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We study analytically a simple game theoretical model of heterogeneous interacting agents. We show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz. Such a stationary state differs from the Nash equilibrium where each agent maximizes her own utility. The latter turns out to be characterized by a replica symmetry broken structure. Numerical results fully agree with our analytic findings.Statistical mechanics of disordered systems provides analytical and numerical tools for the description of complex systems, which have found applications in many interdisciplinary areas [1]. When the precise realization of the interactions in an heterogeneous system is expected not to be crucial for the overall macroscopic behavior, then the system itself can be modeled as having random interactions drawn from an appropriate distribution. Such an approach appears to be very promising also for the study of systems with many heterogeneous agents, such as markets, which have recently attracted much interest in the statistical physics community [2,3]. Indeed it provides a workable alternative to the so called representative agent approach of micro-economic theory, where assuming that agents are identical, one is lead to a theory with one single (representative) agent [4].In this Letter we present analytical results for a simple model of heterogeneous interacting agents, the so called minority game (MG) [3,5], which is a toy model of N agents interacting through a global quantity representing a market mechanism. Agents aim at anticipating market movements by following a simple adaptive dynamics inspired at Arthur's inductive reasoning [6]. This is based on simple speculative strategies that take advantage of the available public information concerning the recent market history, which can take the form of one of P patterns. Numerical studies [3,[7][8][9] have shown that the model displays a remarkably rich behavior. The relevant control parameter [3,7] turns out to be the ratio α = P/N between the "complexity" of information P and the number N of agents, and the model undergoes a phase transition with symmetry breaking [8] independently of the origin of information [9].We shall limit the discussion on the interpretation of the model -which is discussed at some length in refs. [3,7] -to a minimum and rather focus on its mathematical structure and to the analysis of its statistical properties for N ≫ 1. Our main aim is indeed to show that the model can be analyzed within the framework of statistical mechanics of disordered system [1].

We investigate further several properties of the minority game we have recently introduced. We explain the origin of the phase transition and give an analytical expression of σ 2 /N in the N ≪ 2 M region. The ability of the players to learn a given payoff is also analyzed, and we show that the Darwinian evolution process tends to a self-organized state, in particular, the life-time distribution is a power-law with exponent -2. Furthermore, we study the influence of identical players on their gain and on the system's performance. Finally, we show that large brains always take advantage of small brains.Recently we have studied a simple model of a minority game [1] that captures the essential features of the "barproblem" of Arthur ( [2] and [3]). N players compete with each other and act by induction and adaptation. They must choose one side between the two at each time step and those who happen to be in the minority side win. They receive a reward when making the right choice, keep in memory the M last sides which were the right ones and use this knowledge to act at the next time step. Each player possesses a finite set of S strategies and uses the one which would have been the most rewarding if it had been used since the beginning. A strategy is a behavior rule that stipulates an action for every information possible (see [1] or [4] for more details). Recently, Savit, Manuca and Riolo [5] have studied the step payoff case where each player has the same memory M and only two strategies. They have found a phase transition with parameter ρ = 2 M /N .Here, we continue the study of the payoff learning and the Darwinism's process initiated in [1] and introduce other interesting sights of the model. We also give a geometrical explanation of the phase transition and find an analytical expression of σ 2 /N in the N ≪ 2 M region.

We discuss a model of heterogeneous, inductive rational agents inspired by the El Farol Bar problem and the Minority Game. As in markets, agents interact through a collective aggregate variable -which plays a role similar to price -whose value is fixed by all of them. Agents follow a simple reinforcement-learning dynamics where the reinforcement, for each of their available strategies, is related to the payoff delivered by that strategy. We derive the exact solution of the model in the "thermodynamic" limit of infinitely many agents using tools of statistical physics of disordered systems. Our results show that the impact of agents on the market price plays a key role: even though price has a weak dependence on the behavior of each individual agent, the collective behavior crucially depends on whether agents account for such dependence or not. Remarkably, if the adaptive behavior of agents accounts even "infinitesimally" for this dependence they can, in a whole 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.⋆ We acknowledge J. Berg, S. Franz and Y.-C. Zhang for discussions and useful suggestions. On the economic side we profited greatly from critical discussions with A. Rustichini and F. Vega-Redondo on learning and dynamic games.

We address the question of market efficiency using the Minority Game (MG) model. First we show that removing unrealistic features of the MG leads to models which reproduce a scaling behaviour close to what is observed in real markets. In particular we find that (i) fat tails and clustered volatility arise at the phase transition point and that (ii) the crossover to random walk behaviour of prices is a finite-size effect. This, on one hand, suggests that markets operate close to criticality, where the market is marginally efficient. On the other it allows one to measure the distance from criticality of real markets, using cross-over times. The artificial market described by the MG is then studied as an ecosystem with different species of traders. This clarifies the nature of the interaction and the particular role played by the various populations.