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本文提出了一个两家企业竞争环境下的广告投入最优控制模型,模型中将信息性广告和竞争性广告分开考虑,由于这两种广告对销售的作用不尽相同,所以用不同的变量来分别描述这种作用更符合市场实际。采用动态规划的Bellman方程求得模型的闭环Nash均衡解,由于闭环解是状态反馈策略,能根据竞争对手的行动作出反应,所以在实际运用中,更能被企业接受并采纳。针对计算得到的Nash均衡解,通过分析得出满足经济意义的参数范围和闭环Nash均衡解最终收敛的条件。最后用算例分析了和解释了Nash均衡下企业应采取的广告策略。
This paper proposes an optimal control model of advertising investment under the competitive environment of two companies. In the model, informational advertising and competitive advertising are considered separately. Since these two kinds of advertising have different effects on sales, different variables Describe this role respectively more in line with market reality. The closed-loop Nash equilibrium solution of the model is obtained by the Bellman equation of dynamic programming. Because the closed-loop solution is a state feedback strategy and can react according to the competitor’s actions, it can be accepted and adopted by the enterprises in practice. Aiming at the calculated Nash equilibrium solution, the final convergent conditions of the parameter range satisfying the economic significance and the closed-loop Nash equilibrium solution are obtained through analysis. Finally, an example is used to analyze and explain the advertising tactics that enterprises should adopt under Nash equilibrium.