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

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case lead to a very wide class of functions and this makes that the properties obtained for probabilities are no longer valid.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

A characterization of the 2-additive Choquet integral through cardinal information

In the context of multiple criteria decision analysis, we present the necessary and sufficient conditions to represent a cardinal preferential information by the Choquet integral w.r.t. a 2-additive capacity. These conditions are based on some complex cycles called cyclones.MCDA; Choquet integral; 2-Additive capacity; MACBETH methodology

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

Valorisation et reporting du goodwill : enjeux théoriques et empiriques.

Le goodwill est, par nature, un objet complexe en ce qu’il englobe deux perspectives : la première est économique – il s’agit de la valeur présente des profits économiques espérés – la seconde est comptable – excès du prix payé par un acquéreur sur la juste valeur des actifs nets acquis. Ces deux acceptions du goodwill étant imbriquées, nous les analysons en décomposant le goodwill comptable – surpaiement, goodwill généré en interne (GGI) et synergies externes. Notre analyse fait émerger, d’une part, des questions de valorisation du goodwill économique, et de l’autre, des enjeux de reporting du goodwill comptable. Sur la première question, nous avançons une proposition sur les fondements de la mesure du GGI : un modèle alternatif de valorisation, de nature combinatoire, fondé sur la théorie des capacités (Choquet, 1953) et relaxant le postulat standard d’additivité, est testé sur un échantillon d’entreprises cotées américaines. Les performances de ce modèle non-additif s’avèrent supérieures à celles du modèle additif d’Ohlson (1995). Sur la question du reporting, nous analysons l’impact sur le coût des capitaux propres de l’information comptable communiquée au titre des tests de dépréciation du goodwill (IAS 36, FAS 142). Une association négative est établie entre information pertinente et engagement de transparence, et coût du capital. Nous examinons les déterminants et les conséquences de la communication de la valorisation initiale du goodwill et de l’affectation du prix d’acquisition (IFRS 3, FAS 141). L’identification des immatériels réduit l’asymétrie informationnelle entre managers et investisseurs et les erreurs de valorisation de ces derniers.The goodwill is, by nature, complex since this notion encompasses two realities: the first is an economic concept – the expected present value of economic profits – the second is an accounting aggregate – the excess of the acquirer’s purchase price over the fair values of the target’s net assets. As these two concepts overlap, we clarify these notions by disaggregating the accounting goodwill between overpayment, internally generated goodwill (IGG) and external synergies. Our contribution relates to both the valuation of economic goodwill and the analysis of accounting goodwill. With regard to the first subject, we propose a new method to explain the creation and measure the value of IGG. The method is based on the idea that firm value is affected by interactions between assets used in combination to conduct business. Based on Choquet (1953) capacities, i.e. non-additive aggregation operators, we explain the creation of IGG, and demonstrate from a sample of U.S. listed firms that this model over-performs the traditional additive Ohlson (1995)-type model on accuracy in forecasting enterprise value. Regarding the second subject, we examine the effects on the equity cost of capital implied by goodwill impairment disclosures. We document a negative association between the relevant information disclosed and commitment to transparency, and the cost of capital. We also investigate the economic consequences of purchase price allocation disclosures (IFRS 3, FAS 141) on bid-ask spreads as well as on cumulated abnormal returns. We provide evidence that firms disclosing more about purchase price allocation reduce their bid-ask spread and investors’ valuation errors.Synergy; Accounting Valuation Model; Impairment Test; Cost of Capital; Business Combination; Intangible Assets; Goodwill; Synergies; Modèle de valorisation comptable; Test de dépréciation; Coût du capital; Regroupements d’entreprises; Actifs incorporels;

Axiomatic structure of k-additive capacities

In this paper we deal with the problem of axiomatizing the preference relations modeled through Choquet integral with respect to a k-additive capacity, i.e. whose Mobius 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