On the Minority Game : Analytical and Numerical Studies

We investigate further several properties of the minority game we have recently introduced. We explain the origin of the phase transition and give an analytical expression of $\sigma^2/N$ in the $N\ll2^M$ region. The ability of the players to learn a given payoff is also analyzed, and we show that the Darwinian evolution process tends to a self-organized state, in particular, the life-time distribution is a power-law with exponent -2. Furthermore, we study the influence of identical players on their gain and on the system's performance. Finally, we show that large brains always take advantage of small brains.