MULTI-ATTRIBUTE TARGET-BASED UTILITIES AND EXTENSIONS OF FUZZY MEASURES FABIO FANTOZZI AND FABIO SPIZZICHINO ABSTRACT. We introduce a formal description of the Target-Based approach to utility theory for the case of n> 1 attributes and point out the connections with aggregation- based extensions of capacities. Our discussion provides economic interpretations of dif- ferent concepts of the theory of fuzzy measures. In particular, we analyze the meaning of extensions of capacities based on n-dimensional copulas. The latter describes stochastic dependence for random vectors of interest in the problem. We also trace the connections between the case of {0, 1}-valued capacities and the analysis of “coherent” reliability sys- tems. KEYWORDS. Decision Analysis, Stochastic dependence, Copulas, M¨ obius transform, Owen extension, Lov´ asz extension, Reliability-structured utilities, Correlation Aversion. 1. I NTRODUCTION A rich literature has been devoted in the last decade to the Target-Based Approach (TBA) to utility functions and economic decisions (see [4, 5, 8, 9, 34, 35]). This litera- ture is still growing, with a main focus on applied aspects (see, for example, [2, 37, 38]). Even from a theoretical point of view, however, some issues of interest demand for fur- ther analysis. In this direction, the present paper will consider some aspects that emerge in the analysis of the multi-attribute case. Generally TBA can provide probabilistic interpre- tations of different notions of utility theory. Here we will in particular interpret in terms of stochastic dependence the differences among copula-based extensions of a same fuzzy measure. In order to explain the basic concepts of the TBA it is, in any case, convenient to start by recalling the single-attribute case. Let Ξ := {X α } α∈A be a family of real-valued random variables, that are distributed according to probability distribution functions F α respectively. Each element X α ∈ Ξ is seen as a prospect or a lottery and a Decision Maker is expected to conveniently select one element out of Ξ (or, equivalently, α ∈ A). Let U : R → R be a (non-decreasing) utility function, that describes the Decision Maker’s attitude toward risk. Thus, according to the Expected Utility Principle (see [36]), the DM’s choice is performed by maximizing the integral E [U (X α )] =  R U (x) dF α (x). In the Target-Based approach one in addition assumes U to be right-continuous and boun- ded so that, by means of normalization, it can be seen as a probability distribution function over the real line. This approach suggests looking at U as at the distribution function F T of a random variable T . This variable will be considered as a target, stochastically independent of all the prospects X α . If T is a (real-valued) random variable stochastically independent of X α in fact, one has E (F T (X α )) =  P (T ≤ x) F α (dx)= P (T ≤ X α ) , DEPARTMENT OF MATHEMATICS -UNIVERSITY LA SAPIENZA,ROME E-mail addresses: fantozzi@mat.uniroma1.it , fabio.spizzichino@uniroma1.it . 1