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In this paper, it is discussed a framework combining traditional expected utility and weighted entropy (EU-WE)—also named mean contributive value index—which may be conceived as a decision aiding procedure, or a heuristic device generating compositional scenarios, based on information theory concepts, namely weighted entropy. New proofs concerning the maximum value of the index and the evaluation of optimal proportions are outlined, with emphasis on the optimal value of the Lagrange multiplier and its meaning. The rationale is a procedure of maximizing the combined value of a system expressed as a mosaic, denoted by characteristic values of the states and their proportions. Other perspectives of application of this EU-WE framework are suggested.

Shannon entropy has been widely used in ecological studies as a measure of diversity at different scales in space, from local community level to landscapes and regions. Guiasu and Guiasu [

The core of the rationale that relates utility and information theory concepts can be summarized as it was stated by Bernardo [

Under the scope of a general theory of communication, Shannon [

Rènyi [

Weighted entropy was first proposed by Bellis and Guiasu [

Casquilho et al. [

Ricotta [

The work presented here has some similarity with a decision aiding procedure based on expected utility and Shannon entropy [

In what follows a new set of the results and proofs are presented, equivalent, but different, from those presented before, e.g., [

We will be dealing with proportions, defining a normalized measure space. Proportions are relative extension measures—as well as relative frequencies and probabilities—the difference is that proportions reflect the extension of actual, or presumably effective, states of a system, and probabilities are possibility measures of events compatible with Kolmogorov’s axiomatic definition. Nevertheless, the two concepts are intimately linked under the scope of objective or physical probabilities, which often uses probability practically as a synonym for proportion [

Assume that a system is characterized by a scenario of the world defined as the set of n elementary states, or sample space:

Next we outline an information based family of utility functions:

The EU-WE framework here to be discussed, denoted index

where

Building auxiliary Lagrange function defined as

The numeric solution of this equation will be denoted

if

Thus, the critical point of index

As expression (2) depends on the value of

It can also be proven that each optimal coordinate

Next, let us prove that the critical point is a maximum in analogy with Guiasu procedure [

and eventually get the equivalent mathematical expression for the auxiliary function

Using the auxiliary result

Retrieving auxiliary function

Formula (4) is the weighted entropy of the optimal solution of index

Optimal proportions are indifferent to a linear positive transformation in the utilities, such as a change of scale or units of measure. In particular, if we replace the utilities

The minimum value of index

Thus, since

As an example, we retrieve characteristic economic values of forest habitats from [

Using these optimal values and evaluating Formula (4) we obtain 310. 43 which is quite similar to the value of

Other utilities could have been used besides the neutral, either convex, risk-taking utilities, as it would be the case with

The EU-WE framework here discussed emphasizes the notion of contributive value of each component of a mosaic—or stable state of a simultaneous multi-state system—depending both on context and utility values. Others seem to identify contributive value with utility itself (see [

The static nonlinear optimization procedure presented here may be applied with focus on compositional scenarios generated under active adaptive ecosystem management paradigm sensu Gunderson et al. [