In this paper, we study two-person bargaining problems represented by a space of alternatives, a status quo point, and the agents’ preference relations on the alternatives. The notion of a family of increasing sets is introduced, which reflects a particular way of gradually expanding the set of alternatives. For any given family of increasing sets, we present a solution which is Pareto optimal and monotonic with respect to this family, that is, it makes each agent weakly better off if the set of alternatives is expanded within this family. The solution may be viewed as an expression of equal-opportunity equivalence as defined in Thomson [Soc. Choice Welf. 11 (1994) 137–156]. It is shown to be the unique solution that, in addition to Pareto optimality and the monotonicity property mentioned above, satisfies a uniqueness axiom and unchanged contour independence. A noncooperative bargaining procedure is provided for which the unique backward induction outcome coincides with the solution.