Strategic Bargaining Theory

Instead of axioms, reformulate as a game and use equilibriums. Unfortunately, equilibriums are not unique. Still, this models humans/smart agents better since it does not impose an artificial fairness requirement.
It consists of successive bids until a deadline is reached. We assume that when the deadline arrives the other guy is willing to take 0.
If agents were to take turns bidding on splitting the dollar, the first one would wait until the end and then bid .9999 and keep the (almost) whole dollar.

To make the problem more realistic (think perishable or time-sensitive products) we can discount the dollar by some amount each time, e.g., multiply it by .9. Such a setting would look like:

Offerer's maximal acceptable claims in a finite game.
Round	1's share	2's share	Total value	Offerer
t-3	.819	.181	.9^t-4	2
t-2	.91	.09	.9^t-3	1
t-1	.9	.1	.9^t-2	2
t	1	0	.9^t-1	1

For example, at t-1, 2 knows that the next round the total value will lose 10% (.1) so if he gets 10% then 1 will get the same as next time. That is, 90% at t-1 is the same as 100% at t.