The satisfiability problem for Boolean set theory with a choice correspondence (Extended version)

Given a set $U$ of alternatives, a choice (correspondence) on $U$ is a contractive map $c$ defined on a family $\Omega$ of nonempty subsets of $U$. Semantically, a choice $c$ associates to each menu $A \in \Omega$ a nonempty subset $c(A) \subseteq A$ comprising all elements of $A$ that are deemed selectable by an agent. A choice on $U$ is total if its domain is the powerset of $U$ minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on $U$ by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol $\mathtt{c}$, the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of $\mathtt{c}$ satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to WARP. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness under the additional assumption that the number of choice terms is constant

The Satisfiability Problem for Boolean Set Theory with a Choice Correspondence

Given a set U of alternatives, a choice (correspondence) on U is a contractive map c defined on a family Omega of nonempty subsets of U. Semantically, a choice c associates to each menu A in Omega a nonempty subset c(A) of A comprising all elements of A that are deemed selectable by an agent. A choice on U is total if its domain is the powerset of U minus the empty set, and partial otherwise. According to the theory of revealed preferences, a choice is rationalizable if it can be retrieved from a binary relation on U by taking all maximal elements of each menu. It is well-known that rationalizable choices are characterized by the satisfaction of suitable axioms of consistency, which codify logical rules of selection within menus. For instance, WARP (Weak Axiom of Revealed Preference) characterizes choices rationalizable by a transitive relation. Here we study the satisfiability problem for unquantified formulae of an elementary fragment of set theory involving a choice function symbol c, the Boolean set operators and the singleton, the equality and inclusion predicates, and the propositional connectives. In particular, we consider the cases in which the interpretation of c satisfies any combination of two specific axioms of consistency, whose conjunction is equivalent to WARP. In two cases we prove that the related satisfiability problem is NP-complete, whereas in the remaining cases we obtain NP-completeness under the additional assumption that the number of choice terms is constant.Comment: In Proceedings GandALF 2017, arXiv:1709.01761. "extended" version at arXiv:1708.0612

Semantics meets attractiveness: Choice by salience

We describe a context-sensitive model of choice, in which the selection process is shaped not only by the attractiveness of items but also by their semantics ('salience'). All items are ranked according to a relation of salience, and a linear order is associated to each item. The selection of a single element from a menu is justified by one of the linear orders indexed by the most salient items in the menu. The general model provides a structured explanation for any observed behavior, and allows us to to model the 'moodiness' of a decision maker, which is typical of choices requiring as many distinct rationales as items. Asymptotically all choices are moody. We single out a model of linear salience, in which the salience order is transitive and complete, and characterize it by a behavioral property, called WARP(S). Choices rationalizable by linear salience can only exhibit non-conflicting violations of WARP. We also provide numerical estimates, which show the high selectivity of this testable model of bounded rationality

Choice resolutions

AbstractWe describe a process to compose and decompose choice behavior, called resolution. In the forward direction, resolutions amalgamate simple choices to create a complex one. In the backward direction, resolutions detect when and how a primitive choice can be deconstructed into smaller choices. A choice is resolvable if it is the resolution of smaller choices. Rationalizability, rationalizability by a preorder, and path independence are all preserved (backward and forward) by resolutions, whereas rationalizability by a weak order (equivalently, ) is not. We characterize resolvable choices, and show that resolvability generalizes

Necessary and possible hesitant fuzzy sets: A novel model for group decision making

We propose an extension of Torra’s notion of hesitant fuzzy set, which appears to be well suited to group decision making. In our model, indecisiveness in judgements is described by two nested hesitant fuzzy sets: the smaller, called necessary, collects membership values determined according to a rigid evaluation, whereas the larger, called possible, comprises socially acceptable membership values. We provide several instances of application of our methodology, and accordingly design suitable individual and group decision procedures. This novel approach displays structural similarities with Atanassov’s intuitionistic fuzzy set theory, but has rather different goals. Our source of inspiration comes from preference theory, where a bi-preference approach has proven to be a useful extension of the classical mono-preference modelization in the fields of decision theory and operations research

On resolutions of linearly ordered spaces

[EN] We define an extended notion of resolution of topologicalspaces, where the resolving maps are partial instead of total. To showthe usefulness of this notion, we give some examples and list severalproperties of resolutions by partial maps. In particular, we focus ourattention on order resolutions of linearly ordered sets. Let X be a setendowed with a Hausdorff topology τ and a (not necessarily related)linear order . A unification of X is a pair (Y, ı), where Y is a LOTSand ı : X →֒ Y is an injective, order-preserving and open-in-the-rangefunction. We exhibit a canonical unification (Y, ı) of (X,, τ ) such thatY is an order resolution of a GO-space (X,, τ ∗), whose topology τ ∗refines τ . We prove that (Y, ı) is the unique minimum unification ofX. Further, we explicitly describe the canonical unification of an orderresolution.Caserta, A.; Giarlotta, A.; Watson, S. (2006). On resolutions of linearly ordered spaces. Applied General Topology. 7(2):211-231. doi:10.4995/agt.2006.1925.SWORD21123172R. Engelking, General Topology (Heldermann Verlag, Berlin, 1989).V. V. Fedorcuk, Bicompacta with noncoinciding dimensionalities, Soviet Math. Doklady, 9/5 (1968), 1148–1150.K. P. Hart, J. Nagata and J.E. Vaughan (Eds.), Encyclopedia of General Topology (North-Holland, Amsterdam, 2004)

An axiomatic approach to finite means

In this paper we analyze the notion of a finite mean from an axiomatic point of view. We discuss several axiomatic alternatives, with the aim of establishing a universal definition reconciling all of them and exploring theoretical links to some branches of Mathematics as well as to multidisciplinary applications