We obtain then (using the Boltzmann distribution) $$\begin{equation*} \frac{A(j\rightarrow i)}{A(i\rightarrow j)}= \exp{(-\beta(E_i-E_j))} . \end{equation*}$$ We are in this case interested in a new state $E_j$ whose energy is lower than $E_i$, viz., $\Delta E = E_j-E_i \le 0$. A simple test would then be to accept only those microstates which lower the energy. Suppose we have ten microstates with energy $E_0 \le E_1 \le E_2 \le E_3 \le \dots \le E_9$. Our desired energy is $E_0$.