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Social Choice and Individual Values
Originally published in 1951, Social Choice and Individual Values introduced “Arrow’s Impossibility Theorem” and founded the field of social choice theory in economics and political science. This new edition, including a new foreword by Nobel laureate Eric Maskin, reintroduces Arrow’s seminal book to a new generation of students and researchers.
"Far beyond a classic, this small book unleashed the ongoing explosion of interest in social choice and voting theory. A half-century later, the book remains full of profound insight: its central message, ‘Arrow’s Theorem,’ has changed the way we think.”—Donald G. Saari, author of Decisions and Elections: Explaining the Unexpected
"Far beyond a classic, this small book unleashed the ongoing explosion of interest in social choice and voting theory. A half-century later, the book remains full of profound insight: its central message, ‘Arrow’s Theorem,’ has changed the way we think.”—Donald G. Saari, author of Decisions and Elections: Explaining the Unexpected
144 pages, Paperback
First published January 1, 1951
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December 18, 2012
Arrow made his PhD thesis into this book in 1951. It remains one of the most remarkable works in the strange field of economics in the twentieth century. This is the book that launched a thousand puzzled explanations by troubled neo-classical economists in the years following its publication.
The book starts off with the voting paradox, which Arrow attributes to E.J. Nansen (1896). This goes as follows: if A, B, and C are three alternatives and 1, 2, and 3 are three individuals and supposing that 1 prefers A to B and C to A, 2 prefers B to C and A to B, and 3 prefers C to A and B to C, then the majority (of two) prefer A to B but also B to C. A "rational" conclusion would be that the community prefers A to C, but in fact, a majority prefers C to A. So the voting procedure has led to a result that is not "rational". Unfortunately, this paradox is not resolved, but rather deepened as Arrow proceeds.
Starting from Bentham's hedonistic calculus (refuted forcefully by Plato 2500 years ago in the Philebus) and the voting paradox (attributed to Condorcet by many), Arrow proves in math / logical notation that, given five conditions that most neo-classical economists would think should be honored in any theory of social choice, any solution to the social welfare question in which there are at least three alternative choices must either be imposed or dictatorial, thus contradicting two of the assumed conditions. His "possibility" theorem (in the original; it has been referred to as his "impossibility theorem" by many) "shows that, if no prior assumptions are made about the nature of individual orderings, there is no method of voting which will remove the paradox of voting . . . neither plurality voting nor any scheme of proportional representation, no mater how complicated. Similarly, the market mechanism does not create a rational social choice." (Chapter V, Section 4).
In his last chapter Arrow shows that for "single-peaked preferences" majority decision does lead to an outcome satisfying five revised conditions, including the revision to Condition 1 to the effect that "the tastes of individuals fall within certain prescribed realms of similarity" and "provided the number of individuals is odd."(!) Arrow then asks "Do these or possibly other mathematical restrictions have any social significance?" His answer is "I do not pretend to have any definite answer" but that uniformity of social attitudes actually is a condition of one prominent school of political philosophy, namely the "idealist school" of Rousseau, Kant, and T.H.Green. In a footnote he quotes Rousseau to the effect that the law of plurality is established by agreement, which must have been unanimous, at least in the beginning. Rousseau's unanimous consent would, of course, satisfy Arrow's revised Condition 1.
Arrow's conclusion, however, is that even if some degree of consensus is taken as true, "Any view which depends on consensus as the basis for social action certainly implies that the market mechanism cannot be taken as the social welfare function since that mechanism cannot take account of the altruistic motives which must be present to secure that consensus."
So there you have it; the smartest (oddest?) economist of his generation has proven that his "science" cannot be placed on a rational basis given the assumptions that it requires for a market economy. A market economy cannot lead to rational social welfare! My guess is that the only economists that are not worried about this are the ones that remain blind to the issues that Arrow has raised or aren't talking because their salaries depend on keeping silent.
The book starts off with the voting paradox, which Arrow attributes to E.J. Nansen (1896). This goes as follows: if A, B, and C are three alternatives and 1, 2, and 3 are three individuals and supposing that 1 prefers A to B and C to A, 2 prefers B to C and A to B, and 3 prefers C to A and B to C, then the majority (of two) prefer A to B but also B to C. A "rational" conclusion would be that the community prefers A to C, but in fact, a majority prefers C to A. So the voting procedure has led to a result that is not "rational". Unfortunately, this paradox is not resolved, but rather deepened as Arrow proceeds.
Starting from Bentham's hedonistic calculus (refuted forcefully by Plato 2500 years ago in the Philebus) and the voting paradox (attributed to Condorcet by many), Arrow proves in math / logical notation that, given five conditions that most neo-classical economists would think should be honored in any theory of social choice, any solution to the social welfare question in which there are at least three alternative choices must either be imposed or dictatorial, thus contradicting two of the assumed conditions. His "possibility" theorem (in the original; it has been referred to as his "impossibility theorem" by many) "shows that, if no prior assumptions are made about the nature of individual orderings, there is no method of voting which will remove the paradox of voting . . . neither plurality voting nor any scheme of proportional representation, no mater how complicated. Similarly, the market mechanism does not create a rational social choice." (Chapter V, Section 4).
In his last chapter Arrow shows that for "single-peaked preferences" majority decision does lead to an outcome satisfying five revised conditions, including the revision to Condition 1 to the effect that "the tastes of individuals fall within certain prescribed realms of similarity" and "provided the number of individuals is odd."(!) Arrow then asks "Do these or possibly other mathematical restrictions have any social significance?" His answer is "I do not pretend to have any definite answer" but that uniformity of social attitudes actually is a condition of one prominent school of political philosophy, namely the "idealist school" of Rousseau, Kant, and T.H.Green. In a footnote he quotes Rousseau to the effect that the law of plurality is established by agreement, which must have been unanimous, at least in the beginning. Rousseau's unanimous consent would, of course, satisfy Arrow's revised Condition 1.
Arrow's conclusion, however, is that even if some degree of consensus is taken as true, "Any view which depends on consensus as the basis for social action certainly implies that the market mechanism cannot be taken as the social welfare function since that mechanism cannot take account of the altruistic motives which must be present to secure that consensus."
So there you have it; the smartest (oddest?) economist of his generation has proven that his "science" cannot be placed on a rational basis given the assumptions that it requires for a market economy. A market economy cannot lead to rational social welfare! My guess is that the only economists that are not worried about this are the ones that remain blind to the issues that Arrow has raised or aren't talking because their salaries depend on keeping silent.
May 13, 2025
It is extremely interesting to read this masterpiece in the origins of Social Choice. As a soon-to-be economist, I really loved to read this.
Kenneth Arrow was, indeed, a true Master of Mathematical Economics. We owe him so much.
Kenneth Arrow was, indeed, a true Master of Mathematical Economics. We owe him so much.
December 23, 2024
A fascinating work of relatively simple preference ordering that demonstrates that under generally accepted rules of welfare economics, no choice between 3+ alternatives can be made without being imposed or under dictatorial rule. Yet the consequences of such probably are profound, striking at the very heart of the mathematics of choice and welfare economics. Whether this speaks to a limit of rationality, choice, or the formalism of modern economics remains under debate.
January 30, 2018
This book launched the modern study of social choice theory with its famous proof of the General Possibility Theorem, better known as "Arrow's Impossibility Theorem." Arrow's theorem shows the impossibility of a class of social welfare functions (roughly, rules that generates a social ranking of social states from a group of individual ordering of social states). Arrow showed that it is not possible for a social welfare function to simultaneously satisfy the conditions of universal domain (all possible individual orderings are admissible), Pareto efficiency (if everyone prefers one social state x over another y, the SWF ranks x higher than y), independence of irrelevant alternatives (the social choice between x and y depends only on x and y), and non-dictatorship (no one person always decides the social choice between any two social states). However, the proof of this remarkable theorem only takes up one chapter of this book, and in the remainder of the text Arrow takes up a host of questions regarding social choice theory itself and the implications of the impossibility result. For example, one implication of the Arrow result is that no theory of compensation, such as those explored in welfare economics, is satisfactory. Arrow also takes up questions of interpretation, such as the political significance and meaning of the social welfare function itself. Arrow devotes attention to one possible response to the impossibility result--relaxing the requirement of universal domain--and connects the role of convergence with the role of political philosophy in social choice. In this sense, Arrow directly anticipates the work of social contract theorists like Rawls, but also the response of deliberative democrats like Habermas to the impossibility result. This work is a marvelous integration of social science and normative theory and it richly deserves its reputation.
November 5, 2021
Very thought provoking.
I think I disagree with Arrow on some fundamental points. Going from "utility functions are incommensurable" to "ordinal rankings with independence of irrelevant alternatives" seems like a bonkers move to me.
Decent chunks of the book got into the minutiae of semantics or the feasibility of what seemed like totally wrong-headed theories. I also skipped over the actual proofs because I got the gist of them from wikipedia. But I learned a lot, and it was great to get some background on the debate of whether total utilitarianism can be be a practical guide of policy.
Notes:
Apparently Bentham agreed with the incommensurability of different utility functions:
Arrow neither believes that utilities are summable across people, nor in the utility functions of single individuals.
The "Maximin Theorem" is important in game theory.
On p. 112:
Apparently Bergon's "A Reformulation of Certain Aspects of Welfare Economics" is important background on the non-utility-welfare approach Arrow is taking.
On p.27 Arrow lists some problems he sees with not having IIA.
Similarly on p. 110: he says unfeasible alternatives shouldn't factor into decisions.
Arrow refers to total utilitarianism as "Bentham-Edgeworth sum of individual utilities". Who is Edgeworth?
On p.32, Arrow argues that VNM violates IIA, because as you introduce new alternatives you have to recalibrate the utility functions to be between 0 and 1, and this can change the ordering.
On p.10 Arrow rejects VNM as a basis for social welfare because "it would be to assert that the distribution of social income is to be determined by the tastes of individuals for gambling."
On p.48 Arrow proves something like May's Theorem then comments "Theorem 1 is, in a sense, the logical foundation of the Anglo-American two-party system." Intereseting take.
On p.85 in some of the most bizarre post-impossibility floundering, Arrow considers the possibility that perhaps voting should be thought of as using the wisdom of the crowd to figure out what is the one true Platonic welfare function. "Voting, from this point of view, is not a device where each individual expresses his personal interests, but rather where each gives his opinion on the general will."
From p. 110 there's an interesting survey of empirical methods for calibrating utility functions. For example, Dahl asks what if we had a utility which was the same for eveyone? For example, the disutility of having to cast a vote? And Goodman and Markowitz explore the possibility that everyone has the same quantum increment of utility, so we could calibrate on "just noticeable" differences in utility.
I think I disagree with Arrow on some fundamental points. Going from "utility functions are incommensurable" to "ordinal rankings with independence of irrelevant alternatives" seems like a bonkers move to me.
Decent chunks of the book got into the minutiae of semantics or the feasibility of what seemed like totally wrong-headed theories. I also skipped over the actual proofs because I got the gist of them from wikipedia. But I learned a lot, and it was great to get some background on the debate of whether total utilitarianism can be be a practical guide of policy.
Notes:
Apparently Bentham agreed with the incommensurability of different utility functions:
Tis in vain to talk of adding quantities which after the addition will continue distinct as they were before, one man’s happiness will never be another man’s happiness: a gain to one man is no gain to another: you might as well pretend to add twenty apples to twenty pears, which after you had done that could not be forty of any one thing but twenty of each just as there was before.
Arrow neither believes that utilities are summable across people, nor in the utility functions of single individuals.
The "Maximin Theorem" is important in game theory.
On p. 112:
If empirically meaningful interpersonal comparisons have to be based on indifference maps, as we have argued, then the Independence of Irrelevant Alternatives must be violated. The information which enables us to assert that individual A prefers x to y more strongly than B prefers y to x must be based on comparisons by A and B of x and y not only to each other but also to other alternatives.
Apparently Bergon's "A Reformulation of Certain Aspects of Welfare Economics" is important background on the non-utility-welfare approach Arrow is taking.
On p.27 Arrow lists some problems he sees with not having IIA.
Similarly on p. 110: he says unfeasible alternatives shouldn't factor into decisions.
Arrow refers to total utilitarianism as "Bentham-Edgeworth sum of individual utilities". Who is Edgeworth?
On p.32, Arrow argues that VNM violates IIA, because as you introduce new alternatives you have to recalibrate the utility functions to be between 0 and 1, and this can change the ordering.
On p.10 Arrow rejects VNM as a basis for social welfare because "it would be to assert that the distribution of social income is to be determined by the tastes of individuals for gambling."
On p.48 Arrow proves something like May's Theorem then comments "Theorem 1 is, in a sense, the logical foundation of the Anglo-American two-party system." Intereseting take.
On p.85 in some of the most bizarre post-impossibility floundering, Arrow considers the possibility that perhaps voting should be thought of as using the wisdom of the crowd to figure out what is the one true Platonic welfare function. "Voting, from this point of view, is not a device where each individual expresses his personal interests, but rather where each gives his opinion on the general will."
From p. 110 there's an interesting survey of empirical methods for calibrating utility functions. For example, Dahl asks what if we had a utility which was the same for eveyone? For example, the disutility of having to cast a vote? And Goodman and Markowitz explore the possibility that everyone has the same quantum increment of utility, so we could calibrate on "just noticeable" differences in utility.
March 16, 2023
The book investigates the problem of aggregating individual preferences into a collective decision and introduces what is now known as Arrow's Impossibility Theorem. This theorem has had a profound impact on the fields of welfare economics, political science, and decision-making theory.
Arrow's Impossibility Theorem, also known as Arrow's Paradox, states that it is impossible to design a social welfare function that satisfies a set of seemingly reasonable conditions. In particular, Arrow demonstrates that no voting system can simultaneously satisfy the following criteria: (1) Pareto efficiency, (2) independence of irrelevant alternatives, and (3) non-dictatorship. This result implies that there is no perfect method for aggregating individual preferences into a collective decision, and every voting system will have some inherent flaws.
One of the key takeaways from "Social Choice and Individual Values" is the recognition that collective decision-making is inherently complex and cannot be reduced to a simple formula or rule. Arrow's Impossibility Theorem highlights the trade-offs and limitations that arise when trying to aggregate individual preferences and demonstrates the need for a more nuanced understanding of social choice processes.
Another important takeaway from the book is the need to question the assumptions underlying different voting systems and methods of collective decision-making. By challenging the idea that a perfect social welfare function exists, Arrow's work has spurred further research into alternative voting systems and decision-making mechanisms that can better account for the complexities of social choice.
"Social Choice and Individual Values" has been widely praised for its rigorous analysis, innovative approach, and the significant impact it has had on the study of social choice and welfare economics. The book has not only transformed the way economists and political scientists think about collective decision-making but has also influenced the development of public policy and democratic institutions.
Arrow's Impossibility Theorem, also known as Arrow's Paradox, states that it is impossible to design a social welfare function that satisfies a set of seemingly reasonable conditions. In particular, Arrow demonstrates that no voting system can simultaneously satisfy the following criteria: (1) Pareto efficiency, (2) independence of irrelevant alternatives, and (3) non-dictatorship. This result implies that there is no perfect method for aggregating individual preferences into a collective decision, and every voting system will have some inherent flaws.
One of the key takeaways from "Social Choice and Individual Values" is the recognition that collective decision-making is inherently complex and cannot be reduced to a simple formula or rule. Arrow's Impossibility Theorem highlights the trade-offs and limitations that arise when trying to aggregate individual preferences and demonstrates the need for a more nuanced understanding of social choice processes.
Another important takeaway from the book is the need to question the assumptions underlying different voting systems and methods of collective decision-making. By challenging the idea that a perfect social welfare function exists, Arrow's work has spurred further research into alternative voting systems and decision-making mechanisms that can better account for the complexities of social choice.
"Social Choice and Individual Values" has been widely praised for its rigorous analysis, innovative approach, and the significant impact it has had on the study of social choice and welfare economics. The book has not only transformed the way economists and political scientists think about collective decision-making but has also influenced the development of public policy and democratic institutions.
December 10, 2017
Social Choice and Individual Values es en gran medida el punto de partido del estudio moderno de la elección social y la economía del bienestar. Originalmente la tesis de Arrow eventualmente tendría impacto en la economía y la ciencia política. Es el trabajo del cual surge el famoso teorema de imposibilidad. Todo un clásico de la economía de uno de los economistas más importantes desde la segunda mitad del siglo XX.
Todo estudiante de economía aprende sobre el teorema en algún punto dentro de sus clases de microeconomía, sin embargo, suele ser de forma simplificada, traducida para distintos niveles de complejidad (dependiendo de lo avanzada que sea o no la clase). Leerlo en su versión original, siendo la idea desarrollada y demostrada matemáticamente por Arrow deja ver el porque en se le considera uno de los gigantes de la economía.
Todo estudiante de economía aprende sobre el teorema en algún punto dentro de sus clases de microeconomía, sin embargo, suele ser de forma simplificada, traducida para distintos niveles de complejidad (dependiendo de lo avanzada que sea o no la clase). Leerlo en su versión original, siendo la idea desarrollada y demostrada matemáticamente por Arrow deja ver el porque en se le considera uno de los gigantes de la economía.
June 29, 2024
This is my second dive into Social Choice Theory, my first being Amartya Sen's Collective Choice and Social Welfare. Having been prepared by Sen for a book laden with theorems and proofs, I set out to try to further my understanding of Social Choice Theory. I am still very much a novice in the practice of reading the style of argumentation based upon translating concepts into mathematical theorems, and it felt very much like reading Euclid or Spinoza for the first or second time.
Still, individual theorems were graspable if I was patient with the arguments, and I took the time to re-read definitions and conditions, much like engaging in proofs from either math or logic. And the framework Arrow sets out to accomplish i.e. individual choice when examined in aggregate reflect social values, makes sense and was well founded based upon his reasoning and his definitions.
Still, individual theorems were graspable if I was patient with the arguments, and I took the time to re-read definitions and conditions, much like engaging in proofs from either math or logic. And the framework Arrow sets out to accomplish i.e. individual choice when examined in aggregate reflect social values, makes sense and was well founded based upon his reasoning and his definitions.
December 24, 2021
READ DEC 2021
I was intrigued by this one because of how COVID is being dealt with around the world.
This was a short book, but was long as an academic piece. Very technical. The focus was on the tension between individual values and how a social group arrives at a decision on the choices they have. How do individuals make a choice, say about masks and vaccines, and to what extent, if any do they need to consider the impact their chance has on the common good?
Best quote that captures the essence of this work is, social choice is the “aggregation of individual preferences” (p. 103).
Where this aggregation is more nuanced, with no stark variation, we see more divisiveness.
I was intrigued by this one because of how COVID is being dealt with around the world.
This was a short book, but was long as an academic piece. Very technical. The focus was on the tension between individual values and how a social group arrives at a decision on the choices they have. How do individuals make a choice, say about masks and vaccines, and to what extent, if any do they need to consider the impact their chance has on the common good?
Best quote that captures the essence of this work is, social choice is the “aggregation of individual preferences” (p. 103).
Where this aggregation is more nuanced, with no stark variation, we see more divisiveness.
June 22, 2021
I have to give this book 4 stars which I hate about myself because I'm aware it's a groundbreaking, foundational text which would underpin any true understanding of our democratic process but I had a few beers while I was reading it (it was very hot) and I skimmed through the predicate logic and now I am not sure if I could explain, for example, the fact that there exists DICTATORS WALKING AMONG US in language that is at all coherent or compelling. The lost star is a subjective and damning personal experience with this monumental work.
October 14, 2018
Within the scope of todays orthodoxy Arrows work (I read a 3rd printing of the 2nd Edition) seems ontologically reductive but it is more than that. Genuine discourse by a generation of economists to put their
This entire review has been hidden because of spoilers.
May 25, 2024
Interesting to actually read the original impossibility theorem. I found the sections on compensation and single-peaked preferences had useful intuition (e.g., why Kaldor-Hicks orderings aren’t transitive).
December 27, 2017
Dear KJA
Thanks for giving me a field to work in.
Thanks for giving me a field to work in.
March 13, 2019
Great and pioneering presentation on the core problem of social choice. Arrow presented here a dilemma, that has not yet been solved without relaxing some of the requirements presented.
February 16, 2022
Way too mathematically encoded. I believe this book could have been written in a more comprehensive way. The interpretations and conclusions are interesting though.
Read
October 7, 2016A very intellectually stimulating book.
November 14, 2014
"Ken evidently knew more on virtually any subject than any of the rest of us. There was a story at Harvard that a group of junior faculty once concocted a plan by which they could finally appear to outshine their erudite senior colleague. They read up on the most arcane topic they could think of: the breeding habits of gray whales. On the appointed day, they gathered in the coffee room and waited for Ken to come in. Then they started talking about the elaborate theory of a marine biologist named Turner on how gray whales find their way back to the same breeding spot year after year. Ken was silent . . . they had him at last! With a delicious sense of triumph, they continued to discuss Turner, while Ken looked increasingly perplexed. Finally, he couldn't hold back: 'But I thought Turner's theory was discredited by Spenser, who showed that the supposed homing mechanism couldn't possibly work'"
-Eric Maskin, in forward to Kenneth Arrow's Social Choice and Individual Values
"If we exclude the possibility of interpersonal comparisons of utility, then the only methods of passing from individual tastes to social preferences which will be satisfactory and which will be defined for a wide range of sets of individual orderings are either imposed or dictatorial . . .
"[W]e can also phrase the result this way: If consumers' values can be represented by a wide range of individual orderings, the doctrine of voters' sovereignty is incompatible with that of collective rationality."
-Kenneth Arrow, Social Choice and Individual Values
-Eric Maskin, in forward to Kenneth Arrow's Social Choice and Individual Values
"If we exclude the possibility of interpersonal comparisons of utility, then the only methods of passing from individual tastes to social preferences which will be satisfactory and which will be defined for a wide range of sets of individual orderings are either imposed or dictatorial . . .
"[W]e can also phrase the result this way: If consumers' values can be represented by a wide range of individual orderings, the doctrine of voters' sovereignty is incompatible with that of collective rationality."
-Kenneth Arrow, Social Choice and Individual Values
considering
October 3, 2018It was in the context of the on-going search for acceptable formulations of social welfare that Kenneth Arrow presented his well-known ‘‘impossibility theorem’’. His book Social Choice and Individual Values (published in 1951) launched the new subject of social choice theory. Arrow considered a set of very mild-looking conditions relating social choices or judgments to the set of individual preferences. Arrow showed that it is impossible to satisfy those apparently undemanding conditions simultaneously. The ‘‘impossibility theorem’’ precipitated a major crisis in welfare economics, and it is, in fact, a landmark in the history of social and political study as well as economics.
Capabilities and Happiness Pág.20
Capabilities and Happiness Pág.20
April 6, 2016
Classic, lots of set theory and proofs. Firm challenge to the expectation that democratic decision-making can create the best social outcomes without a universally shared set of aims, or an imposed set of aims. Clear writing between the maths.
March 5, 2008
Magnificent book. Arrow's theorem is a phenomenal celebration of the human spirit.
Want to read
August 22, 2008Referenced in http://intermexfreemarket.blogspot.co...
August 2, 2009
One of the foundations of decision theory and choice. Hear Arrow speak once when I was in graduate school, some years before he won the Nobel Prize in economics.
September 21, 2015
Arrow's impossibility theorem
https://en.wikipedia.org/wiki/Arrow%2...
https://en.wikipedia.org/wiki/Arrow%2...
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