Persuasion in Networks: Public Signals and k-Cores
62 Pages Posted: 2 Apr 2019 Last revised: 2 Jun 2021
Date Written: March 9, 2019
We consider a setting where agents in a social network take binary actions, which exhibit local strategic complementarities. The agents are a priori uninformed about an underlying payoff-relevant state. An information designer wants to maximize the expected number of agents who take action 1, and she can commit to a signaling mechanism which upon the realization of the state sends an informative signal to all the agents. We study the structure and design of the optimal public signaling mechanisms.
We establish that given a signal realization, the set of agents who take action 1 correspond to a k-core of the underlying network, for some k. Using this we show that the designer’s payoff is an increasing step function of the posterior mean her signals induce. We provide a convex optimization formulation and an algorithm that obtain the optimal information structure whenever the designer’s payoff exhibits this structure. The latter structural property is prevalent, thereby making our approach useful well beyond network persuasion. The optimal mechanism is based on a double-interval partition of the set of states. In particular, it associates up to two subintervals of the set of possible state realizations with each k-core. When the state is in these intervals the mechanism publicly reveals this, which induces the agents in k-core to take action 1. We also study a class of random networks with known limiting degree distributions, and provide a framework for obtaining asymptotically optimal public signaling mechanisms for these networks. Our approach relies on characterizing the sizes of the cores of such networks (asymptotically) and uses only the degree distribution of the random networks, thereby making it useful even when the network structure is not fully known. Finally, we explore which networks are more amenable to persuasion, and show that more assortative connection structures lead to larger payoffs for the designer. On the other hand, the dependence of the designer’s payoff to the agents’ degrees can be quite counterintuitive. In particular, we focus on networks sampled uniformly at random from the set of all networks consistent with a degree sequence. We show that when the degrees of some nodes increase, this can reduce the designer’s payoff, despite the increase in the extent of the network externalities.
Keywords: Social networks, information design, cores of graphs, public signals
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