Market Dynamics of Best-Response with Lookahead

Abstract

One attractive approach to market dynamics is the level $k$ model in which a level $0$ player adopts a very simple response to current conditions, a level $1$ player best-responds to a model in which others take level $0$ actions, and so forth. (This is analogous to $k$-ply exploration of game trees in AI, and to receding-horizon control in control theory.) If players have deterministic mental models with this kind of finite-level response, there is obviously no way their mental models can all be consistent. Nevertheless, there is experimental evidence that people act this way in many situations, motivating the question of what the dynamics of such interactions lead to. We address this question in the setting of Fisher Markets with constant elasticities of substitution (CES) utilities, in the weak gross substitutes (WGS) regime. We show that despite the inconsistency of the mental models, and even if players' models change arbitrarily from round to round, the market converges to its unique equilibrium. (We show this for both synchronous and asynchronous discrete-time updates.) Moreover, the result is computationally feasible in the sense that the convergence rate is linear, i.e., the distance to equilibrium decays exponentially fast. To the best of our knowledge, this is the first result that demonstrates, in Fisher markets, convergence at any rate for dynamics driven by a plausible model of seller incentives. Even for the simple case of (level $0$) best-response dynamics, where we observe that convergence at some rate can be derived from recent results in convex optimization, our result is the first to demonstrate a linear rate of convergence.