Suzumura Consistency

Binary relations are at the heart of much of economic theory, both in the context of individual choice and in multi-agent decision problems. A fundamental coherence requirement imposed on a relation is the well-known transitivity axiom. If a relation is interpreted as a goodness relation, transitivity postulates that whenever one alternative is at least as good as a second and the second alternative is, in turn, at least as good as a third, then the first alternative is at least as good as the third. However, from an empirical as well as a conceptual perspective, transitivity is frequently considered too demanding and weaker notions of coherence have been proposed in the literature. Two alternatives that have received a considerable amount of attention are quasi-transitivity and acyclicity. Quasi-transitivity demands that the asymmetric factor of a relation (the betterness relation) is transitive, whereas acyclicity rules out the presence of betterness cycles. Quasi-transitivity is implied by transitivity and implies acyclicity. The reverse implications are not valid.