General solutions for choice sets: The Generalized Optimal-Choice Axiom set

Andrikopoulos, Athanasios and Zacharias, Eleftherios (2008): General solutions for choice sets: The Generalized Optimal-Choice Axiom set.

Abstract

In this paper we characterize the existence of best choices of arbitrary binary relations over non finite sets of alternatives, according to the Generalized Optimal-Choice Axiom condition introduced by Schwartz. We focus not just in the best choices of a single set X, but rather in the best choices of all the members of a family K of subsets of X. Finally we generalize earlier known results concerning the existence (or the characterization) of maximal elements of binary relations on compact subsets of a given space of alternatives.

• Item Type: MPRA Paper
• Language: English
• Keywords: Generalized Optimal-Choice Axiom; maximal elements; acyclicity; consistency; ≻-upper compactness
• Subjects: D - Microeconomics > D1 - Household Behavior and Family Economics > D11 - Consumer Economics: Theory

References

Alcantud, J. C., Characterization of the existence of maximal elements of acyclic relations, Economic Theory, 19(2002), 407-416.

Arrow, K., Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, (1954), 265-290.

Bergstrom, T. C., Maximal elements of acyclic relations on compact sets, J. Econ. Theory, 10(1975), 403-404.

Border, K. C., Fixed point theorems with applications to economics and game theory. Cambridge: Cambridge University Press 1985.

Borglin, A., Keiding, H., Existence of equilibrium actions and of equilibrium: A note on the new existence theorems, J. Math. Econ. 3,(1976), 313-316.

Brown, D. J., Acyclic choice. Cowles Foundation Discussion Paper No. 360, Yale University.

Campbell, D. E., Walker, M., Maximal elements of weakly continuous relations, J. Econ. Theory, 50, (1990), 459-464.

Debreu, G., A social equilibrium existence theorem, Proceedings of the National Academy of Sciences of the U.S.A. (1952) 38, 886-893.

Duggan J., A general extension theorem for binary relations, J. Econ. Theory, 86, (1999), 1-16.

Fishburn, P. C., Condorcet social choice function, SIAM J. Appl. Math., 33(1977), 469-489.

Fishburn, P. C., Preference structures and their numerical representations, Theoret. Comput. Sci., 217(1999), 359-383. lyn : Lynn, A. S., J. Arthur Seebach, Jr., Counterexamples in Topology, Campbell, D. E., Springer-Verlag, New York, (1978).

Mehta, G., Maximal elements in Banach spaces, Ind. J. Pure Appl. Math., 20, (1989), 690-697.

Peris, J. E., Subiza, B., Maximal elements of not necessarily acyclic binary relations, Econ. Let., 44, (1994), 385-388.

Shafer, W., Sonnenschein, H., Equilibrium in abstract economies without ordered preferences, J. Math. Econ., 2, (1975), 345-348.

Sloss, J. L., Stable points of directional preference relations. Technical Report No. 71-7, Operations Research House, Stanford University

Sonnenschein, H., Demand Theory without transitive preferences, with applications to the theroy of competitive equilibrium. In: Chipman, J. S., Hurwicz, L., Richter, M. K., Sonnenschein, H. (eds.) Preferences utility and demand, New York: Harcourt Brace Jovanovich 1971.

Subiza, B., Peris, J. E., Numerical representations for lower quasi-continuous preferences, Math. Soc. Sci., 33, (1997),149-156.

Suzumura K., Remarks on the theory of collective choice, Economica, 43, (1976), 381-390.

Suzumura K., Rational choice, collective decisions and social welfare, Cambridge University Press, New York (1983).

Suzumura K., Upper semicontinuous extensions of binary relations, J. Math. Econ., 37, (2002), 231-246.

Schwartz, T., The logic of Collective Choice, New York: Columbia University Press.

Van Deemen M.A., Coalition formation and social choice, Academic Publishers, Dordrecht, (1997).

Walker, M., On the existence of maximal elements, J. Econ. Theory, 16, (1995), 470-474.

Yannelis, N., Prabhakar, N., Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12, (1983), 233-245.

Yannelis, N., Maximal elements over noncompact subsets of linear topological spaces, Economics Letters 17, (1985), 133-136.

Zhou, J., Tian, G., Transfer method for characterizing the existence of maximal elements of binary relations on compact or noncompact sets, SIAM Journal of Optimization 2 (3), (2002), 360-375.