The Rational Core of Preference Relations

We consider revealed preference relations over risky (or uncertain) prospects, and allow them to be nontransitive and/or fail the classical Independence Axiom. We identify the rational part of any such preference relation as its largest transitive subrelation that satisfies the Independence Axiom and that exhibits some coherence with the original relation. It is shown that this subrelation, which we call the rational core of the given revealed preference, exists in general, and under fairly mild conditions, it is continuous. We obtain various representation theorems for the rational core, and decompose it into other core concepts for preferences. These theoretical results are applied to compute the rational cores of a number of well-known preference models (such as Fishburn's SSB model, justifiable preferences, and variational and multiplier modes of rationalizable preferences). As for applications, we use the rational core operator to develop a theory of risk aversion for nontransitive nonexpected utility mod als (which may not even be complete). Finally, we show that, under a basic monotonicity hypothesis, the Preference Reversal Phenomenon cannot arise from the rational core of one's preferences.