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 [11] explicitly stated that the determination of the value of an item should not be based on its price but rather on the utility that it produces, being the marginal utility of money inversely proportional to the amount one already has. Bernoulli approach―considered the first statement on Expected Utility Theory (EUT)―presupposes the existence of a cardinal utility scale, and that remained an obstacle until the theme was revived by the remarkable work of John von Neumann and Oskar Morgenstern in 1944, showing that the expected utility hypothesis could be derived from a set of axioms on preference [12] , considering the utilities experienced by one person and a correspondence between utilities and numbers, involving complete ordering and the algebra of combining. Nevertheless, based on their theorem one is restricted to situations in which probabilities are given (e.g. [13] ).

Since that time there were innumerable contributions on the theme. For instance, Alchian [14] outlines the issue stating that if, in a given context, it is possible to assign numerical values for different entities competing, then a selection process of rational choices is made to maximize the utility and one can say that the normal form of expected utility reduces to the calculation of the average value of a pattern of preferences expressed by li- mited numerical functions. But soon also appeared the objections relative to the adherence of EUT to empirical evidence, namely the Allais paradox which was first published in 1953, showing that individual’s choices in many cases collided with what was predicted by the theory, one reason being because expected utility devices associated with lotteries do not take into account the dispersion of the utilities around their mean values and also because, in general, people overweight positive outcomes that are considered certain compared to outcomes which are merely probable (e.g. [15] for a review).

Subjective expected utility, having a first cornerstone in the works of de Finetti and Ramsey [16] both published in 1931, followed by a further substantial development with Savage in 1954, was regarded by most decision analysts to be the preferred normative model for how individuals should make decisions under uncertainty, eliciting vectors of probabilities given the preferences in outcomes; but that approach was also revealed to be violated in empirical situations, what was illustrated for example by the Ellsberg paradox published in 1961. Other paradoxes were mentioned later such as those referred to by Kahneman and Tversky concerning prospect theory [17] where the carriers of value are changes in wealth or welfare rather than final states. Machina [18] highlighted that the independence axiom of EUT tends to be systematically violated in practice and concluded that the main concepts, results and tools of expected utility analysis may be derived from the much weaker assumption of smoothness of preferences over alternative probability distributions; this remarkable work outlined the scope of generalized expected utility analysis or non-expected utility frameworks. Other methods were pro- posed and used such as mean-variance analysis (e.g. [19] [20] ); based on this type of method and using mean absolute deviation instead of standard deviation, Frosini [21] presented recently a discussion revisiting Borch paradox linked to a general criterion of choice between prospects.

Here we will be focused in lotteries, a concept we shall retain (e.g. [22] ): a lottery is defined as a list or finite collection of simple consequences or outcomes with associated, usually unknown, probabilities stated as for completed with the standard normalization conditions and defining a simplex and denoting a vector. The axioms of EUT with the most usual version―ordering, continuity and independence―allow for preferences over lotteries to be represented by the maximand functional, where is a utility function mapping the set of consequences with image conceived as a set of real numbers with for. For simplicity of notation we shall write a discrete random variable representing a finite lottery denoted as, from what follows that expected utility is therefore evaluated like.

The geometry of, supposed monotonous, has a simple behavioral interpretation under EUTaxioms, whereby being concave implies risk aversion―such as the logarithm function used by Bernoulli―and convexity entails risk prone behavior by an individual agent. The Arrow-Pratt measure (e.g. [23] ) is commonly used to assess the issues of risk-avert or risk-prone behavior. Recently, Baillon et al. [24] presented a general and simple technique for comparing the concavity of different utility functions isolating its empirical meaning in EUT, and an example of a concave utility function used to assess optimal expected utility of wealth under the scope of insurance business can be acknowledged in [25] .

As Shaw and Woodward [26] say, focusing on the issue of natural resources management, the problem in classical utility theory is that the optimization of the models may have to accommodate preferences that are nonlinear in the probabilities. There are many approaches with this perspective, known at least since Edwards in 1955 and 1962 [27] discussed in parallel the theory of Kurt Lewin utility and the theory of subjective probability of Francis Irwin, introducing the concept of decision weights instead of probabilities; the Lewin utility theory was referred to as anchored in the concept that an outcome which has a low probability will, by virtue of its rarity, have a higher utility value than the same outcome would have if it had a high probability.

A substantial review was made by Starmer [28] under the name of non-expected utility theory, the case of subjective probabilities being framed within the conventional strategy approach focused on theories with decision weights, in particular the simple decision weight utility model where individuals concerned with lotteries are assumed to maximize the functional; the probability weighting function transforms the individual probabilities of each consequence into weights, and, in general, it is assumed that is a continuous non-decreasing function with and (e.g. [13] [28] [29] ); decision weights are also named capacities if additionally they satisfy monotonicity with respect to set inclusion. An example of a decision weight procedure concerning assessing preferences for environmental quality of water under uncertainty is outlined in [30] .

There are many other approaches to surpass the limitations of independence axiom, and, for example, Hey and Orme [31] compared traditional expected utility and non-expected utility methods with a total of 11 types of preference functionals, evaluating the trade-off of explaining observed differences of the data relative to the models versus loosing predictive power. Recently, a reasoning of decision-making with catastrophic risks motivated the incorporation of a new axiom named sensitivity to rare events [32] . But simple decision weight utility modeling is the conceptual type of non-expected utility methods that is relevant in this paper.

The quantitative-qualitative measure of information generalizing Shannon entropy characterized by Belis and Guiasu [33] is additive and may be associated with a utility information scheme, anchored in a finite sample space, establishing that an elementary event occurring with (objective) probability entails a positive (sub- jective) utility. Thus, retrieving the discrete random variable we denoted previously as and identifying the event with the outcome of utility, it follows that the mathematical expectation or mean value of is evaluated with the standard formula and is usually referred to as weighted (Shannon) entropy. This mathematical device was further discussed and studied analyti- cally by Guiasu in 1971 [34] , considered as a measure of uncertainty or information supplied by a probabilistic experiment depending both on the probabilities of events and on corresponding qualitative positive weights. The functional was also named useful information (e.g. [35] ). At least since the eighties of last century weighted entropy was used in economic studies concerning investment and risk [36] and that scope of approach used in financial risk assessment and security quantification proceeds until nowadays (e.g. [37] -[40] ). Also, new theoretical developments concerning weighted entropy mathematical properties are available (e.g. [41] ).

In 1976, Emptoz―quoted in Aggarwal and Picard [42] ―introduced the entropy of degree of an experiment outlined with the same premises of the information scheme referred to above and defined as:

Using l’Hôpital’s rule it is easy to prove the result of the limit, thus allowing for the extension by continuity to the case, hence retrieving weighted Shannon entropy. In 1978, Sharma et al. [43] discussed a non-additive information measure generalizing the structural -entropy previously studied by Havrda and Charvat [44] ; they named their mathematical device “generalized useful information of degree”, relative to the utility information scheme stated above and denoted as:

The formula above means exactly the same entity as entropy of degree of Emptoz―what we can check making α = β and rearranging the terms. Hence, the result also holds, and when we get, which we can acknowledge as meaning “double” weighted Gini-Simpson index, as we shall see; the authors interpret the number as a flexibility parameter, exemplifying its semantic content either as an environmental factor or a value of “information consciousness” in aggregating financial accounts.

Weighted Gini-Simpson (WGS) index was outlined by Guiasu and Guiasu [45] in the scope of conditional and weighted measures of ecological diversity denoted as where the positive weights reflect additional information concerning the importance (abundance, economic significance or other relevant quantity) of the species in an ecosystem represented by their proportions or relative frequencies (for) associated with the closure condition, hence defining the standard simplex. Whether we use the objective or physical concept of probability practically as a synonym for proportion of successes in trails governed by large numbers laws as discussed in Ramsey [16] ―outlined subsequently as inter- subjective probabilities by Anscombe and Aumann [46] ―or even as subjective probability measuring the degree of belief of an agent, we can provide the result using the formulas for generalized useful information of degree previously referred to by Sharma et al. [43] or the entropy of degree of Emptoz (see [42] ). Guiasu and Guiasu proceeded with several developments of the WGS index as a relevant measure in the context of ecological diversity assessment (e. g. [47] [48] ).

The formulation and analytical study of weighted Gini-Simpson index was first introduced by Casquilho [49] within a set of indices built as mathematical devices applied to discuss compositional scenarios of landscape mosaics (or ecomosaics), using either ecological or economic weights―there referred to as positive characteristic values of the habitats―in order to assess the relevance of the actual extent of the components, as compared with the optimal solutions of the different indices.

One main feature of WGS index we must keep in mind is that we have as a diversity measure should behave, because WGS index is composed of a sum of positive terms, attaining the null value at each vertex of the simplex (if), meaning that just that component is present and all the others absent. In the context of lottery theory entails there exists only a single consequence, the result of a certain event; whether we should be dealing with prospect theory the result has a semantic shift and would mean that there is a single pure strategy.

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 [50] and other developments concerning ecomosaics composition assessment with forest habitats [51] [52] . In [49] [51] it is shown that weighted Gini-Simpson index can be interpreted as a sum of variances of interdependent Bernoulli variables thus becoming a measure of the variability of the system and enabling a criterion for its characterization.

¹We discard the cases with null utilities considered here not relevant, or equal utilities as that would entail editing the lottery in a more compact and simplified form.

In what follows, the simple lottery is conceived as having fixed strictly positive utilities¹ for and if, while the variables are the probabilities defined in the standard

simplex:.

In this setting used to characterize the discrete random variable we shall denote the EU-WGS mathematical device as in analogy with the formula of the EU-WE framework discussed in [7] , to be defined as:

Equation (1) therefore has the full expression. We see that we can also interpret it as a preference function under the scope of non-expected utility theory with the characteristic formula where the decision weights defined as for verify the standard conditions and; also it is easily seen that the function is strictly increasing in the open interval and concave, as we have that the calcu- lus of derivatives entails: and.

Still, we can proceed with the subsequent interpretation: denoting a random variable with values for, we have an expression for utilities in the sense of Lewin (referred to in [27] ), as when the event is quite rare we have entailing and, on the contrary, when the event is about to be certain we get and; hence, in this approach, utility values are also function of the pro- babilities being enhanced by rarity as claimed in that theory and with this interpretation we compute the mean value as.

The function is differentiable in the open simplex as it is composed of a sum of quadratic elementary real functions, thus being continuous in its domain, which is a compact set (a closed and limited subset contained in), hence Weierstrass theorem ensures that attains minimum and maximum values. Also it is easily seen that is a concave function―the simplest way to show that is recalling that it was previously proved that weighted Gini-Simpson index is a concave function (e.g. [45] [49] ) and adding the bilinear function does not change the geometry.

Concerning the evaluation of the minimum point we see that denoting we have ensured that because and we have at every vertex of the simplex as noticed before, otherwise being strictly positive; hence, is the minimum value attained in the correspondent vertex of the simplex, this vertex being the minimum point. Though we don’t know yet which is the maximum value of we know it exists, so if we denote it as we can write the range―or the image set―of function, a closed real interval defined as:

Searching for the maximum point correspondent to the maximum value we shall proceed in stages: first, we

build the auxiliary Lagrange function and then we compute the

partial derivative(s) as follows:; solving the equation we obtain the critical values of the auxiliary Lagrange function, evaluated as for. As we also have that the equation entails the condition, merging the results and simplifying we get and solving the equation for the Lagrange multiplier we obtain the result:

Using Equation (3) combined with the equation we can substitute, writing the formula for the candidates to optimal coordinates of the maximum point:

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; for the trivial case concerning the certain consequence lottery (with) we get and also Equations (4) solves like as it should be; when we obtain and we can proceed to the evaluation of the critical coordinates, obtaining:

We can also check that the result holds, confirming that the candidates to the maximum point defined in Equations (4) are in the hyperplane of the simplex. But, when we have to be careful; this is because the auxiliary Lagrange function did not include the constraints of non-negativity thus allowing for non-feasible solutions. Explicitly, for and the set of utilities with we can check directly, manipulating Equations (4), that if the following inequality holds, then we get.

Thus, combining in Equation (4) and Equation (3) we obtain the condition to get feasible values, and we deduce the general feasibility conditions which show to be closely related to the harmonic mean of the utilities, Equations (4) being directly applicable to evaluate the optimal point when we have that all the relations defined by Inequalities (6) are verified:

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 with, thus meaning that the maximum point of is insensitive to a positive linear transformation of the original values.

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 [53] in 1957 had already noticed, saying that the negative term has many of the qualitative properties of Shannon’s entropy but has the inherent difficulty arising from the fact that using Lagrange multiplier method entails that the results in general do not satisfy the feasibility conditions relative to.

Here, first we shall choose a forward selection procedure since we know that when it follows that Equations (5) ensure proper optimal solutions. Also, a direct inspection of Equations (4) highlights that the value increases with the correspondent since the term is common to all critical values; in fact, we have that for a fixed j when the limit value in Equation (4) is, and, conversely, when we

get the result.

So, given the set of utilities defined in we begin ordering the set such as we are now dealing with the lottery in the form of an act described as a lottery where the outcomes or utilities are real numbers ordered in a decreasing way (e.g. [54] ):; as it is for sure that and imply strictly positive and

the problem begins with the evaluation of. Does verifies the condition?

Whether not, stop and state and evaluate the optimal solution with Equations (5), all the other components of higher order having null probability or proportion; if it does verify the feasibility condition proceed,

now with the condition (6) restated as and pose the same recurrent ques-

tion until you verify that there is an order for which some of the inequalities (6) fail; then settle as well as all the other proportions until order (n) and evaluate the optimal solutions with the first k ordered utilities using Equations (4). The dimension of the problem is reset as, all the other proportions being null.

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

ning with the lowest utility, then using Equation (4) to compute with

and if observing then setting and proceeding to the evaluation of with, recurring until we obtain; then, all the other subsequent variables will be positive and are evaluated with Equations (4) setting.

Assume that we have the following lottery with n = 5 and ordered utilities:, , , and; then the first doubt is relative to whether it is true or not that; computing the right member of inequality we obtain the numeric value 2.5532 so the condition 3 > 2.5532 holds and we proceed to the next stage: does the condition is true? Computing the right member of the inequality we get 2.3377 so the condition does not hold; thus, stop, set and, hence eva-

luate the optimal point with Equations (4) relative to the set, being the dimension of the problem

reset to, with approximate values:, and.

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

so we have and set; thus, we calcu-

late so we also have and set; next,

we get, a proper value, and the evaluation of the other optimal coordinates proceeds with applying Equations (4).

The maximum value of in itself does not seem to be relevant, because our search was attached to the evaluation of the maximum point as the criterion to find the best lottery according to the framework, but obviously it is possible to evaluate and become clear about the range defined in Inequalities (2); supposing that, for are the proper non-null optimal coordinates previously evaluated we have that

but we also can calculate the maximum value just with the utility values as written in Equa- tion (7), what follows from combining Equation (1) and Equation (4) and simplifying:

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