Continuity properties of preference relations

Abstract

Various types of continuity for preference relations on a metric space are examined constructively. In particular, necessary and sufficient conditions are given for an order-dense, strongly extensional preference relation on a complete metric space to be continuous. It is also shown, in the spirit of constructive reverse mathematics, that the continuity of sequentially continuous, order-dense preference relations on complete, separable metric spaces is connected to Ishihara's principle BD-N, and therefore is not provable within Bishop-style constructive mathematics alone.

Theory of constructive semigroups with apartness -- foundations, development and practice

“…For possible applications within computational linguistic see [40]. Some topics from mathematical economics can be approached constructively too (using some order theory for sets with apartness), [2]. Contrary to the classical case, the applications of constructive semigroups with apartness, due to their novelty, constitute an unexplored area.…”

“…For the classical case see [36], [41]. Examples of applications of these theoretical concepts can be found in [2], [11], [15] [22], [40].…”

Theory of Constructive Semigroups with Apartness – Foundations, Development and Practice

“…For possible applications within computational linguistic see Moshier (1995). Some topics from mathematical economics can be approached constructively too (using some order theory for sets with apartness), Baroni and Bridges (2008).…”

The Weihrauch degree of finding Nash equilibria in multiplayer games

“…The setting of finite games in normal form was considered constructively by Bridges in [13], and already showed that the minmax theorem cannot be proven in that setting as it entails the non-constructive principle LLPO. Bridges and coauthors also explored the construction of utility functions from preferences in a constructive setting, and revealed various obstacles [1,14,15].…”