DICTATORIAL CONSENSUS FUNCTIONS ON N-TREES

The original 'independence of irrelevant alternatives' axiom of K. Arrow has a natural analog when translated from the classical weak order (preference relation) case to n-trees. Using this translated independence axiom for n-trees, it is surprising that Arrow's impossibility Theorem does not follow. Specifically, there exist consensus functions for n-trees that satisfy the independence and Pareto conditions but are not dictatorships. Conversely, a dictatorship must clearly satisfy the Pareto condition but not necessarily independence. In this note it is shown that a consensus function for n-trees is a dictatorship and satisfies independence if and only if it is a projection function.