Most models of the dynamics of economic exchange fall into one of two frameworks. In one approach, the models assume full rationality of the participating agents, who unravel the entire evolution of the exchange and choose in advance all possible reactions optimally. This is unrealistically prescient, contradicts experience, hinders on-the-fly adaptation to unexpected changes, and does not illuminate out-of-equilibrium behavior. An alternative approach examines no-regret reactions that indeed model  reactions to unexpected scenarios out-of-equilibrium, but those reactions have to be damped carefully for a desirable outcome to materialize, and they lack strategic justification.

 

We present a ``control theoretic’’ approach to the dynamics of economic exchange, based on limited lookahead situational analysis of the participating agents. It is motivated by and generalizes 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 the participants 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.

 

This is joint work with Krishnamurthy Dvijotham and Leonard J. Schulman.