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Abstract: In mix-game which is an extension of minority game, there are two groups of agents; group1 plays he majority game, but the group2 plays the minority game. This paper studies the change of the average winnings of agents and volatilities vs.  the change of mixture of agents in mix-game model. It finds that the correlations between the average winnings of agents and the mean of local volatilities are different with different combinations of agent memory length when the proportion of agents in group 1 increases. This  study result suggests that memory length of agents in group1 be smaller than that of agent in group2 when mix-game model is used to simulate the financial markets. 

 Introduction:  

Challet and Zhang's MG model, together with the original bar model of Arthur [1], attracts a lot of following studies. Given the MG's richness and yet underlying simplicity, the MG has also received much attention as a financial market model [2]. The MG comprises an odd number of agents choosing repeatedly between the options of buying  (1) and selling (0) a quantity of a risky asset. The agents continually try to make the minority decision i.e. buy assets when the majority of other agents are selling and sell when the majority of other agents are buying. Neil F. Johnson [3, 4] and coworkers extended MG by allowing a variable number of active traders at each timestep--- they called their modified game as the Grand Canonical Minority Game (GCMG). The GCMG, and to a  lesser extent the basic MG itself, can reproduce the stylized facts of financial markets, such as volatility clustering and fat-tail distributions. However, there are some weaknesses in MG and GCMG. First, the diversity of agents is limited, since agents all have the same memory and time-horizon. Second, in real markets, some agents are tendency-followers,  i.e. “noise traders”  [5, 6, 7, 8, 9, 10, 11, 12], who effectively play a majority game; while others are “foundation traders”, who effectively play a minority game. In order to create an agent-based model which more closely mimics a real financial market I proposed a mix-game model which is a modification of MG [13]. In mix-game model there are two groups of agents: each group has different memory and time-horizon. The most important modification is to make one group plays the minority game and the other plays the majority game. Through simulations, I find out that the fluctuations  of local volatilities change a lot by adding some agents who play majority game into MG, but the stylized features of MG don’t change obviously except agents with memory  length 1 and 2. I also give suggestions about how to use mix-game to model financial markets and show the example of modelling Shanghai stock market by means of mix-game model [13]. In this paper, I further examine the correlations between the average winnings of agents and the local volatilities of systems in mix-game model when the proportion of agents in group1 increases from 0 to 0.4. In section 2,  I describe the mix-game model and the simulation conditions. In section 3, the simulation results and discussion are presented. In section 4, I calculate the quantitative correlations. In section 5, the conclusion is reached.     

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