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

PDF
MPRA_paper_11645.pdf Download (215Kb)  Preview 
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 OptimalChoice 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 

Original Title:  General solutions for choice sets: The Generalized OptimalChoice Axiom set 
Language:  English 
Keywords:  Generalized OptimalChoice Axiom; maximal elements; acyclicity; consistency; ≻upper compactness 
Subjects:  D  Microeconomics > D1  Household Behavior and Family Economics > D11  Consumer Economics: Theory 
Item ID:  11645 
Depositing User:  Eleftherios /E Zacharias 
Date Deposited:  23. Nov 2008 02:12 
Last Modified:  15. Feb 2013 15:02 
References:  Alcantud, J. C., Characterization of the existence of maximal elements of acyclic relations, Economic Theory, 19(2002), 407416. Arrow, K., Debreu, G., Existence of an equilibrium for a competitive economy, Econometrica, 22, (1954), 265290. Bergstrom, T. C., Maximal elements of acyclic relations on compact sets, J. Econ. Theory, 10(1975), 403404. 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), 313316. 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), 459464. Debreu, G., A social equilibrium existence theorem, Proceedings of the National Academy of Sciences of the U.S.A. (1952) 38, 886893. Duggan J., A general extension theorem for binary relations, J. Econ. Theory, 86, (1999), 116. Fishburn, P. C., Condorcet social choice function, SIAM J. Appl. Math., 33(1977), 469489. Fishburn, P. C., Preference structures and their numerical representations, Theoret. Comput. Sci., 217(1999), 359383. lyn : Lynn, A. S., J. Arthur Seebach, Jr., Counterexamples in Topology, Campbell, D. E., SpringerVerlag, New York, (1978). Mehta, G., Maximal elements in Banach spaces, Ind. J. Pure Appl. Math., 20, (1989), 690697. Peris, J. E., Subiza, B., Maximal elements of not necessarily acyclic binary relations, Econ. Let., 44, (1994), 385388. Shafer, W., Sonnenschein, H., Equilibrium in abstract economies without ordered preferences, J. Math. Econ., 2, (1975), 345348. Sloss, J. L., Stable points of directional preference relations. Technical Report No. 717, 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 quasicontinuous preferences, Math. Soc. Sci., 33, (1997),149156. Suzumura K., Remarks on the theory of collective choice, Economica, 43, (1976), 381390. 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), 231246. 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), 470474. Yannelis, N., Prabhakar, N., Existence of maximal elements and equilibria in linear topological spaces, J. Math. Econ. 12, (1983), 233245. Yannelis, N., Maximal elements over noncompact subsets of linear topological spaces, Economics Letters 17, (1985), 133136. 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), 360375. 
URI:  http://mpra.ub.unimuenchen.de/id/eprint/11645 