A characterization and an impossibility of finite length anonymity for infinite generations

In the context of ranking infinite utility streams, the impartiality axiom of finite length anonymity requires the equal ranking of any two utility streams that are equal up to a finite length permutation (Fleurbaey and Michel, 2003). We first characterize any finite length permutation as a composition of a fixed step permutation and an "almost" fixed step permutation. We then show that if a binary relation satisfies finite length anonymity, then it violates all the distributional axioms that are based on a segment-wise comparison. Examples of those axioms include the weak Pareto principle and the weak Pigou-Dalton principle.