Okay, so I've continued to think about bargaining solutions, and it's worth giving a clear example of why the most obvious heuristic fails: why not, when striking a bargain, have both sides benefit equally? Seems simple enough.
Imagine a monk doctor. All they care about is their spiritual life, their inner fulfillment. They believe that if they die, they'll be reincarnated, and be able to continue their spiritual journey, albeit in a slightly less convenient way.
And a father comes to the monk doctor, meditating on top of a mountain, with his child, and says, "Monk, please, save my son!" And the monk thinks, I do need money to keep living my ascetic life, but the monk also wants to be fair, according to the bargaining rules of the material world. So the monk calmly and slowly says, "Okay, I will construct a deal that benefits us both equally. I have no great want for money – indeed, I'd probably give most of it away, and chastise myself for self-satisfaction in doing so, if any arose. And I do not care even if I would perish on his mountain, in my temple. But money is slightly useful to me."
The monk continues, "But your son is irreplaceable for you, of incredibly dear value. You love him with all your heart, and without me, he would surely die, and you would witness his agonizing death, knowing that, if you accepted the bargain, he would live. Because you have not removed yourself from desire, this will bother you immensely for the rest of your days, beyond even the tragic passing of your son.
"So my suggestion is this: that I heal your son, and you pay me one quadrillion dollars."
Of course, the father does not have one quadrillion dollars, but when fairness demands both people benefit equally, it must not also demand that yachts are a nickel – that would be a directly opposing value. Things worth a lot would need to cost a lot, to balance consumer and producer surplus.
I hope this makes clear why equal-gains bargaining solutions are really only good as a first-approximation, like the Newton's laws of bargains. When values are very high, elasticity is very low, or almost any other interesting condition holds, it breaks down, and you must use a more complicated formula for generating a good result.
It's also worth remarking that this bargain would be struck essentially the same way with a normal doctor (it's just an afternoon's worth of work's opportunity cost for an entire child's life – if your child's life is worth at least $10 million, it seems like, if the doctor values money the same as the father, two appointments a day for 200 working days a year would make the selfish, materialist doctor a billionaire in about three months), and therefore even in a competitive marketplace the bargaining solution would lead to highly unintuitive pricing for simple but valuable services.