¹ Following the decisions made in 1990 at the Labour Party Conference, the Labour party's National Executive Committee established a Working Party chaired by Raymond Plant of Southampton University. It was asked to consider which electoral system is appropriate for elections to the Scottish Parliament, the European Parliament, the Assemblies for Wales and the English regions, the Second Chamber, the House of Commons and local governments. The Working Party has issued an interim report (The Plant Report: A Working Party on Electoral Reform, Guardian Studies, Vol. 3 (London: The Guardian, 07 1991)Google Scholar, in which it presented several criteria for evaluating electoral systems – all different from the ones used in this article – and addressed some of the electoral procedures that are also analysed by us.

² It has already been proved that the expected frequency of violation of the Condorcet Winner desideratum is lower under Hare's Single Transferable Vote (STV) procedure than under the plurality voting (PV) procedure. See Felsenthal, Dan S. and Maoz, Zeev, ‘Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures’, Behavioral Science, 37 (1992), 109–28.CrossRefGoogle Scholar

³ See Fishburn, Peter C. and Gehrlein, William V., ‘Majority Efficiencies for Simple Voting Procedures: Summary and Interpretation’, Theory and Decision, 14 (1982), 141–53CrossRefGoogle Scholar; Bordley, Robert F., ‘A Pragmatic Method for Evaluating Election Schemes Through Simulation’, American Political Science Review, 77 (1983), 123–39CrossRefGoogle Scholar; Hoffman, Dale T., ‘Relative Efficiency of Voting Systems’, SIAM Journal of Applied Mathematics, 43 (1983), 1213–19CrossRefGoogle Scholar; Merrill, Samuel, ‘A Comparison of Efficiency of Multicandidate Electoral Systems’, American Journal of Political Science, 28 (1984), 23–48CrossRefGoogle Scholar; Niemi, Richard G. and Frank, Arthur Q., ‘Sophisticated Voting Under the Plurality Procedure: A Test of a New Definition’, Theory and Decision, 19 (1985), 151–62CrossRefGoogle Scholar; Felsenthal, Dan S., Maoz, Zeev and Rapoport, Amnon, ‘The Condorcet Efficiency of Sophisticated Voting Under the Plurality and Approval Procedures’, Behavioral Science, 35 (1990), 24–33.CrossRefGoogle Scholar

⁴ See Felsenthal, Dan S., Rapoport, Amnon and Maoz, Zeev, ‘Tacit Cooperation in Three-Alternative Non-Cooperative Voting Games: A New Model of Sophisticated Behaviour Under the Plurality Procedure’, Electoral Studies, 7 (1988), 143–61CrossRefGoogle Scholar; Rapoport, Amnon, Felsenthal, Dan S. and Maoz, Zeev, ‘Sincere Versus Strategic Voting Behavior in Small Groups’, in Palfrey, Thomas R., ed., Laboratory Research in Political Economy (Ann Arbor: University of Michigan Press, 1991), pp. 201–35.Google Scholar

⁵ See Chamberlin, John R., Cohen, Jerry L. and Coombs, Clyde H., ‘Social Choice Observed: Five Presidential Elections of the American Psychological Association’, Journal of Politics, 46 (1984), 718–33CrossRefGoogle Scholar; Felsenthal, Dan S., Maoz, Zeev and Rapoport, Amnon, ‘Comparing Voting Systems in Genuine Elections: Approval-Plurality Versus Selection-Plurality’, Social Behaviour, 1 (1986), 41–53Google Scholar; Fishburn, Peter C. and Little, John D. C., ‘An Experiment in Approval Voting’, Management Science, 34 (1988), 555–68CrossRefGoogle Scholar; Rapoport, Amnon, Felsenthal, Dan S. and Maoz, Zeev, ‘Proportional Representation: An Empirical Evaluation of Single-Stage Non-Ranked Voting Procedures’, Public Choice, 59 (1988), 151–65CrossRefGoogle Scholar; Felsenthal, Dan S., ‘Proportional Representation Under Three Voting Procedures: An Israeli Study’, Political Behavior, 14 (1992), 159–92.CrossRefGoogle Scholar

⁶ Tideman obtained the data of these thirty-seven elections from the Electoral Reform Society (ERS) of Great Britain and Ireland, which is an organization dedicated to furthering the usage of STV. In addition to direct advocacy, the ERS facilitates the use of STV by serving as an impartial counter of votes in STV elections. For further information and documentation concerning the data, contact T. Nicolaus Tideman, Department of Economics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA (telephone (703) 961–7592).

⁷ See Fishburn, Peter C., ‘Social Choice Functions’, SIAM Review, 16 (1974), 63–90CrossRefGoogle Scholar; Richelson, Jeffrey T., ‘Comparative Analysis of Social Choice Functions’, Behavioral Science, 20 (1975), 331–7CrossRefGoogle Scholar; Nurmi, Hannu, ‘Voting Procedures: A Summary Analysis’, British Journal of Political Science, 13 (1983), 181–208CrossRefGoogle Scholar; Nurmi, Hannu, Comparing Voting Systems (Boston, Mass.: Kluwer Academic Publishers, 1987)CrossRefGoogle Scholar; Felsenthal, Dan S. and Maoz, Zeev, ‘A Comparative Analysis of Sincere and Sophisticated Voting Under the Plurality and Approval Procedures’, Behavioral Science, 33 (1988), 116–30.CrossRefGoogle Scholar

⁸ Felsenthal, and Maoz, , ‘Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures’.Google Scholar

⁹ See Nurmi, , ‘Voting Procedures: A Summary Analysis’Google Scholar; Young, H. P., ‘An Axiomatization of Borda's Rule’, Journal of Economic Theory, 9 (1974), 43–52.CrossRefGoogle Scholar

¹⁰ See Felsenthal, and Maoz, , ‘A Comparative Analysis of Sincere and Sophisticated Voting Under the Plurality and Approval Procedures’Google Scholar; ‘Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures’.

¹¹ We assume that the purpose of most multi-winner elections is to select the best s individuals out of the m competing candidates rather than the best s-membered team out of all possible m!/s!(m-s)! such teams. Consequently, we assume that each voter states his or her preference ordering in terms of the m individual candidates rather than in terms of all possible s-membered teams. (Note that when s > 1 the identity of the s elected candidates may be different under any ranked voting procedure if voters are required to rank the m individual candidates as opposed to ranking all possible s-membered teams that can be composed out of m candidates, even if each voter's preference orderings among the s-membered teams is consistent with his or her preference ordering of the m individual candidates.)

¹² Copeland, A. H., ‘A Reasonable Social Welfare Function’ (manuscript, University of Michigan Seminar on Applications of Mathematics to the Social Sciences, 1951).Google Scholar

¹³ See Felsenthal, Dan S. and Brichta, Avraham, ‘Sincere and Sophisticated Voters: An Israeli Study’, Political Behavior, 7 (1985), 311–24CrossRefGoogle Scholar; Felsenthal, , Maoz, and Rapoport, , ‘Comparing Voting Systems in Genuine Elections: Approval-Plurality Versus Selection-Plurality’Google Scholar; Felsenthal, , ‘Proportional Representation Under Three Voting Procedures: An Israeli Study’Google Scholar; Niemi, Richard G., Whitten, Guy and Franklin, Mark N., ‘Constituency Characteristics, Individual Characteristics and Tactical Voting in the 1987 British General Election’, British Journal of Political Science, 22 (1992), 229–40.CrossRefGoogle Scholar

¹⁴ See Richelson, Jeffrey T., ‘A Comparative Analysis of Social Choice Functions, III’, Behavioral Science, 23 (1978), 169–76CrossRefGoogle Scholar; Nurmi, , ‘Voting Procedures: A Summary Analysis’.Google Scholar

¹⁵ See Dummett, Michael, Voting Procedures (Oxford: Clarendon Press, 1984), pp. 115–22Google Scholar. Although Dummett was concerned only with the case s = 1, his argument can be extended to cases where s > 1 Dummett also suggested (cf. chap. 7) that when s = 1 and there exist several candidates with the same (highest) Copeland score, the one with the highest Borda score ought to be elected.

¹⁶ We are concerned here only with criteria that are relevant, given n, m, s and voters' preference ordering, but not with criteria which are applicable when a hypothetical change is assumed to occur in one of these variables while all the remaining variables are held constant.

¹⁷ See de Condorcet, Marquis, Essai sur l'Application de l'Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix (Paris: Imprimerie Royale, 1785)Google Scholar. For an extensive account of the historical development of social choice theory, see Black, Duncan, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958)Google Scholar. For the sake of historical accuracy it should be mentioned that Condorcet was partly anticipated by Ramon Lull (c. 1235–1315), a native of Palma, Mallorca. In his paper ‘De Arte Eleccionis’, written in 1299, Lull proposes a procedure based on m – 1 pairwise comparisons (where m is the number of candidates), which, if a Condorcet winner exists, leads to his being elected and never elects a Condorcet loser. In an earlier novel by Lull, , Blanquerna (c. 1285)Google Scholar, he proposes a voting procedure which, in our view, is essentially the same as that proposed more than 660 years later by Copeland; cf. McLean, I. and London, John, ‘The Borda and Condorcet Principles: Three Medieval Applications’, Social Choice and Welfare, 7 (1990), 99–108CrossRefGoogle Scholar. Although we think that McLean and London are wrong in interpreting the proposal made by Lull in Blanquerna as equivalent to that proposed almost 500 years later by Borda, they nevertheless state that Condorcet was the first to provide a logicalmathematical justification for his scheme.

¹⁸ It should be noted that there may exist more than one Condorcet Winner and/or more than one Condorcet Loser. For example, if the social preference ordering is a = b > c > d = e, then a and b are Condorcet Winners, whereas d and e are Condorcet Losers.

¹⁹ Note that in terms of satisfying this desideratum Copeland's procedure has an advantage over Condorcet's procedure when the social preference ordering contains cycles. To see this, consider the following example (for which we are grateful to Iain McLean). Suppose there are three voters, 1, 2 and 3, whose (descending) order of preference among four candidates, w, x, y and z, are as follows: voter 1: x, y, z, w; voter 2: y, z, w, x; voter 3: z, w, x, y. Here w is Pareto-inferior because all three voters prefer z to w. Because the social preference ordering is completely cyclical (z > w > x > y > z), the Condorcet set contains all four candidates, whereas Copeland's procedure assigns to y and z a higher score (2) than it assigns to x and w (1).

²⁰ See Brams, Steven J. and Fishburn, Peter C., Approval Voting (Boston, Mass.: Birkhauser, 1983), p. xi.Google Scholar

²¹ Borda's original proposal appears in his 1770 paper ‘Sur la Forme des Élections’ and a revised version of this paper is his ‘Mémoire sur les Élections au Scrutin’ which appears on pp. 657–65Google Scholar of the Histoire et Mémoires de l'Académie Royale des Sciences année 1781, published in 1784Google Scholar; cf. Black, , The Theory of Committees and Elections, pp. 156–7, 178–9Google Scholar. Borda was not in fact the first to propose a ranked voting procedure. Nicolas Cusanus (1401–64) proposed a ranked procedure (for electing the Holy Roman Emperor) in his De Concordantia Catholica (c. 1434) which is essentially identical to that of Borda; see McLean, and London, , ‘The Borda and Condorcet Principles: Three Medieval Applications’Google Scholar. Nevertheless, as stated by McLean and London, Borda, too, was the first to provide a logical-mathematical justification for his scheme.

²² See Dummett, , Voting Procedures, p. 129.Google Scholar

²³ See Arrow, Kenneth J., Social Choice and Individual Values (New York: Wiley, 1951).Google Scholar

²⁴ The Weak Axiom of Revealed Preferences (WARP) requires that if a candidate is elected when there are m competing candidates, then this candidate ought also to be elected, ceteris paribus, if he or she is one of m – 1 candidates. See Richter, M. K., ‘Revealed Preference Theory’, Econometrica, 34 (1966), 635–45.CrossRefGoogle Scholar

²⁵ See Felsenthal, and Maoz, , ‘Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures’.Google Scholar

²⁶ Lakeman, Enid, How Democracies Vote: A Study of Majority and Proportional Electoral Systems (London: Faber and Faber, 3rd revised edn, 1970), p. 298.Google Scholar

²⁷ See Black, , The Theory of Committees and Elections, pp. 176–7.Google Scholar

²⁸ Cited in Black, , The Theory of Committees and Elections, pp. 157–8.Google Scholar

²⁹ See Lakeman, , How Democracies Vote, p. 245.Google Scholar

³⁰ Dummett, , Voting Procedures, p. 268.Google Scholar

³¹ H. R. Droop argued in 1869 that this quota is preferable to Hare's original quota (n/s), because it results in the lowest number such that no more than s candidates can each obtain that number of first-preference votes. Cf. Lakeman, , How Democracies Vote, p. 129.Google Scholar

³² A detailed description of the STV procedure appears in Dummett, , Voting Procedures, pp. 269–73Google Scholar, and in Lakeman, , How Democracies Vote, Appendix IV, pp. 247–75.Google Scholar

³³ See Nurmi, , ‘Voting Procedures: A Summary Analysis’Google Scholar; Felsenthal, and Maoz, , ‘A Comparative Analysis of Sincere and Sophisticated Voting Under the Plurality and Approval Procedures’Google Scholar; ‘Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures’.

³⁴ See Rapoport, Amnon and Felsenthal, Dan S., ‘Efficacy in Small Electorates Under Plurality and Approval Voting’, Public Choice, 64 (1990), 57–71.CrossRefGoogle Scholar

³⁵ Brams, and Fishburn, , Approval Voting, p. 77Google Scholar; Chamberlin, , Cohen, and Coombs, , ‘Social Choice Observed: Five Presidential Elections of the American Psychological Association’.Google Scholar

³⁶ See Felsenthal, , Maoz, and Rapoport, , ‘Comparing Voting Systems in Genuine Elections: Approval-Plurality Versus Selection-Plurality’Google Scholar; Fishburn, and Little, , ‘An Experiment in Approval Voting’.Google Scholar

³⁷ As the AV procedure provides voters with an opportunity to vote for at least as many candidates as they can under the PVM procedure, it is reasonable to assume that voters operating under AV will indeed take advantage of this opportunity. As we assumed that voters under PVM will vote for s candidates, and that voters under AV will vote for approximately m/2 candidates when s ≤ m/2, it is logical to assume that voters under AV will vote for approximately s + (m – s)/k (k is an integer) candidates when s > m/2. We decided to adopt k = 2 because it is the smallest meaningful value that k can assume.

³⁸ As mentioned earlier, all thirty-seven elections were actually conducted under the ALV, RAL or STV procedures. For a ballot to be valid under these procedures it must contain at least one ranked candidate, and all (ranked) candidates must be assigned different ranks. Therefore we ignored in our computations a total of forty-five ballots (found in Elections 2, 3, 4, 5, 8, 9, 10 and 21) that violated the second of these conditions.

³⁹ The Continuity desideratum states that, ceteris paribus, if a candidate is elected when s = x, this candidate should also be elected when s > x (1 ≤ x < m – 1). The article by Felsenthal, and Maoz, , ‘Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures’Google Scholar, shows that the STV procedure may violate this desideratum.

⁴⁰ See Garman, Mark B. and Kamien, Morton I., ‘The Paradox of Voting: Probability Calculations’, Behavioral Science, 13 (1968), 306–17CrossRefGoogle ScholarPubMed; Niemi, Richard G. and Weisberg, Herbert F., ‘A Mathematical Solution for the Probability of the Paradox of Voting’, Behavioral Science, 13 (1968), 317–23CrossRefGoogle ScholarPubMed; Demeyer, Frank and Plott, Charles R., ‘The Probability of a Cyclical Majority’, Econometrica, 38 (1970), 345–54CrossRefGoogle Scholar; May, Robert M., ‘Some Mathematical Remarks on the Paradox of Voting’, Behavioral Science, 16 (1971), 143–51CrossRefGoogle Scholar; Grofman, Bernard, ‘A Note on Some Generalizations of the Paradox of Cyclical Majorities’, Public Choice, 12 (1972), 113–14CrossRefGoogle Scholar; Fishburn, Peter C., ‘Single-Peaked Preferences and Probabilities of Cyclical Majorities’, Behavioral Science, 19 (1974), 21–7.CrossRefGoogle Scholar

⁴¹ See Klahr, David, ‘A Computer Simulation of the Paradox of Voting’, American Political Science Review, 60 (1966), 384–90CrossRefGoogle Scholar; Felsenthal, , Maoz, and Rapoport, , ‘The Condorcet Efficiency Sophisticated Voting Under the Plurality and Approval Procedures’.Google Scholar

⁴² See Riker, William H., ‘The Paradox of Voting and Congressional Rules for Voting on Amendments’, American Political Science Review, 52 (1958), 349–66CrossRefGoogle Scholar; Niemi, Richard G., ‘The Occurrence of the Paradox of Voting in University Elections’, Public Choice, 8 (1970), 91–100CrossRefGoogle Scholar; Blydenburg, John C., ‘The Closed Rule and the Paradox of Voting’, Journal of Politics, 33 (1971), 57–71CrossRefGoogle Scholar; Bowen, Bruce D., ‘Toward an Estimate of the Frequency of the Paradox of Voting in US Senate Roll Call Votes’, in Niemi, Richard G. and Weisberg, Herbert F., eds, Probability Models of Collective Decision Making (Columbus, Ohio: Charles E. Merrill, 1972), pp. 181–203Google Scholar; Weisberg, Herbert F. and Niemi, Richard G., ‘Probability Calculations for Cyclical Majorities in Congressional Voting’Google Scholar, in Niemi, and Weisberg, , eds, Probability Models of Collective Decision Making, pp. 204–31Google Scholar. For two recent articles based on almost the same dataset as the present study (thirty-three out of the thirty-seven elections) that also examine the prevalence of cyclical preference orderings, cf. Feld, Scott L. and Grofman, Bernard, ‘Collectivities as Actors: Consistency of Collective Choices’, Rationality and Society, 2 (1990), 429–48CrossRefGoogle Scholar; Feld, Scott L. and Grofman, Bernard, ‘Who is Afraid of the Big Bad Cycle? Evidence from 36 Elections’, Journal of Theoretical Politics (forthcoming)Google Scholar. However, because we have studied four elections involving large numbers of voters that were not analysed by Feld and Grofman (Elections 4, 7, 8 and 10), whereas they studied three elections (whose database was obtained from another source and hence not available to us) involving relatively few voters, our conclusions as to the prevalence and type of cycles are somewhat different than theirs.

⁴³ Because the STV procedure may violate the Continuity desideratum, we determined under this procedure which candidates were actually elected, given s, in each of the thirty-seven elections, but not also the resultant ranking of all the m candidates in each of these elections. Note also that since PVO and PVM are identical when s = 1, these two versions of the PV procedure have the same r_s, in all the twelve elections where s = 1.

⁴⁴ As noted earlier, the expected frequency with which the STV procedure violates the Condorcet Winner desideratum is indeed lower than the respective frequency under the PVO procedure. It is also interesting to note that the percentage of elected Condorcet Losers under the PVO procedure found in our study (5.5 per cent) is almost identical to the one found by Colman and Pountney in their study of the 1966 British general election. These authors estimate that, of the 266 British MPs elected in the 1966 general election by gaining only a plurality (rather than an absolute majority) of the votes in their constituencies, at least fifteen (that is, 5.7 per cent) were in fact Condorcet Losers (of whom all but one belonged to the Conservative party). See Colman, Andrew M. and Pountney, Ian, ‘Borda's Voting Paradox: Theoretical Likelihood and Electoral Occurrences’, Behavioral Science, 23 (1978), 15–21.CrossRefGoogle Scholar

⁴⁵ See Brams, and Fishburn, , Approval VotingGoogle Scholar; Nurmi, , Comparing Voting SystemsGoogle Scholar; Merrill, Samuel, Making Multi-Candidate Elections More Democratic (Princeton, NJ: Princeton University Press, 1988).Google Scholar

⁴⁶ See, for example, Black, , The Theory of Committees and Elections, pp. 61, 171–5Google Scholar; Schwartz, T., ‘Rationality and the Myth of the Maximum’, Nous, 6 (1972), 97–117CrossRefGoogle Scholar; Kramer, G. H., ‘A Dynamical Model of Political Equilibrium’, Journal of Economic Theory, 16 (1977), 310–34CrossRefGoogle Scholar; Felsenthal, Dan S. and Machover, Moshé, ‘After Two Centuries, Should Condorcet's Voting Procedure Be Implemented?’ (manuscript, King's College London, 1992).Google Scholar

⁴⁷ Felsenthal, and Machover, , ‘After Two Centuries, Should Condorcet's Voting Procedure Be Implemented?’Google Scholar

⁴⁸ It has already been established that all elimination procedures (including ALV-RAL and STV) may violate the Monotonicity, Consistency, Negative Responsiveness and the Weak Axiom of Revealed Preferences (WARP) desiderata, in both single- and multi-winner elections. In addition, the STV procedure may also violate the Continuity desideratum. Of these five desiderata, only the WARP desideratum may also be violated by the PVO, PVM, AV and BR procedures when voters are assumed to vote sincerely. For a discussion of these desiderata under various procedures when voters are assumed to vote sincerely or strategically, as well as in single- or multi-winner elections, see Nurmi, , ‘Voting Procedures: A Summary Analysis’Google Scholar; Felsenthal, and Maoz, , ‘A Comparative Analysis of Sincere and Sophisticated Voting Under the Plurality and Approval Procedures’Google Scholar; ‘Normative Properties of Four Single-Stage Multi-Winner Electoral Procedures’.