RELEVANCIA DE LAS PALABRAS CON SONIDOS COMUNES: LA RIMA Y ALITERACIÓN

In this Section we trace connections between our results about stochas- tic precedence, introduced in the previous sections, and the Target-Based Approach to decision problems under risk.

So far we introduced the Target-Based Model of utility and studied many of its properties, especially in the multi-attribute case and in the case of independence between targets and prospects. Here we concentrate at- tention on the single-attribute case, where (T, X) is a pair of real-valued

random variables. Furthermore, we are interested in the case where there is dependence betweenT and X.

It is clear that the objects of central interest in the TBA are, for a fixed target T , the probabilities P(T _{≤ X) and the analysis developed in the}

previous sections can reveal of interest. Here we assume the existence of regular conditional distributions and, in particular, for any prospect X we

assume that we can determine the function υ_T(X)(x) := P(T _{≤ x|X = x).}

Hence we can write

P(T _{≤ X) =} Z

R

υ(X)_T (x) dFX(x).

Before continuing it is useful to remind the special case whenX and T

are stochastically independent. In this case we can write

P(T _{≤ X) =} Z R υ(X)_T (x) dFX(x) = Z R FT(x) dFX(x).

In such a case, as we already remarked in Chapter 4,P(T ≤ X) can be

seen as the expected value of a utility: by consideringU = FT as the utility

function, we have (see formula (4.2))

E(U(X)) = Z R U(x) dFX(x) = Z R FT(x) dFX(x) = P(T ≤ X).

Under the condition of independence, any bounded and right-continuous utility function can thus be seen as the distribution function of a target T ,

and vice-versa. Such an hypothesis represented a balance point in the study of Target-Based model illustrated in Chapter 4. In this sense, our model can be seen as an extension of classical models for utility, although it adapts

to the expected utility principle. TBA however becomes, in a sense, more general than the expected utility approach by allowing for stochastic de- pendence between targets and prospects. In fact the TBA considers more general decision rules, if we admit the possibility of some correlation be- tween X and T . In this case, υ_T(X)(x) does not coincide anymore with the

distribution function FT(x) of the target. We refer to [19, 28] for further

discussion in this sense.

We now briefly summarize the arguments of previous sections in the perspective of a decision problem where, for a fixed target T , we aim to

rank two different prospectsX1, X2, with marginal distributionsGX1, GX2,

and with connecting copulasC1, C2, corresponding to the pairs(T, X1) and

(T, X2), respectively.

In the case of independence, a prospect X2 should be obviously pre-

ferred to a prospect X1 if X1 st X2. In the case of dependence, on the

contrary, this comparison is not sufficient anymore. In fact the choice of a prospectX should be based not only on the corresponding distribution FX,

but also on the connecting copula of the pair(T, X).

For fixed C, the quantity η(C, GT, GX) = P(T ≤ X) is equal to the

quantity η(C) for all pairs such that GT = GX = G, with G belonging

to the class _{G (See Proposition 5.5) while, for G}T 6= GX, the implication

T _st X ⇒ P(T ≤ X) ≥ γ does not necessarily hold (see Proposition

5.10 and Example 5.15).

For two different prospects X1, X2, Proposition 5.4 guarantees that,

whenC1 = C2 = C, the condition GT st GX1 stGX2 implies

η(C, GT, GX1) = P(T ≤ X1)≤ η(C, GT, GX2) = P(T ≤ X2).

As shown by Example 5.16, when C1 6= C2, we can have both the

conditions η(C1, GT, GX1) > η(C2, GT, GX2) and GT st GX1 st GX2

(GX1 6= GX2).

Concerning the quantitiesη(C1, GT, GX1) and η(C2, GT, GX2), Theo-

rems 5.7 and 5.8 show that, forGT stGXi (i = 1, 2),

P(T ≤ Xi) = η(Ci, GT, GXi)≥ η(Ci).

Finally, let us consider the case when the only available information about C1 andC2 is that η(Ci) ≥ γi (i.e. that Ci belongs to the classLγi).

Then a rough and conservative choice betweenX1andX2suggests to select

Xi with the larger value of γi, provided GX1 st GX2 or thatX1, X2 are

nearly identically distributed.

All these apparently paradoxical results suggest that the criteria for se- lection of random variables based only on stochastic orderings are not suit- able enough for decision-making problems, such as those described by the TBA, when dependance among variables is present. We have shown, in fact, that the usual stochastic orderings can give results in disagreement with the

expected utility concepts expressed by TBA. Furthermore we explicitly pro- vided examples in which the choice of a prospect which is “better” in the stochastic sense may give worse results in the utility context.

In order to describe his preferences to the best, a DM adopting the Target-Based model will then also need to take into account properties of dependence of the random variables involved in his choices, trough the study of their connecting copulas. To this purpose a deeper analysis of the copulas of the classes _Lγ is to be performed, especially for what concerns

In this work we showed the importance of the target-based model in decision making and utility theory. We presented an extension of multi- attribute target-based model, representing preferences according to the von- Neumann Morgenstern utility theory, although built by means of non-addi- tive measures. This model provides, in fact, an analysis of the joint behavior of targets and prospects, describing them in terms of their joint probability distributions, by means of properties of copulas, and by (non-additive) im- portance weights defined in terms of capacities. On this basis, we have pointed out that the theory of multi-attribute target-based utilities can hinge on a formal apparatus, provided by the field of fuzzy measures, extensions of fuzzy measures, and fuzzy, or universal, integrals.

Further improvements can be made to this model, from one side, by deeply investigating the role of capacities in establishing the importance of groups of prospects. On the other side, properties of risk aversion in high dimensions have to be mastered, through the analysis of the connecting copulas of targets and prospects. An overall interaction between copulas and capacities is to be studied in depth, by taking into account the features that these objects jointly assume in our model.

A further direction along which our model is to be extended is the one that takes into account the property of ambiguity. By allowing the exis- tence of such a condition, it can be made a further generalization of the TB model, that considers not only attitudes towards risk of DMs but also attitudes towards uncertainty and lack of information.

In this work we also presented an extension of the concept of stochastic precedence and provided comparison with the usual concepts of stochastic orders, in terms of properties of copulas. We provided some examples in this direction and found link to the target-based model of utility.

Extensions of this topic can be made through a more accurate analy- sis of properties of copulas, especially regarding dependence and asymme- try. Connections with the existing concept of measures of concordance and measures of asymmetry can be improved for this purpose.

Finally, all the results presented in this manuscript can be combined together to give a more general formulation of TB model, with multi-dimen- sional targets and prospects that admit dependence among them. The study of this more complete model could allow to analyze more deep aspects of economic properties, like the ones regarding multi-dimensional risk and

attitude toward risk of DMs, as much as mathematical properties regarding, for example, further extension of fuzzy measures and integrals.