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We present and discuss a conceptual decision-making procedure supported by a mathematical device combining expected utility and a generalized information measure: the weighted Gini-Simpson index, linked to the scientific fields of information theory and ecological diversity analys is. After a synthetic review of the theoretical background relative to those themes, such a device— an EU-WGS framework denoting a real function defined with positive utility values and domain in the simplex of probabilities — is analytically studied, identifying its range with focus on the maximum point, using a Lagrange multiplier method associated with algorithms, exemplified numerically. Yet, this EU-WGS device is showed to be a proper analog of an expected utility and weighted entropy (EU-WE) framework recently published, both being cases of mathematical tools that can be referred to as non-expected utility methods using decision weights, framed within the field of decision theory linked to information theory. This kind of decision modeling procedure can also be interpreted to be anchored in Kurt Lewin utility’s concept and may be used to generate scena rios of optimal compositional mixtures applied to generic lotteries associated with prospect theory, financial risk assessment, security quantification and natural resources management. The epistemological method followed in the reasoned choice procedure that is presented in this paper is neither normative nor descriptive in an empirical sense, but instead it is heuristic and hermeneutical in its conception.

Expected utility theory may be considered to be born in 1738, relative to the general problem that choosing among alternatives imply a consistent set of preferences that can be described by attaching a numerical value to each―designated its utility; also, choosing among alternatives involving risk entails that it is selected that one for which the expected utility is highest (e.g. [

This work is an analogous development of another paper recently published, where it was discussed an expected utility and weighted entropy framework, with acronym EU-WE [

Combining the concepts of expected utility and some measure of variability of the probability score―generat- ing utility functions that are nonlinear in the probabilities―is not an innovative method and we can identify an example concerning meteorology forecasts dating back to 1970 [

First, we shall present a synthetic review of the theoretical background anchored in two scientific fields: expected or non-expected utility methods and generalized weighted entropies or useful information measures. Then, we shall proceed merging the two conceptual fields into a mathematical device that combines tools from each and follow studying it analytically and discussing the main issues that are entailed for such a procedure. The spirit in which this paper is written is neither normative nor descriptive―instead it is conceived as a heuristic approach to a decision procedure tool whose final judge will be the decision maker.

The concept of “expected utility” is one of the main pillars in Decision Theory and Game Theory, going back at least to 1738, when Daniel Bernoulli proposed a solution to the St. Petersburg paradox using logarithms of the values at stake, thereby making the numerical series associated with the calculation of the mean value convergent (there referred to as “moral hope”). Bernoulli [

Since that time there were innumerable contributions on the theme. For instance, Alchian [

Subjective expected utility, having a first cornerstone in the works of de Finetti and Ramsey [

Here we will be focused in lotteries, a concept we shall retain (e.g. [

The geometry of

As Shaw and Woodward [

A substantial review was made by Starmer [

There are many other approaches to surpass the limitations of independence axiom, and, for example, Hey and Orme [

The quantitative-qualitative measure of information generalizing Shannon entropy characterized by Belis and Guiasu [

In 1976, Emptoz―quoted in Aggarwal and Picard [

Using l’Hôpital’s rule it is easy to prove the result of the limit

The formula above means exactly the same entity as entropy of degree

Weighted Gini-Simpson (WGS) index was outlined by Guiasu and Guiasu [

The formulation and analytical study of weighted Gini-Simpson index was first introduced by Casquilho [

One main feature of WGS index we must keep in mind is that we have

Weighted Gini-Simpson index is used in several domains, besides ecological and phylogenetic assessments― focusing in economic applications we have examples such as: estimating optimal diversification in allocation problems [

^{1}We discard the cases with null utilities

In what follows, the simple lottery ^{1}

simplex:

In this setting used to characterize the discrete random variable

Equation (1) therefore has the full expression

Still, we can proceed with the subsequent interpretation: denoting

The function

Concerning the evaluation of the minimum point we see that denoting

Searching for the maximum point correspondent to the maximum value

build the auxiliary Lagrange function

partial derivative(s) as follows:

Using Equation (3) combined with the equation

As it is known, the critical value of the Lagrange multiplier reflects the importance of the constraint in the problem and we can check directly from Equation (3) that we have

We can also check that the result

Thus, combining

Otherwise, we have to proceed using an algorithm, as it will be shown next. But we can already notice, from direct inspection of Equation (4) to Inequality (6) that the candidates to optimal coordinates would be the same if we use a positive linear transformation of the utilities such as

It must be noted that we are certain that the maximum point exists as it is implied by Weierstrass theorem. The problem we are dealing has an old root, as Jaynes [

Here, first we shall choose a forward selection procedure since we know that when

get the result

So, given the set of utilities defined in

the problem begins with the evaluation of

Whether not, stop and state

now with the condition (6) restated as

tion until you verify that there is an order

It must be noted that we could have chosen a backward elimination procedure instead with a faster algorithm, begin-

ning with the lowest utility

and if observing

Assume that we have the following lottery with n = 5 and ordered utilities:

luate the optimal point with Equations (4) relative to the set

reset to

Now, exemplifying the backward elimination procedure with the same utility values: first we evaluate

late

we get

The maximum value of

In the case of the numerical example described above we get the number

Going back to the beginning, we can state that the reasoned choice modeling we introduced was a sequential decision-making procedure that began with a lottery

First, we shall focus the discussion comparing optimal proportions of function

Nevertheless, there are differences, perhaps the most noticeable ones being qualitative, as the fact that the optimal point of

Yet there is another point that deserves an explanation: it was claimed in [

There is also another issue that demands an explanation, or, the least, to be posed straightforwardly: consider the lottery

Weirich [

Discussing the limitations of this modeling approach we have to highlight that the weighting function

Gilboa [

In this work we outlined a reasoned decision-making tool combining traditional expected utility with a generalized useful information measure (a type of weighted entropy) referred to as the weighted Gini-Simpson index― thus becoming a conceptual framework with acronym EU-WGS. This device is original and applies to simple lotteries defined with positive utilities and unknown probabilities, denoted as a real function with domain in the standard simplex.

It was shown that this mathematical device frames into the class of non-expected utility methods relative to the type concerning the use of decision weights verifying standard conditions; also, it was shown that function could be interpreted as a mean value of a finite lottery with utilities conceived in the sense of Kurt Lewin, where the rarity of a component enhances the correspondent utility value by a maximum of a two-fold factor. For each set of fixed positive utilities, the real function is differentiable in the open simplex and concave, having an identifiable range with a unique and global maximum point; we settled the procedure to identify the optimal point coordinates, highlighting the sequence of stages with a numeric example; also, we conclude that the maximum point doesn’t change if utilities are affected by a positive linear transformation.

Such a framework can be used to generate scenarios of optimal compositional mixtures relative to finite lotteries associated with prospect theory, financial risk assessment, security quantification or natural resources management. Nowadays, different entropy measures are proposed to be used to form and rebalance portfolios concerning optimal criteria, and some state that the portfolio values of the models incorporating entropies are higher than their correspondent benchmarks [