The problem comes from an economic and market scenario. We have a function $$\psi(v_i)=v_i-\frac{G(v_i)-F(v_i)}{f(v_i)},$$ where the random variable $v_i$ is any real number (e.g., person $i$'s money). For any person $i$, his money $v_i$ is iid. $F(v)$ and $f(v)$ are the given cumulative distribution function (CDF) and probability density function (PDF) of $v_i$, respectively. However, we don't know the form of $G(v_i)$, we only know that $G(v_i)\ge 1$.
We want to select a person $i^*$ who has the largest $ \psi(v_i)$ comparing with any other person. And more importantly, we also want that the probability that this person $i^*$ has the largest $\psi(v_i)$ should equal to the probability $\frac{1}{G(v_i)}$.
Briefly, we want to find a function $G(v_i)$ such that when we select the largest $\psi(v_i)=v_i-\frac{G(v_i)-F(v_i)}{f(v_i)}$, the maximizer (i.e., person $i^*$) are selected with a probability $\frac{1}{G(v_i)}$. This seems to be a fixed-point problem, so I summarize it into the following style $$ \mathbb{P}\big[{\psi}(G(v_i))\big]= \frac{1}{G(v_i)}. $$ But I'm not sure whether this is correct and how to continue for the computation of $G(v_i)$.
I appreciate your help!