Social Choice and Individual Values - AbsoluteAstronomy.com

Kenneth Arrow

Kenneth 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 have received this award, at 51....

's monograph

Monograph

A monograph is a work of writing upon a single subject, usually 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 theory

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...

, a rigorous melding of social ethics

Ethics

Ethics, also known as moral philosophy, is a branch of philosophy that addresses questions about morality—that is, concepts such as good and evil, right and wrong, virtue and vice, justice and crime, etc.Major branches of ethics include:...

and voting theory with an economic

Economics

Economics is the social science that analyzes 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. 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

Arrow's impossibility theorem

In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...

. 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

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...

, 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 examplify 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

Voting paradox

from use of majority rule

Majority rule

Majority rule is a decision rule that selects alternatives which have a majority, that is, 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.

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

Ordinal utility

Ordinal utility theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles of goods can be ordered such that one is considered by an individual to be worse than, equal to,...

of individual choice to represent behavioral patterns. Cardinal measures

Cardinal utility

In economics, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations...

of individual utility

Utility

In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

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

Abram Bergson

Abram Bergson , born Abram Burk, was an American economist. He was born in New York City....

, whose formulation of a social welfare function

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 the economic welfare of a society...

launched ordinalist welfare economics

Welfare economics

Welfare economics is a branch of economics that uses microeconomic techniques to evaluate economic well-being, especially relative to competitive general equilibrium within an economy as to economic efficiency and the resulting income distribution associated with it...

, 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

Game theory

Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...

, the compensation principle

Compensation principle

In 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 economics

Welfare economics

, extended sympathy

Extended sympathy

Extended 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 indiscernibles

Identity of indiscernibles

The 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...

, logrolling

Logrolling

Logrolling 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...

, and similarity of social judgments through single-peaked preferences

Arrow's impossibility theorem

, Kant’s categorical imperative

Categorical imperative

The Categorical Imperative is the central philosophical concept in the moral philosophy of Immanuel Kant, as well as modern deontological ethics...

, or the decision process

Decision making

Decision making can be regarded as the mental processes resulting in the selection of a course of action among several alternative scenarios. Every decision making process produces a final choice. The output can be an action or an opinion of choice.- Overview :Human performance in decision terms...

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
Vector space
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 distribution among voters of commidities, 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 x

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 allocation

Resource allocation

Resource allocation is used to assign the available resources in an economic way. It is part of resource management. In project management, resource allocation is the scheduling of activities and the resources required by those activities while taking into consideration both the resource...

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

Indifference curve

In microeconomic theory, an indifference curve is a graph showing different bundles of goods between which a consumer is indifferent. That is, at each point on the curve, the consumer has no preference for one bundle over another. One can equivalently refer to each point on the indifference curve...

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

Total order

In 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 economics

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

Consumer theory

Consumer choice is a theory of microeconomics that relates preferences for consumption goods and services to consumption expenditures and ultimately to consumer demand curves. The link between personal preferences, consumption, and the demand curve is one of the most closely studied relations in...

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
Factors of production
In economics, factors of production means inputs and finished goods means output. Input determines the quantity of output i.e. output depends upon input. Input is the starting point and output is the end point of production process and such input-output relationship is called a production function...

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

Consumer theory

with perfect competition

Perfect competition

In economic theory, perfect competition describes markets such that no participants are large enough to have the market power to set the price of a homogeneous product. Because the conditions for perfect competition are strict, there are few if any perfectly competitive markets...

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

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

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....

"). 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 the following "apparently reasonable" conditions to constrain the social ordering(s) of the constitution (pp. 25, 96-97).

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
Universal prescriptivism is the meta-ethical view which claims that, rather than expressing propositions, ethical sentences function similarly to imperatives which are universalizable — whoever makes a moral judgment is committed to the same judgment in any situation where the same relevant facts...

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 ordinalism

Ordinal utility

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 suggests, Pareto unanimity (with universal domain) overrides any social convention selecting some social state.

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
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...

[1]: 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
Arrow's impossibility theorem
In social choice theory, Arrow’s impossibility theorem, the General Possibility Theorem, or Arrow’s paradox, states that, when voters have three or more distinct alternatives , no voting system can convert the ranked preferences of individuals into a community-wide ranking while also meeting a...

[2]: 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. But it is invoked in Arrow (1963, ch. VIII) for a simpler proof than in Arrow (1951). In the latter, Arrow uses 2 other conditions, that with (2) above imply Pareto (1963, p. 97; 1987, p. 127):

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 , ..., 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 for at least one x and y 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 paradox

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

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...

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

Welfare economics

and differentiated Arrow's constitution from the pre-Arrow social welfare function
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 the economic welfare of a society...

. In so doing, it also ruled out any one consistent social ordering to which an agent

Agent (economics)

In 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 might appeal in trying to implement social welfare through the votes of others under the constitution. The result generalizes and deepens the voting paradox

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

William H. Riker

William Harrison Riker was an American political scientist who applied game theory and mathematics to political science....

and Dahl

Robert A. Dahl

Robert Alan Dahl , is the Sterling Professor emeritus of political science at Yale University, where he earned his Ph.D. in political science in 1940. He is past president of the American Political Science Association...

that as an empirical matter intransitivity

Intransitivity

In mathematics, the term intransitivity is used for related, but different, properties of binary relations:- Intransitivity :A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C...

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

Invisible hand

In economics, invisible hand or invisible hand of the market is the term economists use to describe the self-regulating nature of the marketplace. This is a metaphor first coined by the economist Adam Smith...

and Bentham

Jeremy Bentham

Jeremy Bentham was an English jurist, philosopher, and legal and social reformer. He became a leading theorist in Anglo-American philosophy of law, and a political radical whose ideas influenced the development of welfarism...

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

Economic policy

Economic 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 labor market, national ownership, and many other areas of government interventions into the economy.Such policies are often...

and social choice theory

Social choice theory

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
Frank Hahn
Frank Horace Hahn is a British economist whose work has focused on general equilibrium theory, monetary theory, Keynesian economics and monetarism...

, Donald Saari
Donald G. Saari
Donald 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. Buchanan
James M. Buchanan
James McGill Buchanan, Jr. is an American economist known for his work on public choice theory, for which he received the 1986 Nobel Memorial Prize in Economic Sciences. Buchanan's work initiated research on how politicians' self-interest and non-economic forces affect government economic policy...

and Douglass North
Douglass North
Douglass Cecil North is an American economist known for his work in economic history. He is the co-recipient of the 1993 Nobel Memorial Prize in Economic Sciences...

.
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.

H.S. Houthakker
Hendrik S. Houthakker
Hendrik Samuel "Hank" Houthakker was a Dutch Jewish-born American economist.-Life and career:Houthakker was born in Amsterdam. His father was a prominent art dealer...

(1952). [Review], Economic Journal, 62(246), p p. 355-58.

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

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.

Introduction

Terminology

Conditions and theorem

Proof

Summary, interpretation, and aftereffects

See also

External links