Gauss Approximation in the Optimization Problem of the Minority Game Model


  • V.S. Yanishevsky Ivan Franko State Drohobych Pedagogical University





The optimization problem of the well-known minority game model is studied by methods of statistical physics. The problem is reduced to the study of the ground state of some effective system with continuous spin described by a replica Hamiltonian with random parameters. With the use of the central limit theorem of probability theory, the representations of the distribution function for parameters of the Hamiltonian are obtained, and the transition to the Gauss distribution in the case of large P is realized. Within
approximations 1RSB and 2RSB in the replica method, the dependence of the minimum of the quantity under study on the parameter αis determined. It is shown that, in the region of applicability, the proposed method gives a less value of the minimum than that obtained in the cited works.


W. Kinzel, in Proceed. Int. Sympos. on Synergetics, 1985, edited by H. Haken (Springer, Berlin, 1986), p. 107.

I.Ya. Korenblit and E.F. Shender, Uspekhi Fiz. Nauk 157, 2 (1989).

H. Nishimori, Statistical Physics of Spin Glasses and Information Processing. An Introduction (Clarendon Press, Oxford, 2001).

R.N. Mantegna and H.E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge Univ. Press, Cambridge, 1999).

A. De Martino and M. Marsili, Preprint (physics/0606107), (2006).

D. Challet and Y.C. Zhang, Physica A 246, 407 (1997).

D. Challet and Y.C. Zhang, Physica A 256, 514 (1998).

R. Savit, R. Manuca, and R. Riolo, Phys. Rev. Lett. 82, 2203 (1999).

D. Challet, M. Marsili, and R. Zecchina, Phys. Rev. Lett. 84, 1824 (2000) (cond-mat/9904392).

Y.C. Zhang, Physica A 30, 269 (1999), (condmat/9901243).

M. Marsili, D. Challet, and R. Zecchina, Physica A 280, 522 (2000), (cond-mat/9908480).

A. De Martino and M. Marsili, J. Phys. A: Math. Gen. 34, 2525 (2001), (сond-mat/0007397).

H. Moulin, Game Theory with Examples from Mathematical Economics (Mir, Moscow, 1972) (in Russian).

V. Yanishevsky, Zh. Fiz. Dosl., 13, 3602 (2009).

M. Mezard, G. Parisi, and M.A. Virasoro, Spin Glass Theory and Beyond (World Scientific, Singapore, 1987).

R.P. Feynman, Statistical Mechanics (Addison-Wesley, Reading, 1972).

W. Feller, An Introduction to Probability Theory and Its Applications, (Wiley, New York, 1970).

J.R.L. de Almeida and D.J. Thouless, J. Phys. A 11, 983 (1978).

P. Lancaster, Theory of Matrices (Academic Press, New York, 1969).

A. Crisanti, D.J. Amit, and H. Gutfreund, Europhys. Lett. 2, 337 (1986).

H. Steffan and R. K¨uhn, (cond-mat/9404036v1) (1994)

S.K. Ghatak and D. Sherrington, J. Phys. C 10, 3149 (1977).

G.R. Schreiber, (cond-mat/9612189v2) (1999).

F.A. da Costa and J.M. de Ara'ujo, cond-mat/0005029v1 (2000).



How to Cite

Yanishevsky В. (2022). Gauss Approximation in the Optimization Problem of the Minority Game Model. Ukrainian Journal of Physics, 56(1), 80.



General problems of theoretical physics