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Topological Social Choice

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  • Luc Lauwers

Abstract

The topological approach to social choice was developed by Graciela Chichilnisky in the beginning of the eighties. The main result in this area (known as the resolution of the topological social choice paradox) shows that a space of preferences admits of a continuous, anonymous, and unanimous aggregation rule for every number of individuals if and only if this space is contractible. Furthermore, connections between the Pareto principle, dictatorship, and manipulation were established. Recently, Baryshnikov used the topological approach to demonstrate that Arrow's impossibility theorem can be reformulated in terms of the non-contractibility of spheres. This paper discusses these results in a self-contained way, emphasizes the social choice interpretation of some topological concepts, and surveys the area of topological aggregation.

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Paper provided by Katholieke Universiteit Leuven, Centrum voor Economische Studiën in its series Center for Economic Studies - Discussion papers with number ces9912.

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Date of creation: Mar 1999
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Handle: RePEc:ete:ceswps:ces9912

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  1. Baigent, Nick, 1987. "Preference Proximity and Anonymous Social Choice," The Quarterly Journal of Economics, MIT Press, vol. 102(1), pages 161-69, February.
  2. Kelly, Jerry S, 1971. "The Continuous Representation of a Social Preference Ordering," Econometrica, Econometric Society, vol. 39(3), pages 593-97, May.
  3. Debreu, Gerard, 1972. "Smooth Preferences," Econometrica, Econometric Society, vol. 40(4), pages 603-15, July.
  4. Paras Mehta, 1997. "Topological methods in social choice: an overview," Social Choice and Welfare, Springer, vol. 14(2), pages 233-243.
  5. Chichilnisky, Graciela, 1983. "Social choice and game theory: recent results with a topological approach," MPRA Paper 8059, University Library of Munich, Germany.
  6. Graciela Chichilnisky, 1996. "Actions of symmetry groups," Social Choice and Welfare, Springer, vol. 13(3), pages 357-364.
  7. Yuqing Zhou, 1997. "A note on continuous social choice," Social Choice and Welfare, Springer, vol. 14(2), pages 245-248.
  8. Donald G. Saari, 1997. "Informational geometry of social choice," Social Choice and Welfare, Springer, vol. 14(2), pages 211-232.
  9. Chichilnisky, Graciela, 1982. "Structural instability of decisive majority rules," Journal of Mathematical Economics, Elsevier, vol. 9(1-2), pages 207-221, January.
  10. Yuliy M. Baryshnikov, 1997. "Topological and discrete social choice: in a search of a theory," Social Choice and Welfare, Springer, vol. 14(2), pages 199-209.
  11. Chichilnisky, Graciela & Heal, Geoffrey, 1979. "Necessary and sufficient conditions for a resolution of the social choice paradox," MPRA Paper 8495, University Library of Munich, Germany, revised 20 Oct 1981.
  12. Baigent, Nick, 1989. "Some Further Remarks on Preference Proximity," The Quarterly Journal of Economics, MIT Press, vol. 104(1), pages 191-93, February.
  13. Uriarte, I. R., 1987. "Topological structure of a space of continuous preferences as a space of retractions and the aggregation problem," Mathematical Social Sciences, Elsevier, vol. 13(3), pages 259-272, June.
  14. Lauwers, Luc, 1993. "Infinite Chichilnisky rules," Economics Letters, Elsevier, vol. 42(4), pages 349-352.
  15. Efimov, Boris A. & Koshevoy, Gleb A., 1994. "A topological approach to social choice with infinite populations," Mathematical Social Sciences, Elsevier, vol. 27(2), pages 145-157, April.
  16. Campbell, Donald E. & Kelly, Jerry S., 1996. "Continuous-valued social choice," Journal of Mathematical Economics, Elsevier, vol. 25(2), pages 195-211.
  17. E. IndurÂin & J. C. Candeal & G. Chichilnisky, 1997. "Topological aggregation of preferences: the case of a continuum of agents," Social Choice and Welfare, Springer, vol. 14(2), pages 333-343.
  18. Geoffrey Heal, 1997. "Social choice and resource allocation: a topological perspective," Social Choice and Welfare, Springer, vol. 14(2), pages 147-160.
  19. Beth Allen, 1996. "A remark on a social choice problem," Social Choice and Welfare, Springer, vol. 13(1), pages 11-16, January.
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  21. Candeal, Juan Carlos & Indurain, Esteban, 1994. "The Moebius strip and a social choice paradox," Economics Letters, Elsevier, vol. 45(3), pages 407-412.
  22. Ovchinnikov, Sergei, 1996. "Means on ordered sets," Mathematical Social Sciences, Elsevier, vol. 32(1), pages 39-56, August.
  23. McManus, Maurice, 1982. "Some Properties of Topological Social Choice Functions," Review of Economic Studies, Wiley Blackwell, vol. 49(3), pages 447-60, July.
  24. Nitzan, Shmuel, 1989. "More on the Preservation of Preference Proximity and Anonymous Social Choice," The Quarterly Journal of Economics, MIT Press, vol. 104(1), pages 187-90, February.
  25. Chichilnisky, G., 1990. "On Strategic Control," Discussion Papers 1990_30, Columbia University, Department of Economics.
  26. Chichilnisky, Graciela, 1982. "The topological equivalence of the pareto condition and the existence of a dictator," Journal of Mathematical Economics, Elsevier, vol. 9(3), pages 223-233, March.
  27. Chichilnisky, Graciela, 1979. "On fixed point theorems and social choice paradoxes," Economics Letters, Elsevier, vol. 3(4), pages 347-351.
  28. Wilson, Robert, 1972. "Social choice theory without the Pareto Principle," Journal of Economic Theory, Elsevier, vol. 5(3), pages 478-486, December.
  29. Chichilnisky, G., 1993. "Intersecting Families of Sets and the Topology of Cones in Economics," Papers 93-17, Columbia - Graduate School of Business.
  30. Campbell, Donald E, 1992. "Transitive Social Choice in Economic Environments," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 33(2), pages 341-52, May.
  31. Chichilnisky, Graciela, 1986. "Topological complexity of manifolds of preferences," MPRA Paper 8119, University Library of Munich, Germany.
  32. Chichilnisky, Graciela & Heal, Geoffrey, 1984. "Patterns of power: bargaining and incentives in two-person games," Journal of Public Economics, Elsevier, vol. 23(3), pages 333-349, April.
  33. Luc Lauwers, 1997. "Topological aggregation, the case of an infinite population," Social Choice and Welfare, Springer, vol. 14(2), pages 319-332.
  34. Chichilnisky, Graciela, 1982. "Social Aggregation Rules and Continuity," The Quarterly Journal of Economics, MIT Press, vol. 97(2), pages 337-52, May.
  35. Baigent, Nick, 1985. "Anonymity and continuous social choice," Journal of Mathematical Economics, Elsevier, vol. 14(1), pages 1-4, February.
  36. Nitzan, Shmuel, 1976. "On Linear and Lexicographic Orders, Majority Rule and Equilibrium," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 17(1), pages 213-19, February.
  37. Heine Rasmussen, 1997. "Strategy-proofness of continuous aggregation maps (*)," Social Choice and Welfare, Springer, vol. 14(2), pages 249-257.
  38. Chichilnisky, Graciela, 1980. "Social choice and the topology of spaces of preferences," MPRA Paper 8006, University Library of Munich, Germany.
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Citations

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Cited by:
  1. Luc Lauwers, 2009. "The topological approach to the aggregation of preferences," Social Choice and Welfare, Springer, vol. 33(3), pages 449-476, September.
  2. Tanaka, Yasuhito, 2007. "A topological approach to Wilson's impossibility theorem," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 184-191, February.
  3. Yasuhito Tanaka, 2005. "On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem (forthcoming in ``Applied Mathematics and Computation''(Elsevier))," Public Economics 0506012, EconWPA, revised 16 Jun 2005.
  4. repec:ebl:ecbull:v:4:y:2004:i:6:p:1-6 is not listed on IDEAS
  5. Daniel Eckert, 2004. "Proximity Preservation in an Anonymous Framework," Economics Bulletin, AccessEcon, vol. 4(6), pages 1-6.
  6. Yasuhito Tanaka, 2005. "A topological approach to the Arrow impossibility theorem when individual preferences are weak orders (forcoming in ``Applied Mathematics and Compuation''(Elsevier))," Public Economics 0506013, EconWPA, revised 16 Jun 2005.
  7. Pivato, Marcus, 2008. "Sustainable preferences via nondiscounted, hyperreal intergenerational welfare functions," MPRA Paper 7461, University Library of Munich, Germany.
  8. Kari Saukkonen, 2007. "Continuity of social choice functions with restricted coalition algebras," Social Choice and Welfare, Springer, vol. 28(4), pages 637-647, June.
  9. Yasuhito Tanaka, 2005. "A topological proof of Eliaz's unified theorem of social choice theory (forthcoming in "Applied Mathematics and Computation")," Public Economics 0510021, EconWPA, revised 26 Oct 2005.
  10. Tanaka, Yasuhito, 2009. "On the equivalence of the Arrow impossibility theorem and the Brouwer fixed point theorem when individual preferences are weak orders," Journal of Mathematical Economics, Elsevier, vol. 45(3-4), pages 241-249, March.
  11. Mabrouk, Mohamed, 2006. "Allais-anonymity as an alternative to the discounted-sum criterion in the calculus of optimal growth I: Consensual optimality," MPRA Paper 10512, University Library of Munich, Germany.

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