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 Quantitative Finance , 2000, DOI: 10.1103/PhysRevE.63.056121 Abstract: We study the dynamics of the batch minority game, with random external information, using generating functional techniques a la De Dominicis. The relevant control parameter in this model is the ratio $\alpha=p/N$ of the number $p$ of possible values for the external information over the number $N$ of trading agents. In the limit $N\to\infty$ we calculate the location $\alpha_c$ of the phase transition (signaling the onset of anomalous response), and solve the statics for $\alpha>\alpha_c$ exactly. The temporal correlations in global market fluctuations turn out not to decay to zero for infinitely widely separated times. For $\alpha<\alpha_c$ the stationary state is shown to be non-unique. For $\alpha\to 0$ we analyse our equations in leading order in $\alpha$, and find asymptotic solutions with diverging volatility $\sigma=\order(\alpha^{-{1/2}})$ (as regularly observed in simulations), but also asymptotic solutions with vanishing volatility $\sigma=\order(\alpha^{{1/2}})$. The former, however, are shown to emerge only if the agents' initial strategy valuations are below a specific critical value.
 Physics , 2003, DOI: 10.1016/j.physa.2003.10.009 Abstract: To study the interplay between global market choice and local peer pressure, we construct a minority-game-like econophysical model. In this so-called networked minority game model, every selfish player uses both the historical minority choice of the population and the historical choice of one's neighbors in an unbiased manner to make decision. Results of numerical simulation show that the level of cooperation in the networked minority game differs remarkably from the original minority game as well as the prediction of the crowd-anticrowd theory. We argue that the deviation from the crowd-anticrowd theory is due to the negligence of the effect of a four point correlation function in the effective Hamiltonian of the system.
 Physics , 2000, Abstract: We show analytically how the fluctuations (i.e. standard deviation) in the Minority Game (MG) can decrease below the random coin-toss limit if the agents use more general, stochastic strategies. This suppression of the standard deviation results from a cancellation between the actions of a crowd, in which agents act collectively and make the same decision, and an anticrowd in which agents act collectively by making the opposite decision to the crowd.
 Physics , 2000, DOI: 10.1103/PhysRevE.63.017102 Abstract: We show analytically how the fluctuations (i.e. standard deviation) in the Minority Game (MG) can be made to decrease below the random coin-toss limit if the agents use more general behavioral strategies. This suppression of the standard deviation results from a cancellation between the actions of a crowd, in which agents act collectively and make the same decision, and an anticrowd in which agents act collectively by making the opposite decision to the crowd.
 Physics , 2006, DOI: 10.1088/1742-5468/2007/01/P01006 Abstract: Ergodic stationary states of Minority Games with S strategies per agent can be characterised in terms of the asymptotic probabilities $\phi_a$ with which an agent uses $a$ of his strategies. We propose here a simple and general method to calculate these quantities in batch canonical and grand-canonical models. Known analytic theories are easily recovered as limiting cases and, as a further application, the strategy frequency problem for the batch grand-canonical Minority Game with S=2 is solved. The generalization of these ideas to multi-asset models is also presented. Though similarly based on response function techniques, our approach is alternative to the one recently employed by Shayeghi and Coolen for canonical batch Minority Games with arbitrary number of strategies.
 Physics , 2008, Abstract: We study the structure of the underlying network of connections in the Minority Game. There is not an explicit interaction among the agents, but they interact via global magnitudes of the model and mainly through their strategies. We define a link between two agents by quantifying the similarity among their strategies, and analyze the structure of the resulting underlying complex networks as a function of the number of agents in the game and the value of the agents' memory, in games with two strategies per player. We characterize the different phases of this system with networks with different properties, for this link definition. Thus, the Minority Game phase characterized by the presence of crowds can be identified with a small world network, while the phase with the same results as a random decision game as a random network. Finally, we use the Full Strategy Minority Game model, to explicitly calculate some properties of its networks, such as the degree distribution, for the same link definition, and to estimate, from them, the properties of the networks of the Minority Game, obtaining a very good agreement with its measured properties.
 Tobias Galla Statistics , 2004, DOI: 10.1088/1742-5468/2005/01/P01002 Abstract: We study the dynamics and statics of a dilute batch minority game with random external information. We focus on the case in which the number of connections per agent is infinite in the thermodynamic limit. The dynamical scenario of ergodicity breaking in this model is different from the phase transition in the standard minority game and is characterised by the onset of long-term memory at finite integrated response. We demonstrate that finite memory appears at the AT-line obtained from the corresponding replica calculation, and compare the behaviour of the dilute model with the minority game with market impact correction, which is known to exhibit similar features.
 Mathematics , 2003, DOI: 10.1088/0305-4470/36/43/034 Abstract: We present an exact dynamical solution of a spherical version of the batch minority game (MG) with random external information. The control parameters in this model are the ratio of the number of possible values for the public information over the number of agents, and the radius of the spherical constraint on the microscopic degrees of freedom. We find a phase diagram with three phases: two without anomalous response (an oscillating versus a frozen state), and a further frozen phase with divergent integrated response. In contrast to standard MG versions, we can also calculate the volatility exactly. Our study reveals similarities between the spherical and the conventional MG, but also intriguing differences. Numerical simulations confirm our analytical results.
 Quantitative Finance , 2001, DOI: 10.1088/0305-4470/34/40/103 Abstract: We study the dynamics of the `batch' minority game with market-impact correction using generating functional techniques to carry out the quenched disorder average. We find that the assumption of weak long-term memory, which one usually makes in order to calculate ergodic stationary states, breaks down when the persistent autocorrelation becomes larger than c_c=0.772... We show that this condition, remarkably, coincides with the AT-line found in an earlier static calculation. This result suggests a new scenario for ergodicity breaking in disordered systems.
 Physics , 2000, DOI: 10.1016/S0378-4371(00)00095-9 Abstract: In the standard minority game, each agent in the minority group receives the same payoff regardless of the size of the minority group. Of great interest for real social and biological systems are cases in which the payoffs to members of the minority group depend on the size of the minority group. This latter includes the fixed sum game. We find, remarkably, that the phase structure and general scaling behavior of the standard minority game persists when the payoff function depends on the size of the minority group. there is still a phase transition at the same value of z, the ratio of the dimension of the strategy space to the number of agents playing the game. We explain the persistence of the phase structure and argue that it is due to the absence of temporal cooperation in the dynamics of the minority game. We also discuss the behavior of average agent wealth and the wealth distribution in these variable payoff games.
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