Optimal Hierarchical Signaling for Quadratic Cost Measures and General Distributions: A Copositive Program Characterization
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In this paper, we address the problem of optimal hierarchical signaling between a sender and a receiver for a general class of square integrable multivariate distributions. The receiver seeks to learn a certain information of interest that is known to the sender while the sender seeks to induce the receiver to perceive that information as a certain private information. For the setting where the players have quadratic cost measures, we analyze the Stackelberg equilibrium, where the sender leads the game by committing his/her strategies beforehand. We show that when the underlying state space is "finite", the optimal signaling strategies can be computed through an equivalent linear optimization problem over the cone of completely positive matrices. The equivalence established enables us to use the existing computational tools to solve this class of cone programs approximately with any error rate. For continuous distributions, we also analyze the error of approximation, if the optimal signaling strategy is computed for a discretized version obtained through a quantization scheme, and we provide an upper bound in terms of the quantization error.
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