Artificial economic life: a simple model of a stockmarket

We describe a model of a stockmarket in which independent adaptive agents can buy and sell stock on a central market. The overall market behavior, such as the stock price time series, is an emergent property of the agents’ behavior. This approach to modelling a market is contrasted with conventional rational expectations approaches. Our model does not necessarily converge to an equilibrium, and can show bubbles, crashes, and continued high trading volume. 1. Approaches to economic theory In recent years the prevailing rational expectations approach to economic theory has been challenged from several quarters, and increasing interest has been shown in an alternative evolutionary economics viewpoint. In this paper we describe and contrast these paradigms, and discuss our artificial stockmarket model as an example of the evolutionary approach. Our approach is fundamentally based on the inductive theory of learning described in Arthur ( 1992 ) [ 11. This stockmarket model may also be seen as a casestudy in artificial life; from a random soup of simple rules an intelligent system spontaneously organizes, and develops more and more sophisticated behavior as time goes on, rather like life emerging from a prebiotic soup. We emphasize the background and general structure of our model, only indicating results in general terms. Related papers [ 1,2] provide further details. The paper is written for physical scientists, without assuming any background in economics. 1.1. Rational expectations theory In conventional economic theory the standard approach to most problems is fundamentally based on Rational Expectations (RE) theory. According to RE theory, agents deduce their optimum behavior by logical processes from the circumstances of any situation, assuming that other agents do likewise. Here an agent might be an individual, a firm, a state, etc. This seemingly reasonable approach in fact involves several 0167-2789/94/$07.00 @ 1994 Elsevier Science B.V. All rights reserved SSDIO167-2789(94)00078-S R.G. Palmer et al. I Physica D 75 (1994) 264-274 265 strong (and unreasonable) assumptions, and has a number of undesired consequences. Nevertheless it has long been the regnant paradigm, perhaps in part because it leads to very appealing mathematics. Among the assumptions normally made in RE theory are: (i) Complete Information. All agents are assumed to have full knowledge of the problem. (ii) Perfect Rationality. All agents are assumed to be perfectly able to deduce their optimum behavior, no matter how complex the computational problem. (iii) Common Expectations. All agents are assumed to know that all others are working with the same information on the same “perfectly rational” basis. And they know that the others know this too, and that the others know that they know they know, and so on ad infinitum. As a simple example, imagine 20 computer companies who are independently considering the adoption of a new standard 2 for a graphical user interface. A marketing analysis might show that all would benefit if at least 15 adopted Z, but that the adopters would experience a net loss if fewer than 15 adopted it. RE theory predicts that all companies will adopt Z immediately, because they will all reason as follows. If I were the 15th-20th company to consider adoption, it would obviously be in my interest to do so. If I were the 14th, I would adopt because then it would be advantageous for the 15th-20th to do so. If I were the 13th, I would do so because then the 14th would do so, by the preceding argument. As so on, all the way down to the first adopter. Since all will perform this reasoning, all will be ready to be first, and all will jump in immediately (and will expect the others to do so too). Note that the agents figure out the solution initially (“at time O”), and that there is no subsequent dynamics, learning, or evolution. Of course this outcome is not what would be expected in practice, partly because of the failure of the above assumptions, and partly because of other factors not included in our simple model. However it still serves to illustrate both the assumptions and the flavor of an RE argument, although in most applications the mathematical optimization problem is far more complicated. In complex problems, RE theory runs into a number of difficulties, especially because the three assumptions listed above are typically not satisfied. (i) Lack of Complete Information. Agents may have to learn about the context or about the other agents while the “game” is being played out. Problem contexts may themselves not be fully defined initially, only becoming explicit through the choices of the agents. (ii) Lack of Perfect Rationality. Real persons and firms often aren’t clever enough or don’t have enough computational power to compute a true optimum. And even if they have the power, they may not use it, preferring instead rules of thumb that have worked elsewhere. (iii) Lack of Common Expectations. Different agents may well have different information about a situation, and may well use different approaches. They cannot rely on others to duplicate their own reasoning. These difficulties lead in turn to predictions that do not always fit observed outcomes. And even when final outcomes are correctly predicted by RE theory, the theory is silent about the dynamical process (typically involving trial and error, and learning) that agents actually take to reach that solution. 1.2. Evolutionary economics In part because of the difficulties with the standard RE theory, in recent years many researchers have investigated alternative approaches. Some have attempted to perturb away from the perfect rationality ideal with a variety of bounded rationality theories. These theories impose an inten266 R.G. Palmer et al. I Physica D 75 (1994) Z-274 tional limitation on some aspect of an agent’s task, such as the available knowledge, the computational time or complexity, the memory capacity, the forecasting repertoire, etc. One difficulty is that there are many dimensions in which to bound rationality, and no clear guiding principle for how to set the direction and distance from the zenith of perfect rationality. Another approach, into which the current work falls, is to start from the opposite end of the scale with agents who initially have little rationality or specialized knowledge. The agents are then allowed to adapt, or learn, or evolve, eventually becoming reasonably expert in their own domains. There are a number of advantages to this approach, including None of the three assumptions discussed above for RE theory is required. Even the modeler does not need to have the knowledge or computational power to derive an optimum solution for each agent. The evolutionary approach is generally inductive, not deductive; the agents typically generalize patterns observed in the past to guide their behavior in the future. This inductive approach is much closer to normal human behavior than the deductive one of deriving particular choices from general principles [ 1,8 1. The general approach is applicable even in situations where conventional RE theory produces no answers, e.g., due to lack of a single well-defined equilibrium solution. The approach can predict and interpret dynamical behavior, not just final outcomes. Agents can continue to adapt in a changing or ill-defined world (perhaps of their own making) whose characteristics cannot or are not known in advance. The biggest disadvantages of the evolutionary approach are the general lack of analytic methods most work is largely computational and the plethora of possible algorithms for learning and adaptation. The field is presently in an exploratory phase, determining by explicit simulation the potentials and limitations of particular evolutionary models. A narrowing of options and more rigorous results can be expected in the future. 2. An artificial stockmarket 2.1. General framework Turning specifically to financial markets, we first construct the framework of a simple kind of stockmarket, and then consider different approaches (RE and evolutionary) to the agents’ decision problem. Our market will have S kinds of stocks labelled by Q = 1, 2, . . ., S, and N agents labelled by i = 1, 2, . . ., N. The agents are not necessarily homogeneous; they may have quite different operating principles. For simplicity we make time t discrete, so t = 0, 1, 2, . . ., and refer to the interval from t 1 to t as the tth period. There is no predelined time horizon; in principle the market continues for ever. At each time t, each agent i has some number of shares (or holding) hq (t ) of each stock CL There are no complex instruments such as options, and no direct interaction between pairs of agents. The agent’s essential problem is to choose hy (t) at each time t, given various constraints such as a finite net wealth. The goal might be to maximize expected (mean) profit, or might involve a more complicated “utility function” which takes risk into account. The price pa (t) per share of each stock depends mainly on the overall buying and selling behavior of the agents. The companies issuing the stocks may also pay cash dividends d” (t) per share to each stockholder, in an amount depending on company success and policies. Agents can thus make profit in two ways, through the dividend stream and through speculation, relying on price changes of their shares. In addition to stock holdings we need to take into account other assets of each agent, so that not all wealth needs to be invested in stock. For simplicity we regard all other assets collectively R.G. Palmer et al. I Physica D 75 (1994) 264-274 267 as cash, or money A4i (t ) . An agent’s total wealth wi (t) at any time t is thus given by wi(t) = Mi(t) + Chy(t)pa(t). a (1) During the tth period, the price per share of stock cr changes from pa (t 1) to pa (t ) and a dividend d”(t) is declared. We also assume that the agent’s cash is invested in a fixed-rate fund such as a savings account, which pays an interest rate r per period so that Mi (t 1) becomes ( 1 + r )Mi (t 1). Accounting for these changes, the agent’s wealth at the end of a period is given