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Competitive advertising under uncertainty: a stochastic differential game approach. (English) Zbl 1114.90050
Summary: We analyze optimal advertising spending in a duopolistic market where each firm’s market share depends on its own and its competitor”s advertising decisions, and is also subject to stochastic disturbances. We develop a differential game model of advertising in which the dynamic behavior is based on the Sethi stochastic advertising model and the Lanchester model of combat. Particularly important to note is the morphing of the sales decay term in the Sethi model into decay caused by competitive advertising and noncompetitive churn that acts to equalize market shares in the absence of advertising. We derive closed-loop Nash equilibria for symmetric as well as asymmetric competitors. For all cases, explicit solutions and comparative statics are presented.

MSC:
90B60 Marketing, advertising
91A15 Stochastic games, stochastic differential games
91A23 Differential games (aspects of game theory)
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H20 Stochastic integral equations
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