Marta Cardin

Università Ca' Foscari Venezia | UNIVE · Department of Economics

The concept of convexity plays an important role in the studies of economics in general and consumer theory in particular. By removing the unrealistic assumption that consumer preferences are complete on the set X of consumption choices, this chapter, which is mainly based on Forrest et al. (Studia Universitatis “Vasile Goldis” Arad—Economics Serie...

The concept of convexity plays an important role in the study of economics and consumer theory. For the most part, such studies have been conducted on the assumption that consumer preferences are a binary relation that is complete, reflexive and transitive on the set X of consumption choices. However, each consumer is a biological being with multid...

The notion of decisive coalitions of voters with different grades of decisiveness is a part of the mathematical framework for many models in social choice theory. More generally, we study aggregation problems in which a subgroup of decision makers have the right to determine the properties of the aggregate. Then, we introduce property spaces and ri...

Complex situations such as pandemics generally lead to consider different sources of information in the analysis. We propose a general framework for coronavirus risk assessment based on multi-criteria decision aiding (MCDA) where input variables are indicators expressed on the basis of qualitative-ordinal scales. The proposed approach, based on Sug...

The notion of abstract convex structure generalizes the standard notion of convexity in linear spaces. We consider abstract convex structures that are combinatorial objects studied in various areas of mathematics and convex algebras as introduced in [8] and we study a general definition of convex preferences. Relations defined by aggregation of ord...

Ordered Weighted Aggregation operators (OWA) are widely analyzed and applied to real world problems, given their appealing characteristic to reflect human reasoning, but are enable in the basic definition to include importance weights for the criteria. To obviate, some extensions were introduced, but we show how none of them can satisfy completely...

Convexity of preferences is a canonical assumption in economic theory. In this paper we study a generalized definition of convex preferences that relies on the notion of a convex space, that is an abstraction of the standard notion of convexity in a linear space. We introduce also betweenness relations that characterize convex spaces. First we cons...

The notion of betweenness space or of a convex structure is an abstraction of the standard notion of convexity in a linear space. We first consider a ternary betweenness relation that gives rise to an interval space structure and then we propose a more general definition of betweenness. We study morphism between abstract convex spaces and we charac...

Aggregating preferences for finding a consensus between several agents is an important topic in social choice theory. We obtain several axiomatic characterizations of some significant subclasses of voting rules defined on bounded and distributive lattices.

This paper proposes a model for faculty evaluation based on OWA aggregation operators. Our method permits to consider interactions among the criteria in a formal way, and, at the same time, to realize an easy approach to understand, implement and apply.

We deal the problem of aggregation of individual judgments for a global evaluation of a candidate or a product. In our theoretically oriented approach, aggregation operators are compared with each other based on their mathematical properties. We show that any monotone and strategy-proof operators is characterized by a particular collection of decis...

We consider decision problems in which we have to compare and rank a set of alternatives and each alternative is defined by its attributes or properties. We introduce and characterize property-based preference domains. This paper proposes also a characterization and a generalization of Sugeno integral in our framework.

In this short paper, we aim at a qualitative framework for modeling multivariate decision problems where each alternative is characterized by a set of properties. To this extent, we consider convex spaces as underlying universes and make use of lattice operations in convex spaces to formalize the notion of quantiles. We also put in evidence that ma...

We provide an axiomatic characterization of preorders that are defined with respect to a set of properties. Moreover, it is proven that property-based posets are in natural correspondence with topological spaces. This paper propose also a characterization and a generalization of a Sugeno-type integral in our framework.

We provides an axiomatic characterization of preorders in lattices that are representable as benchmarking procedure. We show that the key axioms are related to compatibility with lattice operations. This paper propose also a characterization and a generalization of Sugeno integral in a ordinal framework.

In many decision processes data aggregation is required. In many models the need arises to aggregate data of varying dimension while aggregation operators are considered for a fixed number of arguments. In many contexts inputs to be aggregated are of a qualitative nature. This paper analyzes the evaluation of sequences of ordinal input and of varia...

In this paper we propose a joint grid-based and stochastic search for parameters elicitation in the case of WOWA aggregation functions. The method uses a grid search approach to determine the parameter of a monotonic quantifier, and for each of the values, a stochastic search in the space of the criteria weights minimizes the sum of the quadratic e...

Presented at the Conference “Frontiers in Financial Mathematics” Institute of Bankers, Dublin, Ireland, June 4-7, 2013.

The notion of risk measure arose from the problem of quantifying risk. The coherent risk measures and the insurance risk measures are risk measures that satisfy a set of axioms. In this note we consider a different approach to risk measurement and we study natural risk statistics that are based on data and that are characterized by a new set of axi...

In this paper we analyze the compensation property of a second order fuzzy measure in the context of a multi-attribute problem. In particular, we show that the disjunction/conjunction behavior (andness/orness) changes with the number of criteria to be aggregated. Interpreting the spread between the maximum and the minimum orness as a measure of the...

Purpose Shalit and Yitzhaki presented the mean‐extended Gini (MEG) as a workable alternative to the Markowitz mean‐variance approach in 1984. Since then, the challenge has been to extend the MEG approach. The purpose of this paper is to propose a generalization of the MEG approach for making customized optimal asset allocation to control both down‐...

The present note suggests a theoretic and practical decision rule for auctioneers to set out prices in online auctions in presence of Buy-It-Now options. The minimum starting bid and the Buy-It-Now price are calculated as an intuitive function of seller’s evaluation of the final price stochastic distribution and of seller’s attitude toward the pric...

The aim of this note is to provide a global performance index that allows to evaluate the performance of each faculty member and which is able to consider the multidimensional nature of the academic activity in terms of research, teaching and other activities that academics should/might exercise. In order to model also the case in which there could...

Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necess...

The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. We focus on the notion of quantile of a lattice-valued function which have several properties of its real-valued counterpart and we study a class of aggregation functionals that generalizes Sugeno integrals to the setting o...

The problem of measuring the impact of a scientific output of a researcher has attracted significant interest in recent years. Most of the methodologies actual in use focus the attention to bibliometric indices and features of the journals. In this note we propose a new approach based on class of assignment and a fuzzy extension to asses the resear...

In this short note, we aim at a qualitative framework for modeling multivariate risk. To this extent, we consider completely distributive lattices as underlying universes, and make use of lattice functions to formalize the notion of risk measure. Several properties of risk measures are translated into this general setting, and used to provide axiom...

The aim of this paper is to introduce some classes of aggregation functionals when the evaluation scale is a complete lattice. Two different types of aggregation functionals are introduced and investigated. We consider a target-based approach that has been studied in decision theory and we focus on the equivalence between a utility-based approach a...

The measurement of the quality of research has reached nowadays an increasing interest not only for scientific reasons but also for the critical problem of researchers' ranking, due to the lack of grant assignments. The most commonly used approach is based on the so-called $h$-index, even if the current literature debated a lot about its pros and c...

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied. We propose a new kind of stop-loss transform and a related order in the multivariate setting and some equivalent conditions. In our work there is a characterisation of some particular classes of multivariate and bivariate risk meas...

Since Shalit and Yitzhalit (1984) the Mean-Extended Gini (MEG) has been proposed as a workable alternative to the classical Markowitz mean-variance CAPM. Although MEG keeps under control the risk belonging to the left-tail of the return distribution, small attention is reserved to potential gains belonging to the return right-tail. A generalization...

Bid and ask prices tailored to the traders’ risk-aversion and gain-propension are defined. Risk and gain premia are given by the Extended Gini indices, where the characteristic parameter captures the traders’ perception of the under-performance and over-performance of the asset. Sufficient and necessary conditions for successful trading are set out...

The measurement of the quality of research has reached nowadays an increasing interest not only for scientific reasons but also for the critical problem of researchers’ ranking, due to the lack of grant assignments. The most commonly used approach is based on the so-called h-index, even if the current literature debated a lot about its pros and con...

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and suffi...

Since Shalit and Yitzhaki (1984) the premium principle based on the Extended Gini of an uncertain position has been defined as its expected value minus the extended Gini index. We propose this principle for making capital asset pricing tailored to the investor profile. Bid and ask prices of the counter-parties involved in transactions are defined....

In this paper we are interested in functionals defined on completely distributive lattices and which are invariant under mappings preserving {arbitrary} joins and meets. We prove that the class of nondecreasing invariant functionals coincides with the class of Sugeno integrals associated with $\{0,1\}$-valued capacities, the so-called term function...

We study supermodular property for normalized scalar evaluators on a complete vector lattice, which is not necessarily of finite dimension. Then we consider the particular case of [0,1]^n.

This paper extends the Choquet integral, widely used in multi-attribute decision problems, to the non monotone case in the context of Group Decision Theory. Even if not so often, preference structures which violate the monotonicity axiom can be observed in real applications. Our aim is twofold. First, we propose the Choquet integral with non monoto...

There exist necessary and sufficient conditions on the generating functions of the FGM family, in order to obtain various dependence properties. We present multivariate generalizations of this class studying symmetry and dependence concepts, measuring the dependence among the components of each class and providing several examples.

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied. In our work we propose a characterization of some particular classes of multivariate and bivariate risk measures. Given two random variables we can define an univariate integral stochastic ordering by considering a set of functions...

We want to focus on a class of aggregation functions and present them in the multilinear form with marginal copulae. The aim is to present the copula approach for studying aggregation problems and for the sake of simplicity we consider the case n = 3.

In this paper a set of desirable properties for measures of positive dependence of ordered n-tuples of continuous random variables (n >= 2) is proposed and a class of multivariate positive dependence measures is introduced. We consider the comonotonicity dependence structure as the strong dependency structure and so the class consists of measures t...

In this paper non-monotonic measures and their properties are considered and described. Subsequently we study discrete non-monotonic Choquet integral under the viewpoint of aggregation, and its axiomatic characterization. Moreover, we show that for non-monotonic measures the Shapley index can fail to represent the relative importance of a criterion...

Aggregation operators transform a finite number of inputs, called arguments, into a single output. They are applied in many theoretical and practical domains and in particular ag-gregation operators play important role in different approaches to decision making, where values to be aggregated are typically preference or satisfaction degrees. Many op...

In this paper we present the extension of the copula approach to aggregation functions. In fact we want to focus on a class of aggregation functions and present them in the multilinear form with marginal copulae. Moreover we will define also the joint aggregation density function.

In past years the study of the impact of risk attitude among risks has become a major topic, in particular in Decision Sciences. Subsequently the attention was devoted to the more general case of bivariate random variables. The first approach to multivariate risk aversion was proposed by de Finetti ([2]), and Richard ([15]) and it is related to the...

This paper presents the role of copula functions in the theory of aggregation operators. In this context we are focusing our attention about several properties of aggregation functions, like supermodularity and Schur-concavity, studying also a decomposition of supermodular binary aggregation operators and copulae.

This paper presents the role of copula functions in the theory of aggregation operators. In this context we are focusing our attention about several properties of aggregation functions, like supermodularity and Schur-concavity, studying also a decomposition of supermodular binary aggregation operators and copulae.

This paper deals with non-monotonic Choquet integral, a generalization of the regular Choquet integral. The discrete non-monotonic Choquet integral is considered under the viewpoint of aggregation. In particular we give an axiomatic characterization of the class of non-monotonic Choquet integrals.We show how the Shapley index, in contrast with the...

In actuarial literature the properties of risk measures or insurance premium principles have been extensively studied. We propose a characterization of a particular class of coherent risk measures defined in [P. Artzner, F. Delbaen, J.-M. Eber and D. Heath, Math. Finance 9, No. 3, 203–228 (1999; Zbl 0980.91042)]. The considered premium principles a...

In past years the study of the impact of risk attitude among risks has become a major topic, in particular in Decision Sciences. Subsequently the attention was devoted to the more general case of bivariate random variables. The first approach to multivariate risk aversion was proposed by de Finetti (1952) and Richard (1975) and it is related to the...

This paper presents the role of copula functions in the theory of aggregation operators and an axiomatic characterization of Archimedean aggregation functions. In this context we are focusing our attention about several properties of aggregation functions, like supermodularity and Schur-concavity.

In this note we consider a multicriteria decision problem where the decision maker know the the state of the world but the set of consequences is multidimensional. We suppose that a value function is specified over the attribute of the decision problem and we analyze some classes of non additive functions that can represent interaction between crit...

The properties of risk measures or insurance premium principles have been extensively studied in actuarial literature. We propose an axiomatic description of a particular class of coherent risk measures defined in Artzner, Delbaen, Eber, and Heath (1999). The considered risk measures are obtained by expansion of TVar measures, consequently they loo...

In past years the study of the impact of risk attitude among risks has become a major topic, in particular in Decision Sciences. Subsequently the attention was devoted to the more general case of bivariate random variables. The first approach to multivariate risk aversion was proposed by de Finetti (1952) and Richard (1975) and it is related to the...

In past years the study of the impact of risk attitude among risks has become a major topic, in particular in Decision Sciences. Sub-sequently the attention was devoted to the more general case of bivariate random variables. The first approach to multivariate risk aversion was proposed by de Finetti ([2]) and Richard ([14]) and it is related to the...

The aim of this paper is to generalize the classical definition of a Premium Calculation Principle in a framework based on some non-additive models of decision making under uncertainty. In particular, we focus our attention on models based on the Choquet integral in the simplified framework in which an affine utility function of wealth is assumed,...

In ques to la voro si introduce la classeU delle funzioni reali definite in un reticolo che sono funzioni di utilità di una classe di decisorti detti avversi al rischio e si caratterizza la dominancza stocastica secondo la classeU.

Different characterizations are presented of higher-order convexity which can have interesting interpretations in the field of decision theory and stochastic orders. More precisely, the aim of this paper is to combine some different results on n-convex or Jensen n-convex functions in such a way that is possible to extend some classical and more rec...

Multi-attribute decision problems involve making the correct decision when there is a list of objectives to be optimized. We consider the case where a value function is specified over the attribute of the decision problem, as is typically done in the deterministic phase of a decision analysis. The preference relation between alternatives associated...