Convergence of incentive-driven dynamics in Fisher markets

We study out-of-equilibrium price dynamics in Fisher markets. We develop a general framework in which sellers have (a) a set of atomic price update rules (APU), which are simple responses to a price vector; (b) a belief-formation procedure that simulates actions of other sellers (themselves using the APU) to some finite horizon in the future. Sellers use an APU to respond to a price vector they generate with the belief formation procedure. The framework allows sellers to have inconsistent and time-varying beliefs about each other. Under mild and natural assumptions on the APU, we show that despite the inconsistent and time-varying nature of beliefs, the market converges to a unique equilibrium at a linear rate (distance to equilibrium decreases exponentially in time). If the APU are driven by weak gross substitutes demands, the equilibrium point is the same as predicted by those demands.