Arrow's theorem, indeterminacy, and multiplicity reconsidered

Ethics

Volumn 111, Issue 4, 2001, Pages 706-734

  Risse, Mathias ^a  

^a NONE

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EID: 0035402968     PISSN: 00141704     EISSN: None     Source Type: Journal    
DOI: 10.1086/233570     Document Type: Article

References (65)

1. This episode from the life of Diego de Velázquez is taken from Jonathan Brown, Velázquez: Painter and Courtier (New Haven, Conn.: Yale University Press, 1986), p. 215.
2. See also Frank Easterbrook, "Ways of Criticizing the Court," Harvard Law Review 95 (1982): 811-32, for the view that majority rule for court decisions is dubious. Jules Coleman and John Ferejohn, "Democracy and Social Choice," Ethics 97 (1986): 6-25, speak of an ambiguity problem instead of a multiplicity problem, and of an instability problem rather than an indeterminacy problem. The difference is merely terminological.
3. See also Frank Easterbrook, "Ways of Criticizing the Court," Harvard Law Review 95 (1982): 811-32, for the view that majority rule for court decisions is dubious. Jules Coleman and John Ferejohn, "Democracy and Social Choice," Ethics 97 (1986): 6-25, speak of an ambiguity problem instead of a multiplicity problem, and of an instability problem rather than an indeterminacy problem. The difference is merely terminological.
4. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
5. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
6. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
7. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
8. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
9. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
10. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
11. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
12. Arrow first proved the theorem in his Social Choice and Individual Values (New York City: Wiley, 1951). Hardin's claim can be found in his "Public Choice versus Democracy," in The Idea of Democracy, ed. David Copp, Jean Hampton, and John Roemer (Cambridge: Cambridge University Press, 1993); for Wolff's discussion, see his In Defense of Anarchism (New York City: Harper & Row, 1970), p. 59; and for Riker's views, see his Liberalism against Populism (San Francisco: Freeman, 1982). Authors who have tried to mitigate the impact on democratic theory of Arrow's theorem and the two problems mentioned in the text include Jules Coleman and John Ferejohn; Thomas Christiano ("Social Choice and Democracy," in Copp, Hampton, Roemer, eds.); Gerry Mackie ("All Men Are Liars: Is Democracy Meaningless?" in Deliberative Democracy, ed. Jon Elster [Cambridge: Cambridge University Press, 1998]); David Miller ("Deliberative Democracy and Social Choice," Political Studies 40 [1992]: 46-67); Richard Pildes and Elizabeth Anderson ("Throwing Arrows at Democracy: Social Choice Theory, Value Pluralism, and Democratic Politics," Columbia Law Review 8 [1990]: 2121-216); and Joshua Cohen ("An Epistemic Conception of Democracy," Ethics 97 [1986]: 26-38).
13. My conclusion with respect to the impact of the multiplicity thesis on so-called populist conceptions of voting (where voting is interpreted as expressing "the will of the people") agrees in principle with Jules Coleman and John Ferejohn and with Joshua Cohen, but it differs in particular from Cohen by not privileging majority rule via the Condorcet Jury Theorem. Thereby I make room for a greater multiplicity of voting methods within such conceptions than Cohen allows. As opposed to Gerry Mackie, considerations of the empirical relevance of Arrow's theorem are irrelevant to my argument. This is fortunate, since it is difficult to use such considerations fruitfully (see James Enelow, "Cycling and Majority Rule," in Perspectives on Public Choice, ed. Dennis Mueller [Cambridge: Cambridge University Press, 1997] for an overview). As opposed to David Miller, my argument does not depend on the dubious idea that deliberation allows us to circumvent Arrow's theorem (by creating "single-peaked" preferences). My article shows strong affinities with Coleman and Ferejohn's article, and one may even regard it as a belated companion piece. Sections III and IV make contributions to the literature on majoritarian decision making that are not discussed in Coleman and Ferejohn, but which I take to be entirely consistent with (and even to some extent supported by) their argument. Section V approaches the question of the impact of the multiplicity problem on democratic theory from the point of view of how the existence of those losers produced by a specific choice of an aggregation method affects different justifications of democracy. Coleman and Ferejohn approach this question in terms of a discussion of William Riker, and to the extent that these two treatments overlap, my argument coincides with their earlier and more extensive discussion (see, in particular, Sec. VC). The "Condorcet proposal" submitted in Section III is also put forth in Peyton Young, "Condorcet's Theory of Voting," American Political Science Review 82 (1988): 1231-44, and "Group Choice and Individual Judgements," in Mueller, ed. However, Young does not seem to be interested in investigating whether the procedures he reconstructs from Condorcet's ideas are faithful to majoritarianism. The new contribution of this essay is to show that this method is indeed a majoritarian method that can be justified on its own terms, and I present two very different lines of argument for this claim in Sec. III: on the one hand, I explore formal properties of the proposal and, on the other hand, I show how the proposal is implied by traditional arguments for majority rule, once they are generalized beyond the usual scenario of two options.
14. My conclusion with respect to the impact of the multiplicity thesis on so-called populist conceptions of voting (where voting is interpreted as expressing "the will of the people") agrees in principle with Jules Coleman and John Ferejohn and with Joshua Cohen, but it differs in particular from Cohen by not privileging majority rule via the Condorcet Jury Theorem. Thereby I make room for a greater multiplicity of voting methods within such conceptions than Cohen allows. As opposed to Gerry Mackie, considerations of the empirical relevance of Arrow's theorem are irrelevant to my argument. This is fortunate, since it is difficult to use such considerations fruitfully (see James Enelow, "Cycling and Majority Rule," in Perspectives on Public Choice, ed. Dennis Mueller [Cambridge: Cambridge University Press, 1997] for an overview). As opposed to David Miller, my argument does not depend on the dubious idea that deliberation allows us to circumvent Arrow's theorem (by creating "single-peaked" preferences). My article shows strong affinities with Coleman and Ferejohn's article, and one may even regard it as a belated companion piece. Sections III and IV make contributions to the literature on majoritarian decision making that are not discussed in Coleman and Ferejohn, but which I take to be entirely consistent with (and even to some extent supported by) their argument. Section V approaches the question of the impact of the multiplicity problem on democratic theory from the point of view of how the existence of those losers produced by a specific choice of an aggregation method affects different justifications of democracy. Coleman and Ferejohn approach this question in terms of a discussion of William Riker, and to the extent that these two treatments overlap, my argument coincides with their earlier and more extensive discussion (see, in particular, Sec. VC). The "Condorcet proposal" submitted in Section III is also put forth in Peyton Young, "Condorcet's Theory of Voting," American Political Science Review 82 (1988): 1231-44, and "Group Choice and Individual Judgements," in Mueller, ed. However, Young does not seem to be interested in investigating whether the procedures he reconstructs from Condorcet's ideas are faithful to majoritarianism. The new contribution of this essay is to show that this method is indeed a majoritarian method that can be justified on its own terms, and I present two very different lines of argument for this claim in Sec. III: on the one hand, I explore formal properties of the proposal and, on the other hand, I show how the proposal is implied by traditional arguments for majority rule, once they are generalized beyond the usual scenario of two options.
15. If an agent weakly prefers x to y, she either prefers x toy or is indifferent between x and y.
16. This assumption may be misnamed. Genuine dictatorship cannot be properly expressed in this framework. Andranick Tanguiane, "Arrow's Paradox and Mathematical Theory of Democracy," Social Choice and Welfare 11 (1994): 1-82, discusses what happens if one enriches the formalism to distinguish between agents whose preference ranking coincides with the group ranking and who are representatives of the group preferences in some properly denned sense and those who are not. The latter are dictators. However, as should become clear soon, the Condorcet Paradox already raises the most important questions from the point of view of political philosophy. Questions of aggregation arise in many contexts; it is unclear whether the most interesting interpretation of Arrow's theorem is in political philosophy.
17. A profile of rankings contains one ranking for each agent.
18. There is a broad discussion of Arrow's theorem in the economics literature that I cannot even begin to address. See Amartya Sen "Social Choice Theory," in Handbook of Mathematical Economics, vol. 3, ed. K. Arrow and M. Intrilligator (Amsterdam: North Holland, 1986); and Pildes and Anderson for surveys
19. Would it not do to have a top-ranked option, rather than a ranking? Much of the voting literature addresses that simplified case; see Peter Ordeshook, "The Spatial Analysis of Elections and Committees: Four Decades of Research," in Mueller, ed. Suffice it to say that this move does not much mitigate the theoretical challenges.
20. This idea has also been used in Young, "Group Choice and Individual Judgments"; Young also uses the "support"-terminology. However, he merely employs it in connection with a graphic method designed to illustrate how the generalized Condorcet Jury Theorem works (see below).
21. Although Young puts forth the Condorcet proposal, he only uses its computational core in connection with a graphic method to illustrate the workings of a generalization of the Condorcet Jury Theorem. He does not seem to be interested in investigating whether the procedures he reconstructs from Condorcet's ideas are faithful to majoritarianism (which appears to be because Young simply assumes that they are). What Young is concerned with instead is showing how this procedure can be understood as a statistical method whose outcome is most likely to be "correct" in a specified sense, and how it can be justified within social choice theory.
22. Here is why: suppose no cycles arise from the pairwise votes. Suppose ranking R is selected by the Condorcet proposal. We need to show then that R does not place an option B before an option A if a majority prefers A to B in a pairwise vote. There are two cases. First case: we have a pair of options (A, B), which are adjacent in R (i.e., no option is ranked between them). Suppose B is ranked before A even though a majority prefers A to B in a pairwise vote. Then we find a ranking S which differs from R only with respect to the relative position of A and B. Since A is preferred to B by a majority and since R and S only differ with respect to the relative ranking of the adjacent options A and B, S will gain more support in pairwise votes than R, which contradicts the assumption on R. So this case cannot arise. Second case: we have a pair of nonadjaceni options (A, B) in R, and suppose that B is ranked before A in R while a majority prefers A to B in a pairwise vote. Following the argument in the first case, it is true for any two adjacent options E and F that a majority prefers E to F if E is ranked before F in R. Since in addition A is preferred to B by a majority, we obtain a voting cycle arising from pairwise majority votes, which was ruled out. So this case cannot arise either. So there can be no ranking R that places B before A if A is preferred to B by a majority in pairwise voting, provided that no cycles arise in pairwise votes.
23. Example: suppose a group of 120 needs to rank A, B, C. Forty-six rank them (A, B, C), 34 (B, C, A), 4 (B, A, C), 20 (C, A, B), and 16 (C, B, A). Then we have a cycle. But only (B, C, A) has maximal support.
24. I assume from here on that agents are not indifferent between any two options. Nothing depends on that, but it facilitates the notation.
25. Readers familiar with cooperative game theory may be reminded of the core; and, indeed, a connection between the core of games and winners in majority voting has been drawn; see Peter Ordeshook, Game Theory and Political Theory (Cambridge: Cambridge University Press, 1986), sec. 8.2.
26. The proof depends on the choice of the preference relation between rankings. One may wonder why I do not choose the relation in which agent k prefers S to R if k's ranking agrees with S in at least half the pairs with respect to which R and S differ. That is, "all things considered," S is better than R. This provides a complete order for each agent. Yet I do not think that this is a good preference relation between rankings. Suppose somebody's ranking is (A, B, C, D, E, F), and the rankings to be compared are (A, C, B, E, D, F) and (F, B, C, D, E, A). According to this suggestion, the agent prefers the latter to the former. However, this is not straightforward: (F, B, C, D, E, A) improves on (A, C, B, E, D, F) by switching C and B and E and D, but it also switches F and A, thereby placing what the agents want most last and vice versa. The point is that the kind of weighing involved there is not sanctioned by the information delivered by preference rankings. We should not define relations on rankings that trigger questions involving more information than provided for in those rankings.
27. i)} contains pairs of options not adjacent in R, we end up with a voting cycle, which was excluded by assumption. To see that this converse does not hold if there are voting cycles, see the example in n. 13 (the group of 120): (C, A, B) is undominated, but it does not have maximal support.
28. See George Cornwall Lewis, On the Influence of Authority in Matters of Opinion (London: Parker, 1849), p. 207. Brian Barry, "Is Democracy Special?" in his Democracy and Power (Oxford: Clarendon, 1991), p. 27, points out that "by something akin to the principle of insufficient reason" it should be majorities rather than minorities ruling.
29. See George Cornwall Lewis, On the Influence of Authority in Matters of Opinion (London: Parker, 1849), p. 207. Brian Barry, "Is Democracy Special?" in his Democracy and Power (Oxford: Clarendon, 1991), p. 27, points out that "by something akin to the principle of insufficient reason" it should be majorities rather than minorities ruling.
30. For this argument, see, for instance, Robert Dahl, Democracy and Its Critics (New Haven, Conn.: Yale University Press, 1989), p. 138.
31. This argument is defended, for instance, by Jeremy Waldron in "Legislation, Authority, and Voting," Georgetown Law Journal 84 (1996): 2185-214.
32. For discussion of the epistemic conception of democracy, which most prominently needs that assumption, see Cohen; David Copp, "Could Political Truth Be a Hazard for Democracy?" in Copp, Hampton, and Roemer, eds., pp. 101-18; and David Estlund, "Making Truth Safe for Democracy," in Copp, Hampton, and Roemer, eds., pp. 71-101. A different way of generalizing pairwise voting within an epistemic framework is pursued by Christian List and Robert Goodin, "Epistemic Democracy: Assaying the Options," working paper in Social and Political Theory (Australian National University, Canberra, 2000). They show how the Condorcet Jury Theorem can be generalized from pairwise majority voting to plurality voting over many options. Moreover, Goodin and List also explore the epistemic merits of various group decision rules. They do not only show that standard decision rules are "equally good" from an epistemic point of view but also that, at least in large electorates, those rules do indeed have great epistemic merits.
33. For discussion of the epistemic conception of democracy, which most prominently needs that assumption, see Cohen; David Copp, "Could Political Truth Be a Hazard for Democracy?" in Copp, Hampton, and Roemer, eds., pp. 101-18; and David Estlund, "Making Truth Safe for Democracy," in Copp, Hampton, and Roemer, eds., pp. 71-101. A different way of generalizing pairwise voting within an epistemic framework is pursued by Christian List and Robert Goodin, "Epistemic Democracy: Assaying the Options," working paper in Social and Political Theory (Australian National University, Canberra, 2000). They show how the Condorcet Jury Theorem can be generalized from pairwise majority voting to plurality voting over many options. Moreover, Goodin and List also explore the epistemic merits of various group decision rules. They do not only show that standard decision rules are "equally good" from an epistemic point of view but also that, at least in large electorates, those rules do indeed have great epistemic merits.
34. For discussion of the epistemic conception of democracy, which most prominently needs that assumption, see Cohen; David Copp, "Could Political Truth Be a Hazard for Democracy?" in Copp, Hampton, and Roemer, eds., pp. 101-18; and David Estlund, "Making Truth Safe for Democracy," in Copp, Hampton, and Roemer, eds., pp. 71-101. A different way of generalizing pairwise voting within an epistemic framework is pursued by Christian List and Robert Goodin, "Epistemic Democracy: Assaying the Options," working paper in Social and Political Theory (Australian National University, Canberra, 2000). They show how the Condorcet Jury Theorem can be generalized from pairwise majority voting to plurality voting over many options. Moreover, Goodin and List also explore the epistemic merits of various group decision rules. They do not only show that standard decision rules are "equally good" from an epistemic point of view but also that, at least in large electorates, those rules do indeed have great epistemic merits.
35. i)); see Lloyd Shapley and Bernhard Grofman, "Optimizing Group Judgmental Accuracy in the Presence of Interdependencies," Public Choice 43 (1984): 329-43.
36. i)); see Lloyd Shapley and Bernhard Grofman, "Optimizing Group Judgmental Accuracy in the Presence of Interdependencies," Public Choice 43 (1984): 329-43.
37. i)); see Lloyd Shapley and Bernhard Grofman, "Optimizing Group Judgmental Accuracy in the Presence of Interdependencies," Public Choice 43 (1984): 329-43.
38. i)); see Lloyd Shapley and Bernhard Grofman, "Optimizing Group Judgmental Accuracy in the Presence of Interdependencies," Public Choice 43 (1984): 329-43.
39. See Kenneth May, "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision," Econometrica 10 (1952): 680-84; see also Bruce Ackerman, Social Justice and the Liberal State (New Haven, Conn.: Yale University Press, 1980), chap. 9; and Douglas Rae and Eric Schickler, "Majority Rule" in Mueller, ed., for a discussion of majority rule in light of May's Theorem.
40. See Kenneth May, "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision," Econometrica 10 (1952): 680-84; see also Bruce Ackerman, Social Justice and the Liberal State (New Haven, Conn.: Yale University Press, 1980), chap. 9; and Douglas Rae and Eric Schickler, "Majority Rule" in Mueller, ed., for a discussion of majority rule in light of May's Theorem.
41. See Kenneth May, "A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision," Econometrica 10 (1952): 680-84; see also Bruce Ackerman, Social Justice and the Liberal State (New Haven, Conn.: Yale University Press, 1980), chap. 9; and Douglas Rae and Eric Schickler, "Majority Rule" in Mueller, ed., for a discussion of majority rule in light of May's Theorem.
42. For what follows, see John Kemeny, "Mathematics without Numbers," Daedalus 88 (1959): 571-91; and also Young, "Group Choice and Individual Judgements." For a discussion of the Kemeny rule from a technical point of view, see Donald Saari and Vincent Merlin, "A Geometric Examination of Kemeny's Rule," unpublished ms., 1998 (to be found at www.math.nwu.edu/~d_saari/vote/pre-vote.html), and the literature references therein; that work itself draws on Saari, Geometry of Voting (New York: Springer, 1994). It should be noted, however, that Saari and Merlin's purpose is to discredit the Kemeny rule. Since their discussion is conducted in a technical framework too demanding to be introduced here, let me simply make a quick remark aimed at those readers familiar with their work to show why I am not worried by their objections. In sec. 3.2.2. of their piece, Saari and Merlin point out in summary that they dismiss the Kemeny rule (and its close relatives, such as what they call the Condorcet profile), because the ranking singled out by these methods (and thus the ranking sorted out by the Condorcet proposal above) is also the ranking that is created if the votes of nonexistent, irrational voters (which enter their discussion for technical reasons) are aggregated. Yet, surely, a ranking reached by rational individuals by means of an aggregation method backed up by reasonable considerations (which I claim the Condorcet proposal/the Kemeny rule is) may also be reached in different ways, for instance, by aggregation of the preferences of nonexistent, irrational voters. But this insight discredits such methods no more than the fact that a decision reached by expected-utility reasoning may also have been reached by drawing lots (if only the right physical conditions hold that make one lot come up rather than another) or by following the Roman technique of observing the flight of birds (if only we have the right kind of weather for their flight to suggest a certain decision) discredits expected-utility theory. That is, the strength of the arguments for the Condorcet proposal/the Kemeny rule is by no means undermined by the fact that nonexistent irrational voters act in accordance.
43. For what follows, see John Kemeny, "Mathematics without Numbers," Daedalus 88 (1959): 571-91; and also Young, "Group Choice and Individual Judgements." For a discussion of the Kemeny rule from a technical point of view, see Donald Saari and Vincent Merlin, "A Geometric Examination of Kemeny's Rule," unpublished ms., 1998 (to be found at www.math.nwu.edu/~d_saari/vote/pre-vote.html), and the literature references therein; that work itself draws on Saari, Geometry of Voting (New York: Springer, 1994). It should be noted, however, that Saari and Merlin's purpose is to discredit the Kemeny rule. Since their discussion is conducted in a technical framework too demanding to be introduced here, let me simply make a quick remark aimed at those readers familiar with their work to show why I am not worried by their objections. In sec. 3.2.2. of their piece, Saari and Merlin point out in summary that they dismiss the Kemeny rule (and its close relatives, such as what they call the Condorcet profile), because the ranking singled out by these methods (and thus the ranking sorted out by the Condorcet proposal above) is also the ranking that is created if the votes of nonexistent, irrational voters (which enter their discussion for technical reasons) are aggregated. Yet, surely, a ranking reached by rational individuals by means of an aggregation method backed up by reasonable considerations (which I claim the Condorcet proposal/the Kemeny rule is) may also be reached in different ways, for instance, by aggregation of the preferences of nonexistent, irrational voters. But this insight discredits such methods no more than the fact that a decision reached by expected-utility reasoning may also have been reached by drawing lots (if only the right physical conditions hold that make one lot come up rather than another) or by following the Roman technique of observing the flight of birds (if only we have the right kind of weather for their flight to suggest a certain decision) discredits expected-utility theory. That is, the strength of the arguments for the Condorcet proposal/the Kemeny rule is by no means undermined by the fact that nonexistent irrational voters act in accordance.
44. For what follows, see John Kemeny, "Mathematics without Numbers," Daedalus 88 (1959): 571-91; and also Young, "Group Choice and Individual Judgements." For a discussion of the Kemeny rule from a technical point of view, see Donald Saari and Vincent Merlin, "A Geometric Examination of Kemeny's Rule," unpublished ms., 1998 (to be found at www.math.nwu.edu/~d_saari/vote/pre-vote.html), and the literature references therein; that work itself draws on Saari, Geometry of Voting (New York: Springer, 1994). It should be noted, however, that Saari and Merlin's purpose is to discredit the Kemeny rule. Since their discussion is conducted in a technical framework too demanding to be introduced here, let me simply make a quick remark aimed at those readers familiar with their work to show why I am not worried by their objections. In sec. 3.2.2. of their piece, Saari and Merlin point out in summary that they dismiss the Kemeny rule (and its close relatives, such as what they call the Condorcet profile), because the ranking singled out by these methods (and thus the ranking sorted out by the Condorcet proposal above) is also the ranking that is created if the votes of nonexistent, irrational voters (which enter their discussion for technical reasons) are aggregated. Yet, surely, a ranking reached by rational individuals by means of an aggregation method backed up by reasonable considerations (which I claim the Condorcet proposal/the Kemeny rule is) may also be reached in different ways, for instance, by aggregation of the preferences of nonexistent, irrational voters. But this insight discredits such methods no more than the fact that a decision reached by expected-utility reasoning may also have been reached by drawing lots (if only the right physical conditions hold that make one lot come up rather than another) or by following the Roman technique of observing the flight of birds (if only we have the right kind of weather for their flight to suggest a certain decision) discredits expected-utility theory. That is, the strength of the arguments for the Condorcet proposal/the Kemeny rule is by no means undermined by the fact that nonexistent irrational voters act in accordance.
45. For what follows, see John Kemeny, "Mathematics without Numbers," Daedalus 88 (1959): 571-91; and also Young, "Group Choice and Individual Judgements." For a discussion of the Kemeny rule from a technical point of view, see Donald Saari and Vincent Merlin, "A Geometric Examination of Kemeny's Rule," unpublished ms., 1998 (to be found at www.math.nwu.edu/~d_saari/vote/pre-vote.html), and the literature references therein; that work itself draws on Saari, Geometry of Voting (New York: Springer, 1994). It should be noted, however, that Saari and Merlin's purpose is to discredit the Kemeny rule. Since their discussion is conducted in a technical framework too demanding to be introduced here, let me simply make a quick remark aimed at those readers familiar with their work to show why I am not worried by their objections. In sec. 3.2.2. of their piece, Saari and Merlin point out in summary that they dismiss the Kemeny rule (and its close relatives, such as what they call the Condorcet profile), because the ranking singled out by these methods (and thus the ranking sorted out by the Condorcet proposal above) is also the ranking that is created if the votes of nonexistent, irrational voters (which enter their discussion for technical reasons) are aggregated. Yet, surely, a ranking reached by rational individuals by means of an aggregation method backed up by reasonable considerations (which I claim the Condorcet proposal/the Kemeny rule is) may also be reached in different ways, for instance, by aggregation of the preferences of nonexistent, irrational voters. But this insight discredits such methods no more than the fact that a decision reached by expected-utility reasoning may also have been reached by drawing lots (if only the right physical conditions hold that make one lot come up rather than another) or by following the Roman technique of observing the flight of birds (if only we have the right kind of weather for their flight to suggest a certain decision) discredits expected-utility theory. That is, the strength of the arguments for the Condorcet proposal/the Kemeny rule is by no means undermined by the fact that nonexistent irrational voters act in accordance.
46. Here is an example that shows how Independence may be violated. To use a situation used earlier (n. 13), suppose a group of 120 needs to rank A, B, C. Forty-six rank them (A, B, C), 34 (B, C, A), 4 (B, A, C), 20 (C, A, B), and 16 (C, B, A). Then (B, C, A) has maximal support. Suppose now a few people change their minds such that we now have 46 rank the options (A, B, C), 34 (B, C, A), 12 (B, A, C), 20 (C, A, B), and 8 (C, B, A). That is, 8 agents who previously had the ranking (C, B, A) changed to (B, A, C). So the relative standing of A vs. B has changed in no agent's ranking. However, now the ranking (A, B, C) has maximal support, in which the relative standing of A vs. B has reversed compared to the original winner (B, C, A). This is a violation of Independence.
47. The discussion of social choice in so-called economic environments to be found in John Roemer, Theories of Distributive Justice (Cambridge, Mass.: Harvard University Press, 1996), contains a similar claim. Roemer expresses worries about the abstractness introduced by Arrow-style social choice theory and Nash-style bargaining theory, both of which date from the early 1950s. The theory of economic environments is intended to allow for more concrete investigations.
48. See Riker, in particular, for undesirable consequences of a failure of Independence, which may be a good starting point for such arguments.
49. It should also be noted that Young, following a theorem proved in Peyton Young and Arthur Levenglick, "A Consistent Extension of Condorcet's Election Principle," SIAM Journal of Applied Mathematics 35 (1978): 283-300, shows that the Condorcet proposal satisfies a local independence condition, which Young calls "local stability. Roughly speaking, this condition makes sure that the ranking of options A, B, and C cannot be influenced by introducing an option D, which is inferior to each one of them. This should be good news for all those who disagree with my discussion of Arrow's more general condition. For this indicates that the Condorcet proposal will handle many scenarios that they may want to quote in defense of Arrow's condition in ways that they would deem satisfactory.
50. On the other hand, the arguments in favor of the Condorcet proposal and those to be presented for the Borda count in the next section may also be used to criticize existing voting methods from the respective points of view taken in these two voting procedures.
51. Riker, pp. 76-81, discusses the Condorcet proposal (which he calls the Kemeny rule) as a tie-breaking mechanism, which one may use if no single option beats all other alternatives in pairwise majority rule. He regards it as too complicated, "illustrating, perhaps, the desperation of theorists to discover an adequate Condorcet extension" (p. 79). The Condorcet proposal does involve some computational efforts. However, it is not to be expected that the most reasonable majoritarian position is also computationally the simplest. Also, a simple computer program can easily do the calculations. Another tie-breaking rule Riker discusses is the Copeland rule, which selects the option with the highest Copeland index c(A), where c(A) is the number of times A beats other options less the number of times A loses to other options. Michael Dummett, Principles of Electoral Reform (Oxford: Oxford University Press, 1997), p. 66, says that "Borda's criterion and that of Condorcet, together with Copeland's extension of the second of the two, appear to me to be the only plausible criteria ever suggested for which one candidate, out of a number, would best represent the electors" (where the Condorcet criterion mentioned here is to choose the option that beats all others in majority vote). As Riker (p. 267) shows, the Condorcet proposal may pick a ranking that is not headed by the option with the highest Copeland index. In situations in which a ranking is of no interest, this is an objection against the Condorcet proposal. Whenever the ranking matters, however, the arguments of this section show, pace Dummett, that the Condorcet proposal is superior to a method that ranks options by their Copeland indices. For discussion of the aforementioned Condorcet-criterion and its extensions, see Gerald Kramer, "Some Procedural Aspects of Majority Rule," and Arthur Kuflik, "Majority Rule Procedure," both in Nomos, vol. 18, Due Process, ed. Roland Pennock and John Chapman (New York: New York University Press, 1977).
52. Riker, pp. 76-81, discusses the Condorcet proposal (which he calls the Kemeny rule) as a tie-breaking mechanism, which one may use if no single option beats all other alternatives in pairwise majority rule. He regards it as too complicated, "illustrating, perhaps, the desperation of theorists to discover an adequate Condorcet extension" (p. 79). The Condorcet proposal does involve some computational efforts. However, it is not to be expected that the most reasonable majoritarian position is also computationally the simplest. Also, a simple computer program can easily do the calculations. Another tie-breaking rule Riker discusses is the Copeland rule, which selects the option with the highest Copeland index c(A), where c(A) is the number of times A beats other options less the number of times A loses to other options. Michael Dummett, Principles of Electoral Reform (Oxford: Oxford University Press, 1997), p. 66, says that "Borda's criterion and that of Condorcet, together with Copeland's extension of the second of the two, appear to me to be the only plausible criteria ever suggested for which one candidate, out of a number, would best represent the electors" (where the Condorcet criterion mentioned here is to choose the option that beats all others in majority vote). As Riker (p. 267) shows, the Condorcet proposal may pick a ranking that is not headed by the option with the highest Copeland index. In situations in which a ranking is of no interest, this is an objection against the Condorcet proposal. Whenever the ranking matters, however, the arguments of this section show, pace Dummett, that the Condorcet proposal is superior to a method that ranks options by their Copeland indices. For discussion of the aforementioned Condorcet-criterion and its extensions, see Gerald Kramer, "Some Procedural Aspects of Majority Rule," and Arthur Kuflik, "Majority Rule Procedure," both in Nomos, vol. 18, Due Process, ed. Roland Pennock and John Chapman (New York: New York University Press, 1977).
53. A more precise definition of "ordinal" rankings is hard to give outside a theoreti-cal context more technical than the one used here (see Amartya Sen, Collective Choice and Individual Welfare [San Francisco: Holden Day, 1970]; and Roemer, chap. 1, for more precise treatments). Some readers may be tempted to think that the Borda count is not a merely ordinal aggregation rule because it invokes summing over numbers. But, still, the Borda count is merely ordinal, since indeed all the information contained in the individual rankings prior to the summation and in the group ranking after the summation is which options are preferred to which others. The numbers serve merely to represent relative positions in rankings, and the summation operation only serves to represent the "average" relative position of options in rankings. For instance, it makes no sense to say that the first option is preferred to the second more than the second is preferred to the third.
54. One may wonder then what the criteria are by which the success of arguments for aggregation rules in this section are to be assessed, in particular, one may wonder whether one important criterion is whether these decision rules succeed in determining the will of the people. A complete theory of aggregation methods would have to discuss such criteria (see Sec. IVC). However, the discussion in Sec. IVB tries to remain neutral with regard to this question and asks on what grounds majority rule/the Condorcet proposal can be defended (or have been defended) and whether arguments on those grounds would also convince a defender of the Borda rule. Different arguments may appeal to different criteria of success for aggregation methods, and as it turns out in Sec. IVB, arguments for majority rule/the Condorcet proposal that are based on very different grounds fail to convince the defender of the Borda rule.
55. The Borda count also violates Independence. If in the example above 2 switches C and A, then A will be ranked before B in the social ranking, although the relative standing of A and B has not changed in any ranking. Note that the discussion in this article applies only if the Borda count is defined precisely as done above. Deviants that (e.g.) assign positive numbers only to the first several options and 0 to all others have different features and thus are not covered by the above discussion. In particular, the appealing features of the Borda count discussed above that make the Borda count interesting from a normative point of view do not carry over to such deviants.
56. As an editor of Ethics has pointed out, there is pressure to argue the other way round. That is, since May's Theorem does not distinguish between Condorcet and Borda, it suggests that both have some claim to be specifications of majority rule. Still, the point remains that May's Theorem cannot be invoked to argue against Borda.
57. The ease with which a defender of the Borda count can handle standard majoritarian arguments should make defenders of any version of majority rule pause. For, on the one hand, their arguments already presuppose that we move in a preference aggregation context (i.e., in a merely ordinal model), and so do nothing to convince defenders of other aggregation frameworks (see Sec. IVC). On the other hand, the above discussion shows that they also do very little against a defender of the Borda count once we are moving inside a preference aggregation model.
58. A difference between Borda and Condorcet that I omitted is the extent to which they are manipulable, i.e., prone to "strategic voting." Both are prone to manipulation in the sense that persistently voting against one's second-ranked sometimes favors one's first-ranked option. As Young shows, Borda is also "manipulable" in a way in which Condorcet is not, i.e., by being more sensitive to additions of new options. (This is the condition of "local stability" mentioned in n. 28 above.) Young insists that this makes Borda inferior to Condorcet. However, this kind of "manipulability" seems a bearable consequence of considering all agents' full rankings. Put differently: the Borda count evaluates the average standing of an option within rankings, and, evidently, such average standing will change if new options are added to the rankings. Somebody who is convinced that options should be evaluated by their average standing, i.e., by their relative position within a pool of options, should be willing to embrace such consequences that strike Young as counterintuitive. For discussion of manipulability, see Christiano. It should be noted, however, that a treatment of "strategic voting" easily involves questions of, say, the moral evaluation of what is referred to as "manipulation" or "strategic voting" and its importance in settings in which the "will of the people" (or something similar) is supposed to be determined. The complexity of these questions goes beyond what I can adequately address in this article. My point here is only to say that it is not clear to begin with that such a discussion will entail a decisive problem for Borda. However, to the extent that the present discussion depends on the outcome of such a discussion, a lacuna remains in the present article.
59. However, I should address one concern. One may think that merely ordinal rankings are generally not suitable for deciding on rankings involving more than two candidates. For such rankings do not allow voters to make clear that they prefer candidate A to B much more than they prefer B to C, and ignoring all information of that kind distorts the aggregation result. However, there are two problems with this viewpoint. On the one hand, the kind of comparisons involved in such assessment must be intersubjectively intelligible, and that might be very hard in particular in large electorates. On the other hand, using more information in the aggregation also means making the process more vulnerable to manipulation/strategic voting. For naturally, the less information is used in an aggregation mechanism, the less can be used in strategic voting. (This is a concern independently of the Gibbard/Satterthwaite result that all nontrivial aggregation methods are susceptible to strategic voting to some extent.)
60. Those losers would normally not be known as such. This may explain that people who heatedly discuss, say, the virtues of candidates do not care much about the virtues of the voting method.
61. "A democratic people," so he writes, "would in this way be provided with what in any other way it would almost certainly miss - leaders of a higher grade of intellect and character than itself. Modern democracy would have its occasional Pericles, and its habitual group of superior and guiding minds" (John Stuart Mill, Considerations on Representative Government [Buffalo: Prometheus, 1991], p. 166)
62. In virtue of being an unfortunate fact, however, this problem does rule out any justification of democracy as an impeccable social ideal. Yet the realization that there can be no such ideal should not come as a surprise.
63. My discussion of the impact of the multiplicity problem on what Riker calls liberal and populist conceptions agrees with Coleman and Ferejohn's earlier and more extensive treatment. The difference is that my discussion is motivated from the point of view of whether the "losers" discussed above create problems for justifications of democracy, rather than in terms of an explicit discussion of Riker's Liberalism against Populism. The historical case study presented above and my disagreement with Cohen in Sec. VC should be taken to be supplementary. As I pointed out earlier, this article may be regarded as a belated companion piece to Coleman and Ferejohn, and the discussion in Sees. VB and VC testifies to this claim.
64. It has been objected (e.g., by a referee) that this epistemic conception is evidently wrong and thus should not even play a minor role in this argument. Clearly, I cannot defend the epistemic conception here, but I should like to sketch how one might start such an attempt. Such a defense could look at moral realism in the version developed by Peter Railton, "Moral Realism," Philosophical Review 95 (1986): 163-207, for inspiration. Just as an individual can be said to have objective interests, so at least some groups can be said to have such objective interests as well. For instance, a departmental hiring decision may be guided by three criteria, namely, expected research output, expected performance at teaching, and expected contribution to the social life of the department. If one does not shy away from resorting to "possible ways in which the world may develop" (i.e., the usual possible worlds), one can (at least in principle) construct truth conditions for sentences of the kind "It is better for the department to hire X than Y," even though it might be hard or impossible for us ever conclusively to verify such statements. In such a way, objective standards of correctness for group decisions could be devised, and from here an epistemic conception of voting could be completed. Matters would be much more complicated for political communities, but this at least is a way to see how one can start making a case for developing an epistemic theory of voting.
65. These considerations apply to Rousseau's On the Social Contract. Yet Rousseau's case is peculiar because he does not only arguably hold an epistemic conception of voting, but he is also committed to the preservation of each person's liberty (bk. I of On the Social Contract). For that reason, Rousseau's theory is troubled by the existence of the type of loser mentioned above. Such losers may have lost some of their freedom after all. I have no space to inquire how serious a problem this is for Rousseau. But one fairly obvious conclusion is that Rousseau's insistence on majority rule whenever unanimity is not required is not suitable to his project of keeping the individuals as free as in the