On Information Design with Spillovers

An information designer has access to a set of experiments and decides which of these to assign to each of the agents in a directed network. The network encodes informational spillovers: an agent has access to the experiments assigned to her, as well as to those assigned to any other agent who has a directed path to her. We establish that the designer's problem in any network can be reduced to an equivalent problem in a directed acyclic network. We show that when in the latter network each agent follows at most one other agent (i.e., each node has in-degree at most one), the optimal information structure can be obtained in a tractable way. The problem becomes intractable if some agents follow multiple other agents. Thus, qualitatively, following multiple information sources is what makes information design problems intractable in the presence of spillovers. We also study a voting game with binary actions in the presence of spillovers. We show that when the followers are more pessimistic (i.e., have higher posterior mean requirements to take action $1$), the network effects do not play a role, and a certain monotone information structure is optimal. When the followers are more optimistic a monotone information structure is still optimal if each agent follows at most one other agent, but not in general. That said, in the latter case an optimal monotone information structure can be obtained in a tractable way by using an algorithm we provide, provided that the underlying network has bounded treewidth.