Suppose a mediator M (a neutral party) is working to facilitate a simple financial agreement between opposing parties A and B: a lump sum $X$ to be paid by A to B. They each make offers, $A_{off}$ and $B_{off}$, with $A_{off} < B_{off}$, and are unwilling to go further. The mediator speaks individually with each party and estimates (for himself/herself) the values $A_{max}$ and $B_{min}$ representing the maximum A is willing to pay and the minimum B is willing to receive. He/She sees that:
$$A_{off} < A_{max} < B_{min} < B_{off}$$
Question 1: What strategies are available to M in this situation?
Question 2: Suppose there is also a financial penalty $Y$, for each party, for failing to reach an agreement, and, luckily, it turns out that:
$$A_{max} < B_{min} - Y \le A_{max} + Y < B_{min}$$
What is the right value of $X$ in this case? Is it simply the average of $B_{min}$ and $A_{max}$, i.e. $\frac{A_{max} + B_{min}}{2}$? Can $A_{off}$ and $B_{off}$ simply be discarded, even though $A_{max}$ and $B_{min}$ were never openly disclosed? Perhaps a rule can be introduced to allow the disclosure if and only if the agreement is now possible.
As an example, we could have $A_{max} = 1000$, $B_{min} = 2000$, and $Y = 600$. We would then get $X = 1500$. For completeness, we could add, e.g., $A_{off} = 800$ and $B_{off} = 2400$.
Question 3: How can the above framework be improved from a game-theory perspective?