arXiv subjects

Gianni Bosi

Publications and source records attributed to Gianni Bosi.

6 recordsLinked to original sources

arXiv2026

Characterizations of continuous adequate objective functions for ordinal or interval scaled data

Objective functions (goodness criteria which have to be optimized) that are considered, for instance, in cluster analysis, factor analysis, (linear) structural equation modeling, (linear) regression, multidimensional scaling, choice theory, and utility theory, must be {\em adequate}, i.e. carefully adapted to the structure of the observed data. Adequateness of an objective function means, in our terminology, that (certain) transformations of its arguments, e.g. changes in the unit measures of the quantities involved, do not influence the solutions of the optimization procedure. In this paper we concentrate our attention on affine strictly increasing transformations and we also incorporate the case of continuous adequate objective functions accordingly. The characterization of adequate dissimilarity coefficients for interval scaled data shows the appropriateness of the concept of adequateness that is developed in this paper.

econ.TH↗

arXiv2024

New characterizations of completely useful topologies in mathematical utility theory

Let $X$ be an arbitrary set. Then a topology $t$ on $X$ is said to be completely useful if every upper semicontinuous linear (total) preorder $\precsim$ on $X$ can be represented by an upper semicontinuous real-valued order preserving function. In this paper, appealing, simple and new characterizations of completely useful topologies will be proved, therefore clarifying the structure of such topologies.

econ.TH↗

arXiv2024

Continuous Representations of Preferences by Means of Two Continuous Functions

Let $\precsim$ be a reflexive binary relation on a topological space $(X, τ)$. A pair $(u,v)$ of continuous real-valued functions on $(X, τ)$ is said to be a {\em continuous representation} of $\precsim$ if, for all $x,y \in X$, [$(x \precsim y \Leftrightarrow u(x) \leq v(y))$]. In this paper we provide a characterization of the existence of a continuous representation of this kind in the general case when neither the functions $u$ and $v$ nor the topological space $(X,τ)$ are required to satisfy any particular assumptions. Such characterization is based on a suitable continuity assumption of the binary relation $\precsim$, called {\em weak continuity}. In this way, we generalize all the previous results on the continuous representability of interval orders, and also of total preorders, as particular cases.

econ.TH↗

arXiv2024

A topological study for the existence of lower-semicontinuous Richter-Peleg multi-utilities

In the present paper we study necessary and sufficient conditions for the existence of a semicontinuous and finite Richter-Peleg multi-utility for a preorder. It is well know that, given a preorder on a topological space, if there is a lower (upper) semicontinuous Richter-Peleg multi-utility, then the topology of the space must be finer than the Upper (resp. Lower) topology. However, this condition does not guarantee the existence of a semicontinuous representation. We search for finer topologies which are necessary for semicontinuity, as well as that they could guarantee the existence of a semicontinuous representation. As a result, we prove that Scott topology (that refines the Upper one) must be contained in the topology of the space in case there exists a finite lower semicontinuous Richter-Peleg multi-utility. However, as it is shown, the existence of this representation cannot be guaranteed.

math.GN↗

arXiv2024

Partial representations of orderings

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.

math.GN↗

arXiv2002

Sublinear and continuous order-preserving functions for noncomplete preorders

We characterize the existence of a nonnegative, sublinear and continuous order-preserving function for a not necessarily complete preorder on a real convex cone in an arbitrary topological real vector space. As a corollary of the main result, we present necessary and sufficient conditions for the existence of such an order-preserving function for a complete preorder.

math.GN↗