Persuasion with Limited Communication

We examine information structure design, also called "persuasion" or "signaling," in the presence of a constraint on the amount of communication. We focus on the fundamental setting of bilateral trade, which in its simplest form involves a seller with a single item to price, a buyer whose value for the item is drawn from a common prior distribution over n different possible values, and a take-it-or-leave-it-offer protocol. A mediator with access to the buyer's type may partially reveal such information to the seller in order to further some objective such as the social welfare or the seller's revenue. We study how a limit on the number of bits of communication affects this setting in two respects: (1) How much does this constraint reduce the optimal welfare or revenue? (2) What effect does constraining communication have on the computational complexity of the mediator's optimization problem? In the setting of maximizing welfare under bilateral trade, we exhibit positive answers for both questions (1) and (2). Whereas the optimal unconstrained scheme may involve n signals (and thus log(n) bits of communication), we show that O(log(n) log 1/e) signals suffice for a 1--e approximation to the optimal welfare, and this bound is tight. This largely justifies the design of algorithms for signaling subject to drastic limits on communication. As our main result, we exhibit an efficient algorithm for computing a M-1/M ⋅ (1--1/e)-approximation to the welfare-maximizing scheme with at most M signals. This result hinges on an intricate submodularity argument which relies on the optimality of a greedy algorithm for solving a certain linear program. For the revenue objective, the surprising logarithmic bound on the number of signals does not carry over: we show that Ω(n) signals are needed for a constant factor approximation to the revenue of a fully informed seller. From a computational perspective, however, the problem gets easier: we show that a simple dynamic program computes the signaling scheme with M signals maximizing the seller's revenue. Observing that the signaling problem in bilateral trade is a special case of the fundamental Bayesian Persuasion model of Kamenica and Gentzkow, we also examine the question of communication-constrained signaling more generally. Specifically, in this model there is a sender (the mediator), a receiver (the seller) looking to take an action (setting the price), and a state of nature (the buyer's type) drawn from a common prior. The state of nature encodes both the receiver's utility and the sender's objective as a function of the receiver's action. Our results for bilateral trade with the revenue objective imply that limiting communication to M signals can scale the sender's utility by a factor of O(M/n) in general, where $n$ denotes the number of states of nature. We also show that our positive algorithmic results for bilateral trade do not extend to communication-constrained signaling in the Bayesian Persuasion model. Specifically, we show that it is NP-hard to approximate the optimal sender's utility to within any constant factor in the presence of communication constraints.