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 α=p/N of the number p of possible values for the external information over the number N of trading agents. In the limit N>∞ we calculate the location α_{c} of the phase transition (signaling the onset of anomalous response), and solve the statics for α>α_{c} exactly. The temporal correlations in global market fluctuations turn out not to decay to zero for infinitely widely separated times. For α<α_{c} the stationary state is shown to be nonunique. For α>0 we analyze our equations in leading order in α, and find asymptotic solutions with diverging volatility σ=O(α^{1/2}) (as regularly observed in simulations), but also asymptotic solutions with vanishing volatility σ=O(α^{1/2}). The former, however, are shown to emerge only if the agents' initial strategy valuations are below a specific critical value.
 Publication:

Physical Review E
 Pub Date:
 May 2001
 DOI:
 10.1103/PhysRevE.63.056121
 arXiv:
 arXiv:condmat/0012045
 Bibcode:
 2001PhRvE..63e6121H
 Keywords:

 02.50.Le;
 87.23.Ge;
 05.70.Ln;
 64.60.Ht;
 Decision theory and game theory;
 Dynamics of social systems;
 Nonequilibrium and irreversible thermodynamics;
 Dynamic critical phenomena;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Quantitative Finance  Trading and Market Microstructure
 EPrint:
 15 pages, 6 figures, uses Revtex. Replaced an old version of volatility graph that. Rephrased and updated some references