Social Choice and Individual Values

Social Choice and Individual Values

Kenneth Arrow's monograph "Social Choice and Individual Values" (1951, 2nd ed., 1963) 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". Intuitively, 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 book culminated in what Arrow called the "General Possibility Theorem," thereafter better known as Arrow's (impossibility) theorem. The theorem shows that, absent restrictions on either individual preferences or neutrality of the constitution as to feasible alternatives, there exists no social choice rule that satisfies a set of seemingly plausible requirements. The result is a generalization and extension of the voting paradox, which shows that majority voting may fail to yield a stable outcome.


The Introduction contrasts voting in the political realm and markets in the economic realm with dictatorship and social convention (such as from a religious code). These are all ways of making social decisions. But voting and markets facilitate "social" choice in a sense, as 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.

The earlier definition of an ordering implies that any given ordering entails 1 of 3 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.) Less informally, the social choice function is the function mapping each environment "S" of available social states (at least 2) 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 2 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

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 (especially with no place on the ballot for abstention). The "transitivity" of a social ordering has an advantage over requiring unanimity (or much less) to change between social states if there is a maladapted "status quo" (that is, one subject to "democratic paralysis"). Absent deadlock, transitivity crowds out any reference to the "status quo" as a privileged default blocking the path to a social choice (p. 120).

Arrow proposes conditions to constrain the social ordering(s) of the constitution (pp. 96-97, 25-31). The conditions, presented below, can be interpreted as general, practically necessary, or apparently reasonable.

* 1. Universal (unrestricted) domain U (subsequently so called): Every logically possible set of orderings maps to its own social ordering.

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 (for the subset in question). He ascribes practical advantage to the condition from "every known electoral system" satisfying it (p. 110)

* 3. The (weak) Pareto principle P: For any "x" and "y" in the set of social states, if all prefer "x" over "y", then "x" is socially selected over "y".#
*Arrow's Theorem: The constitution is impossible, that is, the 4 conditions of a constitution imply a contradiction.

# Pareto is stronger than necessary in the proof of the theorem that follows above. Arrow (1951) uses 2 other conditions, instead of Pareto, that with (2) above imply Pareto (Arrow, 1987, p. 126):
* 3a. Monotonicity M (Positive Association of Individual and Social Values), as in Arrow (1987, p. 125): For a given set of orderings with social ordering "R", such that state "x" is socially preferred to state "y", if the preference for "x" rises in some individual ordering(s) and falls in none, "x" is also socially preferred to "y" in the social ordering for the new set of orderings. Arrow (1951, p.26) describes social welfare here as at least not negatively related to individual preferences.
* 3b. As defined by Arrow (1951, pp. 28-29), an imposed constitution is a constitution such that for some alternative social states "x" and "y" and for any set of orderings "R_1", ..., "R_n" in the domain and corresponding social ordering "R", the social ranking is "x R y". :Non-imposition N (Citizens' Sovereignty): A constitution is not to be imposed.

Under imposition, for every set of orderings in the domain, the social ranking is only "x R y". The vote makes no difference to the outcome.


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.

Summary, context, 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 is contradictory. Hence, the "paradox of social choice".

The set of "conditions" across different possible votes of "values" refined welfare economics and differentiated Arrow's constitution from the pre-Arrow social welfare function. Thus, one dictator "across every possible vote" on social alternatives eliminates "any single" non-vacuous ordering as the social ordering. It also makes redundant an agent or official intent on implementing the values of other"s" in the society through the constitution. The remaining alternative, nondictatorship, excludes a pre-Arrow social welfare function as a consistent voting machine. 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. It also elaborates on advantages of the conditions and cites studies of Riker (1958) and Dahl (1956, pp. 39-41) 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 others might take his result "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.

See also

*Arrow's impossibility theorem
*Kenneth Arrow, Section 1 (the theorem & a distributional difficulty of intransitivity + majority rule)
*Abram Bergson
*Buchanan and Tullock,
*Independence of irrelevant alternatives
*Pareto efficiency, strong and weak
*Path dependence, contrasted in Arrow with path independence, which a social ordering assures
*Political argument
*Public choice theory
*Social choice theory
*Social welfare function
*Voting paradox
*Voting system
*Welfare economics
* by scrolling down for Social Choice

*Kenneth J. Arrow, 1951, 2nd ed., 1963, "Social Choice and Individual Values" ISBN 0-300-01364-7
*_____, 1983, "Collected Papers of Kenneth J. Arrow", v. 1, "Social Choice and Justice" ISBN 0-674-13760-4
*_____, 1987, “Arrow’s Theorem," "", v. 1, pp. 124-26.
*_____, 2008. "Arrow's theorem." "The New Palgrave Dictionary of Economics, 2nd Edition [ Abstract.]
*Amartya K. Sen, 1970 [1984] , "Collective Choice and Social Welfare", ch. 1-7.1. LCCN|83|0|27163. ISBN 0-444-85127-5.

External links

* [ Table of Contents] with links to chapters.

* [ Link to text] of Nobel prize lecture with Section 8 on the theory and background.

* [ Comments] of Frank Hahn, Donald Saari, and Nobelists James M. Buchanan and Douglass North./

* [ Economic-justice high theory ] with Arrow’s framework, context, and references in Sections 1 & 4.

* James M. Buchanan (1954). "Social Choice, Democracy, and Free Markets", "Journal of Political Economy", 62(2), p [ p. 114] -123.

* I. M. D. Little (1952). "Social Choice and Individual Values," "Journal of Political Economy", 60(5) , p [ p. 422] -432.

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