TheInfoList

Social choice theory or social choice is a
theoretical A theory is a rational Rationality is the quality or state of being rational – that is, being based on or agreeable to reason Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek, G ...

framework for analysis of combining individual opinions, preferences, interests, or welfares to reach a ''collective decision'' or ''social welfare'' in some sense.
Amartya Sen Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom ( ...

(2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd Edition
Abstract & TOC.
/ref> Whereas choice theory is concerned with individuals making choices based on their preferences, social choice theory is concerned with how to translate the preferences of individuals into the preferences of a group. A non-theoretical example of a collective decision is enacting a law or set of laws under a
constitution A constitution is an aggregate of fundamental principles A principle is a proposition or value that is a guide for behavior or evaluation. In law, it is a rule Rule or ruling may refer to: Human activity * The exercise of political ...

. Another example is voting, where individual preferences over candidates are collected to elect a person that best represents the group's preferences. Social choice blends elements of
welfare economics Welfare economics is a branch of economics that uses microeconomics, microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level. Attempting to apply the principles of welfare economics gives rise to the fiel ...
and
public choice Public choice, or public choice theory, is "the use of economic An economy (from Greek language, Greek οίκος – "household" and νέμoμαι – "manage") is an area of the Production (economics), production, Distribution (economics), ...
theory. It is methodologically individualistic, in that it aggregates preferences and behaviors of individual members of society. Using elements of
formal logic Logic is an interdisciplinary field which studies truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to re ...
for generality, analysis proceeds from a set of seemingly reasonable
axiom An axiom, postulate or assumption is a statement that is taken to be , to serve as a or starting point for further reasoning and arguments. The word comes from the Greek ''axíōma'' () 'that which is thought worthy or fit' or 'that which comm ...

s of social choice to form a ''
social welfare functionIn welfare economics, a social welfare function is a function (mathematics), function that ranks social states (alternative complete descriptions of the society) as less desirable, more desirable, or indifference curve, indifferent for every possible ...
'' (or ''constitution''). Results uncovered the logical incompatibility of various axioms, as in Arrow's theorem, revealing an
aggregation problem An ''aggregate'' in economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consum ...
and suggesting reformulation or theoretical
triage In medicine Medicine is the science Science () is a systematic enterprise that builds and organizes knowledge Knowledge is a familiarity, awareness, or understanding of someone or something, such as facts ( descriptive knowledge) ...

in dropping some axiom(s).

# Overlap with public choice theory

"Public choice" and "social choice" are heavily overlapping fields of endeavor. Social choice and
public choice Public choice, or public choice theory, is "the use of economic An economy (from Greek language, Greek οίκος – "household" and νέμoμαι – "manage") is an area of the Production (economics), production, Distribution (economics), ...
theory may overlap but are disjoint if narrowly construed. The Journal of Economic Literature classification codes place Social Choice under
Microeconomics Microeconomics is a branch of mainstream economics Mainstream economics is the body of knowledge, theories, and models of economics, as taught by universities worldwide, that are generally accepted by economists as a basis for discussion. Als ...
at JEL D71 (with Clubs, Committees, and Associations) whereas most Public Choice subcategories are in JEL D72 (Economic Models of Political Processes:
Rent-Seeking In public-choice theory, as well as in economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumpt ...
, Elections, Legislatures, and Voting Behavior). Social choice theory (and public choice theory) dates from
Condorcet Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet (; 17 September 1743 – 29 March 1794), known as Nicolas de Condorcet, was a French philosopher A philosopher is someone who practices philosophy. The term ''philosopher'' comes from ...

's formulation of the
voting paradox The Condorcet paradox (also known as the voting paradox or the paradox of voting) in social choice theory Social choice theory or social choice is a theoretical A theory is a rational Rationality is the quality or state of being rational ...
, though it arguably goes back further to
Ramon Llull Ramon Llull (; c. 1232 – c. 1315/16) was a philosopher A philosopher is someone who practices philosophy Philosophy (from , ) is the study of general and fundamental questions, such as those about reason, Metaphysics, existence, ...

's 1299 publication.
Kenneth Arrow Kenneth Joseph Arrow (23 August 1921 – 21 February 2017) was an American economist, mathematician, writer, and . He was the joint winner of the with in 1972. In economics, he was a major figure in post-World War II . Many of his former gradu ...
's ''
Social Choice and Individual Values #REDIRECT Social Choice and Individual Values#REDIRECT Social Choice and Individual Values Kenneth Arrow's monograph ''Social Choice and Individual Values'' (1951, 2nd ed., 1963) and a theorem within it created modern social choice theory Soc ...
'' (1951),
Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
and often acknowledged as the basis of the modern social choice theory and public choice theory. In addition to Arrow's theorem and the voting paradox, the
Gibbard–Satterthwaite theorem In social choice theory, the Gibbard–Satterthwaite theorem is a result published independently by philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic Ranked voting, ordinal electoral systems that ...
, the
Condorcet jury theoremCondorcet's jury theorem is a political science theorem about the relative probability of a given group of individuals arriving at a correct decision. The theorem was first expressed by the Marquis de Condorcet in his 1785 work ''Essay on the Applica ...

, the
median voter theoremThe median voter theorem is a proposition relating to ranked preference voting put forward by Duncan Black in 1948.Duncan Black, 'On the Rationale of Group Decision-making' (1948). It states that if voters and policies are distributed along a one-d ...

, and May's theorem are among the more well known results from social choice theory.
Amartya Sen Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom ( ...

's Nobel Prize winning work was also highly influential. See the #Interpersonal utility comparison section below for more about Sen's work. Later work also considers approaches to compensations and
fairness Fairness or being fair can refer to: * Justice * The character in the award-nominated musical comedy ''A Theory of Justice: The Musical!, A Theory of Justice: The Musical.'' * Equity (law), a legal principle allowing for the use of discretion a ...
, liberty and rights, axiomatic
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function *Doma ...
restrictions on
preferences In psychology, economics and philosophy, a preference is a technical term usually used in relation to choosing between wikt:alternative, alternatives. For example, someone prefers A over B if they would rather choose A than B. Preference can also b ...
of agents, variable populations,
strategy-proof In game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a uni ...
ing of social-choice mechanisms,
natural resources , Malaysia Malaysia ( ; ) is a country in Southeast Asia. The federation, federal constitutional monarchy consists of States and federal territories of Malaysia, thirteen states and three federal territories, separated by the South China Se ...
, capabilities and functionings, and
welfare Welfare (or commonly, social welfare) is a type of government support intended to ensure that members of a society can meet basic human needs Maslow's hierarchy of needs is an idea in psychology Psychology is the science of mind and ...
,
justice Justice, in its broadest sense, is the principle that people receive that which they deserve, with the interpretation of what then constitutes "deserving" being impacted upon by numerous fields, with many differing viewpoints and perspectives, ...
, and
poverty Poverty is the state of having little material possessions or income In microeconomics, income is the Consumption (economics), consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expresse ...

.

# Interpersonal utility comparison

Social choice theory is the study of theoretical and practical methods to aggregate or combine individual preferences into a collective social welfare function. The field generally assumes that individuals have
preference In psychology Psychology is the science of mind and behavior. Psychology includes the study of consciousness, conscious and Unconscious mind, unconscious phenomena, as well as feeling and thought. It is an academic discipline of immense scop ...

s, and it follows that they can be modeled using
utility function As a topic of economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumpti ...
s. But much of the research in the field assumes that those utility functions are internal to humans, lack a meaningful unit of measure and ''cannot'' be compared across different individualsLionel Robbins (1932, 1935, 2nd ed.). ''An Essay on the Nature and Significance of Economic Science'', London: Macmillan. Links fo
1932 HTML
an
1935 facsimile
Whether this type of ''interpersonal utility comparison'' is possible or not significantly alters the available mathematical structures for social welfare functions and social choice theory. In one perspective, following
Jeremy Bentham Jeremy Bentham (; 15 February 1748 Old_Style_and_New_Style_dates">O.S._4_February_1747.html" ;"title="Old_Style_and_New_Style_dates.html" ;"title="nowiki/>Old Style and New Style dates">O.S. 4 February 1747">Old_Style_and_New_Style_dates.htm ...

,
utilitarians Utilitarianism is a family of normative Normative generally means relating to an evaluative standard. Normativity is the phenomenon in human societies of designating some actions or outcomes as good or desirable or permissible and others as ba ...
have argued that preferences and utility functions of individuals are interpersonally comparable and may therefore be added together to arrive at a measure of aggregate utility. Utilitarian ethics call for maximizing this aggregate. In contrast many twentieth century economists, following
Lionel Robbins Lionel Charles Robbins, Baron Robbins, (22 November 1898 – 15 May 1984) was a British economist An economist is a professional and practitioner in the social science Social science is the Branches of science, branch of science devot ...

, questioned whether mental states, and the utilities they reflect, can be measured and, ''a fortiori'', interpersonal ''comparisons'' of
utility As a topic of economics Economics () is a social science Social science is the Branches of science, branch of science devoted to the study of society, societies and the Social relation, relationships among individuals within thos ...

as well as the social choice theory on which it is based. Consider for instance the
law of diminishing marginal utility In economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods a ...
, according to which utility of an added quantity of a good decreases with the amount of the good that is already in possession of the individual. It has been used to defend transfers of wealth from the "rich" to the "poor" on the premise that the former do not derive as much utility as the latter from an extra unit of income. Robbins (
1935 Events January * January – Emperor Haile Selassie of Ethiopia creates a military school at Holeta.positive science; that is, one cannot measure changes in the utility of someone else, nor is it required by positive theory. Apologists of the interpersonal comparison of utility have argued that Robbins claimed too much.
John Harsanyi John Charles Harsanyi ( hu, Harsányi János Károly; May 29, 1920 – August 9, 2000) was a Hungarian- American Nobel Prize laureate economist An economist is a practitioner in the social sciences, social science discipline of economics. Th ...
agrees that full comparability of mental states such as utility is never possible but believes, however, that human beings are able to make some interpersonal comparisons of utility because they share some common backgrounds, cultural experiences, etc. In the example from
Amartya Sen Amartya Kumar Sen (; born 3 November 1933) is an Indian economist and philosopher, who since 1972 has taught and worked in the United Kingdom The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom ( ...

(1970, p. 99), it should be possible to say that
Emperor Nero Nero ( ; Nero Claudius Caesar Augustus Germanicus; 15 December 37 – 9 June 68 AD) was the fifth Roman emperor The Roman Emperor was the ruler of the Roman Empire during the History of the Roman Empire, imperial period (starting in 27 BC) ...
's gain from burning Rome was outweighed by the loss incurred by the rest of the Romans. Harsanyi and Sen thus argue that at least partial comparability of utility is possible, and social choice theory proceeds under that assumption. Sen proposes, however, that comparability of interpersonal utility need not be partial. Under Sen's theory of informational broadening, even complete interpersonal comparison of utility would lead to socially suboptimal choices because mental states are malleable. A starving peasant may have a particularly sunny disposition and thereby derive high utility from a small income. This fact should not nullify, however, his claim to compensation or equality in the realm of social choice. Social decisions should accordingly be based on immalleable factors. Sen proposes interpersonal utility comparisons based on a wide range of data. His theory is concerned with access to advantage, viewed as an individual's access to goods that satisfy basic needs (e.g., food), freedoms (in the
labor market Labour economics seeks to understand the functioning and dynamics of the markets Market may refer to: *Market (economics) *Market economy *Marketplace, a physical marketplace or public market Geography *Märket, an island shared by Finlan ...
, for instance), and capabilities. We can proceed to make social choices based on real variables, and thereby address actual position, and access to advantage. Sen's method of informational broadening allows social choice theory to escape the objections of Robbins, which looked as though they would permanently harm social choice theory. Additionally, since the seminal results of Arrow's impossibility theorem and the
Gibbard–Satterthwaite theorem In social choice theory, the Gibbard–Satterthwaite theorem is a result published independently by philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic Ranked voting, ordinal electoral systems that ...
, many positive results focusing on the restriction of the domain of preferences of individuals have elucidated such topics as optimal voting. The initial results emphasized the impossibility of satisfactorily providing a social choice function free of
dictatorship A dictatorship is a form of government A government is the system or group of people governing an organized community, generally a state State may refer to: Arts, entertainment, and media Literature * ''State Magazine'', a month ...
and inefficiency in the most general settings. Later results have found natural restrictions that can accommodate many desirable properties.

# Empirical social choice studies

Since Arrow social choice analysis has primarily been characterized by being extremely theoretical and formal in character. However, since ca. 1960 attention began to be paid to empirical applications of social choice theoretical insights, first and foremost by American political scientist
William H. Riker William Harrison Riker (September 22, 1920 – June 26, 1993) was an American political scientist who is prominent for applying game theory and mathematics to political science. He helped to establish Rochester school, University of Rochester as a ...
. The vast majority of such studies have been focused on finding empirical examples of the
Condorcet paradox The Condorcet paradox (also known as the voting paradox or the paradox of voting) in social choice theory Social choice theory or social choice is a Theory, theoretical framework for analysis of combining individual opinions, preferences, inter ...
. A summary of 37 individual studies, covering a total of 265 real-world elections, large and small, found 25 instances of a Condorcet paradox, for a total likelihood of 9.4% (and this may be a high estimate, since cases of the paradox are more likely to be reported on than cases without). On the other hand, the empirical identification of a Condorcet paradox presupposes extensive data on the decision-makers' preferences over all alternatives—something that is only very rarely available. While examples of the paradox seem to occur occasionally in small settings (e.g., parliaments) very few examples have been found in larger groups (e.g. electorates), although some have been identified.

# Social choice rules

Let $X$ be a set of possible 'states of the world' or 'alternatives'. Society wishes to choose a single state from $X$. For example, in a single-winner election, $X$ may represent the set of candidates; in a
resource allocation In economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviou ...

setting, $X$ may represent all possible allocations. Let $I$ be a finite set, representing a collection of individuals. For each $i \in I$, let $u_i:X\longrightarrow\mathbb$ be a ''
utility function As a topic of economics Economics () is the social science that studies how people interact with value; in particular, the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumpti ...

'', describing the amount of happiness an individual ''i'' derives from each possible state. A ''social choice rule'' is a mechanism which uses the data $\left(u_i\right)_$ to select some element(s) from $X$ which are 'best' for society. The question of what 'best' means is the basic question of social choice theory. The following rules are most common: *The ''utilitarian rule'' - also called the ''max-sum rule'' - aims to maximize the sum of utilities, thus maximizing the efficiency. *The ''egalitarian rule'' - also called the ''max-min rule'' - aims to maximize the smallest utility, thus maximizing the fairness. *The '' proportional-fair rule'' - sometimes called the ''max-product rule'' - aims to balance between the previous two rules, attaining a balance between efficiency and fairness.

# Social choice functions

A social choice function or a voting rule takes an individuals' complete and transitive preferences over a set of candidates (also called alternatives), and returns some subset of (possible singular) the candidates. We can think of this subset as the winners of an election. This is different from social welfare function, which returns a linear order of the set of alternatives as opposed to simply selecting some subset. We can compare different social choice functions based on which axioms or mathematical properties they fulfill. For example,
Instant-runoff voting Instant-runoff voting (IRV) is a type of Ranked voting, ranked preferential electoral system, vote counting method used in single-seat elections with more than two candidates. IRV is also sometimes referred to as the alternative vote (AV), pre ...
satisfies the Independence of clones criterion, whereas the
Borda count The Borda count is a family of single-winner election method An electoral system or voting system is a set of rules that determine how elections and referendums A referendum (plural: referendums or less commonly referenda) is a direct and ...
does not.

## Theorems concerning social choice functions

Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem#REDIRECT Arrow's impossibility theorem {{Redirect category shell, 1= {{R from other capitalisation ...
{{Redirect category shell, 1= {{R from other capitalisation ...
is what often comes to mind when one thinks about impossibility theorems in voting. However, Arrow was concerned with social welfare functions, not social choice functions. There are several famous theorems concerning social choice functions. The
Gibbard–Satterthwaite theorem In social choice theory, the Gibbard–Satterthwaite theorem is a result published independently by philosopher Allan Gibbard in 1973 and economist Mark Satterthwaite in 1975. It deals with deterministic Ranked voting, ordinal electoral systems that ...
states that all non-dictatorial voting rules that is resolute (it always returns a single winner no matter what the ballots are) and non-imposed (every alternative could be chosen) with more than three alternatives (candidates) is manipulable. That is, a voter can cast a ballot that misrepresents their preferences to obtain a result that is more favorable to them under their sincere preferences. The Campbell-Kelley theorem states that, if there exists a
Condorcet winner An electoral system satisfies the Condorcet criterion (; also known as the Condorcet winner criterion) if it always chooses the Condorcet winner when one exists. Any voting method conforming to the Condorcet criterion is known as a Condorcet method ...
, then selecting that winner is the unique resolute, neutral,
anonymous Anonymous may refer to: * Anonymity, the state of an individual's identity, or personally identifiable information, being publicly unknown ** Anonymous work, a work of art or literature that has an unnamed or unknown creator or author Organiza ...
, and non-manipulable voting rule. May's theorem states that when there are only two candidates,
Simple majority vote Plurality voting is an electoral system An electoral system or voting system is a set of rules that determine how elections and Referendum, referendums are conducted and how their results are determined. Political electoral systems are orga ...
is the unique neutral,
anonymous Anonymous may refer to: * Anonymity, the state of an individual's identity, or personally identifiable information, being publicly unknown ** Anonymous work, a work of art or literature that has an unnamed or unknown creator or author Organiza ...
, and positively responsive voting rule.

*
Compensation principle In welfare economics Welfare economics is a branch of economics that uses microeconomic Microeconomics (from Greek prefix ''mikro-'' meaning "small" + ''economics'') is a branch of economics that studies the behavior of individuals and Theo ...
* Computational social choice *
Condorcet paradox The Condorcet paradox (also known as the voting paradox or the paradox of voting) in social choice theory Social choice theory or social choice is a Theory, theoretical framework for analysis of combining individual opinions, preferences, inter ...
* Emotional choice theory *
Extended sympathyExtended sympathy in welfare economics Welfare economics is a branch of economics that uses microeconomics, microeconomic techniques to evaluate well-being (welfare) at the aggregate (economy-wide) level. Attempting to apply the principles of wel ...
*
Game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
*
Group decision-makingGroup decision-making (also known as collaborative decision-making or collective decision-making) is a situation faced when individuals An individual is that which exists as a distinct entity. Individuality (or self-hood) is the state or quality of ...
*
Justice (economics)Justice Justice, one of the four cardinal virtues, by Vitruvio Alberi, 1589–1590. Fresco, corner of the vault, studiolo of the Virgin of Mercy, Madonna of Mercy, Palazzo Altemps, Rome Justice, in its broadest sense, is the principle that peopl ...
*
Liberal paradox The liberal paradox, also Sen paradox or Sen's paradox, is a logical paradox A paradox, also known as an antinomy, is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despit ...
*
Mechanism design function f(\theta) maps a type profile to an outcome. In games of mechanism design, agents send messages M in a game environment g. The equilibrium in the game \xi(M,g,\theta) can be designed to implement some social choice function f(\theta). Me ...
*
Nakamura numberIn cooperative game theory and social choice theory, the Nakamura number measures the degree of rationality of preference aggregation rules (collective decision rules), such as voting rules. It is an indicator of the extent to which an aggregation r ...
*
Rational choice theory Rational choice theory refers to a set of guidelines that help understand economic and social behaviour. The theory postulates that an individual will perform a cost-benefit analysis to determine whether an option is right for them. It also sugge ...
*
Rule according to higher law The rule according to a higher law is a statement which expresses that no law may be enforced by the government unless it conforms with certain universal principles (written or unwritten) of fairness, morality, and justice. Thus, ''the rule accor ...
*
Voting system An electoral system or voting system is a set of rules that determine how elections and referendums A referendum (plural: referendums or less commonly referenda) is a direct and universal vote in which an entire electorate is invited to vote ...

# References

* Arrow, Kenneth J. (1951, 2nd ed., 1963). ''
Social Choice and Individual Values #REDIRECT Social Choice and Individual Values#REDIRECT Social Choice and Individual Values Kenneth Arrow's monograph ''Social Choice and Individual Values'' (1951, 2nd ed., 1963) and a theorem within it created modern social choice theory Soc ...
'', New York: Wiley. *_____, (1972). "General Economic Equilibrium: Purpose, Analytic Techniques, Collective Choice", Nobel Prize Lecture
with Section 8 on the theory and background. *_____, (1983). ''Collected Papers'', v. 1, ''Social Choice and Justice'', Oxford: Blackwell * Arrow, Kenneth J., Amartya K. Sen, and Kotaro Suzumura, eds. (1997). ''Social Choice Re-Examined'', 2 vol., London: Palgrave Macmillan & *_____, eds. (2002). ''Handbook of Social Choice and Welfare'', v. 1. Chapter-previe
*_____, ed. (2011). ''Handbook of Social Choice and Welfare'', v. 2, Amsterdam: Elsevier. Chapter-previe
* Bossert, Walter and John A. Weymark (2008). "Social Choice (New Developments)," ''
The New Palgrave Dictionary of Economics ''The New Palgrave Dictionary of Economics'' (2018), 3rd ed., is an twenty-volume reference work on economics Economics () is a social science that studies the Production (economics), production, distribution (economics), distribution, ...
'', 2nd Edition, London: Palgrave Macmilla
Abstract.
* Dryzek, John S. and Christian List (2003). "Social Choice Theory and Deliberative Democracy: A Reconciliation," ''British Journal of Political Science'', 33(1), pp. 1-28, https://www.jstor.org/discover/10.2307/4092266?uid=3739936&uid=2&uid=4&uid=3739256&sid=21102056001967, 2002 PD
* Feldman, Allan M. and Roberto Serrano (2006). ''Welfare Economics and Social Choice Theory'', 2nd ed., New York: Springer ,
Arrow-searchable chapter previews.
* (1996). Théories économiques de la justice, Paris: Economica. * * Harsanyi, John C. (1987). "Interpersonal Utility Comparisons," ''The New Palgrave: A Dictionary of Economics'', v. 2, London: Palgrave, pp. 955–58. * * * * Robbins, Lionel (1935). ''
An Essay on the Nature and Significance of Economic Science Lionel Robbins Lionel Charles Robbins, Baron Robbins, (22 November 1898 – 15 May 1984) was a British economist, and prominent member of the economics department at the London School of Economics (LSE). He is known for his leadership at LSE, hi ...
'', 2nd ed., London: Macmillan, ch. VI * ____, (1938). "Interpersonal Comparisons of Utility: A Comment," ''Economic Journal'', 43(4), 635–41. * (1970 984. ''Collective Choice and Social Welfare'', New York: Elsevier
Description.
*_____, (1998). "The Possibility of Social Choice", Nobel Prize Lectur

* _____, (1987). "Social Choice," ''The New Palgrave: A Dictionary of Economics'', v. 4, London: Palgrave, pp. 382–93. * _____, (2008). "Social Choice,". ''The New Palgrave Dictionary of Economics'', 2nd Edition, London: Palgrav
Abstract.
* . A comprehensive reference from a computational perspective; see Chapter 9

* Kotaro Suzumura, Suzumura, Kotaro (1983). ''Rational Choice, Collective Decisions, and Social Welfare'', Cambridge: Cambridge University Press *