Axiomatic structure of k-additive capacities

In this paper we deal with the problem of axiomatizing the preference relations modelled through Choquet integral with respect to a $k$-additive capacity, i.e. whose Möbius transform vanishes for subsets of more than $k$ elements. Thus, $k$-additive capacities range from probability measures ($k=1$) to general capacities ($k=n$). The axiomatization is done in several steps, starting from symmetric 2-additive capacities, a case related to the Gini index, and finishing with general $k$-additive capacities. We put an emphasis on 2-additive capacities. Our axiomatization is done in the framework of social welfare, and complete previous results of Weymark, Gilboa and Ben Porath, and Gajdos.Axiomatic; Capacities; k-Additivity

Risk measurement with the equivalent utility principles.

Risk measures have been studied for several decades in the actuarial literature, where they appeared under the guise of premium calculation principles. Risk measures and properties that risk measures should satisfy have recently received considerable at- tention in the financial mathematics literature. Mathematically, a risk measure is a mapping from a class of random variables defined on some measurable space to the (extended) real line. Economically, a risk measure should capture the preferences of the decision-maker. In incomplete financial markets, prices are no more unique but depend on the agents' attitudes towards risk. This paper complements the study initiated in Denuit, Dhaene & Van Wouwe (1999) and considers several theories for decision under uncertainty: the classical expected utility paradigm, Yaari's dual approach, maximin expected utility theory, Choquet expected utility theory and Quiggin rank-dependent utility theory. Building on the actuarial equivalent utility pricing principle, broad classes of risk measures are generated, of which most classical risk measures appear to be particular cases. This approach shows that most risk measures studied recently in the financial literature disregard the utility concept (i.e. correspond to linear utilities), causing some deficiencies. Some alternatives proposed in the literature are discussed, based on exponential utilities.Actuarial; Coherence; Decision; Expected; Market; Markets; Measurement; Preference; Premium; Prices; Pricing; Principles; Random variables; Research; Risk; Risk measure; Risk measurement; Space; Studies; Theory; Uncertainty; Utilities; Variables;

The posterity of Zadeh's 50-year-old paper: A retrospective in 101 Easy Pieces – and a Few More

International audienceThis article was commissioned by the 22nd IEEE International Conference of Fuzzy Systems (FUZZ-IEEE) to celebrate the 50th Anniversary of Lotfi Zadeh's seminal 1965 paper on fuzzy sets. In addition to Lotfi's original paper, this note itemizes 100 citations of books and papers deemed “important (significant, seminal, etc.)” by 20 of the 21 living IEEE CIS Fuzzy Systems pioneers. Each of the 20 contributors supplied 5 citations, and Lotfi's paper makes the overall list a tidy 101, as in “Fuzzy Sets 101”. This note is not a survey in any real sense of the word, but the contributors did offer short remarks to indicate the reason for inclusion (e.g., historical, topical, seminal, etc.) of each citation. Citation statistics are easy to find and notoriously erroneous, so we refrain from reporting them - almost. The exception is that according to Google scholar on April 9, 2015, Lotfi's 1965 paper has been cited 55,479 times

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

Value and utility in a historical perspective

Since value and utility are the highest profile abstractions that underlie an epoch’s intellectual climate and ethical principles, their evolution reflects the transformation of socioeconomic conditions and institutions. The “Classical Phase” flourished during the first global system, laissez-faire/metal money/zero multilateralism (GS1); the second, “Subjective/Utilitarian” phase marked the long transition to the current epoch of “Modern Subjectivism/General Equilibrium,” tied to the second and extant global system, mixed economy/minimum reserve banking/weak multilateralism (GS2). History has witnessed the material de-essentialization of value and substantialization of utility. But now the two concepts face a thorough transvaluation as the world’s combined demographic and economic expansion encounters ecological/physical limitations. An extended macrohistoric implosion may lead to a third form of global self-organization: two-level economy/maximum bank reserve money/strong multilateralism (GS3). If history unfolds along the suggested path, not only economics, but also thinking about economics would change. It would be considered an evolving hermeneutic of the human condition expressed through global-system-specific texts. The implied critical alteration, with the recognition of the entropy law’s importance as its focal point, matches the prediction of Swiss thinker Jean Gebser (1905-1973) about the impending mutation of human consciousness into its integral/arational structure. Such extrapolations form the context in which the fourth historical phase of value and utility is hypothesized, leading to the material re-essentialization of value and de-substantialization of utility

Objective and Subjective Rationality in a Multiple Prior Model

A decision maker is characterized by two binary relations. The first reflects decisions that are rational in an “objective” sense: the decision maker can convince others that she is right in making them. The second relation models decisions that are rational in a “subjective” sense: the decision maker cannot be convinced that she is wrong in making them. We impose axioms on these relations that allow a joint representation by a single set of prior probabilities. It is “objectively rational” to choose f in the presence of g if and only if the expected utility of f is at least as high as that of g given each and every prior in the set. It is “subjectively rational” to choose f rather than g if and only if the minimal expected utility of f (relative to all priors in the set) is at least as high as that of g.Rationality, Multiple Priors.

"Development as Freedom"

This chapter offers a historical and analytical introduction to the school of thought that views 'development as freedom'. Since it is impossible to do justice to the depth and complexity of this 'new' approach to development in just one short chapter, I explore a few significant dimensions of the concept beginning with Amartya Sen's pioneering contributions. I also look at clarifications by Nussbaum and others who offer a list of important functionings that can be included in the capabilities set. The subsequent part builds on this by exploring the transition from a utilitarian welfare economics to a fully social capabilities based ethics for development with justice. The present chapter proceeds in the direction of concretizing the idea of 'development as freedom' by discussing the role of a network of social, economic and political institutions in creating social capabilities. This allows us to develop the argument that freedom is to be viewed dynamically as the development of appropriate social capabilities embedded in a specific network of social, economic and political institutions. The following section then presents an example of a concrete area of application--- namely, the problem of developing women's capabilities as an important aspect of global justice--- in order to illustrate the practical relevance of the capabilities approach. The final, concluding section reflects on the future of the social capabilities approach as an evaluative framework for development theory and policy. It turns out that in addition to the usual list of capabilities, and policies driven by them in the field of development, a deep democratic constitution backed up by ongoing democratizing social and political movement is an integral part of the historical project of enhancing social capabilities.