Pareto Optimality on Compact Spaces in a Preference-Based Setting under Incompleteness

Abstract

We characterize the existence of Pareto optimal elements for a family of not necessarily total preorders on a compact topological space. We identify a rather general semi-continuity assumption, called weak upper semicontinuity, under which there exist Pareto optimal elements. We also show that weak upper semicontinuity of each individual preorder is a necessary and sufficient condition for determining the Pareto optimal elements by solving the classical multi-objective optimization problem in case that each function is upper semicontinuous and order-preserving for the respective preorder, and each preorder satisfies a condition of weak separability.

Multi-utility representations of incomplete preferences induced by set-valued risk measures

“…Classical works investigate multi-utility representations of preferences over sets of consumption bundles or lotteries, but there exists a considerable literature focusing on preferences defined on abstract sets by means of suitable axioms. We refer to Aumann [5] and Bewley [8] for two classical references and to Ok [39], Dubra et al [14], Mandler [36], Eliaz and Ok [15], Kaminski [34], Evren [16,17], Evren and Ok [18], Bosi and Herden [10], Galaabaatar and Karni [26], Nishimura and Ok [38], Bosi et al [9] and Bevilacqua et al [7] for an overview of contributions to the theory of incomplete preferences and their multiutility representations in the last 20 years.…”

Multi-utility representations of incomplete preferences induced by set-valued risk measures

“…In the broad field of economics and finance, incomplete preferences arise naturally in the presence of multi-criteria decision making. We refer to Aumann [5] and Bewley [8] for two classical references and to Ok [39], Dubra et al [14], Mandler [36], Eliaz and Ok [15], Kaminski [34], Evren [16,17], Evren and Ok [18], Bosi and Herden [9], Galaabaatar and Karni [26], Nishimura and Ok [38], Bosi et al [10], and Bevilacqua et al [7] for an overview of contributions to the theory of incomplete preferences and their multi-utility representations in the last twenty years.…”