Private Bayesian Persuasion with Monotone Submodular Objectives

We consider a multi-agent Bayesian persuasion problem wher e a sender aims at persuading multiple receivers to maximize a global objective that depends on all the receivers’ actions. We focus on one of the most basic settings in this space where each receiver takes a binary action, conveniently denoted as action 1 and action0. The payoff of the sender is thus a set function, depending on the set of receivers taking action1. Each receiver’s utility depends on his action and a random s tate of nature whose realization is a-priori unknown to receivers. The sender has an informatio n l advantage, namely access to the realized state of nature, and can commit to a policy, a.k.a., a signaling scheme , to send aprivatesignal regarding the realized state to each receiver. Assuming the sender’s utility function is monotone submodular , we examine the sender’s optimization problem under different input models. When the state of nature is binary, we show that a (1 − 1 e )approximate signaling scheme can be explicitly constructe d. This approximation ratio is tight by [5]. Moreover, the constructed signaling scheme has the followi ng d stinctive properties: (i) it signals independently to each receiver, simply to maximize the probab ility of persuading them to take action 1; (ii) it is oblivious in the sense that it does not depend on the sender’s utility function as long as it is monotone submodular! When there are many states of nature, w e present an algorithm that computes a (1− 1 e )-approximate signaling scheme, modulo an additional addit ive loss ofǫ, and runs in time polynomial in the input size and ǫ . Our algorithm here relies on a structural characterizatio n of (approximately) optimal signaling schemes.