Generating Functional Analysis of the Dynamics of the Batch Minority Game with Random External Information

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 $\alpha =p\mathrm{/}N$ of the number $p$ of possible values for the external information over the number $N$ of trading agents. In the limit $N\to \mathrm{\infty }$ 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 $\sigma =\text{order}({\alpha }^{-1\mathrm{/}2})$ (as regularly observed in simulations), but also asymptotic solutions with vanishing volatility $\sigma =\text{order}({\alpha }^{1\mathrm{/}2})$. The former, however, are shown to emerge only if the agents’ initial strategy valuations are below a specific critical value.