  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.
