The Complex Dynamics of Technological Innovation:
A Comparison of Models Using Cellular Automata *
Systems Research and Behavioral Science (forthcoming)
preprint version
Loet Leydesdorff
Science & Technology Dynamics, University of Amsterdam
Amsterdam School of Communications Research (ASCoR),
Kloveniersburgwal 48, 1012 CX Amsterdam, The Netherlands
loet@leydesdorff.net; http://www.leydesdorff.net/
Abstract
Arthur’s (1988 and 1989) model of the ‘lock-in’ predicates the dominance of a single technology, while Kauffman’s (1993 and 1995) NK-model investigates the existence of different sub-optima in a ‘rugged fitness landscape.’ Can the two mechanisms also be offset against each other? Under what conditions can one expect a technological monopoly or an oligopoly to prevail? What can serve as a discursive ground for making comparisons among the results of these sophisticated models? In this study, the two models are first reconstructed using a common language for the coding. The results of the simulations are made visual on the screen by means of a cellular automaton that represents the diffusion of different technologies in terms of different colours. By using a single platform for the interpretation, one can specify the conditions under which the models compete for the explanation. Second, I address the question of how one model interacts with the other. The reduction of the complexity in the coding enables us to focus on the theoretical assumptions. The introduction of a Schumpeterian dynamics of ‘creative destruction,’ for example, can be shown to introduce variation in ‘rugged fitness landscapes.’
keywords: simulation, complexity, interpretation, innovation, cellular automaton
1. Introduction
Complex systems theoreticians have developed a plethora of models that claim to model social behaviour and processes as a result of relatively simple underlying principles (e.g., Epstein and Axtell, 1996; Axelrod, 1997). Algorithmic models enable us to improve our understanding of the components of a complex dynamics as subdynamics. But can they also be helpful in improving the quality of management and/or policy interventions?
In this study I focus particularly on two models from the Santa Fe Institute that claim to help us understand dominance and speciation. However, I abstract the models from their biological and economic contexts in order to apply them to evolutionary issues related to technological innovation. The first model to be discussed is Arthur’s (1988 and 1989) model of the ‘lock-in’ of a single, but potentially sub-optimal technology. The second is Kauffman’s (1993) NK-model of hill-climbing which predicts potentially different sub-optima in a ‘rugged fitness landscape.’
Sub-optimalization is a well-known phenomenon in technological developments. For example, the QWERTY keyboard was invented in order to prevent jamming of the keys in the case of mechanical typewriting. For various reasons (e.g., network externalities and learning curves) the consequent ‘lock-in’ could no longer be redressed after the conditions for the technological fix had disappeared. For example, the use of the QWERTY keyboard in the case of an electronic computer is nowadays sub-optimal, but attempts to reverse the prevailing ‘lock-in’ have failed hitherto (David, 1985).
The thesis of an irrational ‘lock-in’ of a dominant technology has been opposed in economics by Liebowitz & Margolis (1999) who defend the notion of market equilibrium as a basic premise of economic theorizing. Learning curves can be steep, however (Arrow, 1962), and competition under increasing returns tends to amplify small historical events that favour one or the other technological option (David, 1999).
Evolutionary theorizing about population dynamics predicts that alternative technologies can only survive in niche markets because of the hyper-selective conditions that can otherwise be expected to prevail on the open market (Bruckner et al., 1994; Leydesdorff & Van den Besselaar, 1998a). The niches provide specific environments in which resources can be mobilized that enable agents to face the competition. From this perspective, adopters of competing technologies are climbing on different hills. ‘Rugged fitness landscapes’ can then be expected to emerge (Frenken, 2000 and 2001).
Fitness landscapes are rugged when they contain insurmountable barriers for the evolutionary agencies that compete in hill-climbing. By climbing a specific hill, the agencies can then no longer switch preferences if it becomes evident that they have made the wrong choice. In other words, agencies on different hilltops do no longer compete. In the case of technological evolution, this implies that firms are not able to abandon a suboptimal solution without destroying their technological competencies.
Kauffman’s (1993) NK-model explains the dynamics of various sub-optima in neo-evolutionary terms. Each agent contains genotypically a number of alleles (N). The alleles form a network system with K links among them (K ≤ N- 1). The network relations are ‘epistatic:’ the observable state of the population is not a result of the relations among the agents, but of the competitive characteristics of the agents. Various combinations of alleles have different and uncorrelated fitness. Thus, the agents climb individually.
When the hill-climbing dynamic of agents is combined with various network effects among agents, the competitive characteristics may loose or gain in fitness value since the landscapes are evolving. Thus, the NK-models provides us with a means to simulate a niche system with different sub-optima. By reconstructing the model in the same language as Arthur’s model for ‘lock-in’ as a consequence of network externalities over time, one can specify conditions under which the two models compete for the explanation. Furthermore, the two models will be combined below with static network effects in the direct (‘Von Neumann’) neighbourhood of the hill-climbing agencies.
Let me note that the present study elaborates on a previous study (Leydesdorff, 2001a) in which I confronted Arthur’s model analogously with Axelrod’s (1997) model for ‘the dissemination of culture.’ The dissemination of culture can be considered as a problem formally analogous to the diffusion of a new technology. In principle, Kauffman’s thesis of uphill climbing is consistent with Axelrod’s thesis of ‘balkanization.’ Both models lead to a deadlock between different worlds so that parts of the population remain stucked with a suboptimal solution. However, Axelrod’s introduction of more dimensions and parameters for modeling cultural evolution (as opposed to biological evolution) tends to hide the analytical problem of precisely how different network effects can be offset against one another.
2. A second-order appreciation of models
The possibilities for constructing models of parameter spaces in which complex dynamics can be simulated are almost unlimited. As noted, a complex dynamics can be decomposed into subdynamics. The results of the simulations provide one window or another on the complex phenomena that we observe. In this context, Langton (1989) has compared models with ‘genotypes,’ while one is interested in the match with the observable ‘phenotypes,’ that is, the result of the interactions among the analytically distinguished subdynamics.
Whereas the biological analyst is able to construct models on the basis of specific (biological) assumptions, economic phenomena (e.g., price/performance ratios) are not observable without reflexive theorizing (David, 2000; Leydesdorff, 2001b). What does a model teach us once it is up and running? How can sometimes counter-intuitive and/or conflicting results inform us about socio-economic realities? How can different perspectives be recombined in order to assess their relative contributions? Or should the simulation results be considered merely as suggestions for analogies?
Economic and historical realities integrate the different perspectives ‘phenotypically’ by solving production and distribution problems within the social domain under study (cf. Nelson & Winter’s (1975) ‘productivity growth puzzle’). The interaction terms among the various perspectives cannot be ignored in the appreciation of the historical process. In a series of studies (e.g., Malerba et al., 1999), Nelson, Winter, and their coauthors have chosen to begin their simulations of technological developments in industrial sectors with historical analyses which should allow them to specify the dynamics ‘bottom-up.’ These authors in evolutionary economics call their simulations ‘history friendly.’
It seems to me that a ‘phenotypical’ observation cannot teach us analytically about the ‘genotype’ without further reflection. My approach here will be theory-guided or ‘top-down’: I use the models proposed for specific subdynamics in combination with a cellular automaton in order to be able to see what these models and their combinations can teach us about observable socio-economic realities. The cellular automaton makes the results of the simulations visible on the screen. This enables us to test the conditions under which the visualizations exhibit the theoretically expected results. For example, does the screen turn into a single colour when one expects a ‘lock-in’ and therefore a monopoly to emerge?
What can then serve as a discursive ground for making comparisons among the results of the models? The reconstruction of the respective models in standard BASIC highlights the analytical character of the models as windows on the complex dynamics under study. Each step in the simulation can be followed, and the visible effects can be understood analytically by backtracking to the corresponding lines of code. These lines represent theoretical assumptions about subdynamics.
Note that my recodification implies a simplification of the models that are reconstructed. While these models were originally developed in order to search the respective phase spaces systematically, the simplified reconstructions enable us to assess mainly the results and their relevance for appreciative theorizing. Reducing the complexity by using simple computer language and a cellular automaton for the visualization provides us with a focus on crucial assumptions, since the code can be kept relatively simple.
3. The Models
As a baseline for the simulations, I use a screen exhibiting a random pattern on the basis of the program specified in Table 1. This routine first defines (in line 10) the screen in the CGA-mode (320 x 240 pixels). This screen provides a convenient format since the pixels are larger than in other modes and therefore the results are more visible. The pixels can be considered as representations of individual agents arriving on the market with different preferences for two competing technologies.
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10 SCREEN 1: WINDOW (0, 0)-(320, 240): CLS 20 RANDOMIZE TIMER 30 FOR I = 1 TO 500000 40 y = INT(RND * 240) 50 x = INT(RND * 320) 60 IF RND < .5 GOTO 70 ELSE GOTO 80 70 PSET (x, y), 1: GOTO 90 80 PSET (x, y), 2 90 NEXT I 100 END |
Table 1Program for picturing a screen randomly using two colours |
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Fig. 1: Screen (320 x 240) pictured randomly in two colours
(click here or on the picture for running this simulation under Windows)
A random number is generated on the basis of the clock of the system (in line 20). The next two seed numbers for the random number generator (lines 40 and 50) are used for the attribution of an x- and a y-value to a pixel with one of two corresponding colours on the screen. The value for the total number of arrivers (in this case 500,000) is arbitrarily large. The resulting screen (with a random pattern) is exhibited in Figure 1
3.1 Arthur’s Model of ‘Lock-in’
Let me recapitulate the formalization of Arthur’s (1988 and 1989) model: two competing technologies are labeled A and B. These are cross-tabled with two types of agents, R and S, with different ‘natural inclinations’ towards the respective technologies. In Table 2, aR represents the natural inclination of R-type agents towards type A technology, and bR their (in this case, lower) inclination towards B. Analogously, one can attribute parameters aS and bS to S-type agents (bS > aS). The network effects of adoption (r and s) are modeled as coefficients to the number of previous adopters, but differently for R-type agents and S-type agents.
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Technology A |
Technology B |
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R-Agent |
aR + rnA |
bR+ rnB |
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S-Agent |
aS + snA |
bS + snB |
Table 2. Return values for R- and S-type agents to adopting technology A or B, given nA and nB previous adopters of A and B. (The model assumes that aR > bR and that bS > aS. Both r and s are positive.)
The values of the cells in Table 2 indicate the return that an agent receives for adoption of the respective technology. In addition to the satisfaction of obtaining the technology of his preference—that is, following his/her natural inclination—the global appeal of a technology is increased by previous adopters with a term r for each R-type agent, and s for S-type agents. If R-type and S-type agents arrive on the market randomly, the theory of random walks predicts that the competition will then lock-in on either side (A or B) in the case of positive network effects (r and/or s > 0).
Table 3 provides the code for running this model using the cellular automaton. The screen will be turned into a single colour sooner or later. (Leydesdorff & Van den Besselaar (1998b) provided a representation of this same model as a time-series.) ‘Lock-in,’ represented here by a screen with only one single colour, is an unavoidable result given these parameters of the simulation.
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1 REM Arthur model with spatial representation on the screen 10 SCREEN 1: WINDOW (0, 0)-(320, 240): CLS 20 AR = .8: BR = .2: SA = .2: BS = .8 30 NA = 1: NB = 1: S = .01: R = .01 40 RANDOMIZE TIMER
100 DO 110 y = INT(RND * 240) ' draw random case 120 x = INT(RND * 320) 130 IF RND < .5 GOTO 140 ELSE GOTO 170 140 RETURNA = AR + R * NA: RETURNB = BR + R * NB 150 IF RETURNA > RETURNB THEN NA = NA + 1 ELSE NB = NB + 1 160 IF RETURNA > RETURNB GOTO 200 ELSE GOTO 210 170 RETURNA = SA + S * NA: RETURNB = BS + S * NB 180 IF RETURNA > RETURNB THEN NA = NA + 1 ELSE NB = NB + 1 190 IF RETURNA > RETURNB GOTO 200 ELSE GOTO 210 200 PSET (x, y), 1: GOTO 220 210 PSET (x, y), 2 220 LOOP WHILE INKEY$ = "" 230 END |
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Table 3 Arthur (1988 and 1989) model with representation in colours on the screen [1] |
It has been shown that ‘lock-in’ cannot be prevented by local networking, that is, spatial dependence, given these parameter values (Leydesdorff, 2001a). The ‘lock-in’ effect is global. Changes in the parameter values can sometimes cause a return to equilibrium or an avalanche towards ‘lock-in’ into the competing technology, but the conditions for these break-outs can be counter-intuitive.
2.2 Kauffman’s Model
Kauffman (1993: 42f.) has explicated the NK model for the basic case of three nodes (N= 3) and two links per node (K= 2) in the following wording:
The fitness contribution of the allele at the ith locus depends upon itself (in other words, whether it is 1 or 0) and on the alleles, 1 or 0, at K other loci, hence upon K other alleles. The number of combinations of these alleles is just 2K+1. Since we have no idea what the effects of each such combination on the fitness contribution at the ith locus might be, let us model those effects by assigning to each of the 2K+1 combinations at random, a different fitness contributions drawn from the uniform distribution between 0.0 and 1.0. (...)
Having assigned the fitness contributions, we may now define the fitness of the entire genotype as the average of the contributions of all the loci:
W
= (1/N) 
Given this definition, the NK model is fully specified.
Figure 2 provides Kauffman’s (1993: 42) illustration of the NK-model for the case of N= 3 and K= 2. The three alleles (1, 2, and 3) combine into 23 = 8 combinations. In this drawing, this leads to a global optimum at 100 (W = 0.83) and a local optimum at 111 (W = 0.70). It can be shown that the number of expected optima in a random fitness landscape is 2N / (N + 1) at the maximum for K= N - 1 (Kauffmann, 1995: 166f). In the case of N = 3, the maximum number of expected optima is therefore two. (Note that the random walk may always generate only a single (global) optimum or more than two optima for stochastic reasons.)
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Fig. 2: Random drawing in the case of N = 3 and K = 2 may lead to two (one global and one local) optima.
(From: Kauffman, 1993, at p. 42; cf. Frenken, 2000 and 2001.)
Table 4 provides the code of the Kauffman model as specified above, using BASIC as a language. Because of memory constraints in the freeware version of QBasic the screen is limited to 100 x 200 pixels. An array is declared (in line 31) that attributes three alleles (N = 3) to each pixel as representing an individual agent. The fitness values wi and their mean (W) are stored in the second array.
The two arrays are filled randomly in lines 40-47 and 50-57, respectively. One element of the first array is then selected randomly (lines 110-120) and a mutation is generated in one of its three alleles (line 130). This mutation is implemented at the appropriate position in lines 190-200. After the (technical) attribution of colours and the corresponding mean fitness values (in lines 300-470), the mutation is evaluated (in line 500) on its survival value and, if superior, the mutation is retained in line 510. The screen is in that case repainted at the corresponding place.
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1 REM NK model with spatial representation on the screen 10 SCREEN 7: WINDOW (0, 0)-(320, 240): CLS 20 RANDOMIZE TIMER 30 ' $DYNAMIC 31 DIM scrn(201, 101, 3) AS INTEGER, w(4, 8) AS SINGLE 32 REM scrn() contains three allele values for each pixel [2] 33 REM w() contains three random fitness values and their mean
40 FOR x = 0 TO 200 41 FOR y = 0 TO 100 42 FOR z = 0 TO 2 ' fill alleles randomly 43 IF RND < .5 THEN h = 0 ELSE h = 1 44 scrn(x, y, z) = h ' with zeros and ones 45 NEXT z 46 NEXT y 47 NEXT x 50 FOR y = 0 TO 7 ' fill array with fitness 51 mean = 0 ' values w1, w2, and w3 52 FOR x = 0 TO 2 53 w(x, y) = RND 54 mean = mean + w(x, y) 55 NEXT x 56 w(3, y) = mean / 3 ' add mean of weights 57 NEXT y 100 DO 110 y = INT(RND * 100) ' draw random case 120 x = INT(RND * 200) 130 mut = INT(3 * RND) ' generate mutation 140 allelep$ = "" 150 alleleq$ = "" 160 FOR z = 0 TO 2 170 IF z = mut THEN ' p is prior; q posterior 180 allelep$ = allelep$ + LTRIM$(RTRIM$(STR$(scrn(x, y, z)))) 190 IF scrn(x, y, z) = 1 THEN q = 0 ELSE q = 1 'define mutation 200 alleleq$ = alleleq$ + LTRIM$(RTRIM$(STR$(q))) 210 ELSE ' leave unchanged 220 allelep$ = allelep$ + LTRIM$(RTRIM$(STR$(scrn(x, y, z)))) 230 alleleq$ = alleleq$ + LTRIM$(RTRIM$(STR$(scrn(x, y, z)))) 240 END IF 250 NEXT z
300 IF allelep$ = "000" THEN vp = w(3, 0): cp = 0 'attribute colours 310 IF allelep$ = "001" THEN vp = w(3, 1): cp = 1 'vp = prior 320 IF allelep$ = "010" THEN vp = w(3, 2): cp = 2 'mean fitness value 330 IF allelep$ = "011" THEN vp = w(3, 3): cp = 3 340 IF allelep$ = "100" THEN vp = w(3, 4): cp = 4 350 IF allelep$ = "101" THEN vp = w(3, 5): cp = 5 360 IF allelep$ = "110" THEN vp = w(3, 6): cp = 6 370 IF allelep$ = "111" THEN vp = w(3, 7): cp = 7
400 IF alleleq$ = "000" THEN vq = w(3, 0): cq = 0 'vq = posterior 410 IF alleleq$ = "001" THEN vq = w(3, 1): cq = 1 'mean fitness value 420 IF alleleq$ = "010" THEN vq = w(3, 2): cq = 2 430 IF alleleq$ = "011" THEN vq = w(3, 3): cq = 3 440 IF alleleq$ = "100" THEN vq = w(3, 4): cq = 4 450 IF alleleq$ = "101" THEN vq = w(3, 5): cq = 5 460 IF alleleq$ = "110" THEN vq = w(3, 6): cq = 6 470 IF alleleq$ = "111" THEN vq = w(3, 7): cq = 7
500 IF vq > vp THEN ' evaluate survival value 510 scrn(x, y, mut) = q ' retain mutation 520 PSET (x, y), cq + 7 ' exhibit new colour 530 ELSE 540 PSET (x, y), cp + 7 ' leave colour unchanged 550 END IF
600 LOOP WHILE INKEY$ = "" 610 END |
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Table 4 Kauffman’s NK Model reconstructed |
Figure 3 exhibits a solution provided by this program in terms of two dominant colours. The two colours now represent a sub-optimum and the global optimum, respectively. The resulting pattern is random, since the agents (represented as pixels) optimize individually. The screen freezes when each agent has reached its final state.
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Fig. 3: Demonstration of two optima as a result of the program in Table 4
(click here or on the picture for running this simulation under Windows)
2.3 Extension of the Kauffman Model with a Local Network Effect
The epistatic relations affect the genotypical level of the agents who climb individually in the NK model. The observable network (distribution of the population) itself is not operating. The landscape is determined globally when the fitness values (wi)are drawn. The agents affect only the occupation of the landscape.
In the next simulation (Table 5), we assumed—as an example—that a successful innovation spreads locally in the environment of the innovating firm (lines 511-514). The local environment is defined as the four adjacent cells in the so-called Von Neumann neighbourhood, that is, cells with array values at x + 1, x – 1, y + 1, and y – 1. The result is made visible in Figure 4.
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10 SCREEN 7: WINDOW (0, 0)-(320, 200): CLS 20 RANDOMIZE TIMER 30 ' $DYNAMIC 40 DIM scrn(201, 101, 3) AS INTEGER, w(4, 8) AS SINGLE 41 FOR x = 0 TO 200 [....]
500 IF vq > vp THEN 510 scrn(x, y, mut) = q 511 scrn(x + 1, y, mut) = scrn(x, y, mut) 512 scrn(x, y + 1, mut) = scrn(x, y, mut) 513 IF x > 0 THEN scrn(x - 1, y, mut) = scrn(x, y, mut) 514 IF y > 0 THEN scrn(x, y - 1, mut) = scrn(x, y, mut) 520 PSET (x, y), cq + 7 530 ELSE 540 PSET (x, y), cp + 7 550 END IF
600 LOOP WHILE INKEY$ = "" 610 END |
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Table 5 Kauffman model with local network effect in the ‘Von Neumann neighbourhood’ added |
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Fig. 4: Exhibition of the Kauffman model with a local network effect
in the ‘Von Neumann neighbourhood’
(click here or on the picture for running this simulation)
The addition of this network effect to the Kauffman model generates disturbance, but the effects remain within a limited range because of the ongoing up-hill walks which function as selection mechanisms in the background. Cycles are induced in the system which show on the screen as variations in some of the colours, while other colours (e.g., representing the global optimum) may be more stable.
Figure 4, for example, exhibits a solution in which the light-blue coloured pixels—one of the shades of grey in the picture—are relatively stable, while the other three colours (shades) exhibit fluctuations. The system continuously returns to its respective (sub)optima since the hill-climbing rules of the NK-model are not affected by the local effects. The agents are set off-track for a while, but their walk continues to be attracted by the previous specification of the model.
2.4 Combination of Arthur and Kauffman Models
Let us now combine the ‘lock-in’ model of Arthur with the Kauffman model. In Table 6, it is assumed that if hill-climbing fails (ELSE statement in line 530), the mutated gene (or firm) experiences global pressure to accept the dominant technology as modeled by Arthur (lines 540-620).
[ ... ] 500 IF vq > vp THEN 510 scrn(x, y, mut) = q 520 PSET (x, y), cq + 7 530 ELSE 540 IF RND < .5 GOTO 590 ELSE GOTO 620 550 RETURNA = AR + R * NA: RETURNB = BR + R * NB 560 IF RETURNA > RETURNB THEN NA = NA + 1 ELSE NB = NB + 1 570 IF RETURNA > RETURNB GOTO 650 ELSE GOTO 660 580 RETURNA = SA + S * NA: RETURNB = BS + S * NB 590 IF RETURNA > RETURNB THEN NA = NA + 1 ELSE NB = NB + 1 600 IF RETURNA > RETURNB GOTO 650 ELSE GOTO 660 610 PSET (x, y), 7: GOTO 750 620 PSET (x, y), 8 650 END IF 700 LOOP WHILE INKEY$ = "" 810 END |
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Table 6 Arthur and Kauffman model combined |
Under these conditions, the system always locks-in into a dominant technology. Thus, the global development overrules the local hill-climbing. Using the metaphor of hill-climbing, one might say that the individual hill-climbers are confronted with a sky which exhibits the global dynamics of day or night independently of their positions on different hills.
2.5 The Three (Sub)dynamics Combined
By combining three dynamics, one may generate a complex system in which one can no longer oversee, in principle, the consequences upon visual inspection of the code (Leydesdorff, 2000). In the next model, we assume that an innovation spreads in the ‘Von Neumann neighbourhood’ (lines 500-520), but that in case of failure to improve, the Arthur routine for ‘lock-in’ prevails (lines 580-660).
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[ ... ] 500 IF vq > vp THEN 510 scrn(x, y, mut) = q 511 scrn(x + 1, y, mut) = scrn(x, y, mut) 512 scrn(x, y + 1, mut) = scrn(x, y, mut) 513 IF x > 0 THEN scrn(x – 1, y, mut) = scrn(x, y, mut) 514 IF y > 0 THEN scrn(x, y - 1, mut) = scrn(x, y, mut) 520 PSET (x, y), cq + 7 530 ELSE 580 IF RND < .5 GOTO 590 ELSE GOTO 620 590 RETURNA = AR + R * NA: RETURNB = BR + R * NB 600 IF RETURNA > RETURNB THEN NA = NA + 1 ELSE NB = NB + 1 610 IF RETURNA > RETURNB GOTO 650 ELSE GOTO 660 620 RETURNA = SA + S * NA: RETURNB = BS + S * NB 630 IF RETURNA > RETURNB THEN NA = NA + 1 ELSE NB = NB + 1 640 IF RETURNA > RETURNB GOTO 650 ELSE GOTO 660 650 PSET (x, y), 7: GOTO 750 660 PSET (x, y), 8 750 END IF 800 LOOP WHILE INKEY$ = "" 810 END |
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Table 7 Combination of local network effect, Arthur’s routine, and Kauffman’s model |
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Fig. 5: Addition of Kauffman model with both Arthur model and network effect.
(click here or on the picture for running this simulation under Windows)
Table 7 provides the code and Figure 5 exhibits a snapshot of this simulation after a considerable number of runs. The ‘lock-in’ dominates as a dark background colour in this simulation, but the mutations reinforced by the local network effect cause cycles of disturbance on the screen over time (as above). This variation is not suppressed by the lock-in, but only given changes in other environments can the lock-in perhaps be unlocked. Although the system exhibits variation, it cannot endogenously drift into another ‘lock-in’ or a phase transition because of the selection mechanisms that continue to prevail.
4 Schumpterian dynamics:
Kauffman’s Model with ‘Losers’ Added to ‘Winners’
Both the Arthur- and the Kauffman-model provide the system with rigidity because of recursive selection routines which induce a global effect or local stabilizations, respectively. These stabilizing and globalizing effects are a consequence of the assumption that the selection mechanisms are not affected by the variation.
Table 8 shows a case where we introduce Schumpeter’s (1943) assumption that competition can also be considered as a process of creative destruction. In other words, the winners not only win, but the losers suffer the consequences of losing the competition. In this simulation, we assume that failure to improve one’s position leads to a set back in position to a random value (lines 540-560).
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[ ... ] 500 IF vq > vp THEN 510 scrn(x, y, mut) = q 520 PSET (x, y), cq + 7 530 ELSE 540 q2 = INT(RND * 8) 550 scrn(x, y, mut) = q2 560 PSET (x, y), q2 + 7 570 END IF 600 LOOP WHILE INKEY$ = "" 610 END |
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Table 8 Up-hill climbing and down-hill gliding |
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Fig. 6: Randomizer added in the case of failure to climb the hill
(click here or on the picture for running this simulation under Windows)
Figure 6 exhibits that variation prevails in this model: all eight colours are continuously generated. However, if one removes line 560 from this program and exhibit the variation only after a next round of the selection process, the number of main colours exhibited on the screen is reduced to four or so, while the other colours are continuously selected away. The other variation remains exhibited as isolated phenomena which exist for relatively short periods of time.
Only if we allow—under specifiable conditions—also for changes in the selection environments, for example, by redrawing the random fitness contributions wi in lines 50-57 of Table 4, would we observe that these main colours (representing structure) would also begin to change over time. Indeed, cultural and technological evolution adds to our understanding of biological evolution that ‘the landscapes’ are no longer given and may begin to change dynamically (Allen, 1994; cf. Schumpeter, 1943: 82). However, this further extension would lead me beyond the argument developed in this study.
4 Conclusions
My main results are:
1. It is possible to recode the various models in a common language in order (i) to make comparisons among them and (ii) to study the effects of their possible interactions. Furthermore, the use of techniques like cellular automata facilitates a “phenotypical” appreciation since one can observe the results of running the models and their interactions in terms of coloured pixels on a screen. The various steps (e.g., the interaction terms) can be kept transparent as specific lines of code.
2. I demonstrated the reconstruction of code by using Arthur’s (1988) and Kauffman’s (1993) models for “lock-in” and “uphill climbing in rugged fitness landscapes,” respectively, but with the extension of the visualization. The uphill search for (sub)optima in a rugged fitness landscape can be expected to lead to different technological optimizations among global and local optima. However, a lock-in into a single technology cannot be prevented in either case, since the agents are located on a (potentially rugged) occupational landscape, while the dynamics of the Arthur model operate at the global level of search strategies (Scharnhorst, 1998).
3. In both models, the addition of network effects (e.g., in local neighbourhoods) among the individual “optimizers” in itself only disturbs the development. The eventual results, that is, global lock-in and/or local sub-optimalization, are not affected by these disturbances.
4. The interaction of network effects, lock-in, and up-hill searching lead to a complex dynamics in which not only the dominant and the sub-optimal state are represented (observable as different colours), but other states emerge continuously. The dominant colours provide a stable background against which the variation is generated. The movement towards equilibrium is continuously upset as soon as more than two subdynamics disturb one another (cf. Nelson & Winter, 1982).
5. The NK model continuously generated non-optimal variants under the condition of Schumpeter’s (1943) assumption of ‘creative destruction.’ The local optima can then no longer be reached by all competing hill-climbers at the same moment in time.
5 Normative implications
Casti (1989: 44) noted that “unfortunately (for the reductionist), an operation geared to simplify state descriptions will usually do drastic and terrible things to the tangent vectors associated with the original system dynamics.” In other words, the models of complex dynamics commonly used in evolutionary economics provide us
with windows on an economic reality that is different from these representations. The simulation results using different perspectives need to be weighted against each other before they can be appreciated (cf. Rosen, 1978).
Furthermore, a complex dynamics is complex precisely because it can be decomposed into subdynamics. An analytical solution of this decomposition cannot be expected to remain valid because the interactions may change the system under study in a non-trivial way. Thereafter, another decomposition may become more appropriate at a later moment in time. In other words, not only the phenotypes change, but also the genotypes may change.
In the practical contexts of management advice and policy recommendations the interaction effects among subdynamics—e.g., among wealth generation, knowledge production, and control—should be appreciated. The practice is “phenotypical” and evolving, while each analysis provides a reflection on the basis of an idealization (Allen, 1994). The historical contextualization of the analytical results therefore contains the challenge of a potential paradox (Luhmann, 1984). Bridging this gap calls for intellectual work on both sides of the divide between historical appreciation and quantitatively oriented formalization.
References
Allen, P. 1994. Evolutionary Complex Systems: Models of Technology Change, In: Leydesdorff and Van den Besselaar, 1994; 1-17.
Arrow, KJ. 1962. The economic implications of learning by doing. Review of Economic Studies 29: 155-173.
Arthur WB. 1988. Competing technologies. In: Technical Change and Economic Theory. Giovanni Dosi, Chistopher Freeman, Richard Nelson, Gerald Silverberg, and Luc Soete (eds.). London: Pinter; 590‑607.
Arthur WB. 1989. Competing Technologies, Increasing Returns, and Lock-In by Historical Events. Economic Journal, 99: 116-31.
Axelrod R. 1997. The Dissemination of Culture: A Model with Local Convergence and Global Polarization. Journal of Conflict Resolution, 41:203-26.
Bruckner E, Ebeling W, Jiménez Montaño MA, and Scharnhorst A. 1994. Hyperselection and Innovation Described by a Stochastic Model of Technological Evolution. In: Leydesdorff and Van den Besselaar, 1994; 79-90.
Casti, J. 1989. Alternate Realities. New York, etc.: Wiley.
David, P. 1985. Clio and the Economics of QWERTY, American Economic Review 75(2): 332-337.
David, PA. 1999. At last, a remedy for chronic QWERTY-skepticism!, Paper presented at the European Summer School in Industrial Dynamics, Carsège, 5-12 September.
David, PA. 2000. Path dependence and varieties of learning in the evolution of technological practice. In Technological Innovation as an Evolutionary Process, Ziman J (ed.). Cambridge: Cambridge University Press; 118-133.
Epstein JM., Axtell R.1996. Growing Artificial Societies: Social Science from the Bottom Up. Washington, DC: Brookings Institution Press.
Frenken K. 2000. A Complexity Approach to Innovation Networks: The case of the aircraft industry (1909-1997). Research Policy, 29(2): 257-272.
Frenken K. 2001. Understanding Product Innovation using Complex Systems Theory. Unpublished Ph.D. Thesis, University of Amsterdam.
Kauffman SA. 1993. The Origins of Order: Self-Organization and Selection in Evolution. New York: Oxford University Press.
Kauffman SA. 1995. At Home in the Universe. Oxford, etc.: Oxford University Press.
Langton, CG. 1989. Artificial Life. Redwood City, CA: Addison Wesley.
Leydesdorff L. 2000. The Triple Helix: an evolutionary model of innovations. Research Policy, 29(2): 243-255.
Leydesdorff L. 2001a. Technology and Culture: The dissemination and the potential ‘lock-in’ of new technologies, Journal of Artificial Societies and Social Simulation 4 (3), Paper 5, at < http://jasss.soc.surrey.ac.uk/4/3/5.html >
Leydesdorff L. 2001b. A Sociological Theory of Communication: The Self-Organization of the Knowledge-Based Society. Parkland, FL: Universal Publishers. http://www.upublish.com/books/leydesdorff.htm .
Leydesdorff L, Van den Besselaar P (eds.). 1994. Evolutionary Economics and Chaos Theory: New directions in technology studies. London: Pinter.
Leydesdorff L. Van den Besselaar P. 1998a. Technological Development and Factor Substitution in a Non-linear Model. Journal of Social and Evolutionary Systems, 21(2): 173-192.
Leydesdorff L., Van den Besselaar P. 1998b. Competing Technologies: Lock-ins and Lock-outs. In Computing Anticipatory Systems: CASYS -- First International Conference, AIP Conference Proceedings 437, American Institute of Physics, Daniel M. Dubois (ed.). New York: Woodbury, pp. 309-23.
Liebowitz SJ, Margolis SE, 1999. Winners, Losers & Microsoft: Competition and Antitrust in High Technology. Oakland, CA: The Independent Institute.
Luhmann, Niklas (1984). Soziale Systeme. Grundriß einer allgemeinen Theorie (Suhrkamp, Frankfurt a. M.). [Social Systems (Stanford University Press, Stanford) 1995.]
Malerba, F, Nelson R, Orsenigo L, Winter S. 1999. ‘History-firendly’ Models of Industry Evolution: The Computer Industry, Industrial and Corporate Change 8: 3-35; at http://www3.oup.co.uk/indcor/hdb/Volume_08/Issue_01/ .
Nelson RR, Winter SG. 1975. Growth Theory from an Evolutionary Perspective: The Differential Productivity Growth Puzzle. American Economic Review 65: 338-44.
Nelson RR, Winter SG. 1982. An Evolutionary Theory of Economic Change. Cambridge, MA: Belknap Press.
Rosen R. 1978. Fundamentals of Measurement and Representation of Natural Systems. New York: Elsevier.
Scharnhorst A. 1998. Citation-Networks, Science Landscapes and Evolutionary Strategies, Scientometrics 43(1): 95-106.
Schumpeter J.1943. Socialism, Capitalism and Democracy, London: Allen & Unwin.
* I would like to acknowledge the comments of Koen Frenken, Cooper Langord, and the anonymous referees of previous versions of this paper. The first version was presented at the Fifth International Conference of Computing Anticipatory Systems (CASYS’01), Liège, Belgium, August 2001.
[1] “SA” is used as the variable name for declaring the natural inclination “a” for an S-type agent, aS, since “AS” is a reserved term in BASIC.
[2] Because of memory limitations in the freeware version of Qbasic I use only a screen of 200 x 100 pixels in the following simulations.