Nontransitive-nontotal consumer theory

We show the parallel nature of two approaches to nontransitive or nontotal consumers: through “weak” preferences and through “strict” preferences. This yields specific results (e.g., a new equilibrium existence theorem with weak preferences), general results (a metatheorem translating between the two approaches), and general concepts (a new notion of “rationality”). We give revealed preference axioms to characterize preferences, and prove equivalences among several axiom systems, showing that the apparent weakness of some axiom systems is illusory. We resolve a Weak Axiom conjecture, and we introduce a new axiom. We prove that our nonclassical consumers generalize classical equilibrium theory.