Game Theory - Rubinstein bargaining

Question

may I ask you how to solve this problem?

What will be the outcome of infinite Rubinstein bargaining if the first mover (player A) has outside option with utility M and the second mover (player B) has zero outside option.

Thank you!

Answer 1

Let $m_i$ be the amount offered by $j$ to $i$. Using the Shaked-Sutton shortcut in the standard symmetric version (same discount factor, zero outside option), we derive the two equations $$m_1 = \delta (1-m_2) \mbox{ and } m_2 = \delta (1 - m_1)$$ and find $m_i = \delta/(1+\delta)$.

If $1$ has an outside option $M$, you need to change the first equation in $$m_1 = \max\{ \delta (1-m_2), M \}$$ and solve accordingly. (Distinguish two cases, depending on $\delta (1-m_2) \gtrless M$.