Consider the following utility function $$g(x,u)=uxa+(1-u)(1-x)b$$ where $x\in[0,1]$ is the state variable, $u\in{0,1}$ is the action, and $a,b>0$ are known constants.
The problem of choosing the maximizing action for a given state (and constants) is simply: choose $u=0$ if $(1-x)b>xa$ and $u=1$ otherwise.
Now consider the case where we do not know $x$ with certainty. Instead assume we have a known probability distribution on $x$ denoted by $f_X(x)$. The question I’m interested in is with what probability, call it $p=Pr(u=0)$, do we mix between $u=0$ and $u=1$?
Attempt: I drew a couple diagrams to try to illustrate the setup. As one varies $p$, imagine a line, pivoting at the intersection of the two functions (lines), moving between $g(x,0)$ and $g(x,1)$.
It seems to me based on this (adhoc) figure, there should be some $p$ that maximizes the expected utility, but I can’t seem to visualize it. At what “angle” is the aforementioned imaginary line?
