.
Introduction 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,Abram Bergson, 1938. "A Reformulation of Certain Aspects of Welfare Economics," 1938. Quarterly Journal of Economics, 52(2), p p. 310-334 ([HTML.) 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.
Terminology The book defines a few terms and logical symbols used thereafter and their applied empirical interpretation (pp. 11–19, 23). Key among these is the "vote" ('set of orderings') of the society (more generally "collectivity") composed of individuals ("voters" here) in the following form: Voters, a finite set with at least two members, indexed as i = 1, 2, ... n. Commodities, the objects of choice (things that voters might want, goods and services), both private and public (municipal services, statecraft, etc.). A social state is a specification (formally, an element of a vector) of a distribution among voters of commodities, labor, and resources used in their productions. The set of social states, the set of all 'social states', indexed as x, y, z, . ., with at least three members. A (weak) ordering, a ranking by a voter of all 'social states' from more to less preferred, including possible ties. The set of 'orderings', the set of all n orderings, one ordering per voter.
Example: Three voters {1,2,3} and three states {x,y,z}. Given the three states, there are 13 logically possible orderings (allowing for ties).* Since each of the individuals may hold any of the orderings, there are 13*13*13 = 2197 possible "votes" (sets of orderings). A well-defined social-decision rule selects the social state (or states, in case of tie) corresponding to each of these "votes."
* Namely, from highest to lowest ranked for each triplet and with 'T's indexing ties: x y z x y T z (x z T y is the same ranking as x y T z, so is omitted, etc.) y x z y T x z z x y z x T y x z y x T z y y z x y z T x z y x z T y x z T y T xThe ordering of each voter ranks social states, including the distribution of commodities (possibly based on equity, by whatever metric, or any other consideration), 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 (all social states here) are:
connectedness (completeness): All the objects in the set are included in the ranking (no "undecideds" nor "abstentions") and transitivity: If, for any objects x, y, and z in the set, x is ranked at least as high as y and y is ranked at least as high as z, then x is ranked at least as high as z.
A standard indifference-curve map for an individual has these properties and so is an ordering. Each ray from the origin ranks (conceivable) commodity bundles from least preferred on up (no ties in the ranking). Each indifference curve ranks commodity bundles as equally preferred (all ties in the ranking).
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 (x, y): better than, as good as, or worse than (in preference ranking). (Here "as good as" is an "equally-ranked," not a "don't know," relation.) The denotations of these three "ballot" options are respectively:
x P y (voter prefers x to y) x I y (voter indifferent between x and y) y P x (voter prefers y to x).
It is convenient for deriving implications to compact the first two of these options on the ballot to one, an "at least as good as" relation, denoted R:
x R y: voter either prefers x to y or is indifferent between x and y but not both.
The above two properties of an ordering are then axiomatized as:
connectedness: For all (the objects of choice in the set) x and y, either x R y or y R x.
transitivity: For all x, y, and z, x R y and y R z imply x R z.
Thus, alternation ('or') and conjunction ('and') of R relations represent both the properties of an ordering for all the objects of choice.
The I and P relations are then defined as:
x I y: x R y and y R x (x as good as y means x at least as good as y and vice versa).
x P y: not y R x (y R x includes one of two options. Negating that option leaves only x P y, the third of the original three options, on the ballot.)
From this, conjunction ('and') and negation ('not') of mere pairwise R relations can (also) represent all the properties of an ordering for all the objects of choice. Hence, the following shorthand. 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 constitution (or social welfare function) is a voting rule mapping each (of at least one) set of orderings onto a social ordering, a corresponding ordering of the set of social states that applies to each voter.
A social ordering of a constitution is denoted R. (Context or a subscript distinguishes a voter ordering R from the same symbol for a social ordering.)
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 is selected over y by the rule).
x I y is "social indifference" between x and y (both are ranked the same by the rule).
x R y is either "social preference" of x over y or "social indifference" between x and y (x is ranked least as good as y by the rule).
A social ordering applies to each ordering in the set of orderings (hence the "social" part and the associated amalgamation). This is so regardless of (dis)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. The social ordering for a given set of orderings as to the set of social states is analogous to an indifference-curve map for an individual as to the set of commodity bundles. There is no necessary interpretation from this that "society" is just a big voter. Still, the relation of a set of voter orderings to the outcome of the voting rule, whether a social ordering or not, is a focus of the book.
Arrow (pp. 15, 26–28) shows how to go from the social ordering R for a given set of orderings to a particular 'social choice' by specifying:
the environment, S: the subset of social states that is (hypothetically) available (feasible as to resource quantity and productivity), not merely conceivable.
The social ordering R then selects the top-ranked social state(s) from the subset as the social choice set. This is a generalization from consumer demand theory with perfect competition on the buyer's side. S corresponds to the set of commodity bundles on or inside the budget constraint for an individual. The consumer's top choice is at the highest indifference curve on the budget constraint. Less informally, the social choice function is the function mapping each environment S of available social states (at least two) for any given set of orderings (and corresponding social ordering R) to the social choice set, the set of social states each element of which is top-ranked (by R) for that environment and that set of orderings.
The social choice function is denoted C(S). Consider an environment that has just two social states, x and y: C(S) = C([x, y]). Suppose x is the only top-ranked social state. Then C([x, y]) = {x}, the social choice set. If x and y are instead tied, C([x, y]) = {x, y}. Formally (p. 15), C(S) is the set of all x in S such that, for all y in S, x R y ("x is at least as good as 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(s), the relation of the identical social choices for R and R' is represented as: C(S) = C'(S).
Conditions and theorem