Pricing for Multinomial Logit Choice Models with Network Effects

Abstract

We are experiencing a time in which advances in information technology change everything around us. Almost every aspect of our work and life, from daily travel to grocery shopping, is now or will soon be connected to the internet. This drastic change has led to the emergence of many new products that display network effects, meaning that individual consumers value a product more when more other consumers use the same product. At the same time, development of data storage/processing tools offers a great chance for many firms to collect large amounts of data and understand their customers' purchasing behavior. For these firms, a significant challenge --- as well as opportunity --- is to develop models of customer choice behavior that incorporate network effects and to use such models to make better pricing decisions. This dissertation addresses the development and analysis of some such models. 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 (MNL) 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 pricing scheme boosts the sales of the single low-price product in comparison to the sales of all other products. The analysis is also extended to settings with heterogeneous products, and we show 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. Subsequently, we study the preceding pricing problem from a robust optimization perspective. Unlike the classical MNL model where products' prices and sales have a one-to-one correspondence, in the MNL model with network effects a fixed set of prices may not uniquely determine sales. This occurs because, for given prices, sales arise as the solution to an equilibrium condition. In some cases, there may be multiple sales levels that satisfy the equilibrium condition. Among those sales equilibria corresponding to a given set of prices, we call the one with the highest revenue the ``optimistic" equilibrium, and the one with the lowest revenue the ``pessimistic" equilibrium. In our initial study mentioned in the previous paragraph, we implicitly took an ``optimistic" approach. We next take the pessimistic attitude and study the revenue-maximizing problem in the pessimistic setting. In the case that there is only one product to sell, the problem has the same pessimistic optimal price as the optimistic one, when the network effect is relatively weak. However, when the network effect is strong, the optimal policy requires the seller to become more conservative and the pessimistic optimal price is lower than the optimistic one. In the case that there are two products, the structure of equations at equilibrium becomes more complicated and we are not able to derive an analytical solution. To numerically solve this problem, two directions for finding the pessimistic optimal solution are proposed: divide and search, and linear relaxation. In addition, our numerical studies show that when the network effect is strong, the revenue from offering exactly one product is almost as good as that from selling an assortment of multiple products. This suggests that a company selling multiple products with strong network effects may be wise to simply offer just one product.