A Static Approximation for Dynamic Demand Substitution with Applications in a Competitive Market

Abstract

We propose a static approximation of dynamic demand substitution behavior based on a fluid network model and a serviceinventory mapping. This approximation greatly enhances our ability to analyze the interdependent inventory/service, price, and product assortment decisions in noncompetitive and competitive scenarios with demand substitution. We demonstrate that the approximation is well behaved and then apply it to two previously intractable applications. First, we study a price and service competition between single-product retailers. After establishing a unique pure-strategy Nash equilibrium, we find that competition results in lower price, higher demand, and a higher level of inventory. We also observe that the aggregate profit and inventory level increase to positive constants as the number of retailers goes to infinity. Second, we study a duopolistic competition on price, service, and product assortment. We establish a pure-strategy Nash equilibrium for the product assortment competition and identify a condition for uniqueness. We find that competition on both price and product assortment results in lower price and less variety for each competitor, but the total number of products and the aggregate inventory level in a duopoly market are both likely to be higher than in a monopolistic market.

Category Management and Coordination in Retail Assortment Planning in the Presence of Basket Shopping Consumers

“…Hopp and Xu (2005) study the impact of product modularity in the optimal product line length from a manufacturer's perspective. Hopp and Xu (2006) study price, service, and assortment competition in a single category between two retailers and find that the retailers provide less variety and lower prices in competition.…”

“…Hopp and Xu (2005) study the impact of product modularity in the optimal product line length from a manufacturer's perspective. Hopp and Xu (2006) study price, service, and assortment competition in a single category between two retailers and find that the retailers provide less variety and lower prices in competition.We use game theory to study competitive interactions in the decentralized regime and between the retailers. Gruca and Sudharshan (1991) and Basuroy and Nguyen (1998) study a market share game based on the multinomial logit (MNL) model and demonstrate that certain conditions are needed for equilibrium to exist.…”

Assortment Planning: Review of Literature and Industry Practice

“…Basket profits are easily evaluated using point-of-sale data, and the proposed method produces near-optimal solutions. Hopp and Xu (2008) consider a static approximation of the assortment planning problem under stock-out substitution. They model demand using fluid networks and obtain a mapping between service and inventory, which allows them to analyze the previously intractable, joint assortment, inventory and pricing problem in both competitive and non-competitive scenarios.…”

Private labels and retail assortment planning: a differential evolution approach

“…As noted earlier, PAP models are based on substitution effects and focus on developing algorithms to define inventory levels by stochastic demand. The most popular approach for estimating demand substitution in assortment planning is multinomial logit models (e.g., van Ryzin and Mahajan, 1999;Mahajan and van Ryzin, 2001;Cachon et al, 2005;Li, 2007;Hopp and Xu, 2008) and exogenous substitution models (e.g., Smith and Agrawal, 2000;Rajaram and Tang, 2001;Kok andFisher, 2007;Shah and Avittathur, 2007;Yucel et al, 2009). The multinomial logit model is a discrete consumer choice model assuming that consumers are rational utility maximizers, while exogenous demand models directly specify the consumer reaction and are mostly used in inventory models.…”