# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Inventory competition under dynamic consumer choice. (English) Zbl 1163.90340
Summary: We analyze a model of inventory competition among $n$ firms that provide competing, substitutable goods. Each firm chooses initial inventory levels for their good in a single period (newsboy-like) inventory model. Customers choose dynamically based on current availability, so the inventory levels at one firm affect the demand of all competing firms. This creates a strategic interaction among the firms’ inventory decisions. Our work extends earlier work on variations of this problem by Karjalainen (1992), Lippman and McCardle (1997) and Parlar (1988). Specifically, we model demand in a more realistic way as a stochastic sequence of heterogeneous consumers who choose dynamically from among the available goods (or choose not to purchase) based on a utility maximization criterion. We also use a sample path analysis, so minimal assumptions are imposed on this demand process. We characterize the Nash equilibrium of the resulting stocking game and prove it is unique in the symmetric case. We show there is a bias toward overstocking caused by competition; specifically, reducing the quantity stocked at any equilibrium of the game increases total system profits, and at any joint-optimal set of stocking levels, each firm has an individual incentive to increase its own stock. For the symmetric case, we show that as the number of competing firms increases, the overstocking becomes so severe that total system (and individual firm) profits approach zero. Finally, we propose a stochastic gradient algorithm for computing equilibria and provide several numerical examples.
##### MSC:
 90B05 Inventory, storage, reservoirs 91B42 Consumer behavior, demand theory