Three phases of the minority game

We define three phases of the minority game according to the value ρ=2M/N, the ratio of the number of entries in a strategy to the number of agents. The time sequence of the population in one room has distinct feature in different phase. It shows quasi-periodic structure in the first phase where ρ<ρc1. In the second phase, ρc1<ρ<ρc2, agents coordinate better and better as ρ gets larger. In the third phase, ρ>ρc2, the system is able to reach its best performance in reducing the population variance. The crucial factor, which hinders the system to reach its best performance in the first and second phases, is the number of agents who switch strategy at the same time. When ρ<ρc2, the constraint that bound 2M predictions in a strategy is small, so that too many agents will switch to a better strategy simultaneously. As a result, they form a majority group and are more likely to lose. We give analytical approximate formulas for the population variance in the first and third phases. We have also determined the phase-changing point ρc1.