William L. Cooper’s research while affiliated with University of Minnesota, Duluth and other places

We consider a pricing problem for a single product that experiences network effects. Demand is described by a consumer choice model in which each individual chooses between purchasing the product and not purchasing the product. We assume that there are multiple segments in the population of potential buyers, and that individuals' intrinsic values for the product and sensitivities to the network effect (ie, the extent to which their values are affected by how many others buy the product) vary across segments. The demand model may be viewed as a version of the mixed multinomial logit model, modified to incorporate network effects. We formulate and analyze an optimization problem that aims to find the seller's revenue‐maximizing price. In settings with an arbitrary number of demand segments, we present a simple, effective heuristic solution approach. In settings with two segments, we obtain a solution method that outputs provably near‐optimal prices. We close with an extensive numerical study.

Should a seller make information about its products readily accessible to customers, so that customers do not have to incur any substantive cost — in terms of time and effort — to learn about those products? To help answer this question, we consider a monopolist selling two substitute products to a population of customers, who have differing tastes about the products. Each customer a priori has uncertainty about whether or not he will like each of the products. The seller may choose to make product information easily accessible, thereby allowing customers to resolve their uncertainties for free. Otherwise, customers may conduct research to resolve their uncertainties by incurring a search cost before making purchase decisions. We consider three “information structures” differing in whether the seller makes information about the products freely accessible or not. Our primary objective is to determine which structure gives the seller the highest revenue, while accounting for the seller's pricing decisions as well as the induced customer responses to both the information structure and prices. We find that if each customer's uncertainties are small in magnitude but highly positively correlated, then withholding both products’ information is the best for the seller. If the uncertainties are small in magnitude and negatively correlated, then providing one product's information but not the other's is the best. If the uncertainties are large in magnitude and positively correlated, then providing both products’ information is the best. We also show that when the correlation is negative, withholding both products’ information cannot be optimal. In addition, we also analyze various extensions of the model. These include a variant in which customers’ research is imperfect and may yield incorrect information to the customers, and a variant in which each customer's uncertainty about a product can be decomposed into multiple uncertainties associated with individual attributes of the product. This article is protected by copyright. All rights reserved.

We consider optimal pricing problems for a product that experiences network effects. Given a price, the sales quantity of the product arises as an equilibrium, which may not be unique. In contrast to previous studies that take a best-case view when there are multiple equilibrium sales quantities, we maximize the seller's revenue assuming that the worst-case equilibrium quantity will arise in response to a chosen price. We compare the best- and worst-case solutions, and provide asymptotic analysis of revenues.

We consider a seller's problem of determining revenue-maximizing prices for an assortment of products that exhibit network effects. Customers make purchase decisions according to a multinomial logit choice model, modified- to incorporate network effects-so that the utility each individual customer gains from purchasing a particular product depends on the market's total consumption of that product. In the setting of homogeneous products, we show that if the network effect is comparatively weak, then the optimal pricing decision of the seller is to set identical prices for all products. However, if the network effect is strong, then the optimal pricing decision is to set the price of one product low and to set the prices of all other products to a single high value. This boosts the sales of the single low-price product in comparison to the sales of all other products. We also obtain comparative statics results that describe how optimal prices and sales levels vary with a parameter that determines the strength of the network effects. We extend our analysis to settings with heterogeneous products and establish that optimal solutions have a structure similar to that found in the homogeneous case: either maintain a semblance of balance among all products or boost the sales of just one product. Based on this structure, we propose an effective computational algorithm for such general heterogeneous settings.

We consider an infinite-horizon single-product pricing problem in which a fraction of customers is patient and the remaining fraction is impatient. A patient customer will wait up to some fixed number of time periods for the price of the product to fall below his or her valuation at which point the customer will make a purchase. If the price does not fall below a patient customer's valuation at any time during those periods, then that customer will leave without buying. In contrast, impatient customers will not wait, and either buy immediately or leave without buying. We prove that there is an optimal dynamic pricing policy comprised of repeating cycles of decreasing prices. We obtain bounds on the length of these cycles, and we exploit these results to produce an efficient dynamic programming approach for computing such an optimal policy. We also consider problems in which customers have variable levels of patience. For such problems, cycles of decreasing prices may no longer be optimal, but numerical experiments nevertheless suggest that such a decreasing cyclic policy (suitably chosen) often performs quite well.

We consider repeated pricing games in which two competing sellers use mathematical models to choose the prices of their products. Over the sequence of games, each seller attempts to estimate the values of the parameters of a demand model that expresses demand as a function only of its own price using data comprised only of its own past prices and demand realizations. Thus, as is often the case in practice, the sellers' models do not explicitly account for other sellers. We study the behavior of the sellers' prices and parameter estimates under various assumptions regarding the sellers' knowledge and estimation procedures. We identify situations in which (a) the sellers' prices converge to the Nash equilibrium associated with knowledge of the correct demand model, (b) the sellers' prices converge to the cooperative solution, and (c) the sellers' prices converge to other values that are neither the Nash equilibrium nor the cooperative solution and that depend on the initial prices.

We consider Markov decision processes with unknown transition probabilities and unknown single-period expected cost functions, and we study a method for estimating these quantities from historical or simulated data. The method requires knowledge of the system equations that govern state transitions as well as the single-period cost functions (but not the single-period expected cost functions). The estimation procedure is based upon taking expectations with respect to the empirical distribution functions of such data. Once the estimates are in place, the method computes a policy by solving the obtained "empirical" Markov decision process as if the estimates were correct. For MDPs that satisfy some conditions, we provide explicit, easily computed expressions for the probability that the procedure will produce a policy whose true expected cost is within any specified absolute distance of the actual optimal expected cost of the true Markov decision process. We also provide expressions for the number of historical or simulated data values that is sufficient for the procedure to produce a policy whose true expected cost is, with a prescribed probability, within a prescribed absolute distance of the actual optimal expected cost of the true Markov decision process. We apply our results to multiperiod inventory models. In addition, we provide a specialized analysis of such inventory models that also yields relative, rather than absolute, accuracy guarantees. We make comparisons with related results that have recently appeared, and we provide numerical examples.