Effective history length of the minority game

It is known that the memory is relevant in the symmetric phase of the minority game. In our previous work we have successfully explained the quasi-periodic behavior of the game in the symmetric phase with the help of the probability theory. Based on this explanation, we are able to determine how the history length affects the variance of the system in this paper. By using some particular types of fake history such as periodic type and random type, we determine how efficient the history length has been used in the standard game. Furthermore, the analysis on the effective history length strongly supports the result we proposed previously that there are three distinct phases in the minority game.