Two{dimensional cellular automata can be seen as a natural approach to model .... If depreciation is introduced we link simulation time to the real life cycle of.
Modelling the Spatial Dimension of Economic Systems with Cellular Automata Max Keilbach
1 Introduction The spatially distribution of economic agents can play an important role when the di usion of information or local external e ects are considered. Examples could be:
consumption habit e ects, location of industrial plants, the di usion of technical progress (or routines in general). Two{dimensional cellular automata can be seen as a natural approach to model such spatially distributed phenomena. Cellular automata (CA) are discrete in time and space and each element can take a nite set of discrete values. The situation in time step t +1 of each element (cell) of the automaton depends on its situation in t and of the situation of its neghbouring cells in t via local rules. This property can be useful when phenomena of the above type are to be modelled.CA are used for modelling in Physics and Biology. Wolfram [16] explored CA as a model for complexity. Bhargava and Mukherjee [2] designed a cellular automaton that describes the di usion of a new technology competing with an old one. Markets of investment goods and/or of durable consumer goods, that require a certain level of connected investment can exhibit a di erent behaviour compared e.g. to markets of freqently consumed goods like food. Where a co-investment is involved, the cost of switching to another technology (product) is higher than if those co-investment would not exist. Therefore, once a consumer purchased a certain technology the willingness to switch to another is relatively low. Such co-investment can be necessary in di erent cases: there are training costs involved, to be able to utilize the technology, some linked products are required, like e.g. computer software. Thus, once a buyer (consumer) has decided to purchase a certain technology it is probable that he will stick to it. The more consumers obtain one technology, the cheaper and easier the co-products are to obtain. This can be the origin of a self-reinforcing process, which can lead to a lock-in into a certain situation (a
2
2 THE MODEL
certain technology). The outcome, i.e. into which state the system will lock-in is not predetermined, there is no variable that assures, that the technology with superior performance will survive. Polya urn-processes exhibit characeristics similar to the type of proces described above. They have been widely used to analyze such self-reinforcing market processes . The model presented here has has the same target as Polya-processes. Yet, it is designed to especially explore the spatial dynamics of an economic system (e.g. a market) that emerges when local rules are applied. 1
2
2 The model 2.1 Cellular automata as models of complexity
2.1.1 Principles of cellular automata
Cellular automata (CA) build a class of discrete dynamical systems. They are discrete in space, time and state. Typically, CA are modelled as an n-dimensional array of sites. A one-dimensional CA is an array that forms a line of sites of nite length. A two-dimensional is an array that can be graphically presented as lattice in the plane which sites arranged within this lattice according to a given pattern of neighbourship. A two-dimensional CA is very often presented as a quadratic lattice. However it can adopt any two-dimensional form. The situation of each site in t +1 is determined by the situation of the system (the CA) in t via local rules, i.e. rules under which neighbouring sites interact and/or global rules, i.e. rules which de ne an in uence of some aggregated variable of the CA on each site . In each time step each site takes one state out of a nite set of states S . S can be considered as a k-dimensional alphabet 3
4
= f0; 1; : : : ; k ? 1g: Let 2r be the number of neighbours then a rule de nes a mapping S
: S 2r+1
(1)
! S:
A speci cation of is given below. Rules can be deterministic or stochastic. The situation of the system in t; t = (1; 2; : : : ; T ) emerges if the rules ( ) are iteratively Such type of processes have been discussed in extend. See QWERTY, [1] etc. references ! See Figures 1 and 2 on page 7 for illustration. 4 This de nition goes beyond the ususal de nition of rules. Generally only local rules are considered. Wolfram [16], p. 470 conceives global rules as well. 1 2 3
2.2 Assumptions
3
applied. If rules are deterministic, the situation of the CA in each time step is perfectly predictable. If rules are stochastic, the system is a multivariate Markovprocess of which the situation in each time step is { in principle { predictable . 5
2.1.2 Two-dimensional cellular automata Each site ai;j in a two dimensional CA has 8 neighbours, given a quadratic lattice structure. A rule is then (t+1) ai;j
=
(t) (t) (t) (t) (t) (t) (t) (t) (t) ai;j ; ai;j +1 ; ai+1;j ; ai+1;j +1 ; ai;j ?1; ai?1;j ; ai?1;j ?1 ; ai?1;j +1; ai+;j ?1 ;
(2)
where () can be a stochastic or deterministic function. Such two-dimensional automata have been used by von Neumann and Conway to simulate turing machines . The iterative process is such that in each time step function 2 is applied on each site, i.e. in each time step the number of neighbours is calculated for each site aij simultaneously and the automaton is updated accordingly. This parallel processing characterizes CA. This property makes them attractive for the modelling of processes where such parallel processing is involved. This is the case for economic systems. The following sections present assumptions and rules for a model of a market based on a CA. 6
2.2 Assumptions
The model is based on the following assumptions: 1. I will consider a market for a consumption good where two technologies compete. The market is modelled by a square lattice. Thus each agent has 8 neighbours (except the agents at the border). 2. The square lattice is of limited size and has N cells, each cell is representing an agent aij (or simply a). Each agent is a potential buyer. 3. The possible Alphabet is S = f0; 1; 2g, i.e. the situation of each agent is characterized by whether he possesses one of the considered technologies or not and { if yes { to which of both technologies it belongs (c or c ). If he does not possess either, he is in situation c . Thus, agents are named aijk , k = (0; 1; 2) or sometimes simply a k . 1
0
2
( )
( )
See [8]. However this could prove to be incalculable due to the high dimension of the markov-process. 6 However von Neumann used only orthogonal relations, i.e. only 4 relevant neighbours. See [5] for further discussion 5
4
2 THE MODEL
4. The utility of the agents in possessing one of the two technologies is determined by a) the number of neighbours of each technology b) the proportion of each respective technology in the market and c) the price of each technology. The latter is again in uenced by the overall number of agents via a cost function. 5. There is a given initial number of agents { the early adopters { with c and c . 6. Goods have limited durability, i.e. they are eliminated from the system after a certain time has elapsed(e.g. by depreciation). The actual durability is chosen at random. To keep this simple the survival function is based on a uniform distribution. Thus the probability, that an agent a who posseses technology k; k 6= 0 in t (denoted a k (t)) still possesses it in t + 1 is 1
2
( )
P
h
a(k)(t + 1)
i
= a k (t) = 1 ? ; ( )
(3)
being the depreciation rate. Once the product is eliminated it depends solely on the rules given below as to what type of technology is purchased next. This simply means that there is no utility in purchasing the same technology again. This assumption can be subject to change. 7. There is no second-hand market.
2.3 Rules
Given these assumptions, the rules under which the automaton is running are the following: 1. The following rules apply to agents in situation c (they possess neither technology). No other agent is considered until its good is eliminated from the system (see assumption 5). 2. For each agent aij count the number of neighbours that possess technology c (c respectively). ba (1) [resp. ba(2)] denotes the number of neighbours of agent aij in situation c (c ). 3. Count the overall number of agents in situations c and c . They are denoted s [resp. s ]. 0
1
2
1
2
1
1
2
2
4. For each agent in situation c aij calculate the potential utility in purchasing c (c ) given by the following utility function: 0
1
(0)
2
Uij (ck )
=
ba (k ) nn
!
s p(ck ); 1
(4)
5
where p(ck ) is the relative price of technology ck and and are elasticities. nn is the number of neighbours, where generally nn = 8, sites at the border have nn = 5, those in the corner have nn = 3. 5. Given these utilities calculate the relative preference RPijk of each site aij for one technology. It is given by Uij (ck ) k RPij = Pk? ; (5) Uij (cl ) l ( )
( )
1 =0
where RPijk 2 [0; 1]. These relative preferences de ne the probability that an agent aij in situation c will be in situation ck in the next time step, hence ( )
0
P (aij (t + 1) = ck kaij (t) = c ) = RPijk ; k = (0; 1; 2; : : : ; k ? 1) (6) ( )
0
The next section describes the approach to estimate emerging situations under di erent parameter constellations.
3 A computational approach: Simulation 3.1 Why using simulation?
The system given by the rules given above de nes an N -dimensional markovprocess. It is a natural candidate to be solved by the master equation. Due to the very complex structure of dependence between neighbours, the probabilities (equation 6) would prove to be incalculable after a few time steps. Therefore I chose a simulation approach. The target of the simulations is to explore which is the in uence of di erent parameter constellations of and ? which is the in uence of locally induced behavour and thus di erent patterns of the automaton? which in uence do di erent cost-price functions have on the demand.
3.2 Initial situation and depth of iteration
To explore the in uence of local rules I designed two di erent input matrices. One has a spatial structure with c and c regularly distributed over the whole lattice. Its dimension is 41 41, hence the number of cells is N = 1681 I will call this initial matrix regular matrix, it is shown in Figure 1. 1
2
6
4 SIMULATION RESULTS 40
30
20
10
0 0
10
20
30
40
Figure 1: Structure of a regular initial matrix with N = 1681 sites
The second inital matrix has a central core of dimension 10 10 of agents in situation c and c with the remaining cells in situation c . The overall dimension of the automation is 40 40, hence N = 1600. I will call this matrix core matrix, it is shown in Figure 2. If depreciation is introduced we link simulation time to the real life cycle of the considered technology. If e.g. depreciation rate is = 0:05 per iteration step then all initially present products are removed from the system after 20 iteration steps. This information can be used as benchmark for real time. Assume the maximum length of life cycle of the considered product is 3 years, this period then equals 20 iteration steps. In the simulation the iteration depth is set to 60 which equals a time horizon of 9 years. 1
2
0
7
4 simulation results In this section results of simulation under di erent parameters are presented. Each parameter constellation has been run 200 times. Thre results are presented as follows 1. The paremeters are enumerated in the section heading as to equation 4 on page 4, 2. one examplary set of two time series, one for c and c respectively, 3. 10 time series of the share of c (i.e. c1c c2 ) are given and 1
1
7
See equation 3 on page 4
1 +
2
4.1 Only market share in uences utility ( = 0; = 0), input is a core matrix
7
40
30
20
10
0 0
10
20
30
40
Figure 2: Structure of a core initial matrix with N = 1600 sites 800
600
400
200
0
10
20
30
40
50
60
Figure 3: One emerging time series if only global variables are relevant
4. a histogramm plotting the nal values of c AND/OR c is plotted, 1
2
5. the automaton at t = 300 is graphically presented.
4.1 Only market share in uences utility ( input is a core matrix
),
= 0; = 0
The following plots shows an example of two time series, one for c (the upper) and one for c . A plot of 10 series of the share of c is given in gure 4. Figure 5 shows the distribution of 200 nal values of c after 60 iteration steps. The mean is 798.76, standard deviation is 71.88. 1
2
1
1
8
4 SIMULATION RESULTS
1 0.8 0.6 0.4 0.2
0
10
20
30
40
50
60
0
10
20
30
40
50
Figure 4: 10 emerging time series of the market share of c1
600
700
800 new[, "c1"]
900
1000
Figure 5: Empirical distribition of nal values of c1 after 60 iteration steps.
4.2 Market share and prices in uences utility ( = 0; = ?1:5), input is a core matrix
9
40
30
20
10
0 0
10
20
30
Figure 6: Structure of a CA after 60 time steps.
40
c1
is grey and c2 is black
Figure 6 shows a nal pattern of one run of the automata. The structure seems to be pretty random (note that the blank line on the top has been introduced for grapical puposes).
4.2 Market share and prices in uences utility ( = 0; = ?1:5), input is a core matrix
This sections present results for an automaton where the utility (equation 4 is in uenced by market share and agents are sensitive to prices (there is a negative price-elasticity = ?1:5) . I assume a hyperbolic average cost function. The relative preference function (equation 5) is then nonlinear. It is plotted in Figure 7. A plot of 10 series of the share of c is given in gure 13. Figure 10 shows the distribution of 200 nal values of c after 60 iteration steps. The mean is 859.6, standard deviation is 752.9. The distribution is typical for a market which \locks-in". Figure 11 shows a nal pattern of one run of the automata. a typical pattern of \lock-in". Note that the white line has been adjusted for graphical purposes. 1
1
4.3 Only local in uences on utility ( = 1; = 0), input is a core matrix
This section presents the results of only local in uences on the utility function 4 i.e there is no response to price.
10
4 SIMULATION RESULTS
D_k^r
(b)
1
0.8
0.6
0.4
0.2
0.2
0.4
0.6
1
0.8
s_k
Figure 7: Relative preference function when average costs are degressively decreasing and = ?1:5
1400 1200 1000 800 600 400 200 0
10
20
30
40
50
60
Figure 8: One emerging time series when market share and price in uence utility
4.3 Only local in uences on utility ( = 1; = 0), input is a core matrix
11
1 0.8 0.6 0.4 0.2
0
10
20
30
40
50
60
0
20
40
60
80
100
Figure 9: 10 emerging time series of the market share of c1 when market share and price in uence utility
0
500
1000 globpric[, "c1"]
1500
Figure 10: Empirical distribition of nal values of c1 after 60 iteration steps when agents are sensitive to market share and prices
12
4 SIMULATION RESULTS
40
30
20
10
0 0
10
20
30
40
Figure 11: Structure of a CA after 60 time steps when market share and price in uence utility. c1 is grey and c2 is black.
1400 1200 1000 800 600 400 200 0
10
20
30
40
50
60
Figure 12: One emerging time series if only local rules are applied
4.4 Only local in uences on utility ( = 1; = 0), input is a regular matrix
13
1 0.8 0.6 0.4 0.2
0
10
20
30
40
50
60
Figure 13: 10 emerging time series of the market share of c1 when only local rules are applied
A plot of 10 series of the share of c is given in gure 13. Figure 14 shows the distribution of 200 nal values of c after 60 iteration steps. The mean is 809.6, standard deviation is 256.9. Figure 15 shows a nal pattern of one run of the automata. The structure has a clear local concentration. 1
1
4.4 Only local in uences on utility ( = 1; = 0), input is a regular matrix The structure f the system considered in this section is identical to the one in teh previous section. However here the input matrix is the regular matrix (see Figure 1 on page 6). A plot of 10 series of the share of c is given in gure 17. Figure 18 shows a nal pattern of one run of the automata. The structure also has a clear local concentration. However it is not as polarized as the structre when input matrix is a core matrix (see Figure 15). 1
5 Discussion of results The simulation results can so far be summarized as follows: 1. if agents are only sensitive to the market share of one technology then the relative demand function is linear and market shares tend to stay at ca. 50%. 8
This result is insensitive to the structure of the input matrix but to the initial share of each technology, a simulation result that has not been presented yet in this paper. 8
5 DISCUSSION OF RESULTS
0
10
20
30
40
50
14
600
700
800 new[, "c1"]
900
1000
Figure 14: Empirical distribition of nal values of c1 after 60 iteration steps when only local rules are applied
40
30
20
10
0 0
10
20
30
Figure 15: Structure of a CA after 60 time steps.
40
c1
is grey and c2 is black
15
1000 800 600 400 200
0
10
20
30
40
50
60
Figure 16: One emerging time series if only local rules are applied and input matrix is regular
1 0.8 0.6 0.4 0.2
0
10
20
30
40
50
60
Figure 17: 10 emerging time series of the market share of c1 when only local rules are applied and input matrix is regular
16
6 FURTHER DEVELOPEMENT
40
30
20
10
0 0
10
20
30
40
Figure 18: Structure of a CA after 60 time steps when only local rules are applied and input matrix is regular. c1 is grey and c2 is black
2. given a certain structure of the cost function, price sensitivity can induce self reinforcing processes i.e. after a certain period the market share is 0 or 1 (to be treated further) 3. if local rules are applied, then the market tends to a local pattern. 4. If local rules are applied and at the beginning the agents are regularly distributed the market shares stay at 0.5 with very low variance. The emerging pattern is locally ploarized. 5. If local rules are applied and at the beginning the agents locally concentrated, the market shares tend to stay at 0.5, however with bigger variance. The polarizatin is more accentuated.
6 Further developement As mentioned above, the target of the present model is to explore the probability distributions that emerge under di erent parameter constellations. The model is of course a mere caricature of reality. I nevertheless think, it highlights some basic properties of systems, where local and/or global interactions play a role. A natural extension of the model would be to enlarge the payo properties and the system, e.g.: towards a characterization of the technologies (di erent properties/utility levels)
REFERENCES
17
towards compatibility problems, i.e. once an agent has chosen a certain
technology, he is committed to this technology since he has purchased some by-products as well (e.g. computer software). Generally speaking, the price of changing to another technology is higher than when those by-products do not exist. towards a more exhaustive characterization of the agents, e.g. by classi er systems. towards a multilayer system, where di erent markets are modelled simultaneously. towards a game theoretical approach, e.g. to explore whether rms have higher payo when they concentrate on one area of the market and what happens if the one rm penetrates the other's area.
References [1] W. Brian Arthur (1988): Self-Reinforcing Mechanisms in Economics. In: Anderson, P.W., K.J. Arrow, D. Pines:The Economy as an Evolving Complex System. New York etc.: Addison Wesley. [2] Bhargava, S.C. and A. Mukherjee (1994): Evolution and technological growth in a model based on stochastic cellular automata. In: Leyersdor /v.d. Besselaar: Evolutionary Economics and Chaos Theory. Pinter Publishers [3] Boccara, N., E. Goles, S. Martinez, P. Picco. (Hrsg.) (1993): Cellular Automata and Cooperative Systems. In: NATO ASI Series C: Mathematical and Physical Sciences, Vol, 396. Dordrecht etc.: Kluwer Academic Publishers. [4] Demongeot, J., E. Goles, M. Tchuente (Hrsg.)(1985): Dynamical Systems and Cellular Automata. London, New York etc.: Academic Press. [5] Gardner, M. (1983): Wheels, Life and other Mathematical Amusements. New York: Freeman. [6] Goles, E. (1994): Cellular Automata, Dynamical Systems and Neural Networks. Dordrecht etc.: Kluwer Academic Publishers. [7] Gutowitz, H. (Hrsg.) (1991): Cellular Automata, Theory and Experiment. Cambridge etc.: The MIT Press. [8] Haken, H. (1982): Synergetik. Berlin etc.: Springer.
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REFERENCES
[9] Levy, S. (1993): KL { K"unstliches Leben aus dem Computer. M"unchen: Droemer Knaur. [10] Peitgen, H.O., H. J"urgens, D. Saupe (1992): Bausteine des Chaos: Fraktale. Berlin und Stuttgart: Springer/Klett-Cotta. [11] Peitgen, H.O., H. J"urgens, D. Saupe (1994): Chaos, Bausteine der Ordnung. Berlin und Stuttgart: Springer/Klett-Cotta. [12] Stewart, I.(1994): The Ultimate in Anty-Particles. In: Scienti c American, Juli, S. 88 . [13] To oli, T., N. Margolus (1987): Cellular Automata Machines. Cambridge etc.: The MIT Press. [14] Tsetlin, M.L. (1973): Automaton Theory and Modeling of Biological Systems. Reihe: Mathematics in Science and Engineering, Vol. 102. London, New York etc.: Academic Press. [15] Wolfram, S. (1984): Cellular Automata as models of Complexity. In: Nature 311, S. 419-424. [16] Wolfram, S. (1994): Cellular Automata and Complexity. New York etc.: Addison Wesley. [17] Wuensche, A., M, Lesser (1992): The Global Dynamics of Cellular Automata. In: Santa Fe Institute Studies in the Sciences of Complexity, Reference Volumes, Vol. 1. New York etc.: Addison Wesley.
