Parallel Search for Information in Continuous Time—Optimal Stopping and Geometry of the PDE

We consider the problem of a decision-maker searching for information on multiple alternatives when information is learned on all alternatives simultaneously. The decision-maker has a running cost of searching for information, and has to decide when to stop searching for information and choose one alternative. The expected payoff of each alternative evolves as a diffusion process when information is being learned. After establishing the well-posedness of the equation, we show that the optimal boundary where search is stopped (free boundary) is star-shaped, and present an asymptotic characterization of the value function and the free boundary. We prove that the distance between the free boundary and each point on the diagonal is logarithmic in the number of alternatives.