  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.
