How can I calculate the following optimization?

Question

How can I calculate the following optimization?

Let $\xi$ be a random variable:

$$\xi\overset{\circ}{=}\begin{cases} \frac{p_{1}\cdot0.3}{p_{1}\cdot0.3+p_{2}\cdot0.7} & \text{with}\;p=p_{1}\cdot0.3+p_{2}\cdot0.7\\ \frac{\left(1-p_{1}\right)\cdot0.3}{\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7} & \text{with}\;1-p=\left(1-p_{1}\right)\cdot0.3+\left(1-p_{2}\right)\cdot0.7 \end{cases}$$

$p_{1},p_{2}\in\left[0,1\right]$. How can I calculate

$$\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)?$$

The motivation of this task comes from the following article: https://www.aeaweb.org/articles?id=10.1257/aer.101.6.2590. I tried to solve the example of the article on my own, but I stucked with this (perhaps trivial) problem. I'm not even sure if it is a good starting, but still...

Answer 1

Let me denote $\frac{0.3p_{1}}{0.3p_{1}+0.7p_{2}}$ as $q_{1}$ and $\frac{0.3\left(1-p_{1}\right)}{0.3\left(1-p_{1}\right)+0.7\left(1-p_{2}\right)}$ as $q_{2}$.

Since $\xi$ can have two values, seemingly there can be four values for $\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)$:

$\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=1$ if $q_{1}>0.5$ and $q_{2}>0.5$,

$\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=1-p$ if $q_{1}\leq0.5$ and $q_{2}>0.5$,

$\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=p$ if $q_{1}>0.5$ and $q_{2}\leq0.5$,

$\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=0$ if $q_{1}\leq0.5$ and $q_{2}\leq0.5$.

We can't reach $\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=1$, since the set $$\left\{ q_{1}>0.5\;\text{and}\;q_{2}>0.5,p_{1},p_{2}\in\left[0,1\right]\right\}$$ is an empty set.

However, we can reach $\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=1-p$. In this case the task is the following optimization

$$\begin{cases} \max_{p_{1},p_{2}}\left(1-p\right)=\max_{p_{1},p_{2}}\left(\left(1-p_{1}\right)0.2+\left(1-p_{2}\right)0.8\right)\\ \text{given: }q_{1}\leq0.5;q_{2}>0.5;p_{1},p_{2}\in\left[0,1\right] \end{cases}$$

which is a simple LP optimization.

Similarly, $\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=p$ can be solved as the same way as above. The case is totally simmetric, so it will give the same final optimization result as above.

There is no point to care about the $\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=0$ case if we can reach $\max_{p_{1},p_{2}}\mathbf{P}\left(\xi>0.5\right)=p>0$.