Continuum time limit and stationary states of the minority game
Abstract
We discuss in detail the derivation of stochastic differential equations for the continuum time limit of the minority game. We show that all properties of the minority game can be understood by a careful theoretical analysis of such equations. In particular, (i) we confirm that the stationary state properties are given by the ground state configurations of a disordered (soft) spin system, (ii) we derive the full stationary state distribution, (iii) we characterize the dependence on initial conditions in the symmetric phase, and (iv) we clarify the behavior of the system as a function of the learning rate. This leaves us with a complete and coherent picture of the collective behavior of the minority game. Strikingly we find that the temperaturelike parameter, which is introduced in the choice behavior of individual agents turns out to play the role, at the collective level, of the inverse of a thermodynamic temperature.
- Publication:
-
Physical Review E
- Pub Date:
- November 2001
- DOI:
- 10.1103/PhysRevE.64.056138
- arXiv:
- arXiv:cond-mat/0102257
- Bibcode:
- 2001PhRvE..64e6138M
- Keywords:
-
- 02.50.Le;
- 05.40.-a;
- 64.60.Ak;
- 89.90.+n;
- Decision theory and game theory;
- Fluctuation phenomena random processes noise and Brownian motion;
- Renormalization-group fractal and percolation studies of phase transitions;
- Other topics in areas of applied and interdisciplinary physics;
- Condensed Matter - Statistical Mechanics;
- Condensed Matter - Disordered Systems and Neural Networks
- E-Print:
- Revised version (several new results added). 12 pages, 5 figures