The graph of y=p(x) in the x-y plane is a symmetric bell-shaped curve centered around the line x=0 with thin infinitely elongated tails
where w_{t} is a mean-zero Gaussian process with finite variance, identically distributed for every t. In all experimental treatments, the variance is set at 0.0743. A numerical value for the exogenously given initial dividend d_{0} is not specified.
Additional quantity restrictions are also imposed in the implementation. Specifically, a trader is not permitted to trade more than 10 shares each period, and is not permitted to sell short more than 5 shares each period.
In all experimental treatments, c=0.005.
Note: The following time-line is for an arbitrary period t with t greater than 1
If trader i has multiple activated forecasting rules j in period t, a rule j' with the smallest actual current forecast variance is used to generate this expectation.
The demand function (5) assumes that trader i's demand for stock A in period t is proportional to the deviation between stock A's period t price p_{t} and stock A's period t fundamental value (the period t present value of its period t+1 price p_{t+1} plus period t+1 dividend payment d_{t+1}.) The proportionality factor is a function of both absolute risk aversion (measured by lambda) and confidence in the forecasted expectation (as measured by the forecast variance var_{tij'}). If stock prices have a Gaussian distribution, this demand function can be shown to maximize trader i's expected utility of period-(t+1) wealth subject to his budget constraint (2). Otherwise, (5) is an approximation for the optimal demand function.
NOTE: See LeBaron et al. (1999) for a more detailed discussion of the design and implementation of GA condition/forecast classifier learning in Arthur et al. (1997).
where r denotes the risk-free net return rate and d denotes the latest dividend paid by the risky stock A. Note that pr/d is greater than 1.000 if and only if p is greater than the risky stock's previous-period fundamental value
The next four descriptors correspond to technical trading measures of the current market state. They indicate whether a price trend is under way. Specifically, they indicate whether the current price p is greater than an n-period moving average of past prices, where n takes on the values (5,10,100,500).
The last two descriptors contain no actual market information and hence serve only as experimental controls. They indicate the degree to which agents are learning to act in response to "useless" information. In all experimental treatments, the 11th descriptor is always set to 1 and the 12th descriptor is always set to 0.
Thus, var_{tij} is a "moving average" of the rule's squared forecast error (i.e., the difference between actual and forecasted price plus dividend). The parameter theta determines the weight (theta) placed on the most recent squared error versus the weight (1 - theta) placed on the sum of past squared errors. The value of theta is set differently in each experimental treatment (see the discussion on experimental design, below). The value of theta determines the horizon length that the traders consider relevant for forecasting purposes; the closer theta is to 1, the shorter this horizon becomes.
NOTE: THE MANNER IN WHICH FORECASTING RULES ARE ACTIVATED AND SELECTED FOR USE IN THE INITIAL PERIOD 1 IS NOT EXPLAINED IN EITHER ARTHUR ET AL. (1997) OR LEBARON ET AL. (1999).
NOTE: LeBaron et al. (1999) apparently re-ran all of the experiments in Arthur et al. (1997) with the crossover probability P always set equal to P = 0.10 and theta always set equal to theta = 1/75. Thus, in LeBaron et al. (1999), the only difference between the two experimental treatments was a single parameter k, the "forecast horizon" dictating the average interval length between invocations of the GA for each trader. The authors apparently found no significant differences with the earlier obtained results in Arthur et al. (1997).