[ColFrancis]

I think the example problem has many issues with it which confuse the issue. There is a pervading assumption that more slices is better. What if Alice only wants 5 slices? Then her fall back is quite close to what she needs but also she doesn't mind how the pie is divided so long as she gets one slice.

It's not clear what everyone's utility function is. If a sales person is nearing a quarterly target, their utility function might have a giant kink in it, or if they're at the start of a period it might be very linear. The marginal pie, more often than not, has multiple dimensions which different agents care different amounts for.

Thankfully I've never been in a negotiation without a decent BATNA and so just chat about it until either we decide it's not going to work out, or we are both happy. People who negotiate too hard by playing stupid games and trying to max out themselves without any quarter given are tiring and push me to the first option. Why would I want to work with you if you're a pain to deal with before we even start?