Bayesian Persuasion with a Risk-Conscious Receiver

We study Bayesian persuasion with a receiver who evaluates actions by Conditional Value-at-Risk. CVaR captures settings in which rare adverse outcomes matter for acceptance, including automated alerts, financial advice, and safety-critical AI-assisted decisions. The receiver's value is nonlinear in posterior beliefs, so direct recommendation by action fails: merging two signals that recommend the same action can change the relevant tail event and violate incentive compatibility. Our main result shows that this failure does not make the explicit finite-state problem hard. CVaR has a finite active-threshold representation, and signals can be refined by both the recommended action and the active CVaR facet. This refined revelation principle turns the sender's problem into an exact polynomial-size linear program. We then develop a posterior discretization scheme for finite-precision implementation and large-scale extensions, including a margin condition under which approximate incentive compatibility becomes strict incentive compatibility.