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A multiperiod model of inventory competition.

Article Excerpt
This paper explores when it is important for firms to consider stockout-based substitution and competitor's inventory levels in making inventory decisions in the context of a duopoly model. To address this question, we consider a model where two newsvendors sell substitutable products in a market with aggregate market demand D. The two firms get a proportion p and (1-p) of this demand, where p is random. We characterize the equilibrium inventory levels of the two firms in a single-period model and show the striking property that, under certain reasonable conditions on the cost parameters, the two firms ignore their competitor's inventory levels and potential substitution demand, i.e., their inventory decisions are decoupled. Furthermore, we show under slightly more restrictive conditions on the cost parameters that the single-period results can be extended to the case where D is random. Finally, we extend the decoupling property to a multiperiod periodic review scenario and show that the resulting Nash equilibrium can be characterized simply as the solution to a single-product dynamic newsvendor problem that ignores substitution demand.

Subject classifications: inventory/production applications; uncertainty.

Area of review. Manufacturing, Service, and Supply Chain Operations.

History: Received April 2006; revisions received October 2006, April 2007, August 2007, December 2007, February 2008; accepted April 2008. Published online in Articles in Advance March 11, 2009.

1. Introduction

When firms with substitutable products compete with each other, an important consideration in making inventory decisions is potential substitution demand arising out of stockouts. Several papers in the inventory literature have addressed this issue (Parlar 1988, Lippman and McCardle 1997, Anupindi and Bassok 1999, Netessine and Rudi 2003). We add to this literature by shedding some new light on this issue using a decentralized duopoly model with inventory substitution. We consider a situation where two retailers (denoted as firms 1 and 2) compete in a market by selling substitutable goods (products 1 and 2, respectively). The aggregate demand for the two products is D, which may be random. The individual retailers face demands pD and (1 - p)D, where p and D (when random) are continuous random variables with finite support. Because the proportion of customers who come in for either item is random, the retailers many face stockouts or end up with unsold inventory. We assume that a fixed proportion [gamma] of unsatisfied customer walks out of a retailer, a penalty is assessed. The revenue and cost trade-offs at each retailer used to determine stocking levels are similar to those in the inventory literature.

First, we characterize the equilibrium decisions of the two players in the single-period model. We show that the equilibrium order quantities have the interesting property that each player ignores the strategy of his opponent under certain reasonable conditions on the cost parameters. Second, we study the multiperiod version of the duopoly model. We show that, under reasonable conditions, the equilibrium quantities in the finite-horizon game are obtained by solving a multiperiod, single-product inventory model for each player! Thus, each retailer essentially acts as a monopolist and ignores substitution demand and the strategy of the other retailer in making inventory decisions.

The competitive version of the single-period substitution problem with multiple players has been well analyzed. Parlar (1988) and Wang and Parlar (1994), respectively, considered a model with two and three players with uncorrected demands and established key properties. Netessine and Rudi (2003) generalize many of the results to an n-player game with correlated demands and provide additional results and insights. Lippman and McCardle (1997) establish the existence of the Nash equilibrium in an n-player game. Anupindi and Bassok (1999) look at the effects of the substitution parameter [gamma] on equilibrium base-stock levels in a two-player decentralized system. Mahajan and Van Ryzin (2001) consider centralized and decentralized models where customers dynamically and probabilistically substitute among retailer assortments.

The literature on multiperiod inventory substitution games is very limited, as pointed out in Cachon and Netessine (2004). Even if the single-period game possesses a unique Nash equilibrium, the dynamic game is not guaranteed to have a unique equilibrium if retailers are allowed to pursue nonstationary policies. Avsar and Baykal-Gursoy (2002) study a periodic review model with two competing players facing uncorrelated demand with substitution. They show that if players are restricted to stationary base-stock policies, the infinite-horizon game is myopic, and inventory levels exhibit some monotonic properties. Netessine et al. (2003) examine a dynamic model with stationary base-stock policies in which they examine consumers' backordering behavior and equilibrium inventory levels. All the above papers--except for Lippman and McCardle (1997) and Netessine and Rudi (2003)--assume correlated retailer demands.

2. The Single-Period Model

We first consider a single-period...

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