The Metropolis choice is to maximize the \( A \) values, that is
$$ \begin{equation} A_{j \rightarrow i} = \min \left ( 1, \frac{p_iT_{i\rightarrow j}}{ p_jT_{j\rightarrow i}}\right ). \tag{9} \end{equation} $$Other choices are possible, but they all correspond to multilplying \( A_{i\rightarrow j} \) and \( A_{j\rightarrow i} \) by the same constant smaller than unity. The penalty function method uses just such a factor to compensate for \( p_i \) that are evaluated stochastically and are therefore noisy.
Having chosen the acceptance probabilities, we have guaranteed that if the \( {\cal w}_i^{(n)} \) has equilibrated, that is if it is equal to \( p_i \), it will remain equilibrated. Next we need to find the circumstances for convergence to equilibrium.