Partial Representations of Orderings

Abstract

In the present paper a new concept of representability is introduced, which can be applied to not total and also to intransitive relations (semiorders in particular). This idea tries to represent the orderings in the simplest manner, avoiding any unnecessary information. For this purpose, the new concept of representability is developed by means of partial functions, so that other common definitions of representability (i.e. (Richter-Peleg) multi-utility, Scott-Suppes representability, … ) are now particular cases in which the partial functions are actually functions. The paper also presents a collection of examples and propositions showing the advantages of this kind of representations, particularly in the case of partial orders and semiorders, as well as some results showing the connections between distinct kinds of representations.

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

“…This is in line with recent works concerning the real representation of nontotal preorders (see e.g. Evren and Ok 9 and Bosi et al 10 ).…”

Short Note: A Correction in “Partial Representations of Orderings”

“…In Ref. 1, the authors studied the idea of partial representability. In order to illustrate some of these ideas, they introduced some examples.…”

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.…”