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 Physics , 2006, DOI: 10.1016/j.physa.2007.02.048 Abstract: It is known that the memory is relevant in the symmetric phase of the minority game. In our previous work we have successfully explained the quasi-periodic behavior of the game in the symmetric phase with the help of the probability theory. Based on this explanation, we are able to determine how the memory affects the variance of the system in this paper. By using some particular types of fake history such as periodic type and random type, we determine how efficient the memory has been used in the standard game. Furthermore, the analysis on the effective memory strongly supports the result we proposed previously that there are three distinct phases in the minority game.
 Andrea Cavagna Physics , 1998, DOI: 10.1103/PhysRevE.59.R3783 Abstract: By means of extensive numerical simulations we show that all the distinctive features of the minority game introduced by Challet and Zhang (1997), are completely independent from the memory of the agents. The only crucial requirement is that all the individuals must posses the same information, irrespective of the fact that this information is true or false.
 Quantitative Finance , 2005, DOI: 10.1088/1742-5468/2005/10/P10009 Abstract: Using generating functional and replica techniques, respectively, we study the dynamics and statics of a spherical Minority Game (MG), which in contrast with a spherical MG previously presented in J.Phys A: Math. Gen. 36 11159 (2003) displays a phase with broken ergodicity and dependence of the macroscopic stationary state on initial conditions. The model thus bears more similarity with the original MG. Still, all order parameters including the volatility can computed in the ergodic phases without making any approximations. We also study the effects of market impact correction on the phase diagram. Finally we discuss a continuous-time version of the model as well as the differences between on-line and batch update rules. Our analytical results are confirmed convincingly by comparison with numerical simulations. In an appendix we extend the analysis of the earlier spherical MG to a model with general time-step, and compare the dynamics and statics of the two spherical models.
 Physics , 2004, DOI: 10.1103/PhysRevE.71.066120 Abstract: Minority game is a simple-mined econophysical model capturing the cooperative behavior among selfish players. Previous investigations, which were based on numerical simulations up to about 100 players for a certain parameter $\alpha$ in the range $0.1 \lesssim \alpha \lesssim 1$, suggested that memory is irrelevant to the cooperative behavior of the minority game in the so-called symmetric phase. Here using a large scale numerical simulation up to about 3000 players in the parameter range $0.01 \lesssim \alpha \lesssim 1$, we show that the mean variance of the attendance in the minority game actually depends on the memory in the symmetric phase. We explain such dependence in the framework of crowd-anticrowd theory. Our findings conclude that one should not overlook the feedback mechanism buried under the correlation in the history time series in the study of minority game.
 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.
 Physics , 2000, DOI: 10.1016/S0378-4371(00)00292-2 Abstract: A minority game whose strategies are given by probabilities p, is replaced by a 'simplified' version that makes no use of memories at all. Numerical results show that the corresponding distribution functions are indistinguishable. A related approach, using a random walk formulation, allows us to identify the origin of correlations and self organization in the model, and to understand their disappearence for a different strategy's update rule, as pointed out in a previous work
 V. Sasidevan Physics , 2015, Abstract: The Minority Game (MG) is a prototypical model for an agent-based complex adaptive system. In MG, an odd number of heterogeneous and adaptive agents choose between two alternatives and those who end up on the minority side win. It is known that if $N$ agents play MG, they self-organize to a globally efficient state when they retain the memory of the minority side for the past $m \sim \log_2(N)$ rounds (Challet & Zhang 1997). However, the global efficiency becomes extremely low when the memory of the agents is reduced i.e, when $m << \log_2(N)$. In this work, we consider an MG in which agents use the information regarding the exact attendance on a side for $m$ previous rounds to predict the minority side in the next round. We show that, when employing such strategies, independent of its size, the system is always in a globally efficient state when the agents retain two rounds of information ($m=2$). Even with other values of $m$, the agents successfully self-organize to an efficient state, the only exception to this being when $m=1$ for large values of $N$. Surprisingly, in our model, providing the agents with a random $m=1$ fake history results in a better efficiency than real histories of any length.
 Physics , 2006, Abstract: We study analytically and numerically Minority Games in which agents may invest in different assets (or markets), considering both the canonical and the grand-canonical versions. We find that the likelihood of agents trading in a given asset depends on the relative amount of information available in that market. More specifically, in the canonical game players play preferentially in the stock with less information. The same holds in the grand canonical game when agents have positive incentives to trade, whereas when agents payoff are solely related to their speculative ability they display a larger propensity to invest in the information-rich asset. Furthermore, in this model one finds a globally predictable phase with broken ergodicity.
 Ricardo Mansilla Corona Physics , 1999, Abstract: In this paper we introduce a new approach for the study of the complex behavior of Minority Game using the tools of algorithmic complexity, physical entropy and information theory. We show that physical complexity and mutual information function strongly depend on memory size of the agents and yields more information about the complex features of the stream of binary outcomes of the game than volatility itself.
 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.
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