Existence of a Dictatorial Subgroup in Social Choice with Independent Subgroup Utility Scales, an Alternative Proof

  • Anna B. KhmelnitskayaEmail author
Part of the Theory and Decision Library C book series (TDLC, volume 43)


Social welfare orderings for different scales of individual utility measurement in distinct population subgroups are studied. In Khmelnitskaya (2000), employing the continuous version of Arrow’s impossibility theorem, it was shown that for combinations of independent subgroups scales every corresponding social welfare ordering depends on the utilities of only one of the subgroups and is determined in accordance with the scale type proper to this dictatorial subgroup. In this article we introduce an alternative completely self-contained proof based on the study of the structure of level surfaces of a social welfare function which provides a real-valued representation of the social welfare ordering.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Arrow, K. J. (1951). Social choice and individual values (2nd ed., 1963). New York: Wiley.Google Scholar
  2. Bossert, W., & Weymark, J. A. (2004). Utility in social choice. In: Barberá, S., Hammond, P. J., & Seidl, C. (Eds.), Handbook of utility theory (Vol. 2, pp. 1099–1178). Boston: Kluwer Academic Publishers.Google Scholar
  3. d’Aspremont, C. (1985). Axioms for social welfare orderings. In: Hurwicz, L., Schmeidler, D., & Sonnenschein, H. (Eds.), Social goals and social organizations: Essays in memory of Elisha Pazner (pp. 19–76). Cambridge: Cambridge University Press.Google Scholar
  4. d’Aspremont, C., & Gevers, L. (1977). Equity and the informational basis of collective choice. Review of Economic Studies, 44, 199–209.CrossRefGoogle Scholar
  5. Debreu, G. (1954). Representation of a preference ordering by a numerical function. In: Thrall, R. M., Coombs, C. H., & Davis, R. L. (Eds.), Decision processes (pp. 159–165). New York: Wiley.Google Scholar
  6. Hammond, P. J. (1979). Equity in two person situations: Some consequences. Econometrica, 47, 1127–1135.CrossRefGoogle Scholar
  7. Khmelnitskaya, A. B. (1996). Social choice problems with different scales of individual welfares measurement for different subgroups of individuals. In: Kleinschmidt, P., Bachem, A., Derigs, U., Fischer, D., Leopold-Wildburger, U., & Möring, R. (Eds.), Operations research proceedings 1995 (pp. 252–257). Berlin: Springer-Verlag.Google Scholar
  8. Khmelnitskaya, A. B. (1999). Social welfare orderings for different subgroup utility scales. Discussion Paper #, Center for Rationality and Interactive Decision Theory at The Hebrew University of Jerusalem.Google Scholar
  9. Khmelnitskaya, A. B. (2002). Social welfare functions for different subgroup utility scales. In: Tangian, A., & Gruber, J. (Eds.), Lecture notes in economics and mathematical systems (vol. 510, pp. 515–530). Berlin: Springer-Verlag.Google Scholar
  10. Khmelnitskaya, A. B., & Weymark, J. A. (2000). Social choice with independent subgroup utility scales. Social Choice and Welfare, 17, 739–748.CrossRefGoogle Scholar
  11. Phanzagl, J. (1971). Theory of measurement (2nd ed.). Würzburg–Wien: Physica-Verlag.Google Scholar
  12. Roberts, K. W. S. (1980a). Possibility theorems with interpersonally comparable welfare levels. Review of Economic Studies, 47, 409–420.CrossRefGoogle Scholar
  13. Roberts, K. W. S. (1980b). Interpersonal comparability and social choice theory. Review of Economic Studies, 47, 421–439.CrossRefGoogle Scholar
  14. Sen, A. K. (1970). Collective choice and social welfare. San Francisco: Holden-Day.Google Scholar
  15. Tsui, K.-Y., & Weymark, J. A. (1997). Social welfare orderings for ratio-scale measurable utilities. Economic Theory, 10, 241–256.CrossRefGoogle Scholar
  16. Yanovskaya, E. B. (1988). Social choice functions for different scales of individual preference measurement (in Russian). Annals of VNIISI (All-Union Research Institute for System Studies, Moscow), 6, 64–76.Google Scholar
  17. Yanovskaya, E. B. (1989). Group choice rules in problems with comparisons of individual preferences. Automation and Remote Control, 50, 822–830 (translated from the 1989 Russian original that appeared in Avtomatika i Telemekhanika, 6, 129–138).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  1. 1.St. Petersburg Institute for Economics and Mathematics, Russian Academy of SciencesSt. PetersburgRussia

Personalised recommendations