Is There a Plausible Theory for Decision under Risk? A Dual Calibration Critique

Theories of decision under risk that assume decreasing marginal utility of money have been critiqued with concavity calibration arguments. Since that critique uses varying payoffs and fixed probabilities, it cannot have implications for calibration of nonlinear probability transformation, which is another way to model risk aversion. The concavity calibration critique also has no implication for theories with variable reference points. This paper introduces a new type of (varying-probabilities, fixed-payoffs) calibration that applies to nonlinear transformation of probabilities. It also applies to theories with constant or variable reference points. The two types of calibrations yield dual paradoxes: a pattern of risk aversion that conforms to the (resp. dual) independence axiom implies implausible risk aversion for theories with functionals that are linear in payoffs (resp. probabilities). Functionals that are nonlinear in both payoffs and probabilities are subject to both types of calibration critique. The dual calibrations make clear why plausibility problems with theories of decision under risk may be fundamental. They are fundamental if their empirical relevance can be demonstrated. This paper reports seven experiments that provide data on the empirical relevance of the dual calibration critique of decision theory. (JEL C90, D81) Can prominent theories of decision under risk rationalize both small-stakes risk aversion and large-stakes risk aversion? How do loss aversion and reference payoffs enter in the answer to this question? Can some existing theories, but not others, rationalize same-stakes (i.e. smallstakes or large-stakes) risk aversion? We offer a theoretical duality approach that addresses these questions. We present two (dual) paradoxes in which patterns of risk aversion that conform to one theory of decision under risk imply implausible risk aversion in the dual to that theory. One wonders then whether data conform to either or both of the dual “calibration patterns” for which prominent theories imply implausible risk aversion. We report seven experiments that address that question. Rabin (2000) sparked the literature on concavity calibration by identifying a varyingpayoffs pattern of small-stakes risk aversion that, through calibration arguments, can be shown to imply implausible large-stakes risk aversion for the expected utility of terminal wealth model. Several subsequent authors extended Rabin’s varying-payoffs, concavity calibration analysis to apply to a class of theories that assume decreasing marginal utility of money. How fundamental is this challenge to the plausibility of theories of decision under risk? We address this question about fundamentality both theoretically and empirically. Our theoretical discussion is based on duality. We explain in this paper that the varyingpayoffs patterns of small-stakes risk aversion used in calibrations by Rabin (2000) and all