title: Generating functional analysis of the dynamics of the batch minority game with random external information.
abstract: 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<alphac the stationary state is shown to be nonunique. For alpha-->0 we analyze our equations in leading order in alpha, and find asymptotic solutions with diverging volatility sigma=O(alpha(-1/2)) (as regularly observed in simulations), but also asymptotic solutions with vanishing volatility sigma=O(alpha(1/2)). The former, however, are shown to emerge only if the agents' initial strategy valuations are below a specific critical value.
authors: Department of Mathematics, King's College London, The Strand, London WC2R 2LS, United Kingdom.
authors: J A
authors: JA
authors: Heimel
authors: A C
authors: AC
authors: Coolen