We make use of a series of choice questions based on the common consequence setup, much like the classic Allais problem, to show that we can infer concavity and convexity properties of how we distort probabilities in decision.

Wu, G., & Gonzalez, R. (1998). Common consequence conditions in decision making under risk. Journal of Risk and Uncertainty, 16, 115-139. DOI: 10.1023/A:1007714509322  PDF

Abstract

We generalize the Allais common consequence effect by describing three common consequence effect conditions and characterizing their implications for the probability weighting function in rank-dependent expected utility. The three conditions—horizontal, vertical, and diagonal shifts within the probability triangle—are necessary and sufficient for different curvature properties of the probability weighting function. The first two conditions, shifts in probability mass from the lowest to middle outcomes and middle to highest outcomes respectively, are alternative conditions for concavity and convexity of the weighting function. The third condition, decreasing Pratt-Arrow absolute concavity, is consistent with recently proposed weighting functions. The three conditions collectively characterize where indifference curves fan out and where they fan in. The common consequence conditions indicate that for nonlinear weighting functions in the context of rank-dependent expected utility, there must exist a region where indifference curves fan out in one direction and fan in the other direction.