On the Tractability of Public Persuasion with No Externalities

Abstract

Persuasion studies how a principal can influence agents' decisions via strategic information revelation -often described as a signaling scheme -in order to yield the most desirable equilibrium outcome. A basic question that has attracted much recent attention is how to compute the optimal public signaling scheme, a.k.a., public persuasion, which is motivated by various applications including auction design, routing, voting, marketing, queuing, etc. Unfortunately, most algorithmic studies in this space exhibit quite negative results and are rifle with computational intractability. Given such background, this paper seeks to understand when public persuasion is tractable and how tractable it can be. We focus on a fundamental multi-agent persuasion model introduced by Arieli and Babichenko [3]: many agents, no inter-agent externalities and binary agent actions, and identify well-motivated circumstances under which efficient algorithms are possible. En route, we also develop new algorithmic techniques and demonstrate that they can be applicable to other public persuasion problems or even beyond.We start by proving that optimal public persuasion in our model is fixed parameter tractable. Our main result here builds on an interesting connection to a basic question in combinatorial geometry: how many cells can n hyperplanes divide R d into? We use this connection to show a new characterization of public persuasion, which then enables efficient algorithm design. Second, we relax agent incentives and show that optimal public persuasion admits a bi-criteria PTAS for the widely studied class of monotone submodular objectives, and this approximation is tight. To prove this result, we establish an intriguing "noise stability" property of submodular functions which strictly generalizes the key result of Cheraghchi et al. [15], originally motivated by applications of learning submodular functions and differential privacy. Finally, motivated by automated application of persuasion, we consider relaxing the equilibrium concept of the model to coarse correlated equilibrium. Here, using a sophisticated primal-dual analysis, we prove that optimal public persuasion admits an efficient algorithm if and only if the combinatorial problem of maximizing the sender's objective minus any linear function can be solved efficiently, thus establishing their polynomial-time equivalence.1 Coarse correlated equilibrium for Bayesian games has been studied in other settings such as auctions [11,13], congestion games [30] and general Bayesian games [22], and was coined Bayesian coarse correlated equilibrium by Hartline et al. [22].

Persuading Voters: It's Easy to Whisper, It's Hard to Speak Loud

“…First, we provide an efficient implementation of the optimal private signaling scheme under a k-voting rule. Then, by generalizing a result by Dughmi and Xu (2017)-later revisited by Xu (2019)-we describe a necessary and sufficient condition for the efficient computation of private signaling schemes for a general class of sender's objective functions. This condition is employed to show that private Bayesian persuasion is fixed-parameter tractable in the number of receivers' actions when sender's utility is anonymous, and to show that an optimal private signaling scheme under plurality voting can be found in polynomial time.…”

“…Moreover, the authors show that it is NP-hard to approximate the sender's value provided by the optimal public scheme, within any constant factor. Finally, Xu (2019) focuses on the complexity of public signaling when there are no inter-agent externalities and the action spaces are binary. Finding an optimal public signal is shown to be fixed-parameter tractable under some non-degeneracy assumptions.…”

Efficient Algorithms for Learning Revenue-Maximizing Two-Part Tariffs

“…The high-level idea is to sequentially add each hyperplane to the list of regions maintained so far (starting with the entire Euclidean space R 2L ), iterating over the current regions and checking whether the added hyperplane intersects each region-updating the list of regions if so. See [Xu, 2020] for a more detailed description of enumerating the regions formed by a collection of hyperplanes (in a totally different context). Our algorithm solves the aforementioned linear program for every such region and picks the solution that yields highest empirical revenue.…”

Signaling in Posted Price Auctions

“…The main one is that, in our model, the seller (sender) is not only required to choose how to send signals, but they also have to take some actions in the form of price proposals. This requires significant extensions to standard approaches based on decomposition lemmas (Cheng et al 2015;Xu 2020;. The framework forms the basis on which we design our PTASs.…”