Does the case of strategic complements always result in a situation where the Stackelberg leader is worse off?

For example, in the case of Bertrand competition, is the Stackelberg leader worse off relative to the follower? If so, why? Is this because the first mover can always undercut until marginal cost is reached?

• Not under full info and rationality assumption. In that case, firms know each others' production functions and best-response functions. In particular, the stronger firm (by virtue of having a lower $MC = p_1$) will know that the weaker firm cannot make a credible threat to reduce price below $p_1$. Since $p_2 > p_1$, whether the weaker firm moves first or second, it will not be able to capture the market on price. – heh Nov 27 '19 at 20:14
• I am having difficulties understanding this last comment. I am not sure what "lower $MC = p_1$" means, since $MC$ is a function of quantity sold, and that is also dependent on $p_2$, so it seems like something is missing here. Also, w.r.t. the last sentence: is capturing the entire market necessary for second mover advantage? Cannot I just be better off compared to the simultaneous equilibrium without capturing the entire market? – Giskard Nov 27 '19 at 21:26