Condorcet Paradox

9 Key excerpts on "Condorcet Paradox"

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  Rational Politics

  Decisions, Games, and Strategy

  • Steven J. Brams(Author)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  The standard example of the paradox of voting involves three voters who rank a set of three alternatives {x,y,z} as shown in Table 4.1. Assume that all voters have transitive individual preference scales: a voter who prefers x to y and y to z will necessarily prefer x to z. The fact that this voter prefers x to y to z is indicated by preference scale (x,y,z). Table 4.1 Paradox of Voting Voter Preference Scale 1 (x,y,z) 2 (y,z,x) 3 (z,x,y) Voting Paradoxes 55 The paradox arises from the fact that if all voters have transitive preference scales, the social ordering nevertheless is intransitive: although a majority (voters 1 and 3) prefer x to y and a majority (voters 1 and 2) prefer y to z, a majority (voters 2 and 3) prefer z to x. This means that given at least three alternatives, there may be no social choice that is a Condorcet candidate— that is, one that can defeat every other alternative in a series of pairwise contests. In the present example, every alternative that receives majority support in one contest can be defeated by another alternative in another contest. For this reason, the majorities that prefer each alternative over some other in a series of pairwise contests are referred to as cyclical majorities: they are in a cycle, x > y > z > x (where > indicates de-feats), that returns to its starting point. The absence of a clear-cut winner, or social choice, suggested by the paradox is not dependent, however, on a specific decision rule like majority rule (here, two votes out of three), as will be shown in subsequent sections. Cyclical majorities may manifest themselves in various forms, in-cluding voting on candidate platforms. Consider a situation in which the voters do not vote on the basis of candidate positions on single issues, as assumed in the earlier one-dimensional spatial analysis (Chapter 3), but instead must choose among candidates who take positions on two or more issues.

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  Public Finance and Public Policy

  A Political Economy Perspective on the Responsibilities and Limitations of Government

  • Arye L. Hillman(Author)
  • 2019(Publication Date)
  • Cambridge University Press

    (Publisher)

  As the number of voters becomes very large, the likelihood that the majority makes the correct decision approaches certainty. These conclusions, known as the Condorcet Jury Theorem, justify democracy as informationally ef fi cient. The Theorem is central to a fi eld of study known as decision theory . The Theorem is not in general applicable to questions that arise in the study of economics because usually in economic questions there is not an agreed common objective; for example, voters usually disagree on the level of public spending and about who should pay taxes to fi nance the public spending. 1 11.3 The Paradox of Voting Voting decisions are subject to the “ paradox of voting. ” The paradox is that there is a positive cost of voting (the time taken to vote) whereas the expected bene fi t from voting through the 1 With dictatorship undesirable, there can nevertheless be an alternative to voting – George Tridimas ( 2012 ) recounts how in the democracy of ancient Athens government of fi cials were randomly selected by lottery. 346 Voting and the Common-Pool Problem probability of being decisive is in usual circumstances effectively zero. Given the paradox of voting, a plausible explanation for people voting when they know their vote is non-decisive, is that people vote expressively. Through expressive voting, people con fi rm their identities. 2 Voting can be viewed as a form of expressive participation, with voters happy or sad according to whether “ their ” team wins. We now proceed on the assumption that the paradox of voting does not deter people from voting, although often signi fi cant numbers of voters do not vote (implying they recognize that their individual vote will not be decisive and they do not obtain suf fi cient expressive “ utility ” from voting to justify taking the time to vote). The paradox of voting is a potential problem for democracy.

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  Encyclopedia of Leadership

  • George R. Goethals, Georgia J. Sorenson, James MacGregor Burns, George R. Goethals, Georgia J. Sorenson, James MacGregor Burns(Authors)
  • 2004(Publication Date)
  • SAGE Publications, Inc

    (Publisher)

  Arrow’s work served as an impetus for later analytical attempts to understand group decision rules and for the development by econo-mists and political scientists of what is called social (or public) choice theory. The Condorcet Paradox suggests two ways in which the outcome of group decision making may be manipulated. One way is by the strategic misrepre-sentation of preferences. Consider again the above example and the voting sequence x versus y first, and then the winner versus z —a sequence that results in z as the group decision. This is A ’s least preferred alter-native, so clearly A would like to avoid such a deci-sion. A might try to do this by misrepresenting his or her preference in the vote between x and y. Instead of voting for x, which is his or her real preference, A could vote for y. Alternative y would thereby obtain a majority against x and, upon being placed against z, would defeat z and emerge as the group decision. In this manner, A would obtain his or her second-preferred, rather than least-preferred, alternative. Of course, to adopt such a strategy of misrepresentation, A would need some knowledge of the preferences of B and C. Furthermore, nothing would prevent B or C from trying to employ a similar strategy. Another way in which A could manipulate the group decision, even if all group members were to vote sincerely, is through agenda setting. If A were in a position of leadership in the group and could deter-mine the group’s agenda, then A could schedule a first vote between y and z, followed by a vote between the winner and x. This would cause x, A ’s most-preferred alternative, to be the group decision. It is worth noting that, in fact, committee voting procedures typically involve pairwise voting. For example, a motion is put forward, and an amendment to the motion is pro-posed. There is then a vote on amending the motion, followed by a vote on whichever wins—the motion or the amended motion.

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  Freedom and Time

  A Theory of Constitutional Self-Government

  • Jed Rubenfeld(Author)
  • 2008(Publication Date)
  • Yale University Press

    (Publisher)

  In the acidulous glare of the Impossibility Theorem, the very concept of majority rule begins to dissolve, for majority will comes to be seen as an incoherent construct, its outcomes “self-contradictory” 11 and “meaningless,” 12 its will opposing every choice that the majority is said to make. As Arrow put it, so long as there is “a wide range of individual order-ings, the doctrine of voters’ sovereignty is incompatible with that of collective ra-tionality .” 13 How frequent are Condorcet Paradoxes in reality? No one knows. Grant, however, that they exist with enough regularity to pose the problem just described. What I want to consider is whether transitive preferences are prop-erly regarded as a requirement of rationality. In part, this question turns on the place that Condorcet Paradoxes are supposed to hold in the case of in-dividuals. As noted above, it is supposed that individuals ordinarily do not hold in-transitive preferences, and it is said that they could not hold intransitive pref-erences without being irrational. In other words, it is generally believed that 10. Condorcet’s account of the voters’ paradox can be found in M. de Condorcet, Essai sur l’Application de l’Analyse a ` la Probabilite ´ des De ´cisions Rendues a ` la Plural-ite ´ des Voix (New York photo reprint 1972) (1795). 11. Arrow, supra note 3, at 5. 12. Riker, supra note 6, at 128, 130. 13. Arrow, supra note 3, at 60 (emphasis added). 106 Being Over Time Condorcet Paradoxes are not problems for ordinary individuals, who are pre-sumed rational in economic modeling as well as in the “social choice” anal-yses powerfully influenced by Arrow’s work. But intransitive preferences can exist in individuals—in rational, ordinary individuals.

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  Philosophy, Politics, and Economics

  An Introduction

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

    (Publisher)

  She thus changes her ordering, and we get figure 8.2. FIGURE 8.2. Double-Peaked Preferences Betty no longer orders her preferences along the x-y-z dimension; we can see that her preference curve is double-peaked : this is not the relevant dimension along which she orders her options. Even though y is closer to her ideal point than is z, she prefers z to y. And now we have a Condorcet Paradox ordering. Thus, when voters do not agree on the relevant dimension of the option space, Condorcet Paradox orderings can arise. How often do cycles actually arise? There is spirited debate about this in the literature on democracy: some think that uncontrived cycles are rare, while others believe they are more common. As we will see, however, the main importance of Condorcet Paradox type cycles may be the way they lend themselves to various sorts of contrived manipulation. Collective Choice Rules It is hard to have faith in any SWF in the light of Arrow’s theorem. To be sure, as we have stressed, SWFs can, under some profile of preferences, satisfy the Pareto principle, non-dictatorship, and the independence of irrelevant alternatives. But when the preference profiles of the citizens display considerable multidimensionality, Arrow-type problems come to the fore. Moreover, as we shall see, minorities can generate voting cycles by misrepresenting their true preferences, thus magnifying the problems to which Arrow points. At this juncture, though, students of democracy may insist that social welfare functions are not really of much interest. After all, we do not want a social ordering, just a social choice : for any set of options, we only want to pick out the best. Our interest should be in collective choice rules, not social welfare functions. At first blush this looks inviting since some CCRs meet all of Arrow’s conditions, so the proof does not preclude all Collective Choice Rules

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  Social Choice and Individual Values

  • Kenneth J. Arrow, Eric S. Maskin(Authors)
  • 2012(Publication Date)
  • Yale University Press

    (Publisher)

  P. Wright of the University of New Bruns-wick called my attention to the work of E. J. Nanson. 9 Nanson, in discussing a proposal of his for a method of election, refers without great emphasis to the possibility of intransitivity arising from majority choice (pp. 213-214) for which he gives no previous reference. It is true, how-ever, that the tone of his remarks does not suggest that this possibility is a discovery of his own, although it is rather difficult to be sure. However, Guilbaud 10 notes that the paradox was known and de-veloped by the Marquis de Condorcet in the eighteenth century, 11 and refers to the paradox therefore as the Condorcet effect. This develop-ment was part of Condorcet's great interest in methods of election and essentially, therefore, in the theory of social choice. His work, in turn, * Voting and the Summation of Preferences: An Interpretive Bibliographic Re-view of Selected Developments During the Last Decade, American Political Science Review, Vol. 55, December, 1961, pp. 900-11. 7 J. Rothenberg, The Measurement of Social Welfare, Englewood Cliffs, New Jersey: Prentice-Hall, 1961. 8 The Theory of Committees and Elections, Cambridge, U.K.: Cambridge University Press, 1958. ' Methods of Election, Transactions and Proceedings of the Royal Society of Victoria, Vol. 19, 1882, pp. 197-240. 10 Op. «'(., pp. 513-15. II Most especially in his Essai sur Vapplication de I 'analyse a la probability des deci-sions rendues a la plurality des voix, Paris, 1785. A thorough study of Condorcet's thought appears in G.-G. Granger, La Mathematique Social du Marquis dt, Condorcet, Paris: Presses Universitaires de France, 1956. The work of Condorcet on voting is mostly analyzed in Chapter 3, especially pp. 94-129, an extensive summary of Con-dorcet's Essai.

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  Rational Choice

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

    (Publisher)

  Snow’s fictional account of a Cambridge University college election in The Masters . Yet as Mackie (2003: 197–309) demonstrates, these accounts depend upon contestable interpretations of politicians’ preference-rankings and ignore alternative and usually more obvious explanations of the same event. Furthermore, and even if we accept that these accounts do actually illustrate cases of major-ity cycles, it is unclear how representative they are. Although there has been more empirical work documenting the presence of cycles, it remains unconvincing. In reviewing the evidence of Condorcet’s paradox, Adrian Van Deemen (2014) concludes that the 47 empirical studies he considers jointly provide little insight into the likelihood of majority cycles. Of the 265 elections (ranging from large elections to small committee decisions) considered in these studies, 25 were categorized by their authors as having no Condorcet winner. What are we to make of these findings by Riker and others? The very obscurity of at least some of Riker’s stories suggests not only that he is extremely erudite but perhaps also that he had to search long and hard for some of his examples. This is at least partially explained by the general inac-cessibility of complete individual preference rankings, but it may also be because democratic cycles are uncommon in real-world elections. Kenneth Arrow and Social Choice Theory 137 Assessment In Chapter 1, we suggested that rational choice theory has polarized political science and that its proponents and opponents have been reluctant to engage with each other. This has been particularly true of social choice theory. Despite a growing body of theoretical work clearly relevant to democratic theory and practice, the insights of social choice theorists have not entered the canon of democratic theory.

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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)

  Recall, however, that the exact opposite happened in Condorcet’s paradox (see Question 3.4): Society preferred A over B and B over C , but also C over A . In fact, Condorcet’s paradox is the classic example of a voting system failing to produce a transitive societal preference order. Question 4.11. * Suppose X , Y , and Z are the three candidates in an election. (a) If you know that society prefers X over Z , Z over Y , and X over Y , can you conclude that the resulting societal preference order would be transitive? Explain. (b) If you know only that society prefers X over Y and Z over X , what would the societal preference between Y and Z have to be in order for the resulting societal preference order to be transitive? (c) If a fourth candidate entered the election, would your answer to part (a) necessarily be the same? Explain. Sometimes when societal preference orders fail to satisfy the property of transitivity, we say that the societal preferences represented are cyclic . Again, Condorcet’s paradox provides a good example of why this wording 60 4. TROUBLE IN DEMOCRACY is appropriate; if we try to combine the results of each of the pairwise com-parisons for Question 3.4, the resulting societal preference order would look something like this: A B C A B C A B C A B C · · · Recall that in each of the pairwise comparisons that make up this strange societal ranking, the margin of victory was two votes to one. In other words, two thirds of the voters preferred A over B , two thirds preferred B over C , and two thirds preferred C over A . Question 4.12. * Consider the cyclic societal preferences shown above.

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  Chance, Strategy, and Choice

  An Introduction to the Mathematics of Games and Elections

  • Samuel Bruce Smith(Author)
  • 2015(Publication Date)
  • Cambridge University Press

    (Publisher)

   We apply the preceding two results to prove Arrow’s Principle: Proof of Theorem 23.6 Suppose that we are given a social welfare method with at least three candidates satisfying Independence and Pareto fairness, a collection of voters V , and a dictating set P . Let Q be a proper subset of P and Q 0 the set of voters in P but not in Q. Since P is a dictating set, by definition, P is decisive for A over B for any pair of candidates A and B. By Theorem 23.9, either Q and Q 0 is decisive for some pair of candidates. By Theorem 23.10, this set (Q or Q 0 ) is then a dictating set, as needed.  The proof again reveals the Condorcet Paradox as the basic obstruction to producing a “fair” social welfare. After Arrow .............................................................................. Arrow’s thesis marked the start of a renaissance in social choice theory. We have seen some of the theorems in the same vein that appeared after the Impossibility Theorem, including Sen’s Impossibility Theorem (Theorem 18.22), Mackay’s Theorem (Theorem 18.12), and notably, the Gibbard-Satterthwaite Theorem (Theorem 6.4). We conclude with two examples of other types of developments in the field. The first concerns conditions on voter preferences that ensure that a fair social welfare can be achieved. We then introduce two voting methods that seem to offer an escape from Arrow’s Impossibility Theorem. Sen Coherence .............................................................................. Recall from Definition 5.13 that the relation of social preference is determined by the Simple Majority Method. We say that candidate A is socially preferred to candidate B if a majority of voters prefer A to B. Transitivity is then the property that, for any three candidates A, B, and C, if A is socially preferred to B and B is socially preferred to C, then A is socially preferred to C. A Condorcet Paradox occurs when social preferences are not transitive (Definition 6.2).

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