Proper scoring rules are scoring methods that incentivize honest reporting of subjective probabilities, where an agent strictly maximizes his expected score by reporting his true belief. The implicit assumption behind proper scoring rules is that agents are risk neutral. Such an assumption is often unrealistic when agents are human beings. Modern theories of choice under uncertainty based on rank-dependent utilities assert that human beings weight nonlinear utilities using decision weights, which are differences between weighting functions applied to cumulative probabilities. In this paper, we investigate the reporting behavior of an agent with a rank-dependent utility when he is rewarded using a proper scoring rule tailored to his utility function. We show that such an agent misreports his true belief by reporting a vector of decision weights. my findings thus highlight the risk of utilizing proper scoring rules without prior knowledge about all the components that drive an agent's attitude towards uncertainty. On the positive side, we discuss how tailored proper scoring rules can effectively elicit weighting functions. Moreover, we show how to obtain an agent's true belief from his misreported belief once the weighting functions are known.
