Our goal instead is to seriously model anew this problem without any procedures whatsoever. We therefore first formulate the game and the notion of subgame perfection in continuous time, with truly unrestricted timing and content of offers. We then gain some power lost by the lack of procedures using a behavioural restriction: We introduce a new solution concept aspirational equilibrium --- a Markovian refinement of subgame perfection where behaviour is governed by the players' aspiration values (expected payoffs). This idea happens to reprise experimental work from 1960 among psychologists, and generates some intuitive aspects of bargaining. Crucially, the proposee and not the proposer is now advantaged. The model happens to produce very tractable analysis that anyone can explore, and generates some intuitive implications. Since we now endogenize timing, we have a richer theory of bargaining too.

We find that discounted aspiration values form a martingale, and thereby deduce bounds on the expected bargaining duration by the simple observation of any pair of alternating offers. We also deduce properties of the time path of offers. Finally, we reverse a traditional comparative static: Ceteris paribus, more impatient players expect to get more of the pie.