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Voting Procedures: A Summary Analysis

Published online by Cambridge University Press:  27 January 2009

Extract

Roughly two centuries ago the Marquis de Condorcet and Chevalier Jean-Charles de Borda originated a research tradition – by no means a continuous one – that over the decades has produced results casting doubt on many widely used collective decision-making procedures. The phenomenon known as the Condorcet effect or the Condorcet paradox is the well-known problem of the simple majority rule. The paradox bearing the name of Borda is less commonly known, but it also relates to a procedure that is widely used, namely the plurality principle. Either one of these paradoxes is serious enough to make these procedures suspect unless one is convinced that the situations giving rise to these paradoxical features are extremely rare. In this article we review some voting procedures that have been introduced in the literature. We aim at giving a synthesis of the assessments of procedures with respect to various criteria.

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Articles
Copyright
Copyright © Cambridge University Press 1983

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References

1 See de Condorcet, Marquis, Essai sur l'application de l'analyse à la probabilité des décisions rendues à la pluralité des voix (Paris, 1785)Google Scholar. For an account of the historical development of the social choice theory, see Black, D., The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958).Google Scholar

2 Colman, A. and Pountney, I., ‘Borda's Voting Paradox’, Behavioral Science, XXIII (1978), 1520.CrossRefGoogle Scholar

3 In some contexts we shall assume a connected, transitive and reflexive preference relation (weak ordering) Ri for all voters. Obviously this assumption is less stringent than the assumption that Pi be connected and transitive. The fact that Pi is connected and transitive implies that Ri is connected and transitive. On the other hand, if Ri is connected and transitive, Pi is also transitive but not necessarily connected. For the explanation of the basic set-theoretic concepts, see the

4 McKelvey, R. D., ‘Intransitivities in Multidimensional Voting Models and Some Implications for Agenda Control’, Journal of Economic Theory, XII (1976), 472–82CrossRefGoogle Scholar, and McKelvey, R. D., ‘General Conditions for Global Intransitivities in Formal Voting Models’, Econometrica, XLVII (1979), 1085–112CrossRefGoogle Scholar. For some recent social choice results in spatial context, see Hinich, M. J., ‘The Mean versus the Median in Spatial Voting Games’Google Scholar and Kramer, G. H., ‘Existence of Electoral Equilibrium’, both in Ordershook, P. C., ed., Game Theory and Political Science (New York: New York University Press, 1978)Google Scholar. See also Hinich, M. J. and Pollard, W., ‘A New Approach to the Spatial Theory of Electoral Competition’, American Journal of Political Science, XXV (1981), 323–41.CrossRefGoogle Scholar

5 Schofield, N., ‘General Relevance of the Impossibility Theorem in Dynamical Social Choice’ (paper given at the annual meeting of the American Political Science Association, New York, 1978)Google Scholar. See also Nurmi, H., ‘Majority Rule: Second Thoughts and Refutations’, Quality and Quantity, XIV (1980), 743–65.CrossRefGoogle Scholar

6 Niemi, R. G. and Riker, W. H., ‘The Choice of Voting Systems’, Scientific American, 234 (1976), 21–7.CrossRefGoogle Scholar

7 Richelson, J. T., ‘A Comparative Analysis of Social Choice Functions II’, Behavioral Science, XXIII (1978), 3844.CrossRefGoogle Scholar

8 Richelson, , ‘A Comparative Analysis of Social Choice Functions II’Google Scholar. By preference change we mean the change from xP iy to yP ix.

9 Schwartz, T., ‘Rationality and the Myth of the Maximum’, Nous, VI (1972), 97117.CrossRefGoogle Scholar

10 Condorcet, , Essai sur l'application de l'analyse.Google Scholar

11 Kramer, G. H., ‘A Dynamical Model of Political Equilibrium’, Journal of Economic Theory, XVI (1977), 310–34CrossRefGoogle Scholar. See also Simpson, P. B., ‘On Defining Areas of Voter Choice’, Quarterly Journal of Economics, LXXXIII (1969), 478–90.CrossRefGoogle Scholar

12 Nurmi, H., ‘On Taking Preferences Seriously’, manuscript, Department of Philosophy, University of Turku, 1982.Google Scholar

13 de Borda, J. C., ‘Mémoire sur les elections au scrutin’, Mémoires de l'Académie Royale des Sciences (1781), 657–65Google Scholar. The English translation of this text can be found in de Grazia, A., ‘Mathematical Derivation of an Election System’, Isis, XLIV (1953), 4251.CrossRefGoogle Scholar

14 Colman, and Pountney, , ‘Borda's Voting Paradox’.Google Scholar

15 Of course the fact that the procedure chooses the Condorcet loser implies this.

16 Borda, , ‘Mémoire sur les elections au scrutin’.Google Scholar

17 Borda used the values a = 1 and b = 1 in his example.

18 Young, H. P., ‘An Axiomatization of Borda's Rule’, Journal of Economie Theory, IX (1974), 4352.CrossRefGoogle Scholar

19 See for example Niemi, and Riker, , ‘The Choice of Voting Systems’.Google Scholar

20 Riker, W. H. and Ordeshook, P. C., Introduction to Positive Political Theory (Englewood Cliffs: Prentice-Hall, 1973), pp. 8890.Google Scholar

21 Brams, S. J. and Fishburn, P. C., ‘Approval Voting’, American Political Science Review, LXXII (1978), 831–47.CrossRefGoogle Scholar

22 Fishburn, P. C. and Brams, S. J., ‘Approval Voting, Condorcet's Principle, and Runoff Elections’, Public Choice, XXXVI (1981), 89114Google Scholar; Fishburn, P. C. and Brams, S. J., ‘Efficacy, Power and Equity under Approval Voting’, Public Choice, XXXVII (1981), 425–34CrossRefGoogle Scholar; Fishburn, P. C. and Brams, S. J., ‘Expected Utility and Approval Voting’, Behavioral Science, XXVI (1981), 136–42.CrossRefGoogle Scholar

23 Fishburn, and Brams, , ‘Approval Voting, Condorcet's Principle and Runoff Elections’.Google Scholar

24 Straffin, P. D. Jr., Topics in the Theory of Voting (Boston: Birkhäuser, 1980).Google Scholar

25 Straffin, , Topics in the Theory of Voting, p. 31.Google Scholar

26 Straffin, , Topics in the Theory of Voting, p. 31.Google Scholar

27 Straffin, , Topics in the Theory of Voting, p. 25.Google Scholar

28 Straffin, , Topics in the Theory of Voting, p. 25.Google Scholar

29 Straffin, , Topics in the Theory of Voting, p. 26.Google Scholar

30 Fishburn, P. C., ‘Axioms for Approval Voting: Direct Proof’, Journal of Economic Theory, XIX (1978), 180–5CrossRefGoogle Scholar; Young, , ‘An Axiomatization of Borda's Rule’.Google Scholar

31 Straffin, , Topics in the Theory of Voting, pp. 27–8.Google Scholar

32 Smith, J. H., ‘Aggregation of Preferences with Variable Electorate’, Econometrica, XLI (1973), 1027–41CrossRefGoogle Scholar; see also Straffin, , Topics in the Theory of Voting, p. 24.Google Scholar

33 Brams, S. J., ‘Approval Voting: a Better Way to Elect a President’, prepared for delivery at the New York Academy of Sciences, New York, 1981.Google Scholar

34 Straffin, , Topics in the Theory of Voting, p. 24.Google Scholar

35 Fishburn, P. C., ‘Condorcet Social Choice Functions’, SIAM Journal on Applied Mathematics, XXXIII (1977), 469–89.CrossRefGoogle Scholar

36 Straffin, , Topics in the Theory of Voting, p. 26.Google Scholar

37 Walker, M., ‘On the Nonexistence of a Dominant Strategy Mechanism for Making Optimal Public Decisions’, Econometrica, XLVIII (1980), 1521–40CrossRefGoogle Scholar; Fishburn, P. C., ‘Symmetrical Social Choices and Collective Rationality’, Mathematical Social Sciences, 1 (1980), 19.CrossRefGoogle Scholar

38 See e.g. Plott, C. R., ‘Axiomatic Social Choice Theory: an Overview and Interpretation’, American Journal of Political Science, XX (1976), 511–96.CrossRefGoogle Scholar

39 McKelvey, , ‘Intransitivities in Multidimensional Voting Models’Google Scholar; McKelvey, , ‘General Conditions for Global Intransitivities’.Google Scholar

40 We shall call the procedure Pareto optimal if and only if it necessarily leads to Pareto- optimal choices.

41 Nurmi, H., ‘On the Properties of Voting Systems’, Scandinavian Political Studies, IV (1981), 1932.CrossRefGoogle Scholar

42 Nurmi, , ‘On the Properties of Voting Systems’, p. 28.Google Scholar

43 Nurmi, , ‘On the Properties of Voting Systems’, p. 28.Google Scholar

44 Plott, , ‘Axiomatic Social Choice Theory’, pp. 549–50.Google Scholar

45 Plott, , ‘Axiomatic Social Choice Theory’, pp. 549–50.Google Scholar

46 Nurmi, , ‘On the Properties of Voting Systems’, p. 27.Google Scholar

47 Nurmi, , ‘On the Properties of Voting Systems’, pp. 27–8.Google Scholar

48 Nurmi, , ‘On the Properties of Voting Systems’, p. 27.Google Scholar

49 Nurmi, , ‘On the Properties of Voting Systems’, p. 26.Google Scholar

50 Fishburn, , ‘Axioms for Approval Voting’Google Scholar; Young, , ‘An Axiomatization of Borda's Rule’.Google Scholar

51 Nurmi, , ‘On the Properties of Voting Systems’, p. 25.Google Scholar

52 Richelson, J. T., ‘A Comparative Analysis of Social Choice Functions’, Behavioral Science, XX (1975), 331–7CrossRefGoogle Scholar; Richelson, , ‘A Comparative Analysis of Social Choice Functions II’Google Scholar; Richelson, , ‘A Comparative Analysis of Social Choice Functions III’, Behavioral Science, XXIII (1978), 169–76CrossRefGoogle Scholar; Richelson, , ‘A Comparative Analysis of Social Choice Functions, I, II, III: A Summary’, Behavioral Science, XXIV (1979), 355.CrossRefGoogle Scholar

53 Nurmi, , ‘On the Properties of Voting Systems’, p. 25.Google Scholar

54 Young, , ‘An Axiomatization of Borda's Rule’, p. 45.Google Scholar

55 Nurmi, , ‘On the Properties of Voting Systems’, p. 26.Google Scholar

56 Richelson, , ‘A Comparative Analysis of Social Choice Functions II’, p. 335.Google Scholar

57 Actually, this amounts to disregarding both implementation criteria as all binary methods can be implemented in a simple way.