Modal preference structures

A total preorder is a transitive and complete binary relation on a set. A modal preference structure of rank $n$ is a string composed of 2 to the exponent $n$ binary relations on a set such that there is a family of total preorders that gives all relations by taking intersections and unions. Total preorders are structures of rank zero, NaP-preferences (Giarlotta and Greco, 2013) are structures of rank one, and GNaP-preferences (Carpentiere et al., 2022) are structures of rank two. We characterize modal preference structures of any rank by properties of transitive coherence and mixed completeness. Moreover, we show how to construct structures of a given rank from others of lower rank. Modal preference structures arise in economics and psychology, in the process of aggregating hierarchical judgements of groups of agents, where each of the $n$ coordinates represents a feature/stage of the decision procedure.