Diffusion Approximations for a Class of Sequential Experimentation Problems

Forthcoming in Management Science

36 Pages Posted: 12 Nov 2019 Last revised: 11 Oct 2021

See all articles by Victor F. Araman

Victor F. Araman

American University of Beirut - The Olayan School of Business

Rene Caldentey

University of Chicago - Booth School of Business

Date Written: November 2, 2019

Abstract

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 diffusion-asymptotic analysis to solve it. For that, we scale our problem in a way that both the average number of experiments that is conducted per unit of time is large and the informativeness of each individual experiment is low. Under such regime, 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 and suggests efficient heuristics for the non-asympototic regime. 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.
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 {\em crowdvoting} 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 demonstrate the effectiveness and robustness of the heuristics derived from the diffusion approximation.

Keywords: assortment selection, dynamic programming, Bayesian demand learning, experiment design, optimal stopping, sequential testing, crowdvoting

JEL Classification: C11, C12, C61, C90

Suggested Citation

Araman, Victor F. and Caldentey, Rene, Diffusion Approximations for a Class of Sequential Experimentation Problems (November 2, 2019). Forthcoming in Management Science, Available at SSRN: https://ssrn.com/abstract=3479676 or http://dx.doi.org/10.2139/ssrn.3479676

Victor F. Araman

American University of Beirut - The Olayan School of Business ( email )

Beirut, 0236
Lebanon

Rene Caldentey (Contact Author)

University of Chicago - Booth School of Business ( email )

5807 S. Woodlawn Avenue
Chicago, IL 60637
United States

HOME PAGE: http://www.chicagobooth.edu/faculty/directory/c/rene-caldentey

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