In this paper we establish three distinct results: a) a unique characterization of the family of non-symmetric Nash choice functions using an Independence of Irrelevant Expansions Assumption; b) a characterization of the entire family of choice functions (containing those determined by weighted hierarchy and those which are 'mongrel' choice functions only) satisfying Pareto Optimality, Scale Translation Covariance and Weak Independence of Irrelevant Expansions; c) as a by-product of (b) we obtain a characterization of the family of choice functions determined by a weighted hierarchy by slightly strengthening the Weak Independence of Irrelevant Expansions assumption.