Finitely additive extensions of distribution functions and moment sequences: The coherent lower prevision approach

We study the information that a distribution function provides about the finitely additive probability measure inducing it. We show that in general there is an infinite number of finitely additive probabilities associated with the same distribution function. Secondly, we investigate the relationship between a distribution function and its given sequence of moments. We provide formulae for the sets of distribution functions, and finitely additive probabilities, associated with some moment sequence, and determine under which conditions the moments determine the distribution function uniquely. We show that all these problems can be addressed efficiently using the theory of coherent lower previsions

Envelopes of conditional probabilities extending a strategy and a prior probability

Any strategy and prior probability together are a coherent conditional probability that can be extended, generally not in a unique way, to a full conditional probability. The corresponding class of extensions is studied and a closed form expression for its envelopes is provided. Then a topological characterization of the subclasses of extensions satisfying the further properties of full disintegrability and full strong conglomerability is given and their envelopes are studied.Comment: 2

Strong laws of large numbers for sub-linear expectations

We investigate three kinds of strong laws of large numbers for capacities with a new notion of independently and identically distributed (IID) random variables for sub-linear expectations initiated by Peng. It turns out that these theorems are natural and fairly neat extensions of the classical Kolmogorov's strong law of large numbers to the case where probability measures are no longer additive. An important feature of these strong laws of large numbers is to provide a frequentist perspective on capacities.Comment: 10 page

On the Extension of Pseudo-Boolean Functions for the Aggregation of Interacting Criteria

The paper presents an analysis on the use of integrals defined for non-additive measures (or capacities) as the Choquet and the \Sipos{} integral, and the multilinear model, all seen as extensions of pseudo-Boolean functions, and used as a means to model interaction between criteria in a multicriteria decision making problem. The emphasis is put on the use, besides classical comparative information, of information about difference of attractiveness between acts, and on the existence, for each point of view, of a ``neutral level'', allowing to introduce the absolute notion of attractive or repulsive act. It is shown that in this case, the Sipos integral is a suitable solution, although not unique. Properties of the Sipos integral as a new way of aggregating criteria are shown, with emphasis on the interaction among criteria.

A Discrete Choquet Integral for Ordered Systems

A model for a Choquet integral for arbitrary finite set systems is presented. The model includes in particular the classical model on the system of all subsets of a finite set. The general model associates canonical non-negative and positively homogeneous superadditive functionals with generalized belief functions relative to an ordered system, which are then extended to arbitrary valuations on the set system. It is shown that the general Choquet integral can be computed by a simple Monge-type algorithm for so-called intersection systems, which include as a special case weakly union-closed families. Generalizing Lov\'asz' classical characterization, we give a characterization of the superadditivity of the Choquet integral relative to a capacity on a union-closed system in terms of an appropriate model of supermodularity of such capacities

The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures

My studies and my Ph.D. thesis deal with topics that recently emerged in the field of decisions under risk and uncertainty. In particular, I deal with the "target-based approach" to utility theory. A rich literature has been devoted in the last decade to this approach to economic decisions: originally, interest had been focused on the "single-attribute" case and, more recently, extensions to "multi-attribute" case have been studied. This literature is still growing, with a main focus on applied aspects. I will, on the contrary, focus attention on some aspects of theoretical type, related with the multi-attribute case. Various mathematical concepts, such as non-additive measures, aggregation functions, multivariate probability distributions, and notions of stochastic dependence emerge in the formulation and the analysis of target-based models. Notions in the field of non-additive measures and aggregation functions are quite common in the modern economic literature. They have been used to go beyond the classical principle of maximization of expected utility in decision theory. These notions, furthermore, are used in game theory and multi-criteria decision aid. Along my work, on the contrary, I show how non-additive measures and aggregation functions emerge in a natural way in the frame of the target-based approach to classical utility theory, when considering the multi-attribute case. Furthermore they combine with the analysis of multivariate probability distributions and with concepts of stochastic dependence. The concept of copula also constitutes a very important tool for this work, mainly for two purposes. The first one is linked to the analysis of target-based utilities, the other one is in the comparison between classical stochastic order and the concept of "stochastic precedence". This topic finds its application in statistics as well as in the study of Markov Models linked to waiting times to occurrences of words in random sampling of letters from an alphabet. In this work I give a generalization of the concept of stochastic precedence and we discuss its properties on the basis of properties of the connecting copulas of the variables. Along this work I also trace connections to reliability theory, whose aim is studying the lifetime of a system through the analysis of the lifetime of its components. The target-based model finds an application in representing the behavior of the whole system by means of the interaction of its components

The Target-Based Utility Model. The role of Copulas and of Non-Additive Measures

My studies and my Ph.D. thesis deal with topics that recently emerged in the field of decisions under risk and uncertainty. In particular, I deal with the "target-based approach" to utility theory. A rich literature has been devoted in the last decade to this approach to economic decisions: originally, interest had been focused on the "single-attribute" case and, more recently, extensions to "multi-attribute" case have been studied. This literature is still growing, with a main focus on applied aspects. I will, on the contrary, focus attention on some aspects of theoretical type, related with the multi-attribute case. Various mathematical concepts, such as non-additive measures, aggregation functions, multivariate probability distributions, and notions of stochastic dependence emerge in the formulation and the analysis of target-based models. Notions in the field of non-additive measures and aggregation functions are quite common in the modern economic literature. They have been used to go beyond the classical principle of maximization of expected utility in decision theory. These notions, furthermore, are used in game theory and multi-criteria decision aid. Along my work, on the contrary, I show how non-additive measures and aggregation functions emerge in a natural way in the frame of the target-based approach to classical utility theory, when considering the multi-attribute case. Furthermore they combine with the analysis of multivariate probability distributions and with concepts of stochastic dependence. The concept of copula also constitutes a very important tool for this work, mainly for two purposes. The first one is linked to the analysis of target-based utilities, the other one is in the comparison between classical stochastic order and the concept of "stochastic precedence". This topic finds its application in statistics as well as in the study of Markov Models linked to waiting times to occurrences of words in random sampling of letters from an alphabet. In this work I give a generalization of the concept of stochastic precedence and we discuss its properties on the basis of properties of the connecting copulas of the variables. Along this work I also trace connections to reliability theory, whose aim is studying the lifetime of a system through the analysis of the lifetime of its components. The target-based model finds an application in representing the behavior of the whole system by means of the interaction of its components

Representation of maxitive measures: an overview

Idempotent integration is an analogue of Lebesgue integration where $\sigma$-maxitive measures replace $\sigma$-additive measures. In addition to reviewing and unifying several Radon--Nikodym like theorems proven in the literature for the idempotent integral, we also prove new results of the same kind.Comment: 40 page

Dominance of capacities by k-additive belief functions

In this paper we deal with the set of $k$-additive belieffunctions dominating a given capacity. We follow the lineintroduced by Chateauneuf and Jaffray for dominating probabilities and continued by Grabisch for general $k$-additive measures.First, we show that the conditions for the general $k$-additive case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid. On the other hand, we show that these conditions cannot be improved.We solve this situation by imposing additional constraints on the dominating functions. Then, we consider the more restrictive case of $k$-additive belief functions. In this case, a similar result with stronger conditions is proved. Although better, this result is not completely satisfactory and, as before, the conditionscannot be strengthened. However, when the initial capacity is a belief function, we find a subfamily of the set of dominating $k$-additive belief functions from which it is possible to derive any other dominant $k$-additive belief function, and such that theconditions are even more restrictive, obtaining the natural extension of the result for probabilities. Finally, we apply these results in the fields of Social Welfare Theory and Decision Under Risk.Linear programming, decision analysis, capacity,dominance, k-additivity, belief functions