On the multiplicative inequality

The multiplicative inequality (MI) introduced by Sattath and Tversky (1976) is a rare example of a simple and intuitively appealing condition relating choice probabilities across choice sets of different sizes. It is also a testable implication of two models of stochastic discrete choice: the Elimination by Aspects model of Tversky (1972b) and the independent random utility model. We prove several results on the multiplicative inequality and its relationship to the regularity condition. One major result illustrates how little the MI constrains binary choice probabilities: it implies that every system of binary choice probabilities on a universe of choice objects can be extended to a complete system of choice probabilities satisfying the MI. In this sense, the MI is complementary to axioms for binary choice probabilities, of which many have been proposed. We also discuss choice environments where the multiplicative inequality is implausible.