I want to create a wealth indicator function used by the logic of a bidder agent, that tells the agent if he's rich (in comparison to others).
Given:
Total number of competitors $n$
Amount of all the money $m$ in the system
mean (displaying money per agent) $a$. (E.g. arithmetic mean etc.)
Searching for: $w(x) \in \mathbb{R}, x \in \mathbb{N}$
so that:
$w(m) = 1$
$w(a) = \frac{1}{2}$
$w(0) = 0$
(doesn't need to be an odd function, although it would be nice)
I had a linear function, but that didn't fit my needs, because most of the values were around ~ 0.5, also had a quadratic and a kubic function that didn't fit (created an equoation and solved it with wolfram alpha).
Exponential growth seems to be the solution (assuming money is normally distributed, the desired function would be a logistic curve), but I'm having problems defining a monotonically nondecreasing function. Would be glad if you could help me out.
I created a function based on root.
May $f_{money}$ be a function that depicts amount of money an agent possesses, so that $f_{money}\left(u\right) \in \mathbb{N}, u \in \mathbb{A}$ where $\mathbb{A}$ is the amount of all agents.
Furthermore money in the system $m = \sum_{i=0}^{n-1}f_{money}\left(u_i\right) = 2500$. and number of agents $n = 5$ Mean of money (harmonic mean) $z = \left(\frac{1}{n} \cdot \sum_{i=0}^{n-1} f_{money}\left(u_i\right)^{-1} \right)^{-1} = 277$.
Then wealth indicator $w(x)=a\sqrt{x} + bx$, where
$a = 3.505259*10^{-2}, b = -3.010518*10^{-4}$
Generic formula for $a$ and $b$
$ \begin{pmatrix} \sqrt{m} & m \\ \sqrt{z} & z \end{pmatrix} \cdot \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} 1 \\ \frac{1}{2} \end{pmatrix}$
$a = \frac{\frac{1}{2}m-z}{m\sqrt{z}-z\sqrt{m}}, b=\frac{\sqrt{z}-\frac{1}{2}\sqrt{m}}{m\sqrt{z}-z\sqrt{m}}$
$m\not=0, m\sqrt{z}-z\sqrt{m} \not=0$