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 Gini-Simpson 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 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 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.
Highlights
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; choosing among alternatives involving risk entails that it is selected that one for which the expected utility is highest (e.g. [1])
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 EU-Weighted Gini-Simpson (WGS) device discussed in this paper is suitably defined as a non-expected utility method with decision weights, a tool built combining expected utility and weighted Gini-Simpson index as claimed in the title
Summary
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; choosing among alternatives involving risk entails that it is selected that one for which the expected utility is highest (e.g. [1]). Combining the concepts of expected utility and some measure of variability of the probability score—generating 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 [8]. Those approaches were later merged under the name of “non-expected utility” methods in the 1980s and consist of different conceptual types, the one we shall be dealing with framing into the category of decision models with decision weights or non-additive probabilities, named capacities. 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
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