The Second-Order Tâtonnement: Decentralized Interior-Point Methods for Market Equilibrium

The tâtonnement process and Smale's process are two classical approaches to compute market equilibrium in exchange economies. While the tâtonnement process can be seen as a first-order method, Smale's process, being second-order, is less popular due to its reliance on additional information from the players and expensive Newton steps. In this paper, we study Fisher exchange market for a broad class of utility functions, where we show that all high-order information required by Smale's process is readily available from players' best responses. Motivated by this observation, we develop two second-order tâtonnement processes, constructed as decentralized interior-point methods, which are traditionally known to work in a centralized manner. The methods here bear the name "tâtonnement", since, in spirit, they demand no more information than the classical tâtonnement process. To address the Newton systems involved, we introduce an explicitly invertible approximation with high-probability guarantees and a scaling matrix that optimally minimizes the condition number, both of which rely solely on best responses as the methods themselves. Using these tools, the first second-order tâtonnement process has O(log(1/$\epsilon$))complexity rate. Under mild conditions, the other method achieves a non-asymptotic superlinear convergence rate. Preliminary experiments are presented to justify the capability of the proposed methods for large-scale problems. Extensions of our approach are also discussed.