Statistical Mechanics Of Stylized Models Of Financial Markets With Many Heterogeneous Adaptive Agents

Agents' heterogeneity, arising from asymmetric information, heterogeneous beliefs, endowments or constraints, calls for theoretical approaches to economic systems that go beyond the so-called representative agent approach.In systems of many interacting agents, such as markets, a detailed microeconomic description of each individual participant is prohibitive. On the other hand, one expects that the collective behavior of a system of many degrees of freedom, because of statistical regularities, is insensitive of the detailed behavior at the micro level.The situation is quite similar to that of physical macroscopic systems, for which a stylized description at the micro level -- capturing its essential features -- is enough to describe appropriately the collective behavior. This holds also for heterogeneous systems, such as disordered solids or glasses, where the detailed description of the microscopic interaction can be replaced by random interactions drawn from an appropriate distribution. Laws of large numbers can be invoked to show that, almost all random realizations of the interactions are characterized by the same collective behavior.A similar approach can be used in economic systems (see also the literature on random games) by drawing at random the strategic interactions among agents. We apply this approach to stylized market models with many states. Agents are heterogeneous either because they have partial and asymmetric information on the state, or because of different behavioral modes, beliefs or constraints. The interaction is repeated over and over again, and agents follow simple learning dynamics, which can also be heterogeneous across agents. The simplest such model - which was inspired by Arthur's El Farol problem -- is the so-called Minority Game. Briefly, agents can take one of two actions such as buy or sell. Agents who take the action taken by the minority are rewarded, whereas the majority of agents loose. This structure is complicated by the presence of states (or public information) and by the fact that, as in the El Farol problem, agents learning occurs not on actions but on a small subset of behavioral rules. The Minority Game reveals an extremely rich emergent collective behavior.The Minority Game can be solved exactly in the limit of infinitely many agents and infinitely many states, with a fixed ratio of states to agents. The solution is based on the derivation of a continuum time limit and on the observation that the learning dynamics admits for a Lyapunov function. Hence the long-run properties of the dynamics can be recovered studying the (local) minima of the Lyapunov function. In order to do that, we resort to tools and ideas of statistical mechanics of disordered systems, which allow us to deal with random interactions (heterogeneity).These results give a clear and complete picture of the system and they allow us to investigate in detail issues of market informational efficiency and Pareto optimality. For example, we find that the market undergoes a phase transition between an efficient phase and an inefficient one as the number of states increases (with a fixed, large, numer of agents). We also find that price taking behavior -- by which agents behave as if they were playing against price and not against other agents -- does not lead to Pareto optimality and it may lead to dramatic raise in market's volatility.As in markets, agents in the Minority Game interact through an aggregate quantity. Yet there are neither assets nor prices in the minority game. The approach however generalizes to much more general model and indeed also to simple models of financial markets. Most importantly many features of the collective behavior qualitatively remains the same, hinting at a remarkable robustness of the statistical laws governing the collective behavior.These show the relevance of methods of statistical physics to deal with economic systems of many heterogeneous agents with many states.