Kenneth ArrowKenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to receive this award, at 51....
's
monographA monograph is a work of writing upon a single subject, usually also by a single author. It is often a scholarly essay or learned treatise, and may be released in the manner of a book or journal article. It is by definition a single document that forms a complete text in itself...
Social Choice and Individual Values (1951, 2nd ed., 1963) and a theorem within it created modern
social choice theorySocial choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
, a rigorous melding of social
ethicsEthics is a branch of philosophy which seeks to address questions about morality, such as what the fundamental semantic, ontological, and epistemic nature of ethics or morality is , how moral values should be determined , how a moral outcome can be achieved in specific situations , how moral...
and voting theory with an
economicEconomics is the social science that studies the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
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) theoremIn social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria with three or more discrete options to choose from...
. 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 paradoxThe voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic , even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other...
, which shows that majority voting may fail to yield a stable outcome.
Introduction
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 paradoxThe voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic , even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other...
from use of
majority ruleMajority rule is a decision rule that selects one of two alternatives, based on which has more than half the votes. It is the binary decision rule used most often in influential decision-making bodies, including the legislatures of democratic nations...
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. |
The book 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 rankingsOrdinal utility theory states that while the utility of a particular good and service cannot be measured using a numerical scale, different alternatives can be ordered into worse, equal or better. Goods are often considered in ‘bundles’ or ‘baskets’...
of individual choice to represent behavioral patterns.
Cardinal measuresIn economics, cardinal utility is a theory of utility under which the utility gained from a particular good or service can be measured and that the magnitude of the measurement is meaningful. Under cardinal utility theory, the util is a unit of measurement much like the metre or second...
of individual
utilityIn economics, utility is a measure of the relative satisfaction from, or desirability of, consumption of various goods and services. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's 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 BergsonAbram Bergson , born Abram Burk, was an American economist. He was born in New York City.In a 1938 paper Bergson defined and discussed the notion of an individualistic social welfare function...
, whose formulation of a
social welfare functionIn economics, a social welfare function is a real-valued function that ranks conceivable social states from lowest to highest. Inputs of the function include any variables considered to affect welfare of the society...
launched ordinalist
welfare economicsWelfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine allocative efficiency within an economy and the income distribution associated with it. It analyzes social welfare, however measured, in terms of economic activities of the individuals that...
, 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 theoryGame theory is a branch of applied mathematics that is used in the social sciences, most notably in economics, as well as in biology, engineering, political science, international relations, computer science, and philosophy...
, the
compensation principleIn welfare economics, the compensation principle refers to a decision rule used to select between pairs of alternative feasible social states. One of these states is the hypothetical point of departure...
in
welfare economicsWelfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine allocative efficiency within an economy and the income distribution associated with it. It analyzes social welfare, however measured, in terms of economic activities of the individuals that...
,
extended sympathyExtended sympathy in welfare economics refers to interpersonal value judgments of the form that social state x for person A is ranked better than, worse than, or as good as social state y for person B...
, Leibniz's principle of the
identity of indiscerniblesThe identity of indiscernibles is an ontological principle which states that two or more objects or entities are identical , if they have all their properties in common. That is, entities x and y are identical if any predicate possessed by x is also possessed by y and vice versa...
,
logrollingLogrolling is the trading of favors or quid pro quo, such as vote trading by legislative members to obtain passage of actions of interest to each legislative member. It is also the "cross quoting" of papers by academics in order to drive up reference counts...
, and similarity of social judgments through
single-peaked preferencesIn social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria with three or more discrete options to choose from...
, Kant’s
categorical imperativeThe categorical imperative is the central philosophical concept in the moral philosophy of Immanuel Kant, as well as modern deontological ethics...
, or the
decision processDecision making can be regarded as an outcome of mental processes leading to the selection of a course of action among several alternatives. Every decision making process produces a final choice...
.
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") comprised of individuals (“voters” here) in the following form:
- Voters, a finite set with at least two members, indexed as .
- Commodities, the objects of choice (things that voters might want, goods and services), including private and collective (municipal services, statecraft, etc.).
- A social state, a complete description (representable as a vector
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
) of a (conceivable) bundle of 'commodities' as to quantity and distribution among voters, labor supplied by each voter, and resources used in each productive activity.
- 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 top to bottom for each possible ranking and with 'T's indexing ties:
x y z x y T z (x T y z repeats
y x z y T x z y T x z, so is omitted, etc.)
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 x T y T z |
The 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 allocationResource allocation is used to assign the available resources in an economic way. It is part of resource management.-Strategic planning:In strategic planning, resource allocation is a plan for using available resources, for example human resources, especially in the near term, to achieve goals for...
is specified in the production of each social state in the ordering.
The comprehensive character 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 In microeconomic theory, an indifference curve is a graph showing different bundles of goods, each measured as to quantity, between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. In other words, they are all equally...
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.
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
orderingIn mathematics and set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
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 economicsWelfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine allocative efficiency within an economy and the income distribution associated with it. It analyzes social welfare, however measured, in terms of economic activities of the individuals that...
, 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 mapConsumer theory is a theory of microeconomics that relates preferences to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most complex relations in economics...
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
In economics, factors of production are the resources employed to produce goods and services. They facilitate production but do not become part of the product or are significantly transformed by the production process...
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 theoryConsumer theory is a theory of microeconomics that relates preferences to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most complex relations in economics...
with perfect competitionIn neoclassical economics and microeconomics, perfect competition describes the perfect being a market in which there are many small firms, all producing homogeneous goods...
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
elementIn mathematics, an element or member of a set is any one of the distinct objects that make up that set.- Sets :Writing A = {1, 2, 3, 4 }, means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example {1, 2}, are...
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
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 paralysisPath dependence explains how the set of decisions one faces for any given circumstance is limited by the decisions one has made in the past, even though past circumstances may no longer be relevant....
"). 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 refers to a constitution satisfying this condition as collective rationality. It can be compared to the rationality of a voter ordering. But the prescription Universal prescriptivism is the meta-ethical view which claims that:# Ethical sentences do not express propositions....
of collective rationality, which Arrow proposes, is distinct from the descriptive use of a voter ordering, which he deploys. Hence, his denial at the end of the book that collective rationality is "merely an illegitimate transfer from the individual to society." (p. 120) |
- 2. Independence of irrelevant alternatives I: Let ', ..., ' and , ..., be 2 sets of orderings in the constitution. Let S be a subset of hypothetically available (not merely conceivable) social states, say x and y, from the set of social states. For each voter i, let the ranking of x and y be the same for ' and for . (Different voters could still have different rankings of the 2 social states.) Then the social orderings for the 2 respective sets of orderings select the same state(s) from the subset as the social choice.
| Condition I: Let ', ..., and , ..., be 2 sets of orderings in the constitution. Let S be any subset of hypothetically available social states from the set of social states. For each voter i and for each pair of x and y in S, let x y if and only if x ' y. Then the social choice functions for the 2 respective sets of orderings map to an identical social choice set: C(S) = C'(S). |
| This identical mapping happens even with differences in rankings of any voter between the two sets of orderings outside that subset of social states. Consider a hypothetical “run-off vote” between say only 2 available social states. The social choice is associated with the sets of rankings for that subset, not with rankings of unavailable social states beyond the subset. Thus, that social choice for the subset is unaffected by say a change in orderings only beyond the subset. |
Arrow describes this condition as an extension of ordinalismOrdinal utility theory states that while the utility of a particular good and service cannot be measured using a numerical scale, different alternatives can be ordered into worse, equal or better. Goods are often considered in ‘bundles’ or ‘baskets’...
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.#
| Condition P: For any x and y in the set of social states, if, for every voter i, x y, then x P y. |
| As Sen (ch. 3.4) suggests, Pareto unanimity (with universal domain) overrides any social convention selecting some social state that is otherwise vote-immune. |
The conditions, particularly the second and third, may seem minimal, but jointly they are harsh:
- Arrow’s Theorem
In social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria with three or more discrete options to choose from...
: 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:
- 4. Nondictatorship D: No voter in the society is a dictator. That is, there is no voter i in the society such that for every set of orderings in the domain of the constitution and every pair of distinct social states x and y, if voter i strictly prefers x over y, x is socially selected over y.
| Condition D: There is no voter i in {1, ..., n} such that for every set of orderings in the domain of the constitution and every pair of social states x and y, x y implies x P y. |
- Arrow's Theorem
In social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria with three or more discrete options to choose from...
: The constitution is impossible, that is, the 4 conditions of a constitution imply a contradiction.
| Each voter has an ordering (by attribution). Yet a set of orderings used as an argument of the voting rule does not carry over to a social ordering, with a corresponding loss of social adaptivity and constitutional generality, whether descriptive or prescriptive. |
- 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):
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 ', ..., ' 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.
| For the special case of all preferring y over x, the vote still makes no difference. If the invariant social ranking applies to only one pair of distinct social states, the constitution would violate N. In this respect, as a representation of excluding convention, N is thorough. |
Proof
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 paradoxThe voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic , even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other...
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
welfaristWelfarism is a form of consequentialism. Like all forms of consequentialism, welfarism is based on the premise that actions, policies, and/or rules should be evaluated on the basis of their consequences. Welfarism is the view that the morally significant consequences are impacts on human welfare...
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 economicsWelfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine allocative efficiency within an economy and the income distribution associated with it. It analyzes social welfare, however measured, in terms of economic activities of the individuals that...
and differentiated Arrow's constitution from the pre-Arrow social welfare functionIn economics, a social welfare function is a real-valued function that ranks conceivable social states from lowest to highest. Inputs of the function include any variables considered to affect welfare of the society...
. 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 agentIn economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....
or official intent on implementing the values of others in the society through the constitution. The remaining alternative, nondictatorship, excludes a pre-Arrow social welfare function as a consistent
voting machineVoting machines are the total combination of mechanical, electromechanical, or electronic equipment , that is used to define ballots; to cast and count votes; to report or display election results; and to maintain and produce any audit trail information...
. The result generalizes and deepens the
voting paradoxThe voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic , even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other...
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
intransitivityIn mathematics, the term intransitivity is used for related, but different, properties of binary relations:- The property of not being transitive :...
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 SmithIn economics, the invisible hand, also known as the invisible hand of the market, the term economists use to describe the self-regulating nature of the marketplace, is a metaphor first coined by the economist Adam Smith in The Theory of Moral Sentiments...
and
BenthamJeremy Bentham was an English jurist, philosopher, and legal and social reformer. He was the brother of Samuel Bentham. He was a political radical, and a leading theorist in Anglo-American philosophy of law...
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 policyEconomic policy refers to the actions that governments take in the economic field. It covers the systems for setting interest rates and government budget as well as the labour market, national ownership, and many other areas of government interventions into the economy.Such policies are often...
and
social choice theorySocial choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
.
See also
- Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a certain set of criteria with three or more discrete options to choose from...
- Kenneth Arrow, Section 1 (the theorem & a distributional difficulty of intransitivity + majority rule)
- Abram Bergson
Abram Bergson , born Abram Burk, was an American economist. He was born in New York City.In a 1938 paper Bergson defined and discussed the notion of an individualistic social welfare function...
- Buchanan and Tullock, The Calculus of Consent: Logical Foundations of Constitutional Democracy
- Independence of irrelevant alternatives
Independence of irrelevant alternatives is a term for an axiom of decision theory and various social sciences. Although exact formulations of IIA differ, intentions of the usages are similar in attempting to provide a rational account of individual behavior or aggregation of individual...
- Pareto efficiency
Pareto efficiency, or Pareto optimality, is an important concept in economics with broad applications in game theory, engineering and the social sciences. The term is named after Vilfredo Pareto, an Italian economist who used the concept in his studies of economic efficiency and income distribution...
, strong and weak
- Path dependence
Path dependence explains how the set of decisions one faces for any given circumstance is limited by the decisions one has made in the past, even though past circumstances may no longer be relevant....
, contrasted in Arrow with path independence, which a social ordering assures
- Political argument
A political argument is an instance of a logical argument applied to politics. Political arguments are used by academics, media pundits, candidates for political office and government officials. Political arguments are also used by citizens in ordinary interactions to comment about and understand...
- Public choice theory
Public choice in economic theory is the use of modern economic tools to study problems that are traditionally in the province of political science....
- Social choice theory
Social choice theory is a theoretical framework for measuring individual interests, values, or welfares as an aggregate towards collective decision. A non-theoretical example of a collective decision is passing a set of laws under a constitution. Social choice theory dates from Condorcet's...
- Social welfare function
In economics, a social welfare function is a real-valued function that ranks conceivable social states from lowest to highest. Inputs of the function include any variables considered to affect welfare of the society...
- Utilitarianism
Utilitarianism is the idea that the moral worth of an action is determined solely by its contribution to overall utility: that is, its contribution to happiness or pleasure as summed among all people...
- Voting paradox
The voting paradox is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic , even if the preferences of individual voters are not. This is paradoxical, because it means that majority wishes can be in conflict with each other...
- Voting system
A voting system or voting theory, allows voters to choose between options, often in an election where candidates are selected for public office. Voting can be also used to award prizes, to select between different plans of action, or by a computer program to find a solution to a problem...
- Welfare economics
Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine allocative efficiency within an economy and the income distribution associated with it. It analyzes social welfare, however measured, in terms of economic activities of the individuals that...
- Welfarism
Welfarism is a form of consequentialism. Like all forms of consequentialism, welfarism is based on the premise that actions, policies, and/or rules should be evaluated on the basis of their consequences. Welfarism is the view that the morally significant consequences are impacts on human welfare...
- JEL D71 by scrolling down for Social Choice
External links
- Link to text of Nobel prize lecture with Section 8 on the theory and background.
- Comments of Frank Hahn
Frank Horace Hahn is a British economist whose work has focussed on general equilibrium theory, monetary theory, Keynesian economics and monetarism...
, Donald SaariDonald Gene Saari is the Distinguished Professor of Mathematics and Economics and director of the Institute for Mathematical Behavioral Sciences at the University of California Irvine...
, and Nobelists James M. BuchananJames McGill Buchanan, Jr. is an American economist known for his work on public choice theory, for which he won the 1986 Nobel Prize in Economics. Buchanan's work initiated research on how politicians' self-interest and non-economic forces affect government economic policy.-Biography:Buchanan...
and Douglass NorthDouglass Cecil North is an American economist known for his work in the history of economic thought. He is the co-recipient of the 1993 Nobel Memorial Prize in Economic Sciences...
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- James M. Buchanan (1954). Social Choice, Democracy, and Free Markets, Journal of Political Economy, 62(2), pp. 114-123.
- H.S. Houthakker
Hendrik Samuel "Hank" Houthakker was a Dutch Jewish-born American economist.Houthakker was born in Amsterdam. His father was a prominent art dealer...
(1952). [Review], Economic Journal, 62{246], pp. 355-58.
- I. M. D. Little (1952). "Social Choice and Individual Values," Journal of Political Economy, 60(5) , pp. 422-432.