Exploration and Stopping

Abstract

We study a decision-maker who explores --- dynamically choosing what to learn --- before stopping to act. We first reduce this dynamic control problem to a static one: any exploration-and-stopping strategy is equivalent to a choice of the joint distribution of the stopped state and the stopping time, subject to one information-budget constraint at each date, and we characterize exactly which distributions are attainable. The reduced problem is a convex program with a linear objective; its dual prices information over time, and the optimal policy concavifies the stopping payoff net of these shadow prices. The curvature of the decision-maker's time preference then governs the shape of optimal exploration: convex time preference induces Poisson exploration, concave time preference confines stopping to a window whose length is controlled by the dispersion of the marginal cost of delay --- forcing an initial phase of pure exploration when the window is short --- and the linear case lies at the boundary between them. We apply the framework to real options, to the speed--accuracy tradeoff in information acquisition, and to a continuous-time exploration contest.