We present a comprehensive study of utility function of the minority game in its efficient regime. We develop an effective description of state of the game. For the payoff function $g(x)=\sgn (x)$ we explicitly represent the game as the Markov process and prove the finitness of number of states. We also demonstrate boundedness of the utility function. Using these facts we can explain all interesting observable features of the aggregated demand: appearance of strong fluctuations, their periodicity and existence of prefered levels. For another payoff, $g(x)=x$, the number of states is still finite and utility remains bounded but the number of states cannot be reduced and probabilities of states are not calculated. However, using properties of the utility and analysing the game in terms of de Bruijn graphs, we can also explain distinct peaks of demand and their frequencies.
Reply to cond-mat/0503325: Comment on `Generating functional analysis of Minority Games with real merket histories' by KH Ho, WC Man, FK Chow and HF Chau
In this paper we study the continuum time dynamics of a stock in a market where agents behavior is modeled by a Minority Game and a Grand Canonical Minority Game. The dynamics derived is a generalized geometric Brownian motion; from the Black & Scholes formula the calibration of both the Minority Game and the Grand Canonical Minority Game, by means of their characteristic parameters, is performed. We conclude that for both games the asymmetric phase with characteristic parameters close to critical ones is coherent with options implied volatility market.
Generalization of the minority game to more than one market is considered. At each time step every agent chooses one of its strategies and acts on the market related to this strategy. If the payoff function allows for strong fluctuation of utility then market occupancies become inhomogeneous with preference given to this market where the fluctuation occured first. There exists a critical size of agent population above which agents on bigger market behave collectively. In this regime there always exists a history of decisions for which all agents on a bigger market react identically.
We study minority games in efficient regime. By incorporating the utility function and aggregating agents with similar strategies we develop an effective mesoscale notion of state of the game. Using this approach, the game can be represented as a Markov process with substantially reduced number of states with explicitly computable probabilities. For any payoff, the finiteness of the number of states is proved. Interesting features of an extensive random variable, called aggregated demand, viz. its strong inhomogeneity and presence of patterns in time, can be easily interpreted. Using Markov th…
Stationary states of a spherical Minority Game with ergodicity breaking
利用生成泛函和复制技术，分别研究了球形少数者博弈（MG）的动力学和静力学性质.与J.Phys A：Math. Gen. 36 11159（2003）中提出的球形MG相比，该模型显示了一个具有破缺遍历性和宏观定态依赖于初始条件的相.因此，该模型与原始MG具有更多的相似性。所有的序参量包括波动率都可以在遍历阶段计算而不需要任何近似。我们还研究了市场影响修正对相图的影响。最后，我们讨论了一个连续时间版本的模型，以及在线和批量更新规则之间的差异。通过与数值模拟结果的比较，我们的分析结果得到了令人信服的证实。在附录中，我们将早期的球形MG的分析扩展到一般时间步长的模型，并比较了两个球形模型的动力学和静力学。
In this paper the extended model of Minority game (MG), incorporating variable number of agents and therefore called Grand Canonical, is used for prediction. We proved that the best MG-based predictor is constituted by a tremendously degenerated system, when only one agent is involved. The prediction is the most efficient if the agent is equipped with all strategies from the Full Strategy Space. Each of these filters is evaluated and, in each step, the best one is chosen. Despite the casual simplicity of the method its usefulness is invaluable in many cases including real problems. The signifi…
We study a variation of the minority game. There are N agents. Each has to choose between one of two alternatives everyday, and there is reward to each member of the smaller group. The agents cannot communicate with each other, but try to guess the choice others will make, based only the past history of number of people choosing the two alternatives. We describe a simple probabilistic strategy using which the agents acting independently, can still maximize the average number of people benefitting every day. The strategy leads to a very efficient utilization of resources, and the average deviat…
Dynamics of the Batch Minority Game with Inhomogeneous Decision Noise
我们研究了一个版本的批量少数博弈的动态，随机外部信息和不同类型的非齐次决策噪声（加性和乘性），使用生成函数技术à la De Dominance。该模型中的控制参数是外部信息的可能值的数量$p$与交易代理的数量$N$的比率$α=p/N$，以及代理的决策噪声参数的统计特性。决策噪声的存在被发现具有阻尼宏观振荡的一般效果，这解释了为什么在某些参数区域，它可以有效地减少市场波动，如在早期的研究中观察到的。在极限$N\to\infty$中，我们（i）求解动力学的前几个时间步长（对于任何$α$），（ii）计算相变的位置$α_c$（表示异常响应的开始），以及（iii）求解$α>α_c$的静力学。我们发现，$α_c$是不敏感的加性决策噪声，但我们到达的情况下，乘性噪声的非平凡的相图。我们的理论结果在数值模拟中得到了很好的证实。
We propose an analytically tractable variation of the minority game in which rational agents use probabilistic strategies. In our model, $N$ agents choose between two alternatives repeatedly, and those who are in the minority get a pay-off 1, others zero. The agents optimize the expectation value of their discounted future pay-off, the discount parameter being $λ$. We propose an alternative to the standard Nash equilibrium, called co-action equilibrium, which gives higher expected pay-off for all agents. The optimal choice of probabilities of different actions are determined exactly in terms o…
Using virtual stock markets with artificial interacting software investors, aka agent-based models (ABMs), we present a method to reverse engineer real-world financial time series. We model financial markets as made of a large number of interacting boundedly rational agents. By optimizing the similarity between the actual data and that generated by the reconstructed virtual stock market, we obtain parameters and strategies, which reveal some of the inner workings of the target stock market. We validate our approach by out-of-sample predictions of directional moves of the Nasdaq Composite Index…
We use the Minority Game as a testing frame for the problem of the emergence of diversity in socio-economic systems. For the MG with heterogeneous impacts, we show that the direct generalization of the usual agents' profit does not fit some real-world situations. As a typical example we use the traffic formulation of the MG. Taking into account vehicles of various lengths it can easily happen that one of the roads is crowded by a few long trucks and the other contains more drivers but still is less covered by vehicles. Most drivers are in the shorter queue, so the majority win. To describe suc…
We study the market impact of a meta-order in the framework of the Minority Game. This amounts to studying the response of the market when introducing a trader who buys or sells a fixed amount h for a finite time T. This perturbation introduces statistical arbitrages that traders exploit by adapting their trading strategies. The market impact depends on the nature of the stationary state: We find that the permanent impact is zero in the unpredictable (information efficient) phase, while in the predictable phase it is non-zero and grows linearly with the size of the meta-order. This establishes…
We use the cavity method from statistical physics for analyzing the transient and stationary dynamics of a minority game that is played by agents performing market arbitrage. On the level of linear response the method allows to include the reaction of the market to individual actions of the agents as well as the reaction of the agents to individual information items of the market. This way we derive a self-consistent solution to the minority game. In particular we analyze the impact of general nonlinear price functions on the amount of arbitrage if noise from external fluctuations is present. …
