Fig. 1 a.
D.B. Berg, V.V. Popkov
International A.Bogdanov Institute,
4a Chebysheva str., Ekaterinburg, 620062, Russia
Urals State Technical University, Ekaterinburg, 620002, Russia
It is clear now that long lasting changes in economy are non-equilibrium and evolutionary [1,2]. Their analysis requires the consideration of the following main features of economic systems.
Hierarchy of scales is the structural principle of any organization. Structural element of the lowest level is a worker. A number of workers forms a working group and so on according to the scheme: worker ® working group ® department ® factory ® industry branch ® national economy ® world economy. There exists a great number of small firms. At the same time one can observe only a few companies like General Motors. It was shown that size distribution of companies is fractal [3,4].
Collective phenomena: result of local interactions of great number of agents. Typical behavior of economics agent was describe by Adam Smith [5] who showed that each agent was trying to use his capital with the highest efficiency. An agent takes into account only his own profit and does not think about the common weal. But studying only his own interests such agent helps the common interests better than in the case of his conscious activity it this area. Clearly organized economic system is the result of great number of such local interactions.
Irreversibility and dissipative structures formation. Economy is developing as result of assimilation of matter and energy from the environment. Dissipative structures are represented not only by manufactures that are processing material resources but also by bank system which exists due to financial flows, and so on. Economy, as well as ecological systems, is an open system which is in constant exchange with environment (matter, energy, information flows) that provides its development in direction of growing complexity [6].
Limited resources. A thesis about limited amount of resources available for economic growth is the basis for the great number of economic theories, for example [7]. Usually only one type of resources leads to the growth limitation. Such resource is the subject of competition in the “game with the zero sum” (GZS) multi-agent models.
Non-linear character of economic processes is one of the main characteristic of transition economy. This fact cuts down possibilities of traditional equilibrium of classic and neo-classic models application and requires new approach.
The main features of economic systems, mentioned above, correspond to the growth phenomenon in physics, for example, aggregation of colloid particles. Such models are well-known, the most popular are based on diffusion-limited aggregation (DLA) rule [8]. Usually they are designed in “cell automata” (CA) [9] media because such technique is suitable for computer simulation of complex hierarchical systems which contain the great number of elements with non-linear interactions.
The goal of the present paper is to design computer simulation model of economics agents’ growth and competition, based on the physical growth process and to show some of it’s features as extension of the traditional GZS model.
Agents growth and competition model is performed on the square lattice (average size is about 5000x5000) that represents the closed economic space with periodic boundary conditions (lattice is closed in torus). Limited resource is presented by the particles migrating (L) from one cell to another according to brownian motion rules (Margolus algorithm [9]) that may be assimilated by the agent and become immovable (S). Agents are presented by single cells (A) that are the "aggregation" centers for resource particles L. Assimilation of resource L by the agent A is performed according to diffusion-limited aggregation (DLA) algorithm (resource assimilation rule)
Xij(t)Î L ® Xij(t+1)Î S, if {Mij(t) Ç {SÈ A}} ¹ Æ , (1)
where Xij(t) is the state of the ij-cell that contains resource particle at the time t,
Mij(t) = {Xi+1,j(t); Xi-1,j(t); Xi,j+1(t); Xi,j-1(t)} is the vicinity of ij-cell.
The rule (1) means that when the migrating resource particle (L) at the time t comes closely to the agent (A) or resource assimilated (S) by this agent it is also assimilated by this agent at the time t+1. Lattice cells occupied by the assimilated particles forms clusters with fractal shape.
In order to satisfy the agents competition and environment stress that result in dissimilation of resource assimilated by the agent earlier well-known irreversible DLA mechanism (1) was extended by the following reversible rule (resource dissimilatiom rule)
Xij(t) Î S ½ Xij(t+1) ® S with probability (1-Pk) (2)
½ Xij(t+1) ® L with probability Pk,
where k is coordination number of S particle in ij-cell {S Ç Mij(t)}, k=1..4. The proposed reversibility makes possible to separate an earlier aggregated cluster surface particle under the probability Pk..
In physics such reversible aggregation takes place in the secondary minimum of particles pair interaction potential in colloidal suspensions [10]. Proposed reversible aggregation together with irreversible diffusion-limited and reaction-limited aggregation mechanisms presents the complete description of aggregation processes in colloidal systems. There are k bonds between cluster and each particle. Pk value is quantitative characteristic of probability that all k bonds will be broken by external forces is
Pk ~ exp(-D Uk/E), (3)
where E is a kinetic energy of particle, D U is the depth of the secondary minimum, D Uk ~ k× D U1 (so P4<P3<P2<P1). For economic agent Pk parameters reflect the stress of environment.
Total amount of resource (the sum of L and S particles) at the lattice is constant. Initially only L particles exist and are distributed at random (with concentration Co ). A certain number of agents (1 or more) also exist initially. Each agent is characterized by it’s own values of Pk parameters. During the growth process other agents may appear at the lattice as well as the Pk parameters shift may take place for one or more agents according to the study goal.
Variety of resource amount (Co) leads to the different growth kinetics (curves 1, 2, 3 and 4, fig 1a). Low values of resource concentration Co lead to decrease of the growth velocity that results in the low number of collisions of L particles with S and A. Coordination number k and fractal dimension values of clusters (agents) increase for the higher values of Co and Pk.
Agent growth velocity decrease for the higher values of Pk. (dissimilation is more intensive), see curves 2, 4, 5, fig. 1a. Cluster (agent) becomes more compact, fig. 1b.
After assimilation of about 95% of resources the dynamic equilibrium of S® L and L® S transitions is observed, agent’s size starts to fluctuate. At the same time the agent structure characteristics k and density are still increasing that is reflecting the “movement” of S particles to the positions with the higher k value via L state.
Calculation results made it possible to design the agents growth scenario diagrams in coordinates “resource concentration - stress intensity”, fig. 2. One can obtain three main zones: growth of fractal clusters; growth of non-fractal (high density) clusters; dissimilation of any cluster.
Variety of Pk values during the agent growth process makes it possible to study agent behavior under the shift of environment conditions.
Agent growth under the changes of environment was also studied. Logistic curves are shown at the fig. 3. Difference between favorable (growth period) and non-favorable (crisis period) conditions of growth was in values of parameters of resource dissimilation Pk. During the crisis period Pk numbers are higher than during the growth one. So in the crisis period resource dissimilation predominated over assimilation and vice versa. In result of external conditions shift agent growth was replaced by dissimilation (in crisis period) that resulted in to agent disappearance (fig. 3, dashed line).
According to the self-organized criticality theory [11] we have generated scenario that leads to growth of the agent which is tolerant to crisis. It was found that agent tolerance to crisis stress may be increased by specific artificial procedure - the number of “crisises” with small amplitude each - that looks like the “vaccination”. During the growth period (zone 0-3) conditions were done worse for a short period of time (small shift of Pk to the higher values), see zone 1-2 for the solid line curve. Intensity of such stress was 1/10 part of the “real” crisis stress. In result the agent dissimilation velocity during the crisis period (zone 3-4) decreases (compare solid and dashed lines, fig. 3). The main result is that such “vaccinated” agent is tolerant to crisis and in the following favorable period (zone after point 4) starts its growth. Non-tolerant agent disappears during the crisis period. At the same time the growth velocity of tolerant agent in favorable period was lower than of non-tolerant one, fig. 3.
The main difference between the proposed model and traditional “game with the zero sum” (GZS) is in consideration of the agents environment. In traditional GZS the sum of resources assimilate by all agents is constant and resource exchange leads to its redistribution. Such conditions really may be observed only in a stable market when segments were distributed between agents earlier. In physics such systems are called “closed systems” [12] and they are very convenient for qualitative and quantitative analysis.
At the same time all systems in economy are “open” (see introduction). In this case in order to deal with the closed system one must take into account the environment. As it is known that open system and its environment all together form the closed system that is proposed in the model described above.
In the last case one is able to study evolution of agents and to observe the primary distribution of the market segments from the very beginning.
The demonstrated above examples of simulation show some new features of the GZS designed as the aggregation model in the cell automata media. In the present paper we pay attention to the growth of single agent. Multi-agent growth and their competition may be easily performed in proposed model by placing of additional A points (agents) to the lattice [13].
Simulation results of multi-agent competition were compared with evolution of Russian bank system after the crush in August 1998 when intensive redistribution process in the bank service market took place. A period since January 1999 till January 2000 was investigated using monthly data of Russian banks (database of Urals Bank of Foreign Trade, Ekaterinburg). Behavior of real banks and model agents was similar and reflected the real hierarchy of bank system due to resources available and is in agreement with the market self-similarity [14].
Results of the present study were used in preparation of “The Base Concept of Russian bank system development" [15] that was approved at the X congress of Association of Russian Banks (May, 2000, Moscow).
Fig. 1. Agent growth under the different Pk and Co values, P3 = P4 = 0:
| 1 | 2 | 3 | 4 | 5 | |
| Co, % | 10 | 50 | 30 | 50 | 50 |
| P1 | 0 | 0 | 0.5 | 0.5 | 0.5 |
| P2 | 0 | 0 | 0 | 0 | 0.1 |
a - kinetics; X - portion of assimilated resource, t
- time;
b - images of corresponding agents (clusters of assimilated resource S is
shown by dark points) after the end of growth process, non-assimilated resource
particles L are not shown.
Fig. 2. Agents growth scenario diagram. Parameter E is environment stress (arb. units). It is controlling Pk values for any k according to equation (3).
Three main zones are observed:
I - growth of fractal clusters;
II - growth of non-fractal (high density) clusters;
II’ - growth of non-fractal clusters and their disintegration to separate subclusters;
III - dissimilation of any cluster.
Shift of conditions during the growth of one and the same agent is shown by sequence of states A® B® A (for dashed line curve, fig 3) and A® A’® B® A (for solid line curve, fig 3).
Fig. 3. Agent growth under the shift of external conditions. Moments of shift are marked by numbers (0-3 - favorable growth period, point “A” at fig. 2; 3-4 - crisis period, point “B” at fig.2; after 4 - favorable growth period, again point “A” at fig. 2):
——— - agent tolerant to crisis (additional 1-2 - “vaccination” period, point “ A’ ” at fig. 2).
— — — - agent non-tolerant to crisis).
X is the quantitative characteristic of agent size in portion of assimilated resource, t - time (in cycles of growth).
References
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