Persuasion in Networks: Public Signals and k-Cores
44 Pages Posted: 2 Apr 2019
Date Written: March 9, 2019
We consider a setting where agents in a social network take binary actions, which exhibit local strategic complementarities. In particular, the payoff of each agent depends on the number of her neighbors who take action 1, as well as an underlying state of the world. The agents are a priori uninformed about the 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. In this paper, we study the structure and design of the optimal public signaling mechanisms.
We establish that to find an optimal mechanism it suffices to restrict attention to public signaling mechanisms where (i) the possible signal realizations correspond to the k-cores of the underlying network, (ii) once the signal realization is k, the agents in the k-core take action 1. Using this observation, we obtain a convex optimization relaxation of the problem of the information designer, and establish that using an optimal solution of this problem together with an algorithm we provide, the information designer can construct an optimal public signaling mechanism. Our optimization problem and algorithm are tractable, and they enable us to characterize optimal mechanisms in large networks, as we illustrate by focusing on subnetworks of the Facebook graph. The optimal mechanism associates up to two subintervals of the set of the possible realizations of the state of the world with each k-core, and sends signal k (which induces the k-core to take action 1) when the state of the world is in one of these intervals. We also study a class of random networks with known 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 establish that our findings extend to settings where the structure of the strategic complementarity can be fairly rich, and in particular our approach can handle weighted networks, and payoffs with a threshold structure (where agents derive additional utility from the actions of their neighbors, only if the fraction of their neighbors who take action 1 exceeds a certain threshold).
Keywords: Social networks, information design, cores of graphs, public signals
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