Dynamical Solution of the On-Line Minority Game

Abstract

We solve the dynamics of the on-line minority game, with general types of decision noise, using generating functional techniques a la De Dominicis and the temporal regularization procedure of Bedeaux et al. The result is a macroscopic dynamical theory in the form of closed equations for correlation- and response functions defined via an effective continuous-time single-trader process, which are exact in both the ergodic and in the non-ergodic regime of the minority game. Our solution also explains why, although one cannot formally truncate the Kramers-Moyal expansion of the process after the Fokker-Planck term, upon doing so one still finds the correct solution, that the previously proposed diffusion matrices for the Fokker-Planck term are incomplete, and how previously proposed approximations of the market volatility can be traced back to ergodicity assumptions.

25 pages LaTeX, no figures

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• Generating functional analysis of batch minority games with arbitrary strategy numbers
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• Incorporating Inertia Into Multi-Agent Systems
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• The cavity method for minority games between arbitrageurs on financial markets
• From Minority Game to Black & Scholes pricing
• Statistical Mechanics of Dilute Batch Minority Games with Random External Information
• Market response to external events and interventions in spherical minority games
• Effective memory of the minority game