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: 7.1$$ \begin{gathered} {{\dot{S}}_1} = {\beta_1}A_1^{{{\alpha_1}}}\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_1}{S_1} \hfill \\ {{\dot{S}}_2} = {\beta_2}A_2^{{\alpha 2}}\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_2}{S_2} \hfill \\ {{\dot{S}}_3} = {\beta_3}A_3^{{{\alpha_3}}}\left( {N - \sum\limits_{{i = 1}}^3 {{S_i}} } \right) - {\delta_3}{S_3} \hfill \\ \end{gathered} $$ The objective of each competitor is to maximize discounted profit: 7.2$$ \begin{gathered} \mathop{{\max }}\limits_{{{A_1}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} ({h_1}{S_1} - {A_1})dt \hfill \\ \mathop{{\max }}\limits_{{{A_2}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} ({h_2}{S_2} - {A_2})dt \hfill \\ \mathop{{\max }}\limits_{{{A_3}}} \int\limits_0^{\infty } {{e^{{ - rt}}}} ({h_3}{S_3} - {A_3})dt \hfill \\ \end{gathered} \dot{S} = ({\beta_1}A_1^{{{\alpha_1}}} + {\beta_2}A_1^{{{\alpha_2}}})(N - S) - \delta S $$ where h _i i = 1, 2, 3, is competitor i’s contribution per unit of sale.