Arrow's Impossibility Theorem

11 Key excerpts on "Arrow's Impossibility Theorem"

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  Economics Paradoxes

  • (Author)
  • 2014(Publication Date)
  • White Word Publications

    (Publisher)

  ____________________ WORLD TECHNOLOGIES ____________________ Chapter 2 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 discrete alternatives (options), no voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a certain set of criteria. These criteria are called unrestricted domain , non-dictatorship , Pareto efficiency , and independence of irrelevant alternatives . The theorem is often cited in discussions of election theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his Ph.D. thesis and popularized it in his 1951 book Social Choice and Individual Values . The original paper was titled A Difficulty in the Concept of Social Welfare. Arrow was a co-recipient of the 1972 Nobel Memorial Prize in Economics. In short, the theorem proves that no voting system can be designed that satisfies these three fairness criteria: • If every voter prefers alternative X over alternative Y, then the group prefers X over Y. • If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change). • There is no dictator: no single voter possesses the power to always determine the group's preference. There are several voting systems that side-step these requirements by using cardinal utility (which conveys more information than rank orders) and weakening the notion of independence. Arrow, like many economists, rejected cardinal utility as a meaningful tool for expressing social welfare, and so focused his theorem on preference rankings.

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  Political Bargaining

  Theory, Practice and Process

  • Gideon Doron, Itai Sened(Authors)
  • 2001(Publication Date)
  • SAGE Publications Ltd

    (Publisher)

  Arrow’s (1951) theorem proved that if we impose minimal constraints such as conditions 2–5 discussed above, on mechanisms that are supposed to aggregate individual preferences into social choices, there would always exist preference profiles that yield cyclical social preference orders. This constitutes a violation of the transitivity condition that is so basic to any pretence of logical decision making, as we explained above. This important result suggests that the common perception of the operation of representative politics is naive. Mechanisms designed to aggregate individual preferences into social choices are bound to be indecisive, inasmuch as they yield cyclical social preference orders, or arbitrary, inasmuch as they allow ‘local dictators’ to manipulate the agenda and political outcomes. In short, social outcomes are not reflective of individual preferences to the extent we previously believed.

  Arrow’s Impossibility Theorem proves that any mechanism that tries to aggregate individual preferences into collective preference must violate at least one of Arrow’s conditions. Below we show how simple majority rule almost always violates the first condition and produces cyclical preference orders in violation of the transitivity assumption. In the remainder of the book we show how institutions and bargaining mechanisms help alleviate this problem by allowing complementary and substitute procedures to take over when simple social choice mechanism, like majority rule, that are supposed to aggregate individual preferences into social choices, fail.

  Meanwhile, it is important to note that Arrow provided a clear theory of the possibility of disagreement. If social order relied solely on mechanisms that aggregate individual preferences into social choices, society would be stuck in an uneasy balance. A mix of arbitrary cycles of choice, arbitrary dictates of local dictators and social choice preferences twisted by the peculiarities of the mechanism that was used to generate them. The process of political bargaining, discussed in this book, originates, among other things, in the need to alleviate the damage caused by these inherent sources of arbitrariness in political decision-making. Governments and constituents realize the damage caused by cyclical social preferences and arbitrary aggregation mechanisms and, therefore, engage in different processes of bargaining that they have designed in order to enable them to attain a more desirable state of affairs.

  2.3 The Chaos Theorem

  Arrow’s Impossibility Theorem provides a general statement concerning the arbitrariness of social decisions. In this section we discuss an important example of how political choices may be affected by this profound observation. The typical mechanism used to aggregate individual preferences into political choices is the two-stage scheme that asks adult citizens to elect representatives to a legislative body and then requires the elected members of legislative bodies to make social choices using majority rule. How then does the problem presented by Arrow and discussed in the previous section manifest itself in the context of this commonly used political institution?

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  Welfare Economics

  Towards a More Complete Analysis

  • Y. Ng(Author)
  • 2003(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  5.1 Arrow’s impossibility theorem Arrow (1951b) calls his theorem the general possibility theorem since he first proves another theorem, the possibility theorem, for the special case of two alternatives. However as the answer to the general possibility theorem is neg- ative, it is also called the impossibility theorem or the general impossibility theorem. In simple terms the theorem states that a rule (or a constitution) for deriving – from individual orderings of social states – a social ordering that is consistent with reasonable conditions cannot be found in general. 92 A first glimpse at the content of the theorem can be provided by the well- known ‘paradox of voting’. Assume that a three-person group is faced with a choice of three alternatives. A simple and obvious way of arriving at a collective ordering is to say that one alternative is preferred to another if the majority hold such preference. Now suppose that the preferences of the three individuals are as follows: xP 1 yP 1 z; zP 2 xP 2 y; yP 3 zP 3 x where x, y and z are the three alternatives and P i stands for ‘is preferred to’ for the ith individual. It can be seen that a majority prefers x to y and a majority prefers y to z. According to the rule of majority voting we can say that the group prefers x to y and y to z. By transitivity, which is usually accepted as a condition of logical consistency (and a necessary condition for being an ordering), x should be preferred to z. But actually a majority of the group prefers z to x. This shows that the rule of majority voting when making a social or collective choice can result in cyclicity (xPyPzPx) and a fortiori may fail to satisfy the requirement of transitivity. What Arrow proves in his theorem is that the difficulty illustrated by the paradox of voting is general, not just applicable to the majority rule, that is we cannot find any method or rule for passing from individual to collective ordering while satisfying reasonable conditions.

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  Mathematical Modeling in the Social and Life Sciences

  • Michael Olinick(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  After a 4-year stint in the U.S. Army Air Force during World War II, Arrow was a research associate with the Cowles Commission at the University of Chicago from 1947 to 1949. The Impossibility Theorem was part of his Ph.D. thesis and in finished form was published as a book, Social Choice and Individual Values, in 1951. Arrow began his work on the social welfare problem by trying to develop a rea- sonably fair function that took a collection of individual preference rankings and produced a group ranking. “I just started playing around,” he told one interviewer. “It took me about two days to decide I was on the wrong track because I was looking for some solution. It didn’t occur to me that there was no solution.” In 1949, Arrow joined the faculty of Stanford University where he taught for almost 20 years and was a major force in developing at Stanford an outstanding group of economic theorists and mathematical model builders. He also worked briefly with the Council of Economic Advisers during the administration of President John F. Kennedy. In 1968 Arrow moved to Harvard University where he became the James Bryant Conant University 218 CHAPTER 6 Social Choice and Voting Procedures Professor in 1974. In 1979, he returned to Stanford University with the position of Joan Kenney Professor of Economics and Professor of Operations Research. Arrow formally retired in 1991 but continues to be an active participant in economics conferences and was a vigorous bicyclist well into his mid-eighties. Arrow has written or edited many books and dozens of papers whose topics include the mathematical theory of inventory and production, time series analysis of interindustry demands, linear and nonlinear programming, public investment and optimal fiscal policy, the theory of risk bearing, and general competitive analysis.

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  Democratic Planning and Social Choice Dilemmas

  Prelude to Institutional Planning Theory

  • Tore Sager(Author)
  • 2020(Publication Date)
  • Routledge

    (Publisher)

  They hold that unrestricted scope and IIA are conditions to the effect that the institution should be a ‘real-world’ decision rule. (Most election procedures satisfy these conditions.) Furthermore, they regard the Pareto principle and non-dictatorship as conditions to the effect that the decision rule should be ‘reasonable’. Applied to voting, they can then reinterpret Arrow’s theorem to say that: ‘Every reasonable real-world voting rule is manipulable through strategic candidacies and similar maneuvers’ (ibid. 184). In the context of dialogical decision-making, this means that the outcome can be manipulated by altering the set of planning alternatives under consideration. This links Arrow’s theorem to the problem of manipulation discussed in Part III. It will now be investigated whether it is really necessary to limit the argumentative impossibility theorem to debates involving intangibles. Generalization of the Impossibility Result The aim of this section is to show that the scope of the impossibility theorem of dialogical decision-making is much wider than one is led to believe from the direct analogy with Arrow’s reasoning. It is argued that the theorem is valid in nearly all cases of making recommendations from a societal point of view in, e.g., land-use and transport planning. Arrow (1963) took individual utility to be ordinal and non-comparable. The last assumption implies that utility levels or changes in utility cannot be compared across individuals. It has been known for some years that the possibility of aggregating the individual preference relations is critically dependent on their measurability and comparability. Research in social choice theory after the establishment of Arrow’s theorem has explored the exact nature of this dependence (Roberts 1980, Blackorby et al. 1984, Kelsey 1987). There is little room for sensible social choice when interpersonal comparison of utility is not allowed

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  Choice, Preferences, and Procedures

  A Rational Choice Theoretic Approach

  • Kotaro Suzumura(Author)
  • 2016(Publication Date)
  • Harvard University Press

    (Publisher)

  Several weak conditions imposed on the resultant choice functions are each shown to contradict one or more of the same democratic requirements, even if the choices have no binary rationalization. 5.4. References [1] Arrow, K. J., “Rational Choice Functions and Orderings,” Economica NS 26 , 1959, 121–127. [2] Arrow, K. J., Social Choice and Individual Values, New York: Wiley, 1951; 2nd edn., 1963. [3] Blair, D. H., “Possibility Theorems for Non-binary Social Choice Functions,” unpublished manuscript. [4] Blair, D. H., “Path-Independent Social Choice Functions: A Further Result,” Econometrica 43 , 1975, 173–174. [5] Bordes, G., “Alpha-Rationality and Social Choice: A General Possibility Theorem,” unpub-lished manuscript. [6] Brown, D. J., “Acyclic Aggregation over a Finite Set of Alternatives,” unpublished manuscript, 1973. [7] Brown, D. J., “Aggregation of Preferences,” Quarterly Journal of Economics 89 , 1975, 456– 469. [8] Chernoff, H., “Rational Selection of Decision Functions,” Econometrica 22 , 1954, 423–443. [9] Fishburn, P. C., The Theory of Social Choice, Princeton, N.J.: Princeton University Press, 1973. [10] Gibbard, A., “Social Choice and the Arrow Conditions,” unpublished manuscript. Published subsequently in Economics and Philosophy 30 , 2014, 269–284. [11] Hansson, B., “Choice Structures and Preference Relations,” Synthese 18 , 1968, 443–458. [12] Herzberger, H. G., “Ordinal Preference and Rational Choice,” Econometrica 41 , 1973, 187– 237. [13] Kelly, J. S., “Two Impossibility Theorems on Independence of Path,” unpublished manu-script. 5. IMPOSSIBILITY THEOREMS WITHOUT COLLECTIVE RATIONALITY 201 [14] Mas-Colell, A., and H. Sonnenschein, “General Possibility Theorems for Group Decision Functions,” Review of Economic Studies 39 , 1972, 185–192. [15] Parks, R. P., “Choice Paths and Rational Choice,” unpublished manuscript.

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  Philosophy, Politics, and Economics

  An Introduction

  • Gerald Gaus, John Thrasher(Authors)
  • 2021(Publication Date)
  • Princeton University Press

    (Publisher)

  Arrow’s proof is sometimes said to reveal a conflict between “representation” and “coherence.” We can obtain coherent social preferences if we are willing to abandon the requirement that the social preference reflects the preferences of the body of citizens—we can accept a dictator or abandon the Weak Pareto principle—or we can ensure that the social preference reflects the preferences of the citizens, but then, as the Condorcet Paradox shows, we can end up with an incoherent social preference. One can easily see why Arrow’s theorem is seen as a challenge to the rationality of democracy. If the aim of democracy is to generate a social decision that (1) represents the preferences of the citizens no matter what their preferences, and yet (2) is coherent, it seems that democracy aims at the impossible. Not all the conditions can be met; the set is contradictory.

  The Importance of Arrow’s Theorem

  Does Arrow’s Theorem Challenge Democracy?

  Arrow’s theorem shows that there is no way to construct a Social Welfare Function that is guaranteed to meet his conditions. Democracy can be seen as a way of aggregative preferences (notions of betterness) into a social decision. So then does Arrow’s theorem undermine the rationality of democracy? Interestingly, some insist that it must cause us to question whether democracy can be said to be a way to generate a reasonable social choice, while others dismiss the theorem as interesting, but not crucial. There are four important ways to challenge it.

  First, we might dispute whether Arrow’s conditions are really intuitively compelling; to the extent that we do not mind dropping one of the conditions, the proof should not cause concern. The pairwise independence and unrestricted domain conditions both have been subject to considerable debate.

  Second, it is sometimes argued that Arrow’s theorem is concerned with mere “preferences,” but democratic decision-making pertains to rational judgments about what is in the common good; so, it is said, Arrow’s problem of how to aggregate individual preferences into a social preference is irrelevant to democratic decision- making. This challenge is, we think, misguided, for at least two reasons.

  (a)   As we have stressed throughout, a “preference” is simply a ranking of one option over another—it does not necessarily involve a liking, any sort of selfishness, etc. If democratic politics is about asking people to choose among candidates or policies (for whatever reasons), the idea of a preference is entirely appropriate.

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  Rational Choice

  • Andrew Hindmoor, Brad Taylor(Authors)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  Moving beyond a characterization theorem, we may impose further constraints and find that there is no possible rule which sat-isfies them. We then have an impossibility theorem such as Arrow’s. Thus, the procedure for proving that only one voting rule (or family Kenneth Arrow and Social Choice Theory 123 thereof) is consistent with certain normative requirements must, in a mathematical sense, always be a step along the way to proving that no voting rule is acceptable on slightly more restrictive grounds. As Sen (1999: 354) puts it in his Nobel speech, ‘a full axiomatic determina-tion of a particular method of making social choice must inescapably lie next door to an impossibility – indeed just short of it’. Conversely, slightly weakening the conditions of an impossibility result can pro-duce a characterization theorem. As we will see later in this chapter (with respect to Donald Saari’s argument for the Borda count) it is possible to argue that Arrow’s conditions are slightly too restrictive and that with appropriate revision his theorem shows one voting rule to be uniquely preferable to all other possibilities. Arrow shows that there is no logically possible way out of Condorcet’s paradox which does not violate one or more of the six conditions listed. Arrow argues informally that the conditions are appropriate, but, being axioms, they are the unproved premises of his deductive argument rather than the conclusion. To consider the relevance of Arrow’s theorem, we need to move beyond strictly axio-matic reasoning by considering the empirical likelihood of cycles, and whether the requirements he imposes on a reasonable social choice rule are acceptable. In the remainder of this chapter we consider how other social choice theorists – including economists, political scientists, mathematicians and philosophers – have tackled these topics.

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  Advances In Interdisciplinary Applied Discrete Mathematics

  • Hemanshu Kaul, Henry Martyn Mulder(Authors)
  • 2010(Publication Date)
  • World Scientific

    (Publisher)

  In doing so, my goal is to go beyond the formal proofs to determine “why” the various conclusions hold. By answering the “why” question, information is obtained about how to circumvent the negative conclusions. Basic ideas are developed and discussed in this paper; a more extensive report will appear elsewhere. 10.2. Arrow’s Theorem and surprising extensions Arrow’s result is among the most influential theorems in the areas of social choice and consensus theory. The statement of his theorem is simple; each voter has a complete transitive ranking of the alternatives; there are no other restrictions. The societal outcome also must be a complete transitive ranking. Only two conditions are imposed upon the rule: (1) Pareto : If everyone ranks some pair in the same manner, then that unanimous choice is the pair’s societal ranking. (2) Independence : The ranking of each pair is strictly determined by the voters’ relative rankings of that pair. Specifically, if p 1 and p 2 are any two profiles where each voter ranks a specified pair in the same manner, then the societal ranking for the pair is the same for both profiles. Arrow’s striking conclusion is that with three or more alternatives, only one rule always satisfies these condition: a dictatorship. In other words, the rule is equivalent to identifying a particular person (or, in the context of a decision problem where voters are replaced with criteria, a specific criterion) so that for all possible profiles the rule’s ranking always agrees with the preferences of the identified person. The dictatorial assertion underscores the true conclusion that no reasonable decision rule can always satisfy these conditions. To analyze Arrow’s result, notice how the conditions imposed on the rule require it to concentrate on what happens with each pair. Thus, differing from standard in-terpretations, I prefer to treat Arrow’s theorem as describing a “divide-and-conquer” methodology.

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  The Mathematics of Voting and Elections: A Hands-On Approach

  Second Edition

  • Jonathan K. Hodge, Richard E. Klima(Authors)
  • 2018(Publication Date)
  • American Mathematical Society

    (Publisher)

  Chapter 5 Explaining the Impossible No one pretends that democracy is perfect or all-wise. Indeed it has been said that democracy is the worst form of government except for all those other forms that have been tried from time to time. – Winston Churchill Focus Questions In this chapter, we’ll explore the following questions: • What is the basic idea behind the proof of Arrow’s Theorem? • Can Pareto’s unanimity condition be weakened to solve the problems revealed by Arrow’s Theorem? • What is approval voting? Does it solve any of the problems revealed by Arrow’s Theorem? • What is the intensity of binary independence criterion? How is it related to Arrow’s Theorem? Warmup 5.1. Consider the following mathematical claim: It is impossible for a whole number to be divisible by 2 , 11 , and 23 and not be greater than 500 . Is this claim true or false? Give a convincing argument or example to justify your answer. Suppose we wanted to prove that the claim from Warmup 5.1 is true. How could we do it? One method would be to simply check all of the whole numbers one by one and verify that none of them are divisible by 2, 11, and 23 and not greater than 500. However, this would take quite a long time, wouldn’t it? Actually, since there are infinitely many whole numbers, the truth is we’d never be able to check them all. Of course, we could reduce our work quite a bit if we only considered the whole numbers that are not greater than 500. Then we’d just have to 71 72 5. EXPLAINING THE IMPOSSIBLE show that none of these numbers are divisible by 2, 11, and 23. But even this seems like an awful lot of work. Fortunately, there’s a much better way to prove that the claim from Warmup 5.1 is true. What if, instead of considering numbers one by one, we constructed some sort of logical argument to establish the truth of the claim? For instance, we might say something like this: The numbers 2 , 11 , and 23 are all prime numbers.

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  The Portfolio Theorists

  von Neumann, Savage, Arrow and Markowitz

  • C. Read(Author)
  • 2011(Publication Date)
  • Palgrave Macmillan

    (Publisher)

  We have failed to mention who won the 1972 Nobel Prize with Hicks – it was Kenneth Arrow, the scholar credited with originating and describing the phenomenon of moral hazard, creating Arrow’s Impossibility Theorem, formulating the Arrow Securities so essential in derivatives markets, and, in parallel with Gerard Debreu, establishing the definitive proof that a general equi- librium exists. Arrow and Debreu developed their general equilibrium theories on the foundation of the von Neumann-Morgenstern expected utility hypothesis and the existence proofs von Neumann pioneered in 1938. 148 15 Arrow’s Great Idea Often, brilliant inspirations come quite by chance. Kenneth Arrow’s contribution came through serendipity. His concern that he could not forge a postgraduate career in economics induced him to consider a career in statistics or insurance. As he studied, he found work in these two areas and, through his work and study, gained a valuable perspec- tive that would influence his life’s work and contribution in finance. Arrow had the tools. Well trained in mathematics, statistics, and economics, and well inspired through his provocative work in the insur- ance industry, he had a broad and all-encompassing understanding of the important issues emerging in finance at that time. Of course, as an economist, Arrow was classically trained in the tools of marginal analysis and the calculus approach. However, he was also exposed to the hotbed of the emerging personal probability approach, as championed by Leonard Jimmie Savage, his colleague at both the Statistical Research Group in New York and the Cowles Commission in Chicago, the expected utility hypothesis of John von Neumann and Oskar Morgenstern, and the general equilibrium and set theoretic results of von Neumann.

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