So consider if the strong player proposes first. He can reason 'If the weak player fights, he'll only get 2, so I can offer him 3, and then I'll get 7'. Similarly, the weak player can reason 'If he fights, he'll only get 3, so I can offer him 4, and then I'll get 6.' In theory they could both keep making proposals and counter proposals indefinitely, never able to come to an agreement. So to prevent infinite-loop-land, we'll attach a cost of .1 for every proposal which is offered but not accepted (.1 from the original 10, with the half the remainder being split 60/40 if they fight). Now the question is twofold. Where's the flaw in their reasoning, and how should it be split? Clearly, neither should accept anything less than what they would get if they fought (now that we've added a cost to negotiations, we need to add the option for either of them to start fighting when they want to, so the pile doesn't get negotiated away). One idea would be to split the pile as it would be split in a fight, but without fighting, so the strong player gets 6, and the weak player gets 4. I suspect that, in the absence of non-linear payoff functions, this is the rational conclusion, but I'm not sure.