A unified view of the existence of maximals

Undominated Maximals: General Definition and Characterizations

“…An analogous statement does not hold true, in general, in the case of undominated maximals. (A similar observation extends to the Fishburn shading as defined in [18]: see Section 5.4 in [19] for a longer and more detailed discussion). The following Remark clarifies the previous point.…”

“…The following Remark clarifies the previous point. (If needed, see part 2 of Proposition A.1 in [20] for proof).…”

On the Existence of Greatest Elements and Maximizers

“…The point raised in [15] was in fact implicit in Alcantud [1,Theorem 4], where the author provided a characterization of the existence of unconstrained maximals of an acyclic relation by showing the necessity of the sufficient topological conditions of a theorem dubbed in [1] as the "Bergstrom-Walker Theorem." An implicit 1 presence of that same point can be found in the more recent Bosi and Zuanon [8,Corollary 3.2] and Quartieri [22,Theorem 4], where the authors provide, respectively, characterizations of the existence of unconstrained maximals of a preorder and of the existence of maximals of an arbitrary relation on every nonempty compact subset of its ground set. Somehow relatedly, the same point is tacitly present also in Andrikopoulos and Zacharias [5], where Smith and Schwartz sets are considered with respect to fixed choice sets.…”

“…A known result of the literature states that a preorder has a maximal element on every nonempty compact subset of its ground set provided that each of its upper sections is closed. This known result is-in a sense that will be clear in the course of the papera generalization of the Weierstrass extreme value theorem that must be essentially credited to Wallace [26]: see also [22] for a brief historical account. An immediate Communicated by Juan-Enrique Martinez Legaz.…”

“…Certainly, the point holds true not only when one considers unconstrained optimization problems-as in the just mentioned paper-but also when one considers families of constrained optimization problems whose constraint sets are expressed with respect to some starting natural topology. On the importance of the last-mentioned type of problems for Economic Theory, see Walker [25] and the Introduction in [22].…”