Finite-Dimensional Approach to Optimal Dynamic Information Acquisition

Zhong (2022) characterizes, in continuous time, how a decision maker should acquire information before acting when the state is binary. For more than two states, his characterization assumes that an associated partial differential equation has a solution with suitable regularity and shape. I study the same problem with time in discrete periods and beliefs restricted to a finite grid. The decision maker’s problem is then a finite linear or concave program, and each property of its solution follows from an elementary fact about such programs: duality, the number of positive entries in a basic solution, the greedy solution of a knapsack roblem, and the mediant inequality. For two states the discrete model reproduces every qualitative feature of Zhong’s characterization: the value function is quasi-convex, learning is confirmatory, effort rises with the value, targets move toward the current belief as that value rises, and the decision maker acts immediately after a breakthrough. For any number of states, an optimal strategy exists; it uses at most one more posterior belief than there are states, effort rises with the stakes, and the value is no lower at beliefs between the current belief and the posteriors it moves to. The last is the discrete form of Zhong’s result on the direction of learning, which he obtains for more than two states assuming the existence of a desired solution for the associated partial differential equation.