Social choice and the closed convergence topology
This paper revisits the aggregation theorem of Chichilnisky (1980), replacing the original smooth topology by the closed convergence topology and responding to several comments (N. Baigent (1984, 1985, 1987, 1989), N. Baigent and P. Huang (1990) and M. LeBreton and J. Uriarte (1900 a, b). Theorems 1 and 2 establish the contractibility of three spaces of preferences: the space of strictly quasiconcave preferences Psco, its subspace of smooth preferences Pssco, and a space P1 of smooth (not necessarily convex) preferences with a unique interior critical point (a maximum). The results are proven using both the closed convergence topology and the smooth topology. Because of their contractibility, these spaces satisfy the necessary and sufficient conditions of Chichilnisky and Heal (1983) for aggregation rules satisfying my axioms, which are valid in all topologies. Theorem 4 constructs a family of aggregation rules satisfying my axioms for these three spaces. What these spaces have in common is a unique maximum (or peak). This rather special property makes them contractible, and thus amenable to aggregation rules satisfying anonymity and unanimity, Chichilnisky (1980 1982). The results presented here clarify an erroneous example in LeBreton and Uriarte (1990a, b) and respond to Baigent (1984, 1985, 1987) and Baigent and Huang (1990) on the relative advantages of continuous and discrete approaches to Social Choice.
|Date of creation:||12 Jul 1990|
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- Chichilnisky, Graciela & Heal, Geoffrey, 1979.
"Necessary and sufficient conditions for a resolution of the social choice paradox,"
8495, University Library of Munich, Germany, revised 20 Oct 1981.
- Chichilnisky, Graciela & Heal, Geoffrey, 1983. "Necessary and sufficient conditions for a resolution of the social choice paradox," Journal of Economic Theory, Elsevier, vol. 31(1), pages 68-87, October.
- Debreu, Gerard, 1972.
Econometric Society, vol. 40(4), pages 603-15, July.
- Chichilnisky, Graciela, 1979. "On fixed point theorems and social choice paradoxes," Economics Letters, Elsevier, vol. 3(4), pages 347-351.
- Baigent, Nick, 1989. "Some Further Remarks on Preference Proximity," The Quarterly Journal of Economics, MIT Press, vol. 104(1), pages 191-93, February.
- Chichilnisky, Graciela, 1990. "General equilibrium and social choice with increasing returns," MPRA Paper 8124, University Library of Munich, Germany.
- Chichilnisky, Graciela, 1990. "On the mathematical foundations of political economy," MPRA Paper 8123, University Library of Munich, Germany.
- Chichilnisky, Graciela, 1983. "Social choice and game theory: recent results with a topological approach," MPRA Paper 8059, University Library of Munich, Germany.
- Baigent, Nick, 1987. "Preference Proximity and Anonymous Social Choice," The Quarterly Journal of Economics, MIT Press, vol. 102(1), pages 161-69, February.
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