Persuasion with Limited Communication

Abstract: 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. In the setting of maximizing welfare under bilateral trade, we show that $O(\log(n) \log \frac{1}{\epsilon})$ signals suffice for a $1-\epsilon$ approximation to the optimal welfare, and this bound is tight. As our main result, we exhibit an efficient algorithm for computing a $\frac{M-1}{M} \cdot (1-1/e)$-approximation to the welfare-maximizing scheme with at most M signals. For the revenue objective, we show that $\Omega(n)$ 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. 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. 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.