The study of complexity and optimization in decision theory involves both partial and complete characterizations of preferences over decision spaces in terms of real-valued monotones. With this motivation, and following the recent introduction of new classes of monotones, like injective monotones or strict monotone multi-utilities, we present the classification of preordered spaces in terms of both the existence and cardinality of real-valued monotones and the cardinality of the quotient space. In particular, we take advantage of a characterization of real-valued monotones in terms of separating families of increasing sets in order to obtain a more complete classification consisting of classes that are strictly different from each other. As a result, we gain new insight into both complexity and optimization, and clarify their interplay in preordered spaces.
翻译:对决策理论的复杂性和优化的研究既涉及对决定空间的偏好的局部和完整定性,也涉及对实际价值单质的偏好进行局部和完整定性。有了这一动机,并在最近引入新的单质类,如注入单质单质或严格的单质单质多功能,我们从实际价值单质的存在和基本性以及商质空间的根本性的角度来介绍预先划定的空间的分类。特别是,我们利用对实际价值单质的定性,将不断增长的家庭分离,以便获得由严格不同的类别组成的更完整的分类。 结果,我们获得了对复杂性和优化性的新认识,并澄清了在预先排序空间的相互作用。