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Computational Criteria for Voting Systems

Published online by Cambridge University Press:  27 January 2009

Donald E. Campbell
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Extract

Condorcet's paradox of voting and Arrow's impossibility theorem are by now well known. Inspired by Arrow's treatment of social choice, others have presented alternative proofs of his theorem and different impossibility results. Professor Fishburn has recently treated us to some interesting new voting paradoxes. It is important to have the area of inconsistency among the various treatments explored and clearly mapped out. It is equally important to come to terms with the known inconsistencies in order to construct a solid social choice edifice on safe ground. Coming to terms with the inconsistencies must surely mean deciding between alternative normative conditions when all of them cannot be satisfied simultaneously. This paper attempts to do just that by adding some computational criteria to the standard list of normative criteria and then singling out a subset as being more important (to the author at least) than the rest. Since the ‘important’ criteria are mutually consistent they can be used to derive some properties of democratic decision processes. Simple majority rule, applied in sequential elimination, is distinguished as the best collective decision method. It fails to satisfy the Pareto, or unanimity, criterion – one often regarded as a sine qua non of social choice – but when this condition is added to the author's list an impossibility result obtains. An argument is proposed to counter the suggestion that Pareto optimality be added to the list and some other condition removed.

Type
Research Article
Information
British Journal of Political Science , Volume 7 , Issue 1 , January 1977 , pp. 85 - 98
Copyright
Copyright © Cambridge University Press 1977

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References

1 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; Arrow, K. J., Social Choice and Individual Values, 2nd edn. (New York: Wiley, 1963).Google Scholar

2 The following papers contain significant impossibility theorems: Blau, J. H., ‘Arrow's Theorem with Weak Independence’, Economica, XVIII (1971), 413–20CrossRefGoogle Scholar; Blau, J. H., ‘A Direct Proof of Arrow's Theorem’, Econometrica, XL (1972), 61–8CrossRefGoogle Scholar; Fishburn, P. C., ‘Arrow's Impossibility Theorem: Concise Proof and Infinite Voters’, Journal of Economic Theory, II (1970), 103–6CrossRefGoogle Scholar; Hansson, B., ‘Group Preferences’, Econometrica, XXXVII (1969), 50–4CrossRefGoogle Scholar; Kirman, A. P. and Sondermann, D., ‘Arrow's Theorem: Many Agents and Invisible Dictators’, Journal of Economic Theory, v (1972), 267–77CrossRefGoogle Scholar; MasCollel, A. and Sonnenschein, H., ‘General Possibility Theorems for Group Decisions’, Review of Economic Studies, xxxix (1972), 185–92CrossRefGoogle Scholar; Sen, A. K., ‘The Impossibility of a Paretian Liberal’, Journal of Political Economy, LXXVIII (1970), 152–7CrossRefGoogle Scholar; Wilson, R. B., ‘Social Choice Theory without the Pareto Principle’, Journal of Economic Theory, v (1972), 478–86.CrossRefGoogle Scholar

3 Fishburn, P. C., ‘Paradoxes of Voting’, American Political Science Review, LXVIII (1974), 537–46.CrossRefGoogle Scholar

4 The assumption that n is odd and that each P i is a linear order is not uncommon. See for example Fishburn, , ‘Paradoxes of Voting’.Google Scholar A binary relation > is a linear order if it is transitive and irreflexive and for all x ≠ y either x > y or y > x (transitivity and irreflexivity together rule out the possibility that x > y and y > x).

5 > is a suborder on X if for all x 1, x 2,…, x t in x the statement x 1 > x 2 … > x t implies not x t > x 1 (t varies over all integers). For a discussion of the relation between choice functions and binary relations see: Herzberger, H. G., ‘Ordinal Preference and Rational Choice’, Econometrica, XLI (1973), 187–238CrossRefGoogle Scholar; Sen, A. K., ‘Choice Functions and Revealed Preference’, Review of Economic Studies, XXXVIII (1971), 307–18CrossRefGoogle Scholar; and Sen, A. K., Collective Choice and Social Welfare (San Francisco, Calif.: Holden-Day, 1970), Chap. 1*.Google Scholar

6 Moon, J. W., Topics on Tournaments (New York: Holt, Rinehart and Winston, 1968), p. 21.Google Scholar

7 Fishburn, , ‘Paradoxes of Voting’, p. 538.Google Scholar

8 Hint – the only case of the paradox for n = 3 is the one given above. It cannot be preserved by adding a fourth person. But if xMPy when n = 4 we still have xMPy when any one person is removed.

9 Arrow, Social Choice and Individual Values.

10 These are discussed for example in Fishburn, P. C., ‘Subset Choice Conditions and the Computation of Social Choice Sets’, Quarterly Journal of Economics, LXXXVIII (1974), 320–9.CrossRefGoogle Scholar

11 Borda, Jean-Charles, ‘Mémoire sur les élections au scrutin’, Histoire de l' Académie Royale des Sciences, 1781Google Scholar (translated, with commentary, by de Grazia, Alfred, ‘Mathematical Derivation of an Election System’, Isis, XLIV (1953), 42–51)CrossRefGoogle Scholar; Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958)Google Scholar; Copeland, A. H., ‘A “Reasonable” Social Welfare Function’ (mimeographed, University of Michigan, 1951)Google Scholar; Dodgson, C. L., A Method of Taking Votes on More than Two Issues (Oxford: Clarendon Press, 1876)Google Scholar, reprinted in Black, , The Theory of Committees and Elections.Google Scholar

12 Fishburn, , ‘Paradoxes of Voting’, p. 544.Google Scholar

13 A superb axiomatic treatment of weighted voting procedures is given in Smith, J. H., ‘Aggregation of Preferences with Variable Electorate’, Econometrica, XLI (1973), 1027–42.CrossRefGoogle Scholar I have been unable to determine if there exists a counterpart to ‘Condorcet's Other Paradox’ (Fishburn, ‘Paradoxes of Voting,’ p. 544) for (M2).Google Scholar

14 In a private communication Professor Fishburn has suggested to me that the paradox can be strengthened so that there is no iteration prior to the final one in which the winner has the largest plurality. Fishburn discusses another type of sequential-elimination method, which satisfies neither the weak Condorcet criterion nor the Pareto, criterion, in his book, The Theory of Social Choice (Princeton, N.J.: Princeton University Press, 1973), p. 96.Google Scholar

15 The states x k, for each iteration k, are comprehensive and specify every feature of collective activity. In practice a party espousing x k would only announce the changes required to convert x k−1 into x k

16 Arrow, Social Choice and Individual Values, defines these criteria. It is easy to verify that they are implied by C and C*.

17 Note that Arrow permitted individual indifference between alternatives whereas we have not.

18 A referee pointed out that sequential elimination can take forever if the set of alternatives is a continuum. For example, suppose the problem is to determine the fraction of the government's budget to be devoted to hospitals and everyone agrees that more is better. Let x 1 = 1 − (I/t)(t = 1,2, …). Then x t+1EPx t for all t but the unique best alternative x* = 1 is never reached, even after an arbitrarily long time has passed. One has to assume that very small changes are not considered. If a proposed change must be some minimum distance away from the previous proposal, the alternative set is rendered finite.