 A1  and  A2  together entail that \(V(\cdot)\) assigns a real-valued index to prospects such that \(P \succeq Q\) if and only if \(V(P) \geq V(Q)\). The independence axiom,  A3,  encodes a separability property for choice, one that ensures that expected utilities are linear in probabilities. Motivations for dropping the independence axiom stem from difficulties in applying expected utility theory to describe choice behavior, including an early observation that humans evaluate possible losses and possible gains differently. Although expected utility theory can represent a person who either gambles or purchases insurance, Friedman and Savage remarked in their early critique of von Neumann and Morgenstern’s axiomatization, it cannot simultaneously do both (M. Friedman &amp; Savage 1948).