This paper characterizes a class of twice continuously differentiable objective-probability preference representations exhibiting endogenous reference dependence under risk. Weak rank-dependent utility (WRDU) preserves objective probabilities, partitions outcomes at an endogenous reference point, and evaluates lotteries through a gainloss representation in which the reference point maximizes a penalized functional. The first-order condition yields a virtual loss-aversion index equal to the ratio of marginal utilities across the loss and gain domains, recovering both the utility-based index of Köbberling and Wakker (2005) and the slope ratio of Tversky and Kahneman (1992) as special cases. The main theorem shows that, within a class satisfying affine admissibility, loss-factorization, dispersion monotonicity, and attenuation, the derivative-ratio form is uniquely admissible. In this class, WRDU generates the modal Allais pattern on an admissible region and blocks the Rabin calibration implication through range-dependent attenuation. The result is conditional and does not claim uniqueness over all behavioral models of risky choice.

翻译：暂无翻译