Consistency and Its Converse: An Introduction

This essay is an introduction to the recent literature on the "consistency principle" and its "converse". An allocation rule is consistent if for any problem in its domain of definition and any alternative that it selects for it, then for the associated "reduced problem" obtained by imagining the departure of any subgroup of the agents with their "components of the alternative" and reassessing the options open to that subgroup. Converse consistency pertains to the opposite operation. It allows us to deduce that the rule chooses an alternative for some problem from the knowledge that for all two-person subgroups, it chooses its restriction to the subgroup for the associated reduced problem this subgroup faces. We present two lemmas, the Elevator Lemma and the Bracing Lemma, involving these properties. These lemmas have been found useful in the analysis of a great variety of models. We also describe some of their applications. Finally, we illustrate the versatility of consistency and of its converse by means of a sample of characterizations based on these principles