Kenneth Arrow's monograph Social Choice and Individual Values (1951, 2nd ed., 1963) and a theorem within it created 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. Kenneth Arrow Kenneth Joseph Arrow (born August 23, 1921) is an American economist, winner of the Bank of Sweden Prize in Economic Sciences in 1972. ... A monograph is a scholarly book or a treatise on a single subject or a group of related subjects. ... Social choice theory studies how individual preferences are aggregated to form a collective preference. ... Ethics (from the Ancient Greek ethikos, meaning arising from habit), a major branch of philosophy, is the study of value or quality. ... Voters at the voting booths in the US in 1945 Voting systems are methods (algorithms) for groups of people to select one or more options from many, taking into account the individual preferences of the group members. ... Face-to-face trading interactions among on the New York Stock Exchange trading floor Economics, as a social science, studies the production, distribution, and consumption of commodities. ...

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. In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ... The voting paradox (also known as Condorcets paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i. ...

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 paradox from accepting majority rule as a value. 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, he proposes removing the distinction between voting and markets in favor of a more general category of collective social choice. The voting paradox (also known as Condorcets paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i. ... Majoritarianism (often also called majority rule) is a political philosophy or agenda which asserts that a majority (sometimes categorized by religion, language or some other identifying factor) of the population is entitled to a certain degree of primacy in society, and has the right to make decisions that affect the...

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). Ordinal utility theory states that while the utility of a particular good and service cannot be measured using an objective scale; a consumer is capable of ranking different alternatives available. ... Cardinal utility theory states that the utility (satisfaction) gained from a particular good or service can be measured in the same way as distance, temperature and time can. ... In economics, utility is a measure of the happiness or satisfaction gained consuming good and services. ...

Following Abram Bergson, whose formulation of a social welfare function launched ordinalist welfare economics, Arrow avoids locating a social good as independent of individual values. Rather, social values inhere in actions from social-decision rules (hypostatized as constitutional conditions) using individual values as input. Then 'social values' means "nothing more than social choices." (p.106) Abram Bergson, born Abram Burk (April 21, 1914, New York City - April 23, 2003), was an American economist. ... A social welfare function, in welfare economics, is a function which gives a measure of the material welfare of society, given a number of economic variables as inputs. ... Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ...

Topics implicated along the way include game theory, the compensation principle in welfare economics, extended sympathy, Liebniz'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. Game theory is a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ... The compensation principle in welfare economics refers to a decision rule used to select between pairs of alternative feasible social states. ... Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ... 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 (Arrow, 1963, pp. ... The identity of indiscernibles is an ontological principle that states that if there is no way of telling two entities apart then they are one and the same entity. ... Logrolling is a colorful phrase used to describe trading of votes by legislative members to obtain passage of actions of interest to each legislative member. ... In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ... The categorical imperative is the philosophical concept central to the moral philosophy of Immanuel Kant and to modern deontological ethics. ... Decision making is the cognitive process of selecting a course of action from among multiple alternatives. ...

Terminology

The book defines a few terms and logical symbols used thereafter and their applied empirical interpretation. 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 1, 2, ..., i, ..., n.
  • 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) 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.

 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 allocation is specified in the production of each social state in the ordering. In strategic planning, a resource-allocation decision is a plan for using available resources, especially in the near term, to achieve goals for the future. ...

The comprehensive character of commodities, the set of social states, and the set of orderings was noted by early reviewers.

The 2 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.

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 satisfaction). (Here "as good as" is an "equally-ranked," not a "don't know," relation.) An indifference curve is a graph showing combinations of goods for which a consumer is indifferent, that is, it has no preference for one combination versus another, as they render the same level of satisfaction for the consumer. ...

An ordering of a voter is denoted by R. That ordering of voter i is denoted with a subscript as R_i.

If voter i changes orderings, primes distinguish the first and second, say R_i compared to R_i' . The same notation can apply for 2 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 1) 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 2 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).

x R y is either "social preference" of x over y or "social indifference" between x and y.

A social ordering applies to each ordering in the set of orderings (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. Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ...

Arrow shows how to go from the social ordering for a given set of orderings to a particular 'social choice' by specifying: Consumer theory is a theory of economics. ...

• 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 then selects the top-ranked social state(s) from the subset as the 'social choice'. Factors of production are resources used in the production of goods and services in economics. ...

Formally, 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, the top-ranked social state(s) for that environment and that set of orderings. Consumer theory is a theory of economics. ... Perfect competition is an economic model that describes a hypothetical market form in which no producer or consumer has the market power to influence prices. ...

The social choice function is denoted C(S). Consider an environment that has just 2 social states, x and y. S is denoted [x, y], and the social choice function is C([x, y]). Suppose x is the only top-ranked social state. Then C([x, y]) = {x}, the social choice. If x and y are instead tied, C([x, y]) = {x, y}.

The next section invokes the following. Let R and R' stand for different social orderings corresponding to different 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). Path-dependence exists when the outcome of a process depends on its past history, on the entire sequence of decisions made by agents and resulting outcomes, and not just on contemporary conditions. ...

Arrow proposes conditions to constrain the social ordering(s) of the constitution. 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.

• 2. Independence of irrelevant alternatives I: Let S be a subset of hypothetically available (not merely conceivable) social states, say x and y, from the set of social states. Let R₁, ..., R_n and R₁' , ..., R_n' be any 2 sets of orderings with the following property: For every voter i , the ranking of x and y in S is (by construction) the same. (Different voters could still have different rankings of the 2 social states.) Then the subset S for each of the 2 sets of orderings maps to an identical social choice.

Arrow describes this condition as an extension of ordinalism with its emphasis on prospectively observable behavior (based on availability of the subset in question). He ascribes practical advantage to the condition from "every known electoral system" satisfying it (p.110). A philosophy of modified Kantianism, originated by R. M. Hare, who believes that our moral judgments should be of the form I ought to do X in Y situation, whenever all of the relevant, universal properties of the facts that obtain in any similar situation are the same. ... This page is a candidate to be moved to Wiktionary. ... Independence of irrelevant alternatives (IIA) is an axiom often adopted by social scientists as a basic condition of rationality. ...

• 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

The conditions, particularly the second and third, may seem minimal, but jointly they are harsh:

• Arrow’s Theorem: The 3 conditions of the constitution imply a dictator who prevails as to the social choice whatever that individual's preference and those of all else.

An alternate statement of the theorem adds the following condition to the above: In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ...

• 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.

• Arrow's Theorem: The constitution is impossible, that is, the 4 conditions of a constitution imply a contradiction.

In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ...

Proof

The proof is in two parts. The first part considers if someone's ordering prevails ('is decisive') as to the social choice for some pair of social states no matter what the orderings of that voter and those of others. It is shown that the voter would prevail similarly for every pair of social states and thus all social states. So, the voter would be a dictator. Thus, avoiding dictatorship requires postulating that no one would so prevail for some pair of social states. The second part shows that, given the first 3 conditions, someone would so prevail for some pair of social states. This contradicts the postulate (and thus nondictatorship) and so proves the theorem.

Conclusion, challenge, 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 any constitution that allows dictatorship requires it, and any constitution that requires nondictatorship is contradictory. Hence, the paradox of social choice. Welfarism is a form of consequentialism. ...

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 others 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. Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ... A social welfare function, in welfare economics, is a function which gives a measure of the material welfare of society, given a number of economic variables as inputs. ... In economics, an agent is an element of a model who solves an optimization problem. ... A voting machine is a device to record and register votes to be counted as per any voting system, with or without printing a ballot for the voter to verify. ... The voting paradox (also known as Condorcets paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i. ...

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) In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox demonstrates the impossibility of designing a set of rules for social decision making that would meet all of a certain set of criteria. ... William Harrison Riker (September 22, 1920 - June 26, 1993) was an influential political scientist, who advanced the field of political science through his application of game theory and mathematics to the field. ... Robert Alan Dahl (b. ... Intransitivity is a scenario in which weighing several options produces a loop of preference. ...

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." To meet Wikipedias quality standards, this article or section may require cleanup. ... Jeremy Bentham (IPA: or ) (February 15, 1748 O.S. (February 26, 1749 N.S.) – June 6, 1832) was an English jurist, philosopher, and legal and social reformer. ...

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. Voters at the voting booths in the US in 1945 Voting systems are methods (algorithms) for groups of people to select one or more options from many, taking into account the individual preferences of the group members. ... Economic policy refers to the actions that governments take in the economic field. ... Social choice theory studies how individual preferences are aggregated to form a collective preference. ...

See also

• Arrow's impossibility theorem
• Kenneth Arrow, Section 1 (the theorem & a distributional difficulty of intransitivity + majority rule)
• Abram Bergson
• Buchanan and Tullock, The Calculus of Consent: Logical Foundations of Constitutional Democracy
• 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
• Utilitarianism
• Voting paradox
• Voting system
• Welfare economics
• Welfarism
• JEL D71 by scrolling down for Social Choice

In voting systems, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can possibly meet a certain set of reasonable criteria when there are three or more options to choose from. ... Kenneth Arrow Kenneth Joseph Arrow (born August 23, 1921) is an American economist, winner of the Bank of Sweden Prize in Economic Sciences in 1972. ... Abram Bergson, born Abram Burk (April 21, 1914, New York City - April 23, 2003), was an American economist. ... The Calculus of Consent: Logical Foundations of Constitutional Democracy is a book written by economists James M. Buchanan and Gordon Tullock in 1962. ... Independence of irrelevant alternatives (IIA) is an axiom often adopted by social scientists as a basic condition of rationality. ... Pareto efficiency, or Pareto optimality, is an important notion in economics with broad applications in game theory, engineering and the social sciences. ... Path-dependence exists when the outcome of a process depends on its past history, on the entire sequence of decisions made by agents and resulting outcomes, and not just on contemporary conditions. ... A political argument is an instance of a logical argument applied to politics. ... Public choice theory is a branch of economics that studies the decision-making behavior of voters, politicians and government officials from the perspective of economic theory, namely game theory and decision theory. ... Social choice theory studies how individual preferences are aggregated to form a collective preference. ... A social welfare function, in welfare economics, is a function which gives a measure of the material welfare of society, given a number of economic variables as inputs. ... This article or section does not cite its references or sources. ... The voting paradox (also known as Condorcets paradox or the paradox of voting) is a situation noted by the Marquis de Condorcet in the late 18th century, in which collective preferences can be cyclic (i. ... A voting system is a means of choosing between a number of options, based on the input of a number of voters. ... Welfare economics is a branch of economics that uses microeconomic techniques to simultaneously determine the allocational efficiency of a macroeconomy and the income distribution consequences associated with it. ... Welfarism is a form of consequentialism. ... Articles in economics journals are usually classified according to the system used by the Journal of Economic Literature (JEL). ...

References

• 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," The New Palgrave: A Dictionary of Economics, v. 1, pp. 124-26
• Amartya K. Sen, 1970 [1984], Collective Choice and Social Welfare, ch. 1-7.1 ISBN 0-444-85127-5

Kenneth Arrow Kenneth Joseph Arrow (born August 23, 1921) is an American economist, winner of the Bank of Sweden Prize in Economic Sciences in 1972. ... Amartya Sen Dr. Amartya Kumar Sen CH (Hon) (Bengali: Ômorto Kumar Shen) (born 3 November 1933 in Santiniketan, India), is an economist and a winner of the Bank of Sweden Prize in Economic Sciences (sometimes referred to informally as the Nobel Prize for Economics) in 1998, for his work on...

External links

• Acknowledgments

• Table of Contents

• Comments of Frank Hahn, Donald Saari, and Nobelists James M. Buchanan and Douglass North

• Link to text of Nobel prize lecture with Section 8 on the theory and background

• Economic-justice high theory with Arrow’s framework, context, and references in Sections 1 & 4