Knowing What Others Know: Coordination Motives in Information Acquisition Additional Notes

This section shows that the equilibrium of the action game in section 1 of the main text is unique. It does that by adapting an argument first made Angeletos and Pavan (2007, propositions 1 and 3) to our environment. The idea of the proof is that there is a social planner problem such that every equilibrium of our model is also a solution to this planning problem. The planning problem is strictly convex, meaning that it has a unique minimum. Since the planning problem has a unique solution and every equilibrium is a solution to the planning problem, the equilibrium of the model must be unique. We begin by setting up some notation for the proof. We let p̂ (·) denote the candidate equilibrium function characterized by equation (4) in the main text, and will make use of the fact that s = b′ω. We let F (ω) denote the prior distribution of ω, with density f (ω). We let μ denote the distribution of the agents’ information choices, and φ (Xz|ω) the distribution of observed signals, conditional on the state ω. Together, μ and φ determine the distribution F (I|ω)of information sets I = (χ,Xz), conditional on the state ω. The agents’ posterior beliefs conditional on I are defined by the pdf φ̂ (ω|I) = φ (Xz|ω) f (ω) ∫ ω̂ φ (Xz|ω̂) dF (ω̂) .