Dynamic Multinomial Logit Choice Model with Network Effect

In this paper, we study a dynamic multinomial logit choice model with network effect (we call it the "dynamic model'' in short). In our model, a continuum of customers stay in a market for an infinite horizon. We adopt a random utility model to capture customer's valuation of each product in which the expected utility of each product is determined by its intrinsic value, its price and the market share at the current time period. In each time period, for a customer, if the highest utility among products is higher than the utility of the no-purchase option, then this customer will choose the product with the highest current utility to purchase and leave the market. Otherwise this customer will not purchase any product and will stay in the market in the next time period. This process continues until it reaches a steady state. We call the market shares under the steady state the choice probabilities under the dynamic model. We study the properties of the dynamic model, particularly its choice probabilities, and compare the choice probabilities under the dynamic model with those under a previously proposed static MNL model with network effect in the literature (we call it the "static model" in short). We find that under proper dominance conditions, the dominant product will have a higher choice probability than the other products. We also find that under mild conditions, the choice probabilities under the dynamic model tend to be more balanced than those of the static model, and the total choice probability of all products will be smaller than that under the static model. Then we study the operational decision problems under the dynamic model, including the optimal pricing problem and the assortment optimization problem. For the optimal pricing problem, we propose an approximation scheme of the final market shares, and establish an efficient algorithm for the optimal pricing problem with logarithmic utility function. We also propose a gradient descent method that works for general optimal pricing problems under the dynamic model. For the assortment optimization problem, we propose a new class of assortments called k-proximity assortments, and investigate the optimality of these assortments both theoretically and numerically.