Variations of a novel, recently proposed load balancing scheme based on small amounts of choice are investigated. Recently it has been shown that breaking ties in a fixed, asymmetric fashion, actually improves performance, whereas in all previous analyses, ties were broken randomly. The nature of this improvement using fluid limit models, suggest further improvements, and verify and quantify the improvement through simulations.  Finally, practical applications of our improved hashing models are discussed. From the Series:CSE Colloquia - 1999