Decomposition of Relations in Choice Problems: Completely Decomposable Relations and Path Independence

Abstract

Decomposition of a relation means its representation by a set of relations such that successive choice by these relations is the same as the choice by the initial relation. A relation is completely decomposale if any set of relations, whose union is the relation, forms a decomposition. There is close relationship between the decomposability of a relation and the Plott path independence condition. Only partial ordering relations are completely decomposable. Within the framework of the concept of the decomposability, only commutative decompositions are obtained, and commutativity may lead to an exponential growth of the complexity of decompositions. Hence a more general concept of complete decomposability is introduced such that any decomposition can be obtained. An explicit description of completely decomposable relations is given.