Before revision:  The key in our proof is a net gain from revision of the strategy , i.e., the payoff improvement by the revision minus the switching cost . Static stability implies that the aggregate net gain monotonically decreases over time under an economic reasonable dynamic and thus can be used as a Lyapunov function.
After revision:  The key is expected net gains (payoff improvements) from strategy revisions after paying switching costs . Static stability implies that the aggregate net gain monotonically decreases over time under an economic reasonable dynamic and thus can be used as a Lyapunov function.