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.(c) 2020 Elsevier Inc. All rights reserved.