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By L P J. Groenewegen

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1), the advertising response fun ctions f i(a) take pos itive values and are increasing. Usually, ff'(a) ~ 0 is assumed, to represent diminishing or constant marginal effects of advertising. 1) reflects a hypothesis that a firm's advertising efforts only influence the rival firms' customers, not the firm's own customers. Thus, a firm 's advertising is designed to draw away customers from the rivals. The Lanchester model reflects a situation where consumers are not loyal and drift toward the firm that spends the most on advertising.

Erickson assumes concave advertising responses and linear costs . 45 Advertising Models where II and 12 are positive constants that reflect the strengths of brands 1 and 2, respectively. Thus, the sales rate Si(t) of firm i is Si(t) = Ii II + 12 Set) . In the objective functionals, 7ri is the unit margin of firm i and Ci > 0 is a constant 10 • We look for an MNE with stationary strategies (A(S) in the differential game defined above. Suppose that ai > O. J2(S))(m - S) - 6S l} p V2(S) = S - C22a~ + V~(S)[(,Bl¢I(S) + ,B2a2)(m - S) - 6Sl}.

Friedman (1986), Vives (1999) . Inco rporating trigger strategies in differential games causes some t echnicalit ies that are avoided in a repeated game setting. For the differential game case, see Dockner et al. (2000 , Ch . 6) . A Brief Tutorial on Differential Games 25 strategy is that the players threaten each other by (sufficiently) harsh punishments that will be invoked should someone dare to deviate from her part of the agreement. An important issue here is the credibility of threats: It must be in the best interest of a player who issues a threat, to carry out the threat whenever called upon.

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