What is an intuitive interpretation for the softmax transformation? A recent question on this site asked about the intuition of softmax regression.  This has inspired me to ask a corresponding question about the intuitive meaning of the softmax transformation itself.  The general scaled form of the softmax function $\mathbf{S}: \bar{\mathbb{R}}^{n-1} \times \mathbb{R}_+ \rightarrow \Delta^n$ is given by:
$$\mathbf{S}(\mathbf{z}, \lambda) \equiv \Bigg( \frac{1}{1 + \sum_{k=1}^{n-1} \exp(\lambda z_k)}, \frac{\exp(\lambda z_1)}{1 + \sum_{k=1}^{n-1} \exp(\lambda z_k)}, \ \cdots \ , \frac{\exp(\lambda z_{n-1})}{1 + \sum_{k=1}^{n-1} \exp(\lambda z_k)} \Bigg).$$
Is there any intuitive interpretation that describes the mapping from the extended-real value $\mathbf{z} \in \bar{\mathbb{R}}^{n-1}$ to the corresponding probability vector $\mathbf{p} \in \Delta^n$ in simple terms?  I.e., is there some explanation that can describe the mapping in an intuitive way by use of geometry, analogy, etc.?

The presence of the exponentials in this function makes it fairly easy to construct an intuitive meaning for the transformation in terms of exponential growth of a set of quantities.  Consequently, I will give an intuitive description for the function in terms of a simple financial portfolio earning returns over time.  This can be modified or generalised to refer to any similar example involving a set of quantities affected by exponential growth.
A simple intuitive interpretation: Suppose you have an initial investment portfolio consisting of a set of $n$ investment items with the same value.  (Without loss of generality, set the initial values of each item to one.)  The first item always earns zero return, and the remaining items earn fixed continuous rates-of-return of $z_1, ..., z_{n-1}$ in each time period (these returns may be positive or negative or zero).  Now, after $\lambda$ time units the first item will have a value of one, and the remaining items will have respective values $\exp(\lambda z_1), ..., \exp(\lambda z_{n-1})$.  The total value of the portfolio is $1 + \sum_{k=1}^{n-1} \exp(\lambda z_k)$.  Consequently, the softmax function gives you the vector of proportions of the size of each item in the portfolio after $\lambda$ time units have elapsed.
$$\mathbf{S}(\mathbf{z}) = \text{Proportion vector for size of items in portfolio after } \lambda \text{ time units}.$$
This gives a simple intuitive interpretation of the softmax transformation.  It is also worth noting that one can easily construct a corresponding intuitive interpretation for the inverse-softmax transformation.  The latter transformation would take in the proportion vector showing the relative sizes of the items in the portfolio, and it would figure out the continuous rates-of-return that led to that outcome over $\lambda$ time units.

This is just one intuitive interpretation for the softmax function, using a finance context.  One can easily construct corresponding interpretations for any finite set of initial items that are each subject to exponential growth over time (with one item fixed to have zero growth).
