Consistency and its converse: an introduction

This essay is a didactic introduction to the literature on the “consistency principle” and its “converse”. An allocation rule is consistent if for each problem in its domain of definition and each alternative that it chooses for it, then for the “reduced problem” obtained by imagining the departure of an arbitrary subgroup of the agents with their “components of the alternative” and reassessing the options open to the remaining agents, it chooses the restriction of the alternative to that subgroup. Converse consistency pertains to the opposite operation. It allows us to deduce that a rule chooses an alternative for a problem from the knowledge that for each two-agent subgroup, it chooses its restriction to the subgroup for the associated reduced problem this subgroup faces. We present two lemmas that have played a critical role in helping understand the implications of these properties in a great variety of models, the Elevator Lemma and the Bracing Lemma. We describe several applications. Finally, we illustrate the versatility of consistency and of its converse by means of a sample of characterizations based on them.