In this paper, to begin with we present a generalization of the independence of irrelevant expansions assumption to the situation with an arbitrary yet finite number of players, and with the help of a comparatively simpler proof than the one suggested by Thomas (19981), we uniquely characterize the Nash bargaining solution. In a recent paper, Lahiri (1993) introduces the concept of a shift for bargaining problems. A shift for a bargaining problem amounts to a displacement of the origin to a point in the nonnegative orthant of a finite dimensional Euclidean space (in which the bargaining problem is defined) so as to reduce the original problem to a new one consisting only of those points that weakly Pareto dominate the new origin. A characterization of Nash bargaining solution is also obtained in this paper using a convexity assumption. A related version of this convexity assumption and a similar characterization theorem can be found in Chun and Thomson (1990) and Peters (1992). An intermediate property used in the latter characterization called localization, which can be found in Peters (1992) is similar in spirit to the independence of irrelevant alternatives assumption. We also obtain a characterization of the Nash solution, by relaxing this localization property and invoking Pareto continuity.