Exact solution of a modified El Farol's bar problem: Efficiency and the role of market impact

We discuss a model of heterogeneous, inductive rational agents inspired by the El Farol Bar problem and the Minority Game. As in markets, agents inter act through a collective aggregate variable - which plays a role similar to price - whose value is fixed by all of them. Agents follow a simple reinfo rcement-learning dynamics where the reinforcement, for each of their availa ble strategies, is related to the payoff delivered by that strategy. We der ive the exact solution of the model in the "thermodynamic" limit of infinit ely many agents using tools of statistical physics of disordered systems. O ur results show that the impact of agents on the market price plays a key r ole: even though price has a weak dependence on the behavior of each indivi dual agent, the collective behavior crucially depends on whether agents acc ount for such dependence or not. Remarkably, if the adaptive behavior of ag ents accounts even "infinitesimally" for this dependence they can, in a who le range of parameters, reduce global fluctuations by a finite amount. Both global efficiency and individual utility improve with respect to a "price taker" behavior if agents account for their market impact. (C) 2000 Elsevie r Science B.V. All rights reserved.