Analysis of a Lanchester Duopoly

Abstract

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

$$ \mathop {\max }\limits_{{A_1}} \int\limits_0^\infty {{e^{ - rt}}} ({g_1}M - {A_1})dt $$

(3.1)

and for competitor 2

$$ \mathop {\max }\limits_{{A_2}} \int\limits_0^\infty {{e^{ - rt}}} ({g_2}[1 - M]{A_2})dt $$

(3.2)

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.