Agent-based versus macroscopic modeling of competition and business processes in economics

Working on Physics of Risk is very interesting and useful experience. This experience provides valuable insights into the mechanics behind various complex systems, well modeled by macroscopic models. Using our experience we are able to obtain qualitative and quantitative agreements between varying models. In our newest publication [1] we have used one-step formalism [2] to obtain macroscopic treatments of Kirman model [3].

Unidirectional Kirman's model

Bass diffusion model [1] is widely known and very important model in marketing science. This model predicts diffusion, sales, of new successful products inside the market. While previously discussed model, Kirman model [2], has wide range of possible applications - from biology to finance. In [2] it is noted that behavior observed by entomologists is similar to one observed in economic scenarios (ex. popularity of books and restaurants). Thus one might expect that Kirman's model might be modified to work in the marketing scenario.

Newton-Raphson method

Newton-Raphson, sometimes just Newton or Newton-Fourier, method is an approximate method in mathematical analysis for finding local roots of very complex functions (such as polynomials with large powers). Recall that root of the function is defined as a solution of \( f(z) = 0 \). The essence of this method is to linearize function at the guessing point. The point where linearized function passes the abscissa axis is assumed to be a more precise estimate of the actual root.

Bornholdt's heterogeneous agent-based spin model for financial markets

Previously we have discussed ant colony model [1, 2] (see Kirman's agent based and stochastic model of ant colony), which is an interesting example of applying knowledge obtained from one field to another. Human (ex., trader in the financial markets) crowd behavior is ideologically quite similar to the behavior in ant colonies, thus the success and relevancy of the aforementioned model were to be expected. Though the key to success lies in the description of large number of entities.

Interestingly enough one can also create, and thus provide additional backing for the argument above, a successful model for human crowd behavior using classical models of statistical physics as an inspiration. In this text we will discuss agent-based spin model of the financial markets proposed by Bornholdt [3, 4], which is based on widely known Ising model.

Long-range memory stochastic model of return

From the practical point of view price is the most interesting observable of the financial markets. Though modeling and analysis of price fluctuations are hindered by the fact that price itself is non-stationary process - mean price and market volatility constantly change. While price changes, at least at short time scales, behave as stationary process - mean price change is equal, or at least approximately equal, to zero. Thus it is convenient to introduce variable related to the relative price changes, which is known as return