Mixed choice structures, with applications to binary and non-binary optimization

Abstract

We introduce the concept of mixed choice structure in order to propose an alternative model of non-binary choice behavior under certainty. Some general sufficient conditions for optimality on not-necessarily compact sets are proven. The main conclusion is that one single result incorporates as particular cases classical theorems that exemplify different approaches both to binary – [Bergstrom, T.C., 1975, Maximal elements of acyclic relations on compact sets, Journal of Economic Theory 10, 403–404; Mehta, G., 1989, Maximal elements in Banach spaces, Indian Journal of Pure and Applied Mathematics 20, 690–697; Sonnenschein, H., 1971, Demand theory without transitive preferences, with applications to the theory of competitive equilibrium. In: Chipman, J.S., Hurwicz, L., Richter, M.K., Sonnenschein, H. (Eds.), Preferences, Utility and Demand. Harcourt Brace Jovanovich, New York; Walker, M., 1977, On the existence of maximal elements, Journal of Economic Theory 16, 470–474] – and non-binary – [Nehring, K., 1996, Maximal elements of non-binary choice functions on compact sets, Economics Letters 50, 337–340; Alcantud, J.C.R., 2002a, Characterization of the existence of maximal elements of acyclic relations, Economic Theory 19, 407–416] – optimization.