When treating a relative, normalized utility function as a pmf, what is the interpretation of Shannon entropy or Shannon information?
After the exchange with the OP in my other answer, let's work a bit with his approach.  

We have a discrete random variable $X$ with finite support, $X = \{x_1,...,x_k\}$, and probability mass function (PMF), $\Pr(X=x_i)=p_i, i=1,...,k$

The values in the support of $X$ are also inputs in a real-valued  cardinal utility function, $u(x_i) &gt; 0\; \forall i$. We then consider the normalized utility function

$$w(X): w(x_i) = \frac {u(x_i)}{\sum_{i=1}^ku(x_i)},\;\;i=1,...,k \tag{1}$$

and we are told that

$$w(x_i) = p_i \tag{2}$$

Note that we do not just make the observation that a normalized non-negative discrete function of finite domain, satisfies the properties of a probability mass function in general -we specifically assume that $w(x_i)$ has the functional form of the PMF of the random variable whose values $w(x_i)$ takes as inputs.

Since $w(x_i)$ is a measurable function of a random variable, it, too, is a random variable. So we can meaningfully consider things like its expected value. Using the Law of the Unconscious Statistician we have

$$E[w(X)] = \sum_{i=1}^kp_iw(x_i) = \sum_{i=1}^kp_i^2 \tag{3}$$

This is a convex function, and if we try to extremize it over the $p_i$'s under the constraint $\sum_{i=1}^kp_i=1$ we easily obtain

$$\text{argmin} E[w(X)] = \mathbf p^*: p_1=p_2=...=p_k=1/k \tag {4}$$

and we have obtained a general result:


  The normalized utility function as defined above has minimum expected
  value iff the distribution of $X$ is Uniform.


Obviously in such a case $w(X)$ will be a constant function, a degenerate random variable with $E[w(X)]=1/k$ and zero variance.

Let's turn to Shannon's Entropy which is the focus of the OP. To be calculated, Shannon's Entropy needs the probability mass function of the random variable... so we should find the PMF of the random variable $w(X)$...  

But it is my impression that this is not what the OP has in mind. Rather, it views Shannon's Entropy as a metric that has some desirable algebraic properties and perhaps can measure compactly in a meaningful way something of interest.  

This has been done before in Economics, specifically in Industrial Organization, were Indices of Market Concentration ("degree of competition/monopolistic structure of a market") have been constructed. I note two that look particularly relevant here.

A) The Herfindahl Index, has as its arguments the market shares of the $n$ companies operating in a market, $s_i$, so they sum to unity by construction. Its unscaled version is 

$$H = \sum_{i=1}^n s_i^2$$

which is an expression that has the exact same structure with the expected value of $w(X)$ derived above.

B) The Entropy Index
$$R_e = -\sum_{i=1}^n s_i\ln s_i$$
which has the exact mathematical form with Shannon's Entropy.

Encaoua, D., &amp; Jacquemin, A. (1980). Degree of monopoly, indices of concentration and threat of entry. International Economic Review, 87-105., provide an axiomatic derivation of "allowable" concentration indices, i.e. they define the properties that such an index must possess. Since their approach is abstract, I believe it may be useful to what the OP wishes to explore and attach meaning to.