Bayesian Persuasion under Ex Ante and Ex Post Constraints

TL;DR

This paper studies how to design optimal information sharing policies under practical constraints like privacy and discrimination, providing tight bounds on signaling complexity and efficient algorithms for constrained Bayesian persuasion.

Contribution

It introduces a mathematical framework for ex ante and ex post constraints in Bayesian persuasion, tightens bounds on the number of signals needed, and develops an efficient approximation scheme for optimal signaling.

Findings

Tight bounds on the number of signals for ex ante constraints.

Bounded the signals needed under ex post constraints, matching classical results.

Developed an additive FPTAS for optimal constrained signaling with constant states.

Abstract

Bayesian persuasion is the study of information sharing policies among strategic agents. A prime example is signaling in online ad auctions: what information should a platform signal to an advertiser regarding a user when selling the opportunity to advertise to her? Practical considerations such as preventing discrimination, protecting privacy or acknowledging limited attention of the information receiver impose constraints on information sharing. In this work, we propose and analyze a simple way to mathematically model such constraints as restrictions on Receiver's admissible posterior beliefs. We consider two families of constraints - ex ante and ex post, where the latter limits each instance of Sender-Receiver communication, while the former more general family can also pose restrictions in expectation. For the ex ante family, Doval and Skreta establish the existence of an optimal signaling scheme with a small number of signals - at most the number of constraints plus the number of states of nature; we show this result is tight and provide an alternative proof for it. For the ex post family, we tighten a bound of V{\o}lund, showing that the required number of signals is at most the number of states of nature, as in the original Kamenica-Gentzkow setting. As our main algorithmic result, we provide an additive bi-criteria FPTAS for an optimal constrained signaling scheme assuming a constant number of states; we improve the approximation to single-criteria under a Slater-like regularity condition. The FPTAS holds under standard assumptions; relaxed assumptions yield a PTAS. Finally, we bound the ratio between Sender's optimal utility under convex ex ante constraints and the corresponding ex post constraints. This bound applies to finding an approximately welfare-maximizing constrained signaling scheme in ad auctions.