  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.
