The class of bargaining solutions that are defined on the domain of finite sets of alternatives and satisfy Weak Pareto Optimality (WPO), Independence of Irrelevant Alternatives (IIA) and Covariance (COV), is characterized. These solutions select from the set of maximizers of a nonsymmetric Nash product -- i.e., from a nonsymmetric (multi-valued) Nash bargaining solution -- according to a specific decomposition of the indifference curves of this Nash product. We use this characterization in two ways. First, we derive consequences on this domain and on larger domains of compact (non-convex) bargaining problems, and show that most results in the literature are special cases and consequences of our central results -- in particular by adding continuity or symmetry axioms. Second, since the continuity axiom prevents nontrivial selections from the Nash bargaining solutions, we use the Axiom of choice to construct for example non-single-valued discontinuous WPO, IIA and COV bargaining solutions. It is conjectured that, in the case of two-person bargaining problems,the existence of such discontinuous bargaining solutions cannot be shown from the Zermelo-Fraenkel axioms for set theory without using the Axiom of Choice.
