Bayesian Persuasion Meets Mechanism Design: Going Beyond Intractability with Type Reporting
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Bayesian persuasion studies how an informed sender should partially disclose information so as to influence the behavior of self-interested receivers. In the last years, a growing attention has been devoted to relaxing the assumption that the sender perfectly knows receiver's payoffs. The first crucial step towards such an achievement is to study settings where each receiver's payoffs depend on their unknown type, which is randomly determined by a known finite-supported probability distribution. This begets considerable computational challenges, as computing a sender-optimal signaling scheme is inapproximable up to within any constant factor. In this work, we circumvent this issue by leveraging ideas from mechanism design. In particular, we introduce a type reporting step in which the receiver is asked to report their type to the sender, after the latter has committed to a menu defining a signaling scheme for each possible receiver's type. We prove that, with a single receiver, the addition of this type reporting stage makes the sender's computational problem tractable. Then, we extend our framework to settings with multiple receivers, focusing on the case of no inter-agent externalities and binary actions. We show that it is possible to find a sender-optimal solution in polynomial-time by means of the ellipsoid method, given access to a suitable polynomial-time separation oracle. This can be implemented for supermodular and anonymous sender's utility functions. As for the case of submodular sender's utility functions, we first approximately cast the sender's problem into a linearly-constrained mathematical program whose objective function is the multi-linear extension of the sender's utility. Then, we show how to find in polynomial-time an approximate solution to the program by means of a continuous greedy algorithm. This provides a (1 -1/e)-approximation to the problem.
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