Tian, Guoqiang and Zhou, Jianxin (1992): Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets. Published in: SIAM Journal on Optimization , Vol. 2, No. 3 (August 1992): pp. 360-375.
Download (2MB) | Preview
This paper systematically studies the existence of maximal elements for unordered binary relation on compact or noncompact sets in a general topological space. This is done by developing a method, called transfer method, to derive various necessary and sufficient conditions that characterize the existence of maximal elements for a binary relation in terms of:(1) (generalized) transitivity conditions under certain topological assumptions;(2) topological conditions under certain (generalized) transitivity assumptions; and (3) (generalized)convexity conditions under certain topological assumptions. There are two basic approaches in the literature to prove the existence by providing sufficient conditions. One assumes certain convexity and continuity conditions for a topological vector space and the other assumes certain weakened transitivity and continuity conditions for a general topological space. The results unify those two approaches and generalize almost all of the existing results in literature.
|Item Type:||MPRA Paper|
|Original Title:||Transfer Method for Characterizing the Existence of Maximal Elements of Binary Relations on Compact or Noncompact Sets|
|Keywords:||Binary relations, maximal elements, transfer continuities, transfer transitivities, transfer convexities|
|Subjects:||D - Microeconomics > D5 - General Equilibrium and Disequilibrium > D50 - General|
|Depositing User:||Guoqiang Tian|
|Date Deposited:||19 Sep 2012 11:38|
|Last Modified:||22 Apr 2016 08:06|
C.BERGE, Espaces Topologiques et Fonctions Multivoques, Donod, Paris, 1959.
Topological Spaces, E.M. Paterson, trans., Macmillan, New York, 1963.
T.X.Bergstrom,Maximal elements of a cyclic relations on compact sets, J.Econom. Theory,10(1975), pp,403-404.
J.M. Borwein, On the existence of Pareto efficient points, Math. Oper. Res., 8(1983), pp.64-73.
D.E.Campbell and M.WALKER, Optimization with weak continuity, J. Econom. Theory, 50(1990), pp .459-464.
S.S.CHANG AND Y.ZHANG, Generalized KKM theorem and variational inequalities, J.Math.Anal.Appl., 159(1991), pp. 208-223.
K.FAN, A generalization of Tychonoff’s fixed point theorem, Math. Ann., 142(1961), pp. 305-310.
F.FERRO, A minimax theorem for vector-valued functions, J.Optim.Theory Appl., 60(1989), pp.18-31.
T.KIM AND M.K.RICHTER, Nontransitive-Nontotal Consumer Theory, J.Econom. Theory, 38(1986), pp.324-363.
D.T.LUC, An existence theorem in vector optimization, Math. Oper.Res.,14(1989), pp.693-699.
W.SHAFER, The nontransitive consumer, Econometrica, 42(1974), pp. 913-919.
WSHAFER AND H.SONNENSCHEIN, Equilibrium in abstract economies without ordered preferences, J.Math.Econom., 2(1975), pp.345-348.
D.SCHMEIDLER, competitive equilibrium in markets with a continuum of traders and incomplete preferences, Econometrica, 37(1969), pp. 578-585.
H.SONNENSCHEIN, Demand theory without transitive preferences, with application to the theory of competitive equilibrium, in Preferences, Utility, and Demand, J.S. Chipman, L. Hurwicz, M.K. Richter, and H. Sonnenschein, eds., Harcourt Brace Jovanovich,New York, 1971.
T. TANAKA, Some minimax problems of vector-valued functions, J. Optim. Theory Appl., 59(1988), pp.504-524.