Combining Expected Utility and Weighted Gini-Simpson Index into a Non-Expected Utility Device

Abstract

We present and discuss a conceptual decision-making procedure supported by a mathematical device combining expected utility and a generalized information measure: the weighted GiniSimpson index, linked to the scientific fields of information theory and ecological diversity analysis. After a synthetic review of the theoretical background relative to those themes, such a devicean 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 scenarios 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.

A methodology to determine the maximum value of weighted Gini–Simpson index

“…Several theoretical developments were presented in the following, from which stand out, concerning ecological or related biological fields: the application of weighted Gini–Simpson to assess ecomosaics compositional scenarios (Casquilho 2011 ); the application to biodiversity partitioning and measuring of diversity with respect to the pairs of species (Guiasu and Guiasu 2012 ); Ricotta et al ( 2012 ), discussing Rao’s quadratic index under the scope of functional rarefaction, claim that their method is suitable to be extended to any concave diversity measure including WGS index; also, weighted Gini–Simpson index was said to be closely related to a unified framework based on Hill numbers concerning specific, phylogenetic, functional and other diversity measures (Chiu and Chao 2012 ; Chao et al 2014 ); Guiasu and Guiasu ( 2014 ) proceeded with developments concerning the use of the index as a biodiversity assessment tool for interdependent species; Pavoine and Izsák ( 2014 ) formulated a new parametric index of diversity related to Rao’s quadratic entropy and discuss connections relative to other indices including WGS index; last, WGS index was combined with expected utility generating a non-expected utility device (Casquilho 2015 ). Other empirical studies or applications using WGS index will be mentioned in the discussion of results.…”

Discussing Landscape Compositional Scenarios Generated with Maximization of Non-Expected Utility Decision Models Based on Weighted Entropies

“…The real function W 2 displayed in Equation (3) was also previously studied (see [44]), and, analogously to what was stated relative to function W 1 shown in Equation (2), it is also a smooth and concave real function with minimum value evaluated like minW 2 = min i=1,··· ,n u i , while the maximum point is attainable with a Lagrange multiplier method checked for the feasibility of solutions verifying altogether the full set of inequalities u i > (n − 1)/ ∑ n i=1 1/u i , for i = 1, · · · , n. In general, the set of inequalities doesn't hold and we have to proceed with an algorithm obtaining the maximum point coordinates defined with the expression p *…”