Diffusion Approximations for a Class of Sequential Testing Problems

We consider a decision maker who must choose an action in order to maximize a reward function that depends on the action that she selects as well as on an unknown parameter "Theta". The decision maker can delay taking the action in order to experiment and gather additional information on "Theta". We model the decision maker's problem using a Bayesian sequential experimentation framework and use dynamic programming and asymptotic analysis to solve it. In particular, we consider environments in which the average number of experiments that can be conducted per unit of time is large but the informativeness of each individual experiment is low. Under these conditions, we derive a diffusion approximation for the sequential experimentation problem, which provides a number of important insights about the nature of the problem and its solution. First, it reveals that the problems of (i) selecting the optimal sequence of experiments to use and (ii) deciding the optimal time when to stop experimenting decouple and can be solved independently. Second, it shows that an optimal experimentation policy is one that chooses the experiment that maximizes the instantaneous volatility of the belief process. Third, the diffusion approximation provides a more mathematically malleable formulation that we can solve in closed form. Our solution method also shows that the complexity of the problem grows only quadratically with the cardinality of the set of actions from which the decision maker can choose from.We illustrate our methodology and results using a concrete application in the context of assortment selection and new product introduction. Specifically, we study the problem of a seller who wants to select an optimal assortment of products to launch into the marketplace and is uncertain about consumers' preferences. Motivated by emerging practices in e-commerce, we assume that the seller is able to use a crowd voting system to learn these preferences before a final assortment decision is made. In this context, we undertake an extensive numerical analysis to assess the value of learning and highlight the limitations of some commonly used strategies such as a full display assortment policy or a fixed duration crowdvoting campaign