The literature on strategic communication originated with the influential cheap talk model, which precedes the Bayesian persuasion model by three decades. This model describes an interaction between two agents: sender and receiver. The sender knows some state of the world which the receiver does not know, and tries to influence the receiver's action by communicating a cheap talk message to the receiver.

We study a variant of the principal-agent problem in which the principal does not directly observe the agent's effort outcome; rather, she gets a signal about the agent's action according to a variable information structure designed by a regulator. We consider both the case of a risk-neutral and of a risk-averse agent, focusing mainly on a setting with a limited liability assumption. We ask the following question - which actions and utility profiles can be implemented by some information structure? Surprisingly, even though the principal-agent problem with unobserved outcomes has appeared in previous work, ours is the first work to study the implementability of utility profiles and expected transfers.

We study a setting with a decision maker making a binary decision by aggregating information from symmetric agents. Each agent provides the decision maker a recommendation depending on her private signal about the hidden state. We assume that agents are truthful - an agent recommends guessing the more likely state based on her information. This assumption is natural if the agents are unaware of how the decision-maker will aggregate their recommendations. While the decision maker has a prior distribution over the hidden state and knows the marginal distribution of each agent's private signal, the correlation between these signals is chosen adversarially. The decision maker's goal is choosing an information aggregation rule that is robustly optimal.

The celebrated Bayesian persuasion model considers strategic communication between an informed agent (the sender) and uninformed decision makers (the receivers). The current rapidly-growing literature mostly assumes a dichotomy: either the sender is powerful enough to communicate separately with each receiver (a.k.a. private persuasion), or she cannot communicate separately at all (a.k.a. public persuasion). We study a model that smoothly interpolates between the two, by considering a natural multi-channel communication structure in which each receiver observes a subset of the sender's communication channels. This captures, e.g., receivers on a network, where information spillover is almost inevitable. We completely characterize when one communication structure is better for the sender than another, in the sense of yielding higher optimal expected utility universally over all prior distributions and utility functions. The characterization is based on a simple pairwise relation among receivers - one receiver information-dominates another if he observes at least the same channels. We prove that a communication structure $M_1$ is (weakly) better than $M_2$ if and only if every information-dominating pair of receivers in $M_1$ is also such in $M_2$. We also provide an additive FPTAS for the optimal sender's signaling scheme when the number of states is constant and the graph of information-dominating pairs is a directed forest. Finally, we prove that finding an optimal signaling scheme under multi-channel persuasion is, generally, computationally harder than under both public and private persuasion.

Bayesian persuasion, as introduced by Kamenica and Gentzkow in 2011, 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. We propose 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; 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, a result of Doval and Skreta (2018) establishes 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 - and we show this result is tight. For the ex post family, we tighten the previous bound of Vølund (2018), 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 of nature; we improve the approximation to single-criteria under a Slater-like regularity condition. The FPTAS holds under standard assumptions, and more relaxed assumptions yield a PTAS. We then establish a bound on the ratio between Sender's optimal utility under convex ex ante constraints and the corresponding ex post constraints. We demonstrate how this result can be applied to find an approximately welfare-maximizing constrained signaling scheme in ad auctions.

We study a Bayesian persuasion setting with binary actions (adopt and reject) for Receiver. We examine the following question - how well can Sender perform, in terms of persuading Receiver to adopt, when ignorant of Receiver's utility? We take a robust (adversarial) approach to study this problem; that is, our goal is to design signaling schemes for Sender that perform well for all possible Receiver's utilities. We measure performance of signaling schemes via the notion of (additive) regret: the difference between Sender's hypothetically optimal utility had she known Receiver's utility function and her actual utility induced by the given scheme. On the negative side, we show that if Sender has no knowledge at all about Receiver's utility, then Sender has no signaling scheme that performs robustly well. On the positive side, we show that if Sender only knows Receiver's ordinal preferences of the states of nature - i.e., Receiver's utility upon adoption is monotonic as a function of the state - then Sender can guarantee a surprisingly low regret even when the number of states tends to infinity. In fact, we exactly pin down the minimum regret value that Sender can guarantee in this case, which turns out to be at most 1/e. We further show that such positive results are not possible under the alternative performance measure of a multiplicative approximation ratio by proving that no constant ratio can be guaranteed even for monotonic Receiver's utility; this may serve to demonstrate the merits of regret as a robust performance measure that is not too pessimistic. Finally, we analyze an intermediate setting in between the no-knowledge and the ordinal-knowledge settings.