AI-Driven Dynamic Pricing and Advertising in E-Commerce Competition: A Differential Game Analysis of Coordination Reversal and Welfare

Abstract

This paper develops a continuous-time differential game to examine how artificial intelligence shapes strategic pricing and advertising competition among e-commerce sellers. Building on recent reinforcement-learning evidence of counterintuitive lower-price outcomes under consumer search responsiveness, the analysis incorporates AI capability through demand sensitivity and advertising effectiveness within a Nerlove-Arrow (1962) goodwill framework. A closed-form coordination-reversal condition is derived in a static-goodwill benchmark, showing that coordinated equilibrium prices fall below non-cooperative Nash equilibrium prices if and only if the advertising-spillover effect dominates the cross-price substitution effect. The mechanism is then embedded in a continuous-time differential game with endogenous goodwill, and the feedback Nash equilibrium conditions are formulated through a coupled Hamilton-Jacobi-Bellman system. In the static benchmark, higher AI capability is shown to expand the parameter region supporting lower prices under coordination through the advertising-effectiveness channel. Welfare analysis reveals that lower prices do not automatically guarantee higher consumer surplus when advertising and goodwill adjust; the exact demand condition for consumer-surplus improvement is derived. The framework provides an analytical characterization of a mechanism that can generate the lower-price coordination outcome reported in recent computational research, clarifying conditions under which multidimensional coordination can be associated with consumer-beneficial rather than purely anticompetitive outcomes.