Multi-Channel Bayesian Persuasion

TL;DR

This paper introduces a multi-channel Bayesian persuasion model that interpolates between public and private persuasion, providing a complete characterization of optimal communication structures and analyzing their computational complexity.

Contribution

It characterizes when one communication structure is better than another in multi-channel persuasion and offers an approximation scheme for optimal signaling when the structure is a directed forest.

Findings

A simple pairwise relation determines the superiority of communication structures.

An additive FPTAS is provided for constant states with a directed forest structure.

Optimal signaling under multi-channel persuasion is computationally harder than public or private persuasion.

Abstract

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.