To understand this better, assume that we have two energies, $E_i$ and $E_j$, with $E_i < E_j$. This means that the largest acceptance value must be $A(j\rightarrow i)$ since we move to a state with lower energy. It follows from also from the fact that the probability $w_i$ is larger than $w_j$. The trick then is to fix this value to $A(j\rightarrow i)=1$. It means that the other acceptance probability has to be $$\begin{equation*} A(i\rightarrow j)= \exp{(-\beta(E_j-E_i))}. \end{equation*}$$