We consider a persuasion problem between a sender and a receiver whose utility may be nonlinear in her belief; we call such receivers risk-conscious. Such utility models arise when the receiver exhibits systematic biases away from expected-utility-maximization, such as uncertainty aversion (e.g., from sensitivity to the variance of the waiting time for a service). Due to this nonlinearity, the standard approach to finding the optimal persuasion mechanism using revelation principle fails. To overcome this difficulty, we use the underlying geometry of the problem to develop a convex optimization framework to find the optimal persuasion mechanism. We define the notion of full persuasion and use our framework to characterize conditions under which full persuasion can be achieved. We use our approach to study binary persuasion, where the receiver has two actions and the sender strictly prefers one of them at every state. Under a convexity assumption, we show that the binary persuasion problem reduces to a linear program, and establish a canonical set of signals where each signal either reveals the state or induces in the receiver uncertainty between two states. Finally, we discuss the broader applicability of our methods to more general contexts, and illustrate our methodology by studying information sharing of waiting times in service systems.

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