Meaningful aggregation functions mapping ordinal scales into an ordinal scale: a state of the art

We present an overview of the meaningful aggregation functions mapping ordinal scales into an ordinal scale. Three main classes are discussed, namely order invariant functions, comparison meaningful functions on a single ordinal scale, and comparison meaningful functions on independent ordinal scales. It appears that the most prominent meaningful aggregation functions are lattice polynomial functions, that is, functions built only on projections and minimum and maximum operations

The Gini index,the dual decomposition of aggregation functions, and the consistent measurement of inequality

In several economic fields, such as those related to health, education or poverty, the individuals’ characteristics are measured by bounded variables. Accordingly, these characteristics may be indistinctly represented by achievements or shortfalls. A difficulty arises when inequality needs to be assessed. One may focus either on achievements or on shortfalls but the respective inequality rankings may lead to contradictory results. Specifically, this paper concentrates on the poverty measure proposed by Sen. According to this measure the inequality among the poor is captured by the Gini index. However, the rankings obtained by the Gini index applied to either the achievements or the shortfalls do not coincide in general. To overcome this drawback, we show that an OWA operator is underlying in the definition of the Sen measure. The dual decomposition of the OWA operators into a self-dual core and anti-self-dual remainder allows us to propose an inequality component which measures consistently the achievement and shortfall inequality among the poor.Aggregation functions, dual decomposition, OWA operators, Gini index, consistent measures of achievement/shortfall inequality, Sen index, poverty measures.

Construction of aggregation operators with noble reinforcement

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements. <br /

Intangibles mismeasurements, synergy, and accounting numbers : a note.

For the last two decades, authors (e.g. Ohlson, 1995; Lev, 2000, 2001) have regularly pointed out the enforcement of limitations by traditional accounting frameworks on financial reporting informativeness. Consistent with this claim, it has been then argued that accounting finds one of its major limits in not allowing for direct recognition of synergy occurring amongst the firm intangible and tangible items (Casta, 1994; Casta & Lesage, 2001). Although the firm synergy phenomenon has been widely documented in the recent accounting literature (see for instance, Hand & Lev, 2004; Lev, 2001) research hitherto has failed to provide a clear approach to assess directly and account for such a henceforth fundamental corporate factor. The objective of this paper is to raise and examine, but not address exhaustively, the specific issues induced by modelling the synergy occurring amongst the firm assets whilst pointing out the limits of traditional accounting valuation tools. Since financial accounting valuation methods are mostly based on the mathematical property of additivity, and consequently may occult the perspective of regarding the firm as an organized set of assets, we propose an alternative valuation approach based on non-additive measures issued from the Choquet's (1953) and Sugeno's (1997) framework. More precisely, we show how this integration technique with respect to a non-additive measure can be used to cope with either positive or negative synergy in a firm value-building process and then discuss its potential future implications for financial reporting.Financial reporting; accounting goodwill; assets synergy; non-additive measures; Choquet’s framework;

Aggregation on bipolar scales

The paper addresses the problem of extending aggregation operators typically defined on $[0,1]$ to the symmetric interval $[-1,1]$, where the ``0'' value plays a particular role (neutral value). We distinguish the cases where aggregation operators are associative or not. In the former case, the ``0'' value may play the role of neutral or absorbant element, leading to pseudo-addition and pseudo-multiplication. We address also in this category the special case of minimum and maximum defined on some finite ordinal scale. In the latter case, we find that a general class of extended operators can be defined using an interpolation approach, supposing the value of the aggregation to be known for ternary vectors.bipolar scale; bi-capacity; aggregation

Aggregation functions: an approach using copulae

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 define the joint aggregation density function.