Abstract
A discrete analog for a calculus argument about mappings between spaces is developed. This discrete ‘calculus argument’ is used to characterize the ‘kinds of axioms’ that lead to conclusions similar to that of Arrow's Theorem. In this manner, new extensions of Arrow's Theorem are found. In addition, this approach is applied to certain economic allocation and welfare procedures, paradoxes from statistics and probability, the Hurwicz-Schmeidler result about Pareto optimal Nash equilibria, the Gibbard-Satterthwaite Theorem, Nakamura's Theorem about simple games, etc. A new, sharp class of ‘possibility theorems’ are derived that hold not only for transitive preferences, but also for utility functions, probability measures, etc.
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