I am trying to find the form (or even any necessary properties) of a function $f(x;c)$, noting that it may be helpful to think of a family of univariate functions parameterized by a variable $c$ rather than a function defined in $R^2$.
- It only needs to be defined on $x\in[0,1], c\in [0,1]$
- It is continuous on $x\in[0,1]$, and $c\in(0,1)$ (or perhaps even $[0,1]$)
- It is likely convex in $x$, for a given $c$.
- For any $c\in [0,1]$, $\int_0^1 f(x;c) dx=1$.
- As $c\to 1$, $f(0;c) \to \infty$ and $f(\epsilon,c)\to 0, \forall \epsilon >0$.
- $f(1;0) = 2$, and $f(0;0) = 0.5$.
I realize this may be woefully underspecified, and so I would be happy with any properties at all that this function might have. The last two requirements naturally point me toward Dirac-like functions, and what I have so far looks like
$$ f(x;c) = \frac{2}{(1-c)\sqrt{\pi}}\frac{1}{\text{erf}(\frac{1}{1-c})}e^{-(\frac{x}{1-c})^2} $$
This fulfills most of the requirements, except that it is not convex, and the constants it converges to as $c\to 0$ are different (1.339 and 0.4925 instead of 2 and 0.5).
I have attached numerically generated plots of $f(x;c)$ for $c=0.1, 0.25, 0.5, 0.75, 0.9, 0.99, 0.999$ in 2-D. The latter two have their own plots for the sake of scale, noting that the spike does not seem to grow too quickly with regard to $c\to 1$. Also attached is a 3-D scatterplot.
Any progress at all would be greatly appreciated. This function is an attempt at studying Nash equilibrium distributions of a continuous game.
