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.

Adapting to heterogeneous comfort levels

“…The two observables which we will focus on, namely the mean attendance and the fluctuations of the attendance, can be computed from the effective particle problem as follows. Similarly to [14] one finds that…”

“…We here consider the so-called batch process of the learning dynamics, which for the present case reads (3) similar to that of the standard MG [14,2]. We have here introduced the score differences…”

“…The further analysis then assumes the existence of an ergodic stationary state (in which correlation and response functions depend only on time-differences, i.e. C tt ′ = C(t − t ′ ) and similarly for G and G ′ and in which the integrated responses remain finite) and proceeds along the lines of [14]. We will not report these further steps here, but only quote some of the resulting equations describing the persistent order parameters in the appendix.…”

Dynamical mechanisms of adaptation in multiagent systems

“…The interpretation is that when ␣ is large, the distributions of strategies become so sparse that motions in the phase space cannot be achieved by the switching of strategies. Note that the value of ␣ c is close to the value of 0.337 obtained by the continuum approximation [4] or batch update [10].…”

“…System efficiency can be improved by random initial conditions in systems driven by internal information [7], or external information [8], accompanied by hysteresis [9]. The same is valid in models with batch update [10] and their noisy extension [11].…”