Suppose $x = (x_1,x_2,\dots,x_K) \in \mathbb{Z}^K_{\geq 0}$. For $x,y \in \mathbb{Z}^K_{\geq 0}$, we write $x \succ y$ or $y \prec x$ if $x \neq y$ and
\begin{align*} x_{i(x,y)} > y_{i(x,y)},\quad \text{ where } \quad i(x,y) := \max\{ i: x_i \neq y_i\}. \end{align*}
That is, for any two vectors $x$ and $y$ that are not equal, we let $i(x,y)$ be the last position on which they differ and say that $x \succ y$ if the coordinate of $x$ at $i(x,y)$ is larger than the corresponding coordinate of $y$. We write $x \succeq y$ if either $x = y$ or $x \succ y$, and similarly for $x \preceq y$. This is a total order.
For example, if $x = (7,2,1,0,0)$ and $y = (6,3,1,0,0)$ then $y \succ x$ because they are equal on the last three positions and the next position that they differ is the second coordinate, since 3>2 we conclude that $y \succ x$.
Now, let $mx(x) = max\{k: x_k > 0\}$, we are interested in defining a function $f: \mathbb{Z}^K_{\geq 0} \rightarrow [0,K+1)$ that has the following properties:
- $f(0,0,\ldots,0) = 0$
- $mx(x) \leq f(x) < mx(x)+1$ (Note that when one of the coordinates of x is 1 and the rest are 0, then $f(x)= mx(x)$, for example let $x = (0,1,0,0,0)$, then $f(x)=mx(x)=2$)
- $f(.)$ is strictly increasing on $\mathbb{Z}^K_{\geq 0}$ wrt. the total-ordering defined above
- The effect of adding a positive value to coordinate $k$ should be smaller than adding the same value to coordinate $k+1,....,K$, having all the other values fixed, sth like convexity property but I'm not sure if the exact definition of convexity applies here. For example suppose $K=5$, $f(0,3,0,0,0) - f(0,2,0,0,0) \leq f(0,0,3,0,0) - f(0,0,2,0,0)$
I could define a function that has the first three properties, but not the fourth one: For any $x \in \mathbb{Z}^K_{\geq 0}$, let $g_{k}(x) = \prod_{i=k}^{K} (1+i)^{-x_i}$ for $k=2,\dots,K$ and $g_{K+1}(x) = 1$. \begin{align} f(x) := \sum_{k=2}^{K+1} k g_k(x) \big(1 - k^{-x_{k-1}}\big). \end{align} $f(0,3,0,0,0) - f(0,2,0,0,0) = 2.888889 - 2.666667 = 0.222222$ but $f(0,0,3,0,0) - f(0,0,2,0,0) = 3.9375 - 3.75 = 0.1875$
How to define $f(.)$ so that it follows all the 4 properties?
First, note that $f(x)$ cannot be equal to $mx(x)$. Thus the first condition can be stated in strict form, i.e. $$mx(x)<f(x)<mx(x)+1$$ You can also define a function that is $\mathbb{Z}^K_{\geq 0}\to(0,1)$, say, $\hat{h}(x)$ and then set $f(x)=mx(x)+\hat{h}(x)$. Now if $\hat{h}(x)$ satisfies the other two conditions, then $f(x)$ would satisfy all three. This can be easily verified.
We first focus on finding a suitable $h(x)$ and then normalize it. Something like $\hat{h}(x)=\frac{h(x)}{1+h(x)}$. Let $$h(x)=\sum_{i=1}^K (1+i)^{i+x_i/(1+x_i)}$$ You can verify that $h$ is strictly increasing wrt. to the total ordering. It also satisfies the third condition.