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I am gratified that there is sufficient interest in the subject matter so as to support the offering of a second edition of this monograph. The of differential games dynamic interpretation and game theoretic foundation form a powerful and vital methodology for helping us study and understand marketing competition. This second edition offers a blend of what proved to be successful with the first edition and new material. The first two chapters, reviewing empirical and modeling research, have been updated to include contributions in the last decade that have advanced the area. I have not changed the essential content in the duopoly analyses in chapters 3, 4, and 5. A notable addition to the present edition are the new chapters, 6, 7, and 8, which offer analysis of three triopoly models. In the final chapter, I offer my summary view of the area and hope for continued contributions. I want to express my appreciation for the support of Josh Eliashberg, editor of the International Series in Quantitative Marketing, as well as Zachary Rolnik, Director, and David Cella, Publishing Editor, of Kluwer. Their encouragement has provided crucial motivation in this endeavor.



1. Advertising and Competition

A critical aspect of the advertising budgeting process involves competitive issues—anticipated spending levels of major competitors, effects that competitive advertising may have on the firm’s market share, sales, and profit, and the interactive nature of a competitor’s advertising with a firm’s own. Competition is ignored only at the firm’s peril; empirical studies (e.g. Little 1979; also see the empirical survey below) have shown quite clearly that competitive advertising can have a direct, and negative, effect on a company’s market share. Also, management practice appears to recognize the importance of competition; in a survey of leading U.S. advertisers, Lancaster and Stern (1983) reveal that, among various general characteristics describing the advertising budgeting process, “competitive effects” were considered by 52% of the sample (second only to “communication effects” at 55%).
Gary M. Erickson

2. Analytical Models and Strategy Concepts

Eliashberg and Chatterjee (1985) indicate that the analytical study of competition requires certain basic assumptions to be made regarding the number of competitors studied, the nature of competitive interaction, and the information base of the competitors involved. Generally, studies have assumed a game theory framework, in which each of the various competitors is a decision maker, although some (e.g. Horsky 1977) study only the decisions of a single competitor. A game theory framework offers a potentially rich setting for studying the interactive nature of competitive advertising. It usually is assumed in existing research that the competitors make their advertising decisions simultaneously and with complete information. (Bensoussan et al. 1978, on the other hand, view the situation as a sequential decision-making process, and adopt the perspective of the market leader that needs to anticipate the reactions of its competitors to its decisions.) That is, each competitor has full knowledge of the nature of the competitive interaction and the motivations and profit structures of the other competitors, so that each competitor can infer the strategies of its rivals. Also, it is generally assumed that the competitors cannot collude (although see Gasmi et al. 1992 and Vilcassim et al. 1999 for a different perspective). Assuming collusion is not possible, the noncooperative branch of game theory is appealed to, and Nash equilibria are sought to define the advertising strategies of the competitors.
Gary M. Erickson

3. Analysis of a Lanchester Duopoly

Assume we have two competitors in a competition for market share, and that each wishes to maximize its discounted cash flow over an infinite horizon. We have for competitor 1
$$ \mathop {\max }\limits_{{A_1}} \int\limits_0^\infty {{e^{ - rt}}} ({g_1}M - {A_1})dt $$
and for competitor 2
$$ \mathop {\max }\limits_{{A_2}} \int\limits_0^\infty {{e^{ - rt}}} ({g_2}[1 - M]{A_2})dt $$
The parameters g 1 and g 2 represent the economic values of market shares for competitors 1 and 2, respectively. Also, r is the discount rate, assumed equivalent for the two competitors.
Gary M. Erickson

4. Analysis of a Vidale-Wolfe Duopoly

In this chapter, we analyze the Vidale-Wolfe model version introduced in chapter 2. We assume the objectives
$$\begin{array}{*{20}{c}} {\mathop{{\max }}\limits_{{{{A}_{1}}}} \int\limits_{0}^{\infty } {{{e}^{{ - rt}}}\left( {{{h}_{1}}\frac{{{{\gamma }_{1}}}}{{{{\gamma }_{1}} + {{\gamma }_{2}}}}S - {{A}_{1}}} \right)} dt} \hfill \\ {\mathop{{\max }}\limits_{{{{A}_{2}}}} \int\limits_{0}^{\infty } {{{e}^{{ - rt}}}\left( {{{h}_{2}}\frac{{{{\gamma }_{2}}}}{{{{\gamma }_{1}} + {{\gamma }_{2}}}}S - {{A}_{2}}} \right)} dt} \hfill \\ \end{array}$$
and that total industry sales level S is subject to the dynamic constraint
$$\dot{S} = \left( {{{\beta }_{1}}A_{1}^{{{{\alpha }_{1}}}} + {{\beta }_{2}}A_{2}^{{{{\alpha }_{2}}}}} \right)\left( {N - S} \right) - \delta S$$
In the Vidale-Wolfe model we analyze, each competitor’s advertising is used to attract industry sales S, which are divided between the two competitors according to brand-strength parameters γ1, γ2. The industry sales level changes dynamically according to (4.2). The maximum sales level is N, and the difference between the maximum and current sales levels is attractable through advertising. In addition to the buildup in sales through advertising, there is a decay at the rate of δ times the current level of sales. The parameters h 1, h 2, are the unit contributions of the two competitors, and r is the common discount rate.
Gary M. Erickson

5. Analysis of a Diffusion Duopoly

A third model we analyze is the diffusion formulation with repeat purchase. With this model, we assume as before that the two competitors want to maximize discounted profits over an infinite horizon:
$$ \begin{gathered} \mathop{{\max }}\limits_{{{A_1}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} \left( {{h_1}\frac{{{\gamma_1}}}{{{\gamma_1} + {\gamma_2}}}S - {A_1}} \right)dt \hfill \\ \mathop{{\max }}\limits_{{{A_2}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} \left( {{h_2}\frac{{{\gamma_2}}}{{{\gamma_1} + {\gamma_2}}}S - {A_2}} \right)dt \hfill \\ \end{gathered} $$
The dynamic constraint on industry sales S in the diffusion model is
$$ \dot{S} = ({\beta_1}A_1^{{{\alpha_1}}} + {\beta_2}A_1^{{{\alpha_2}}}\varepsilon S)(N - S) - \delta S \dot{S} = ({\beta_1}A_1^{{{\alpha_1}}} + {\beta_2}A_1^{{{\alpha_2}}})(N - S) - \delta S $$
The diffusion model is similar to the Vidale-Wolfe structure analyzed in chapter 4, with the addition of the word-of-mouth parameter ε. As in the Vidale-Wolfe model, r is the common discount rate, h 1 and h 2 are unit contributions, α 1 and α 2 are advertising elasticities, β 1 and β 2 are advertising effectiveness parameters, δ is a decay factor, and N is the maximum market size.
Gary M. Erickson

6. Analysis of a Lanchester Triopoly

In this chapter, and the next two, we analyze competitive advertising situations involving three competitors. The move beyond three competitors generates analytical difficulties in developing closed-loop strategies, due to the presence of multiple market share state variables, and a different analytical approach is needed.
Gary M. Erickson

7. Analysis of a Vidale-Wolfe Triopoly

We consider a Vidale-Wolfe triopoly model in the present chapter. As opposed to the analysis of the Vidale-Wolfe duopoly in chapter 4, here we do not restrict the model to a single state variable, and instead assume a separate sales state variable for each of the three competitors. The sales state variables are assumed to change across time according to the following relationships:
$$ \begin{gathered} {{\dot{S}}_1} = {\beta_1}A_1^{{{\alpha_1}}}\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_1}{S_1} \hfill \\ {{\dot{S}}_2} = {\beta_2}A_2^{{\alpha 2}}\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_2}{S_2} \hfill \\ {{\dot{S}}_3} = {\beta_3}A_3^{{{\alpha_3}}}\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_3}{S_3} \hfill \\ \end{gathered} $$
The objective of each competitor is to maximize discounted profit:
$$ \begin{gathered} \mathop{{\max }}\limits_{{{A_1}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} ({h_1}{S_1} - {A_1})dt \hfill \\ \mathop{{\max }}\limits_{{{A_2}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} ({h_2}{S_2} - {A_2})dt \hfill \\ \mathop{{\max }}\limits_{{{A_3}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} ({h_3}{S_3} - {A_3})dt \hfill \\ \end{gathered} \dot{S} = ({\beta_1}A_1^{{{\alpha_1}}} + {\beta_2}A_1^{{{\alpha_2}}})(N - S) - \delta S $$
where h i i = 1, 2, 3, is competitor i’s contribution per unit of sale.
Gary M. Erickson

8. Analysis of a Diffusion Triopoly

In this chapter, we consider a triopoly with sales dynamics that involve diffusion:
$$ \begin{gathered} {{\dot{S}}_1} = \left( {{\beta_2}A_2^{{{\alpha_2}}} + \varepsilon 2{S_2}} \right)\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_1}{S_1} \hfill \\ {{\dot{S}}_2} = \left( {{\beta_1}A_1^{{{\alpha_1}}} + {\varepsilon_1}{S_1}} \right)\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_2}{S_2} \hfill \\ {{\dot{S}}_3} = \left( {{\beta_3}A_3^{{{\alpha_3}}} + {\varepsilon_3}{S_3}} \right)\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_3}{S_3} \hfill \\ \end{gathered} $$
The ε i , i=1, 2, 3, in (8.1), are coefficients of “internal influence” that indicate that a competitor’s existing customers are assumed to be a positive factor in attracting new customers from the part of the market that are not currently customers of any of the competitors. Advertising is assumed to exert an “external influence” on currently uncommitted customers.
Gary M. Erickson

9. Summary and Final Considerations

We have considered two different approaches to developing dynamic advertising strategies in competitive situations, the Case (1979) perfect equilibria approach for duopolies and for triopolies the dynamic conjectural variation approach of Erickson (1997), in the search for useful alternatives to the unsatisfactory open-loop concept. Each of the alternative approaches has advantages but also limitations.
Gary M. Erickson


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