Best Response Function

A best response function maps each rival strategy to the firm's profit-maximising own strategy. In a price-setting complementary-goods game, Firm $i$'s best response is

derived by taking the FOC $A - P - p_i = 0$. The Nash Equilibrium is the intersection of all firms' best response functions; for the symmetric two-firm complementary case the intersection is $p_1^* = p_2^* = A/3$.

When to use

Use best response functions whenever strategies are continuous — Cournot quantity competition, Bertrand with differentiated products, and complementary-goods pricing. The mechanical recipe: write each firm's FOC, solve for its own choice as a function of the rival's, and intersect. The sign of the slope tells you whether the strategies are strategic substitutes (downward-sloping, Cournot quantities) or strategic complements (upward-sloping, Bertrand prices with differentiation).