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Microcosms and macrocosms: Seat allocation in proportional representation systems

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Abstract

Three alternative methods are proposed to determine a normative standard concerning the fair proportion of seats a party ought to receive in a representative assembly as a function of the voters' preference orderings. The methods differ from one another in their treatment of indifference relations and the assumptions they make about the type of scale underlying voters' preferences. Common to all three methods is the basic idea that the ratio between the number of voters preferring party i over j to the number of voters preferring party j over i can be tested for consistency, in a precisely defined sense, and if sufficiently consistent, can be appropriately scaled to determine the proportion of seats each party ought to receive. The proposed solutions are shown to satisfy several desiderata when the matrix of preference ratios is consistent. When there are cyclical majorities of equal size, the matrix of preference ratios is inconsistent. The main application of the proposed scheme is as a normative benchmark against which actual or proposed voting procedures can be evaluated in proportional representation systems. The theoretical implications of these solutions are briefly discussed.

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References

  • Arrow, K. J.: 1963, Social Choice and Individual Values, New York: Wiley (second edition).

    Google Scholar 

  • Black, D.: 1958, The Theory of Committees and Elections, Cambridge: Cambridge University Press.

    Google Scholar 

  • Budescu, D. V., Zwick, R., and Rapoport, A.: 1986, ‘A Comparison of the Eigen Value Method and the Geometric Mean Procedure for Ratio Scaling’, Applied Psychological Measurement 10, 69–78.

    Google Scholar 

  • Chamberlin, J. R., Cohen, J. L., and Coombs, C. H.: 1984, ‘Social Choice Observed: Five Presidential Elections of the American Psychological Association’, Journal of Politics 46, 479–502.

    Google Scholar 

  • Chamberlin, J. R. and Courant, P. N.: 1983, ‘Representative Deliberations and Representative Decisions: Proportional Representation and the Borda Count’, American Political Science Review 77, 718–733.

    Google Scholar 

  • Concordet, Marquis de: 1785, Essai sur l'Application de l'Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix, Paris: L'Imprimerie Royale.

  • Duverger, M.: 1963, Political Parties: Their Organization and Activity in the Modern State, New York: Wiley.

    Google Scholar 

  • Felsenthal, D. S., Maoz, Z., and Rapoport, A.: 1986, ‘Comparing Voting Systems in Genuine Elections: Approval-Plurality versus Selection-Plurality’, Social Behaviour 1, 41–53.

    Google Scholar 

  • Felsenthal, D. S., Maoz, Z., and Rapoport, A.: 1985, ‘The Condorcet-Efficiency of Sophisticated Approval and Plurality Voting’, Presented at the annual meeting of the American Political Science Association, New Orleans, LA., August 29–September 1.

  • Fishburn, P. C.: 1971, ‘A Comparative Analysis of Group Decision Methods’, Behavioral Science 16, 538–544.

    Google Scholar 

  • Fishburn, P. C. and Gehrlein W. V.: 1982, ‘Majority Efficiencies for Simple Voting Procedures: Summary and Interpretation’, Theory and Decision 14, 141–153.

    Google Scholar 

  • Fishburn, P. C. and Brams, S. J.: 1981, ‘Approval Voting, Concorcet's Principle, and Runoff Elections’, Public Choice 36, 89–114.

    Google Scholar 

  • Jensen, R. E.: 1984, ‘An Alternative Scaling Method for Priorities in Hierarchical Structures’, Journal of Mathematical Psychology 28, 317–332.

    Google Scholar 

  • Merrill, S.: 1984, ‘A Comparison of Efficiency of Multicandidate Electoral Systems’, American Journal of Political Science 28, 23–48.

    Google Scholar 

  • Niemi, R. G. and Frank, A. Q.: 1985, ‘Sophisticated Voting Under the Plurality Procedure: A Test of a New Definition’, Theory and Decision 19, 151–162.

    Google Scholar 

  • Nurmi, H.: 1983, ‘Voting Procedures: A Summary Analysis’, British Journal of Political Science 13, 181–208.

    Google Scholar 

  • Rae, D.: 1971, The Political Consequences of Electoral Laws, New Haven: Yale University Press.

    Google Scholar 

  • Rapoport, A., Felsenthal, D. S., and Maoz, Z.: 1986, ‘Proportional Representation in Israel's General Federation of Labor: An Empirical Analysis of a New Scheme’, Mimeographed, University of Haifa.

  • Riker, W. R.: 1982a, Liberalism Against Populism, San Francisco: W. H. Freeman.

    Google Scholar 

  • Riker, W. R.: 1982b: ‘The Two Party System and Duverger's Law: An Essay on the History of Political Science’, American Political Science Review 76, 753–766.

    Google Scholar 

  • Saaty, T. L.: 1980, The Analytic Hierarchy Process, New York: McGraw Hill.

    Google Scholar 

  • Saaty, T. L.: 1977, ‘A Scaling Method for Priorities in Hierarchical Structures’, Journal of Mathematical Psychology 15, 134–281.

    Google Scholar 

  • Saaty, T. L. and Vargas, L. G.: 1984a, ‘Comparison of Eigenvalue, Logarithmic Least Squares, and Least Squares Methods in Estimating Ratios’, Mathematical Modelling 5, 309–324.

    Google Scholar 

  • Saaty, T. L. and Vargas, L. G.: 1984b, ‘Inconsistency and Rank Preservation’, Journal of Mathematical Psychology 28, 205–214.

    Google Scholar 

  • Straffin, P. D.: 1980, Topics in the Theory of Voting, Boston: Birkhauser.

    Google Scholar 

  • Williams, C. and Crawford, G.: 1985, ‘A Note on the Analysis of Subjective Judgment Matrices’, Journal of Mathematical Psychology 29, 387–405.

    Google Scholar 

Download references

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This research was supported by the United States - Israel Binational Science Foundation (BSF), Jerusalem, Israel. We wish to thank Steven Brams, David V. Budescu, and Joseph Greenberg for many helpful comments.

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Rapoport, A., Felsenthal, D.S. & Maoz, Z. Microcosms and macrocosms: Seat allocation in proportional representation systems. Theor Decis 24, 11–33 (1988). https://doi.org/10.1007/BF00137220

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