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4 The Effect of Irrational Behavior on an Artificial Stock Market Note that equations (2) and (3) are only true for bounded rational agents. For irrational agents, specific algorithms will be executed to form expected price. (3) 6.2 Market Equilibrium In the market equilibrium segment, the program determines the market equilibrium price. It is assumed that each agent has CARA utility function in (4) Each agent is given a fixed amount of cash c(t) and stock q(t) at the initial period. The initial budget equation can be written as (5). The expected next period budget can be represented by (6). The wealth of an agent includes the risk free asset (cash) and the risky asset (stock). The anticipated wealth is conditioned on the? s of the agents. The cash asset has a fixed rate of return (r) and the rate of return of the risky asset is determined by the actual price. Substituting (5) into (6) we get equation (7). To maximize the utility function, equation (4) can be represented by (8). (4) (5) (6) (7) (8) From (7), we can derive (9) and (10). (9) Substitute (9) and (10) into (8), we get (10) (11) From equation (11), we get (12). The individual demand can be represented by (13). (12) (13) 663

5 Yue, Chaturvedi, and Mehta The individual cash demand can be represented by (14). (14) Given a constant X number of outstanding shares, m number informed and n number uninformed agents in the market, the market price is determined in equation (15). (15) Using the market price, the actual individual demand can be represented by (16). (16) In the end of the period, given that all agents realize the actual value of the stock, individual wealth can be represented by (17). (17) 6.3 Agent Evaluation The main task in this segment is to update the agent status. Individual agent wealth is evaluated once the actual wealth is realized. Agents with negative wealth are considered bankrupt and will be eliminated from the market. In future experiments, agent s belief, agent learning, agent strategies effectiveness, etc will be evaluated and updated in this stage. 6.4 Market Value Evaluation This segment makes sure the market value stays constant. The gain or loss of the stock market value is determined by (18). (18) if S(t) > 0, there will be a lump sum tax imposed on all agents. If S(t) < 0, new agents will be added into the economy and their agent type (informed, uninformed, rational, or irrational) will identical to the most successful agent. 6.5 Parameters Simulations enable modelers to construct multiple scenarios. Parameters dictate the overall structure of the model. Table 1 provides some of the general parameters of the model; specific parameters related to the design of the individual irrational agents will be provided in the future. 664

6 The Effect of Irrational Behavior on an Artificial Stock Market Table 1. Model Parameters Parameters Number of period Total number of agent Rational agent percentage Initial number individual shares Initial stock price Initial cash for informed agents Initial cash for uninformed agents Individual absolute risk aversion Fixed asset rate of return Variance of Mean of Descriptions Total number of periods in simulation Give the number of informed and uninformed agents All agents are assumed to have the same number of stocks initially. Individual initial wealth is the sum of initial cash and the initial security asset value All agents possess the same preference in risk Interest rate remains constant for all periods Variables that dictate the actual market and expected individual prices. Variance of? 1 Mean of? 1 Variance of? 2 Mean of? 2 7. CONCLUSION AND FUTURE DIRECTION This research encompasses areas in economics and finance, psychology and human behavior, and agent modeling and artificial intelligence. This paper lays down the framework of the market structure; the next phase of the project is to design and implement the agents. Individual agent behavior will be applied by specific algorithm. Part of the appeal of using the simulation approach is that we can model multiple agent behaviors that resemble the real-life traders. The goal is to learn about the potential market performance in the presence of aggregation of individual behaviors. So far, this research has laid down a market framework with simple agents. The next focus is to build more detailed irrational agents. Some of the irrational agents can be easily modeled by incorporating simple rules in them. A few of the irrational agent examples are: Momentum agents: They form their expectation of prices based solely on the movement of past prices. As a result, private market information does not affect their decisions. Noise agents: They form their price expectation randomly based on uniform distribution. The boundary of the distribution sets up the randomness range in the expectation. Generic agents: They form their price expectation based on the market information they receive. Informed agents make decisions based on more market information than the uninformed agents because they receive more private market information. Learning agents: They are the extension of generic agents that possess certain intelligence with the ability to remember past information. This type of agent is important in cases where the distribution of the information that constitutes the actual market price is different than the distribution of the private information received by the agents. By forming their price expectation based on past public market information and current private information, they can slowly learn about the distribution of the signals. One easy way to implement this is to average out past and current information to form price expectation. Other ways include using genetic algorithms to learn about the distribution of the market information. 665

7 Yue, Chaturvedi, and Mehta Overconfident agents: They over-weigh their private market information they receive from the market. This can be implemented by assuming they underestimate noise variance while other agents do not. As a result, they are over-weighing their private market information. Two approaches can be used to implement adaptive or dynamic elements into the model. One is by considering the agent behaviors described above as potential trading strategies of the agent. Classifier systems and genetic algorithms can be applied to agents to learn the more effective strategy or combination of strategies. Another approach is to assume that each agent displays only one type of behavior defined above. An ineffective agent is eliminated from the market and replaced by an agent that belongs to the same type as the wealthiest agent. We will use the latter approach for now and may apply the first approach in the future. We are currently designing specific agents, and hopefully we will have the results soon. Reference Alemdar, N. M., and Ozyildirim, S. A Genetic Game of Trade, Growth and Externalities, Journal of Economic Dynamics and Control (22), 1998, pp Arthur, B., Holland, J., LeBaron, B., Palmer, R., and Tayler, P. Asset Pricing Under Endogenous Expectations in an Artificial Stock Market, Economic Notes (26), 1997, pp Bloomfield, R., and O Hara, M. Market Transparency: Who Wins and Who Losses? The Review of Financial Studies (12), 1999, pp Bullard, J., and Duffy, J. A Model of Learning and Emulation with Artificial Adaptive Agents, Journal of Economic Dynamics and Control (22), 1998, pp Daniel, K., Hirshleifer, D., and Subrahmanyam, A. Investor Psychology and Security Market Under- and Overreactions, The Journal of Finance (53:6), 1998, pp Fama, E. F. The Behavior of Stock Market Prices, Journal of Business (38), 1965, pp Flood, M. D., Huisman, R., Koedijk, K. G., and Mahieu, R. J. Quote Disclosure and Price Discovery in Multiple-Dealer Financial Markets, The Review of Financial Studies (12), 1999, pp Friedman, M. The Case for Flexible Exchange Rates, Essays in Positive Economics, Chicago: University of Chicago Press, Grossman, S. On the Efficiency of Competitive Stock Markets Where Trades Have Diverse Information, The Journal of Finance (31), 1976, pp Kahneman, D., and Tversky, A. Prospect Theory: An Analysis of Decision under Risk, Econometrica (47:2), 1979, pp LeBaron, B., Arthur, B., and Palmer, R. Time Series Properties of an Artificial Stock Market, Journal of Economic Dynamics and Control (23), 1999, pp Lettau, M. Explaining The Facts With Adaptive Agents: The Case of Mutual Fund Flows, Journal of Economic Dynamics and Control (21), 1997, pp Midgley, D. F., Marks, R. E., and Cooper, L. G. Breeding Competitive Strategies, Management Science (43), 1997, pp Odean, T. Volume, Volatility, Price, and Profit When All Traders Are Above Average, The Journal of Finance (53:6), 1998, pp Shleifer, A., and Summers, L. H. The Noise Trader Approach to Finance, Journal of Economic Perspectives (4), 1990, pp

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