Agent Based Models

1 1 BUT INDIVIDUALS ARE NOT GAS MOLECULES A. Chakraborti TOPICS TO BE COVERED IN THIS CHAPTER: • Agent based models: going beyond the simple statistical mechanics of colliding particles • Explaining the hidden hand of economy: self-organization in a collection of interacting "selfish" agents • Bio-box on A Smith • Minority game and its variants (evolutionary, adaptive etc) • Box on contributions of: WB Arthur, YC Zhang, D Challet, M Marsili et al • Agent-based models for explaining the power law for price fluctuations 1.1 Agent based models: going beyond the simple statistical mechanics of colliding particles One might actually wonder how the mathematical theories and laws which try to explain the physical world of electrons, protons, atoms and molecules could be applied to understand the complex social structure and economic behavior of human beings. Specially so, since human beings are complex and widely varing in nature, properties or characteristics, whereas e.g., the electrons are identical (we do not have to electrons of different types) and indistinguishable. Each electron has a charge of 1.60218 × 10−19C, mass of 9.10938 × 10−31kg, and is a spin-1/2 particle. Moreover, such properties of Econophysics. Sinha, Chatterjee, Chakraborti and Chakrabarti Copyright c 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-XXXXX-X 2 1 BUT INDIVIDUALS ARE NOT GAS MOLECULES electrons are universal (identical for all electrons), and this amount of information is sufficient to explain many physical phenomena concerning electrons, once you know the inter actions. But is such little information sufficient to in- fer the complex behavior of human beings? Is it possible to quantify the nature of the interactions between human beings? The answers to these questions are on the negative. Nevertheless, during the past decade physicists have made attempts in studying problems in Economics, the “Social science that analyzes and describes the consequences of choices made concerning scarce resources”. We have already seen in the previous chapters, models of wealth exchange between individuals or economical entities, referred to as kinetic wealth exchange models (KWEMs), since they provide a description of wealth flow in terms of stochastic wealth exchange between agents, resembling the energy transfer between the molecules of a fluid. We do realize that even though KWEMs have been the subject of intensive investigations, their economical interpretation is still an open problem. It is important to keep in mind that in the framework of a KWEM the agents should not be related to the “rational” agents of neoclassical economics: an interaction between two agents does not represent the effect of decisions taken by two economic agents who have full information about the market and behave rationally in order to maximize their utility. The description of wealth flow provided by KWEMs takes into account the stochastic element, which does not respond by definition to any rational criterion. Also some terms employed in the study of KWEMs, such as saving propensity, risk aversion, etc., should be taken with caution since they might seem to imply a decisional aspect behind the behavior of agents. How- ever, it is interesting to note that very recently, Chakrabarti and Chakrabarti have put forward a microeconomic formulation of the above models, using the utility function as a guide to the behavior of agents in the economy, which might actually bring the cross-exchange of ideas even more. Of course, the behaviour of most of the complex systems found in natural and social environments can be characterized by the competition among interacting agents for scarce resources and their adaptation to the environment [17, 18]. The different agents could be diverse in form and in capability, for example, cells in an immune system to great firms in a business centre accord- ing to the system considered. In these dynamically evolving complex systems the nature of agents and their manners also differ. In order to have a deeper understanding of the interactions of the large number of agents, one should study the capabilities of the individual agents. For simplicity, a n agent’s behaviour may be thought of as a collection of rules governing “responses” to “stimuli”. For example, a typical reponse of an animal (prey) when it sees a predator, is that it should run, or if the stock indices fall then a financial agent should take immediate action (sell/buy), and so on. Therefore, in order to model any complex dynamically adaptive system, a major concern is the selec- 1.1 Agent based models: going beyond the simple statistical mechanics of colliding particles 3 tion and representation of the stimuli and responses, since the behaviour and strategies of the component agents are determined thereby. In a behavioural model, the rules of action are a straightforward way to describe agents’ strategies. One studies the behaviour of the agents by looking at the rules acting sequentially. Then one considers “adaptation”, which is described in biology as a process by which an organism tries to fit itself into its environment. Usually, we see that an organism’s experience guides it to change its structure so that as time passes, the organism makes better use of the environment for its own benefit. This is actually the basis beteen Charles Darwin’s theory of evolution: the survival of the fittest, and natural selection (the process by which heritable traits that make it more likely for an organism to survive and successfully reproduce become more common in a population over successive generations). The timescales over which the agents adapt, will of course vary from one system to another. Thus, for example, adaptive changes in the immune system take hours to days, whereas adaptive changes in a financial firm take usually months to years or the adaptive changes in the ecosystem require years to several millennia. It is noteworthy that in complex adaptive systems, a major part of the environment of a particular agent includes other adaptive agents. Thus, a con- siderable amount of an agent’s effort goes in adaptation to the other agents. This feature is the main source of the interesting temporal patterns that these complex adaptive systems produce. For example, in financial markets, human beings react with strategy and foresight by considering outcomes that might result as a consequence of their behaviour.This brings in a new dimension to the system, namely “rational” actions, which are not innate to agents in natural environments. To handle this new dimension, the use of game theory has become quite natural. It helps in making decisions when a number of rational agents are involved under conditions of conflict and competition. However, game theory and other conventional theories in economics, study patterns in behavioural equilibrium that induce no further interaction. These consistent patterns are quite different from the temporal patterns that the complex adaptive systems produce. Hence, models of physical systems that have self-organization and cooperative phenomena, have also been used such complex systems. In order to model these features of complex systems, one therefore goes beyond the use of simple statistical mechanics, and devise in many cases multi-agent models to mimic the behaviour of complex systems. In- terestingly, in economic models there is usually the “representative agent” who has un-limited foresight and capability of deliberation (“perfect rational- ity”), and uses the “utility maximization” principle to act economically, tak- ing into account all potential future events with the correct probabilities. The “rational-agent” paradigm is also coupled with another reductionism: Since there is a single way to act perfectly rationally, all the agents should display 4 1 BUT INDIVIDUALS ARE NOT GAS MOLECULES exactly the same behaviour. And so a “representative agent” would be sufficient. Thus, the typical format of current economic models is that of a single agent or firm maximizing its utility or profit over a finite or infinite period. The multi-agent models that have originated from simple statistical physics considerations have allowed one to go beyond the prototype theories with a representative agent in traditional economics. The recent failure of economists to anticipate the collapse of markets worldwide since 2007, over a short period of time has now led to some voices from within the field of economics itself, suggesting that new foundations for the discipline are required. In this Chapter, we study a few models which have some of these features and are attempting to address such issues. 1.2 Explaining the hidden hand of economy: self-organization in a collection of interacting "selfish" agents In 1759, Adam Smith published his first work, “The Theory of Moral Senti- ments”, which he continued to revise the work throughout his life until his death in 1790. Published in 1776, “An Inquiry into the Nature and Causes of the Wealth of Nations” is Smith’s magnum opus. It is an account of political economy written at the dawn of the Industrial Revolution. It is widely considered to be the first modern work in the field of economics. It was written for the average educated individual of the 18th century rather than for special- ists and mathematicians. There are three main concepts that Smith expands upon in this work that forms the foundation of free market economics: divi- sion of labour, pursuit of self interest, and freedom of trade. Although “The Wealth of Nations” is widely regarded as Smith’s most influential work, Smith himself considered the former as a much superior work. It was in this book (Theory of Moral Sentiments) that Smith first referred to the "invisible hand" to describe the apparent benefits to society of people behaving in their own interests.

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