Heimel J A, Coolen A C
Department of Mathematics, King's College London, The Strand, London WC2R 2LS, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2001 May;63(5 Pt 2):056121. doi: 10.1103/PhysRevE.63.056121. Epub 2001 Apr 24.
We study the dynamics of the batch minority game, with random external information, using generating functional techniques introduced by 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-->infinity we calculate the location alphac of the phase transition (signaling the onset of anomalous response), and solve the statics for alpha>alphac exactly. The temporal correlations in global market fluctuations turn out not to decay to zero for infinitely widely separated times. For alpha
我们使用由德多米尼斯引入的生成泛函技术，研究具有随机外部信息的批量少数者博弈的动力学。该模型中的相关控制参数是外部信息可能值的数量(p)与交易代理数量(N)的比值(\alpha = p/N)。在(N\to\infty)的极限情况下，我们计算相变的位置(\alpha_c)（标志着异常响应的开始），并精确求解(\alpha>\alpha_c)时的静态情况。结果表明，对于无限远分离的时间，全球市场波动中的时间相关性不会衰减到零。对于(\alpha<\alpha_c)，稳态被证明是非唯一的。对于(\alpha\to0)，我们在(\alpha)的主导阶分析我们的方程，发现具有发散波动率(\sigma = O(\alpha^{(-1/2)}))的渐近解（如模拟中经常观察到的），但也有波动率消失(\sigma = O(\alpha^{(1/2)}))的渐近解。然而，只有当代理的初始策略估值低于特定临界值时，前者才会出现。
