Well, in my premise there is no need for agreement. If side A thinks it's worth $N and side B thinks it's worth $M, they each get ($N-$M)/2 to work with. This asymmetry is what makes it a bad choice to try to overpower a less wealthy opponent by simply outspending them.

I haven't really thought through the consequences of requiring some kind of agreement in advance, but my instinct is it couldn't really work just because being obstinate would be heavily rewarded.