Stuart A. Kauffman
Chapter 9: The Persistently Innovative Econosphere
Oxford University Press, 2000, 211-241
IT IS NO ACCIDENT that the words for economics and ecology have the same I Greek root, “house?’ Ecology and economics are, at root, the same. The economy of Homo habilis and Homo erectus, the stunning flaked flint tools of the Magdalinian culture of the magnificent Cro-Magnon in southern France 14,000 years ago when the large beasts had retreated southward from the glaciation, the invention and spread of writing in Mesopotamia, the Greek agora, and today’s global economy are all in the deepest sense merely the carrying on of the more diversified forms of trade that had their origins with the first autonomous agents and their communities over four billion years ago.
Economics has its roots in agency and the emergence of advantages of trade among autonomous agents. The advantages of trade predate the human economy by essentially the entire history of life on this planet. Advantages of trade are found in the metabolic exchange of legume root nodule and fungi, sugar for fixed nitrogen carried in amino acids. Advantages of trade were found among the mixed microbial and algal communities along the littoral of the earth’s oceans four billion years ago. The trading of the econosphere is an outgrowth of the trading of the biosphere.
Economics has considered itself the science of allocation of scarce resources. In doing so, it shortchanges its proper domain. Indeed, if we stand back and squint, it
is easy to see the most awesome feature of an economy and its roots in autonomous agents: The most awesome feature of the econosphere, as of the biosphere - both built by communities of autonomous agents in their urgent plunging, lunging, sliding, gliding, hiding, trading, and providing - has been a blossoming diversity of molecular and organismic species and of novel ways of making a living that has persistently burgeoned into the adjacent possible. From tens of organic molecular species to tens of trillions; from one or a few species of autonomous agents to a standing diversity of some hundred million species and a total diversity some hundred to thousandfold larger of those creatures come and gone.
Homo erectus had fire and early tools. Homo habilis traded stone axe parts 1.6 million years ago. The diversity of Cro-Magnon goods and services in the south of France some 14,000 years ago may have numbered in the several hundreds to a few thousands. Today, surf the web and count the diversity of goods and services, the ways of making a living; it is in the millions.
Neither the biosphere nor the econosphere are merely about the distribution of limited resources, both are expressions of the immense creativity of the universe, and in particular; of autonomous agents as we exapt molecularly, morphologically, and technologically in untold, unforetellable ways persistently into the adjacent possible. Jobs and job holders jointly coevolve into existence in the econosphere in an ever-expanding web of diverse complexity.
One of the most striking facts about current economic theory is that it has no account of this persistent secular explosion of diversity of goods, services, and ways of making a living. Strange, is it not, that we have no theory of these overwhelming facts of the biosphere and econosphere? Strange, is it not, that we pay no attention to one of the most profound features of the world right smack in front of our collective nose? And consistent with this strangeness, the most comprehensive theory, the rock foundation of modern economics, the beautiful “competitive general equilibrium” theory of Arrow and Debreu, does not and cannot discuss this explosion, perhaps the most important feature of the economy.
So we begin with an outline of competitive general equilibrium as the cornerstone conceptual framework of modern economics. Ken Arrow is a friend. As one of the inventors of the framework, Ken is more at liberty to be a critic than the two generations of economists who have followed in his footsteps. My best reading of Ken’s view, which I share, is that competitive general equilibrium is, at present, the only overarching framework we have to think about the economy as a whole. Yet Ken suspects that that framework is incomplete; I agree.
Competitive general equilibrium grows out of a conceptual framework in which the core question is how prices form such that markets clear. Recall the fa-
mous supply-and-demand curves for a single good (Figure 9.1) [HHC: figure not reproduced). As a function of increasing price, plotted on the x-axis, supply, plotted on the y-axis, increases from low to high. More companies are willing to create widgets as the price per widget increases. On the other hand, demand, where demand is also plotted on the y-axis, decreases from high to low as prices increase. Fewer customers are willing to buy widgets as the price per widget increases.
As the figure shows, the supply-and-demand curves cross at some price. At that price, the markets “clear,” that is, all the widgets supplied are purchased. The price at which markets clear is the “equilibrium price.”
For a single good, the problem is simple. But consider bread and butter. Since many of us like butter on our bread, the demand for butter depends not only on the price and hence the supply of butter, but also on the price of bread, hence on the supply of bread. And vice versa, the demand for bread depends upon the price of butter and hence on the supply of butter. For thousands of goods, where the demand for any one good depends upon the price of many goods and the supply of any one good depends upon the price of many goods, it is not so obvious that there is a price for each good such that all markets clear.
But worse, the supply or demand for bread today may be different than the supply or demand for bread tomorrow. And still worse, the supply or demand for bread tomorrow may depend on all sorts of odd contingent facts. For example, if severe cold kills the winter wheat next month, the supply of bread will drop; if a
HHC: Figure 9.1 not reproduced
FIGURE 9.1 Textbook supply and demand curves for a single good, the supply curve increasing as a function of price, the demand curve decreasing as a function of price. Crossing point is the equilibrium price at which supply equals demand and markets clear. For an economy with many coupled goods, the requirements for market clearing are more complex (see text).
bumper crop of winter wheat comes available globally because of weather or suddenly improved irrigation and farming practices worldwide, the supply will go up.
Arrow and Debreu made brilliant steps. First, they consider a space of all possible dated and contingent goods. One of their examples of a “dated contingent good” is “1 ton of wheat delivered in Chicago on May 2, 2008, under the condition that the average rainfall in Nebraska for the six preceding months has been 10 percent less than normal for the past fifty years and that the Boston Red Sox won the World Series the previous year.”
In the Arrow-Debreu theory, we are to imagine an auctioneer, who at a defined beginning of time, say, this morning, holds an auction covering all possible dated contingent goods. All suppliers and customers gather at this imaginary auction, bidding ensues for all possible dated contingent goods, with values calculated under different hypotheses about the probabilities that the different dated contingencies will come about. At the end of an imaginary hour of frantic bargaining, the auction closes. All participants now have contracts to buy or sell all possible dated contingent goods, each at a fixed price. Everybody hustles home to watch Good Morning America. And, wondrously, however the future unfolds, whether there’s rain, sun, or snow in Nebraska, the dated contingent contracts that are appropriate come due, the contracts are fulfilled at the preestablished price for each contract, and all markets clear.
It is mind-boggling that Arrow and Debreu proved these results. The core means of the solution depends upon what mathematicians call “fixed-point theorems.” A beginning case is your hair, particularly for males, where short hair makes the fixed point easy to see. When you comb your hair in a normal fashion, there is a point roughly on top of your head, slightly to the back, where a roughly circular swirl of hair occurs (ignoring baldness) around a fixed point where typically a bit of scalp shows through.
A general theorem considers hair on a spherical surface and combing the hair in any way you want. You cannot avoid a fixed point. More generally, replace hair by arrows, with tails and heads, where each arrow is a line with an arrow head at one end, drawn on the surface of the sphere. The arrows may bend if you like . Each arrow can be thought of as mapping the point on the sphere at the tail of that arrow to the point of the sphere at the tip of that arrow. So the arrows are a mapping of the surface of the sphere onto itself. For a continuous mapping, such that there is a mapping from each point on the sphere, a general fixed-point theorem proves that there must be at least one point on the surface of the sphere that maps onto itself - that point is a fixed point under the arrow mapping.
The wonderful Arrow-Debreu general competitive equilibrium theorems depend on such a fixed point. In a space where all possible dated contingent goods can be prestated and all possible markets for trade of all such goods exist, a fixed point also exists that corresponds to a price at which all such markets clear, how-
ever the future unfolds. Arrow and Debreu won the Nobel Prize for their work, and won it deservedly. It is a beautiful theory. Yet there are important critiques of general competitive equilibrium. For example, the theorem depends on “complete markets,” that is, markets to trade all possible dated contingent goods, and fine economists have raised issues about how well the theory works if markets are incomplete, as indeed they are.
I pose, however, a wider set of issues. The overarching feature of the economy over the past million years or so is the secular increase in the diversity of goods and services, from a dozen to a dozen million or more today. Nowhere does general competitive equilibrium speak about this. Nor can the theory speak about the growth in diversity of goods and services, for it assumes at the outset that one can finitely prestate all possible dated contingent goods. Then the theory uses complete markets and a fixed-point theorem to prove that a price exists such that markets clear.
But we have seen grounds to be deeply suspicious of the claim that we can finitely prestate all possible exaptations - whether they be new organic functionalities or new goods - that arise in a biosphere or an econosphere, such as Gertrude learning to fly in her terrified leap from the pine tree 60 million years ago last Friday or the engineers working on the tractor suddenly realizing that the engine block itself could serve as the chassis.
I do not believe for a moment that we can finitely prestate all possible goods and services. Indeed, economists intuitively know this. They distinguish between normal uncertainty and “Knightian uncertainty.” Normal uncertainty is the kind we are familiar with in probability theory concerning flipping coins. I am unsure whether in 100 flips there will be 47 heads and 53 tails. Thanks to lots of work, I can now calculate the probability of any outcome in the finitely prestated space of possible outcomes.
Knightian uncertainty concerns those cases where we do not yet know the possible outcomes. Knightian uncertainty has rested in an epistemologically uncomfortable place in economics and elsewhere. Why? Because we have not realized that we cannot finitely prestate the configuration space of a biosphere or an econosphere; by contrast, Newton, Laplace, Boltzmann, Einstein, and perhaps Bohr have all more or less presupposed that we can finitely prestate the configuration space of any domain open to scientific enquiry. After all, as I have noted, we can and do prestate the 6N configuration space for a liter of N gas particles.
From this point of view, the wonderful Arrow-Debreu theory is fundamentally flawed.
Moreover, general competitive equilibrium, seen as a culmination of one central strand of economic theory, is too limited. Insofar as economics is concerned with understanding the establishment of prices at which markets clear, general competitive equilibrium was a masterpiece. But insofar as economics is or should
be concerned with how and why economies increase the diversity of goods and services, the reigning theory is a nonstarter. And since the growth in wealth per capita over the past million years is deeply related to the growth in the diversity of technology and goods and services, contemporary economics is clearly inadequate.
We need a theory of the persistent coming into existence of new goods and services and extinction of old goods and services, rather like the persistent emergence of new species in an ecosystem and extinction of old species. In the previous chapter, we discussed ecosystems as self-organized critical. We discussed the biosphere and econosphere as advancing into the adjacent possible in self-organized critical small and large bursts of avalanches of speciation and extinction events. We discussed the power law distribution of extinction events in the biological record. And we discussed the power law distribution of lifetimes of species and genera.
But the econosphere has similar extinction and speciation events. Consider my favorite example: The introduction of the automobile drove the horse, as a mode of transport, extinct. With the horse went the barn, the buggy, the stable, the smithy, the saddlery, the Pony Express. With the car came paved roads, an oil and gas industry, motels, fast-food restaurants, and suburbia. The Austrian economist, Joseph Schumpeter, called these gales of creative destruction, where old goods die and new ones are born. One bets that Schumpeterian gales of creative destruction come in a power law distribution, with many small avalanches and few large ones. More, if species and genera have a power law distribution of lifetimes, what of firms? Firms do show a similar power law distribution of lifetimes. Most firms die young, some last a long time. We may bet that technologies show similar avalanches of speciation and extinction events and lifetime distributions.
The parallels are at least tantalizing, and probably more than that. While the mechanisms of heritable variation differ and the selection criteria differ, organisms in the biosphere and firms and individuals in the econosphere are busy trying to make a living and explore new ways of making a living. In both cases, the puzzling conditions for the evolutionary cocreation and coassembly of increasing diversity are present. The biosphere and econosphere are persistently transforming, persistently inventing, persistently dying, persistently getting on with it, and, on average, persistently diversifying. And into a framework of such a diversifying set of goods and services we must graft the central insights of general competitive equilibrium as the approximate short-timescale mechanism that achieves a rough-and-ready approximate clearing of markets at each stage of the evolution of the economy.
A rough hypothetical biological example may help understand market clearing in a more relaxed formal framework than general competitive equilibrium. Consider two bacterial species, red and blue. Suppose the red species secretes a red metabolite, at metabolic cost to itself, that aids the replication rate of the blue species. Conversely, suppose the blue species secretes a different blue metabolite,
at metabolic cost to itself, that increases the replication rate of the red species. Then the conditions for a mutualism are possible. Roughly stated, if blue helps red more than it costs itself, and vice versa, a mixed community of blue and red bacteria may grow. How will it happen? And is there an optimal “exchange rate” of blue-secreted metabolite to red-secreted metabolite, where that exchange rate is the analogue of price?
Well, it can and does happen. Here is the gedankenexperiment: Imagine an ordered set of blue mutant bacteria that secrete different amounts of the blue metabolite that helps the red bacteria. Say the range of secretion is from 1 to 100 molecules per minute per blue bacterium, with metabolic cost to the blue bacteria proportional to the number of molecules secreted. Conversely, imagine mutant red bacteria that secrete from 1 to 100 of the red molecules valuable to the blue bacteria, at a similar cost proportional to the number of molecules secreted.
Now create a large, square petri plate with 100 rows and columns drawn on the plastic below to guide your experimental hands. Arrange the rows, numbered 1 to 100 to correspond to blue bacteria secreting 1 to 100 molecules a second. Arrange the columns, numbered 1 to 100 to correspond to the red bacteria secreting 1 to 100 molecules a second. Into each of the 100 x 100 cells on your square petri plate, place exactly one red and one blue bacterium with the corresponding secretion rates. Thus, in the upper left, bacteria that are low blue and low red secretors are coplated onto each square. On the lower left, high blue and low red secretors are coplated. On the upper right, low blue and high red secretors are coplated. And in the lower right corner, high red and high blue bacteria are coplated.
Go for lunch, and dinner, and come back the next day. In general, among the 10,000 coplated pairs of bacteria, while all 10,000 colonies will have grown, a single pair will have grown to the largest mixed red-blue bacterial colony. Say the largest mixed red-blue colony corresponds to red secreting 34 molecules per second, blue secreting 57 molecules per second.
This gedankenexperiment is important, for the exchange ratio of red and blue molecules is the analogue of price, the ratio of trading of oranges for apples. And there exists a ratio, 34 red molecules to 57 blue molecules per second, that maximizes the growth of the mixed red-blue bacterial colony. Since the fastest growing mixed red-blue colony will exponentially outgrow all others and dominate our gedankenexperiment petri plate, this red-blue pair establishes “price” in the system at 34 red to 57 blue molecules. Further, in the fastest growing red-blue colony, where red secretes 34 molecules and blue secretes 57 molecules per second, both the red and blue bacteria in that mixed colony are replicating at the identical optimum rate. As discussed in chapter 3, using a rough mapping of biology to economics, that rate of replication of a bacterium corresponds to economic utility and the increased the rate of replication corresponds to increased economic utility. The red and blue bacteria not only establish price, but they also share equally the
advantages of trade present along the Pareto-efficient contract curve in the Edgeworth box discussed in chapter 3.
Mutualists in the biosphere have been hacking out rough price equilibria for millions of years and have done so without foresight and without the Arrow-Debreu fixed-point theorems. Indeed, critters have been hacking out rough price equilibria even as exaptations and new ways of living have come into existence and old ways have perished. Presumably, these rough biological price equilibria are reached because in the short and intermediate term they optimize the fitness of both of the mutualists. And the markets clear, in the sense that all the 34 red molecules are exchanged for 57 blue molecules per second. But it’s a self-organized critical world out there, with small and large avalanches of speciation and extinction events in the biosphere and econosphere, and equilibrium price or no, most species and technologies, job holders and jobs, are no longer among us to mumble about advantages of trade.
I confess I am happier with this image of prices established in local, rough-and-ready ways at many points in an ecosystem or economy than with the beautiful fixed-point theorems of general competitive equilibrium. Bacteria and horseshoe crabs keep establishing rough price equilibria in their mutualisms without a prespecified space of ways of making a living. If they can do it, so can we mere humans. Getting on with it in the absence of predefined configuration spaces has been the persistent provenance of autonomous agents since we stumbled into existence.
Actually, there has been a major extension of general competitive equilibrium called “rational expectations.” Like general competitive equilibrium, this theory too is beautiful but, I think, deeply flawed.
Rational expectations grew, in part, out of an attempt to understand actual trading on stock exchanges. Under general competitive equilibrium, little trading should occur and stock prices should hover in the vicinity of their fundamental value, typically understood as the discounted present value of the future revenue stream from the stock. But, in fact, abundant trading does occur, and speculative bubbles and crashes occur. Rational expectations theory is built up around another fixed-point theorem. Rational expectations theory assumes a set of economic agents with beliefs about how the economy is working. The agents base their economic actions on those beliefs. A fixed point can exist under which the actions of the agents, given their beliefs about the economy, exactly create the expected economic behavior. So, under rational expectations one can understand bubbles. It is rational to believe that prices are going above fundamental value and thus to invest, and the investments sustain the bubble for a period of time.
Meanwhile, Homo economicus has been thought to be infinitely rational. In the
Arrow-Debreu setting, such infinitely rational agents bargain and achieve the best equilibrium price for each dated contingent good. In rational expectations, the agents figure out how the economy is working and behave in such a way that the expected economic system is the one that arises. The theories and actions of the agents self-consistently create an economy fitting the theories under which the agents operate.
But beautiful as these fixed-point theorems are, there are two troubles in the rational expectations framework. First, the beautiful fixed points may not be stable to minor fluctuations in agent behaviors. Under fluctuations, the economic system may progressively veer away from the fixed point into a feared conceptual no-man’s-land. Second, achieving the fixed points seems to demand excessive rationality to fit real human agents. So it appears necessary to extend rational expectations.
One direction was broached thirty years ago, when economist Herb Simon introduced the terms “satisficing” and “bounded rationality.” Both seem sensible but have been problematic. Satisficing suggests that agents do not optimize but do well enough; yet it has been hard to make this concept pay off. It has also been hard to make advances with the concept of bounded rationality for the simple reason that there is one way, typically, of being infinitely smart and indefinitely many ways of being rather stupid. What determines the patterns of bounded stupidity? How should economic theory proceed?
I suspect that there may be a natural extension to rational expectations applicable to human and any strategic agents, and I report a body of work suggested by me but largely carried out by Vince Darley at Harvard for his doctoral thesis. Two virtues of our efforts are to find a natural bound to infinite rationality and a natural sense of satisficing.
The core ideas stated for human agents are these: Suppose you have a sequence of events, say, the price of corn by month, and want to predict next month’s price of corn. Suppose you have data for twenty months. Now, Fourier invented his famous decomposition, which states that any wiggly line on a blackboard can be approximated with arbitrary accuracy by a weighted sum of sine and cosine waves of different wavelengths and phase offsets, chosen out of the infinite number of possible sine and cosine functions with all possible wavelengths.
Now, you could try to “fit” the data on the corn prices with the first Fourier “mode’ namely the average price. But presumably if the twenty prices vary a fair amount, that average will not predict the twenty-first month’s price very well. You have “underfit the data.” Or you could use twenty or more Fourier modes, all different wavelengths, with different phase offsets, and you would, roughly, wind
up drawing a straight line between the adjacent pairs of points in the twenty-period series. This procedure will not help too much in predicting the twenty-first period. You have “overfit” the data by using too many Fourier modes.
Typically, optimal prediction of the twenty-first period price will be achieved by using two to five Fourier modes, each of different wavelength and different phase offset. As is well known in the art, you have neither underfit nor overfit your data.
This fact suggests that the optimal prediction of a short sequence of data is obtained by a model of intermediate complexity - a few Fourier modes, neither a single one nor very many. The sense of bounded rationality Vince and I want to advocate is that optimal prediction of a limited time series is achieved with models using only a few Fourier modes, or their analogs in other basis sets - models of modest, or bounded, complexity.
The rest of the theory Vince and I have developed goes to show that agents who have theories of one another and act selfishly based on those theories will typically create a persistently changing pattern of actions. Therefore, they persistently create a nonstationary world in which only the relatively recent past has valid data. Thus, there is always only a limited amount of valid data on which to base theories, and the agents, in turn, must always build models of intermediate, bounded complexity to avoid over- or underfitting the meager valid data.
Natural rationality is, in this sense, bounded. It is bounded because we mutually create nonstationary worlds. What happens is that the agents act under their theories. But in due course some agent acts in a way that falsifies the theories of one or more other agents. These agents either are stubborn or change their theories. If they change their theories of the first agent, then typically they also change their actions. In turn, those changes disconfirm the theory of the first agent, and perhaps still other agents. So the agents wind up in a space of coevolving theories and actions with no fixed-point, stable steady states, which means that past actions are a poor guide to future actions by an agent since his theories, and hence his action plans, have changed. But this means that the agents mutually create a “nonstationary” time series of actions (nonstationary just means that the statistical characteristics of the time series keep changing because the agents keep changing their theories and actions). In turn, the agents typically have only a modest amount of relatively recent data that is still valid and reliable on which to base their next theories of one another. Given only a modest amount of valid and reliable data, the agents must avoid overfitting or underfitting that smallish amount of data, so they must use theories of intermediate complexity - for example, four Fourier modes to fit the data, not one or twenty.
Vince and I want to say that natural rationality is bounded to models of intermediate complexity because we collectively and persistently create nonstationary worlds together. In the agent-based computer models Vince has created for his thesis, just this behavior is seen. Indeed, we allow agents to evolve how much of the
past history of the interactions they will pay attention to and how complex their models of one another will be - one, four, or fifty Fourier modes. Agents evolve in a history and complexity space to find currently optimal amounts of history and complexity to use to optimally predict their neighbors. In our little world, the agents evolve to use a modest history, ignoring the distant past, and only modestly complex theories of one another.
We have found evidence of a further, perhaps generic, property that appears to drive such systems to settle down, then change in a sudden burst. As the system of agents and actions settles down to some repeatable behavior, an increasingly wide range of alternative theories, simple and very complex, fit the same data. But the complex theories, with many Fourier modes, attempt to predict fine details of the repeatable behavior. As those theories become more complex, they are more fragile because they can be disconfirmed by ever more minor fluctuations in the repeatable behavior. Sooner or later such a fluctuation happens, and the agents with the complex disconfirmed theories change theories and actions radically, setting up a vast avalanche of changes of theories and actions that sweeps the system, driving the collective behavior far from any repeatable pattern. In these new circumstances, only a small subset of theories fits the current facts, so the diversity, and complexity, of theories in the population of agents plummets, and the system finds its way back to some repeatable pattern of behavior.
In short, there appears to be not only a bounded complexity in our rationality, but a fragility-stability cyclic oscillation in our joint theories and actions as well. In these terms, the system of agents and theories never settles down to a fixed-point equilibrium in which markets clear. Instead, the system repeatedly fluctuates away from the contract curve then returns to new points in the vicinity of the contract curve. Hence, in precisely the sense of repeatedly fluctuating away from a contract curve then returning to its vicinity, the system does not achieve an optimizing price equilibrium, but satisfices.
The bounded complexity issues would seem to apply to any coevolving autonomous agents that are able to make theories of one another and base actions on those theories. The tiger chasing the gazelle and the starfish predating the trilobite are, we suppose, Popperian creatures able to formulate hypotheses about their worlds that may sometimes die in their stead. Presumably all such autonomous agents, under persistent mutation and selection, would opt for changeable models of one another of bounded complexity.
While these steps are only a beginning to go beyond rational expectations in economics, they seem promising. Whatever natural, or unnatural, games autonomous agents are playing as they and we coevolve in a biosphere or econosphere, nonstationarity arises on many levels. Here we see it at the level of the agents’ theories of one another and the actions based on those theories. Perhaps this is just part of how the world works. Given the semantic import of yuck and
yum, and the reality of natural games for fox and hare, for E. coli and paramecium, these changing theories and actions are part of the fabric of history of the market, the savannah, and the small pond.
Natural rationality is bounded by the very nonstationarity of the worlds we cocreate as we coexapt.
Life takes its unexpected turns. I have been an academic scientist, a biologist, for thirty years at the University of Chicago, the National Institutes of Health, the University of Pennsylvania, then twelve stunningly exciting years at the Santa Fe Institute. After thirty years, I’ve written the canonical hundred or more scientific articles, was fortunate enough to receive a MacArthur Fellowship, during whose five years my IQ went up and then slumped back to normal as the funding ended, invented and patented this and that, and published two previous books of which I am proud, Origins of Order and At Home in the Universe, both by Oxford University Press.
I thought Origins and At Home were largely about the challenge of extending Darwinism to an account of evolution that embraced both self-organization and natural selection in some new, still poorly understood marriage. One hundred and forty years after Darwin, after all, we still have only inklings about the kinds of systems that are capable of adaptation. What principles, if any, govern the coevolutionary assembly of complex systems such as ecosystems or British common law, where a new finding by a judge alters precedent in ways that ricochet in small and large avalanches through the law? If new determinations by judges did not have any wider impact, the law could not evolve. If every new determination altered interpretation of precedents throughout the entire corpus of common law, the law also could not evolve.
My rough bet is that systems capable of coevolutionary construction, such as British common law, can evolve and accumulate complexity because they are somehow self-organized critical, and a power law distribution of avalanches of implications of new precedent ricochet in the law and in other complex coevolving systems to allow complexity to accumulate. Indeed, based on self-organized criticality, and more particularly on the analysis of the NK fitness landscape model discussed in Origins and At Home and the “patches” version of the NK model discussed in At Home, I am rather persuaded that adapting systems can best exploit the trade-off between exploitation and exploration at a rough phase transition between order and chaos. Here power law distributions of small and large avalanches of change can and do propagate through the system as it adapts.
So saying, and having published Origins and then At Home, I was rather surprised to find business people approaching me. The consulting companies of
McKinsey, Coopers and Lybrand, Anderson, and Ernst and Young began visiting the Santa Fe Institute to learn about the “new sciences of complexity.” In due course, Chris Meyer at the Center for Business Innovation at Ernst and Young asked me if I might be interested in forming a fifty-fifty partnership with E and Y to discover if complexity science, that nuanced term, could be applied in the practical world. I found myself deeply intrigued. Was the work of my colleagues and myself mere theory or did it have application in real biospheres and econospheres? Why not plunge in and try my best to find out, to do it right, even knowing how early was the stage of the science we had been inventing.
Bios Group Inc., the partnership with Ernst and Young, is now just three-and-a-half years old. We have grown to over seventy people, heading, we hope, for a hundred. Our annual revenues are running at $6 million. We have a hopeful eye on $7 to $8 million this year with clients ranging from Texas Instruments, for whom we invented a novel adaptive chip, to the U.S. Marine Corps with its concern for adaptive combat, to Unilever, the NASDAQ stock market, Honda, Boeing, Johnson and Johnson, Procter & Gamble, Kellogg, Southwest Airlines, the Joint Chiefs of Staff, and others. We have spun out a biotechnology company, CIStem Molecular, that aims to clone the small cis acting DNA regions that control turning genes on and off in development and disease; a European daughter company, Euro-Bios; as well as EXA, a company spun out with NASDAQ to make tools for financial markets. I’m deeply glad to be chairman of the board and chief scientist of Bios, to be working with a very creative group of colleagues, and to be finding routes in the practical world where our ideas do, in fact, apply.
I mention Bios and my involvement because some of the science we have done bears on diverse aspects of practical economics and even begins to suggest pathways beyond the limitations of the Arrow-Debreu theory. I begin with Boeing, which came to Bios wondering how to design and build airplanes in a year rather than seven years. Out of what modular parts and processes, wonder Boeing folks, might it be possible to assemble a family of related aircraft for a diversity of markets?
The obvious approach was to invent “Lego World?’ As founding general partner and chief scientist, I duly authorized the expenditure of $23.94 to buy a largish box of Lego parts. (In truth, I fib. I actually won the Lego box at our first Bios Christmas party.)
Most of the readers of this book will be familiar with Lego. It is a construction game consisting of snap-together, plastic parts, based on square blocks that graduate in size, for example, 1 x 1, 1 x 2, 2 x 3, and 3 x 4 blocks. The blocks can be assembled into wonderfully complex structures, as many delighted children and adults have discovered.
But what might Lego World be? What, indeed. Well, consider a large pile of Lego blocks on a bare wooden table. Consider these blocks the “primitive parts.” Now consider all the primitive construction or deconstruction operations, pressing two
parts together or adding a primitive part to a growing assemblage, or taking a part off another part or off an assemblage.
Consider the pile of unassembled bare Lego parts as the founder set, and place in “rank 1” all the unique Lego objects that can be constructed from the founder set in a single construction step. Thus, a 1 x 3 can be attached in a specific overlapping way to a 2 x 4. Now place in rank 2 all the unique Lego objects that can be constructed from the founder set in two construction (or deconstruction) steps. Similarly, consider ranks 3, 4, 5,.., 20, 100, 10,000, 11,343,998…
A set of primitive parts and the transformations of those parts into other objects is a “technology graph.” In fact, a technology graph is deeply similar to a chemical-reaction bipartite graph from a founder set of organic molecules, where the molecules are the objects and the reaction hyperedges linking substrate and products are the transformations among the objects. The graph is “bipartite” because there are two types of entities, nodes, and hyperedges, representing objects and transformations.
The first thing to notice about the Lego World technology graph is that it might extend off to infinity, given an infinite number of primitive Lego parts.
The second thing to notice is that within Lego World an adjacent possible relative to any actual set of primitive and more complex Lego structures is perfectly definable. The adjacent possible is just that set of unique novel objects, not yet constructed, that can be constructed from the current set of Lego objects in a single construction step. Of course, within the limited world of Lego we can think of the technologically adjacent possible from any actual. A Lego economy might flow persistently from simple primitive objects into the adjacent possible, building up evermore complex objects.
A third feature is that we might consider specific Lego machines, made of Lego parts, each able to carry out one or more of the primitive gluing or ungluing operations. Lego World could build up the machine tools to build other objects including other tools.
Indeed, in Origins of Order and At Home in the Universe, I borrowed theoretical chemist Walter Fontana’s algorithmic chemistry and defined a mathematical analogue of Lego World, namely a “grammar model” of an economy. In that model, binary symbol strings represented goods and services, as do Lego objects in Lego World. In a grammar model, “grammar” specifies how symbol strings act on symbol strings, rather like machines on inputs, to produce new symbol strings. In Lego World, the grammar is specified by the ways primitive blocks can be attached or unattached and by any designation of which Lego objects can carry out which primitive construction operations. The grammar in question may be simple and “context-insensitive” or a far richer “context-sensitive” grammar in which what objects can be added in what ways to different objects depends upon the small or
large context surrounding those blocks. In short, in a context-sensitive grammar, the objects and transformations rules are sensitive to the context of the objects and previous transformations themselves.
Before proceeding with current uses of Lego World and its intellectual children, notice that Lego World, like the grammar models in Origins and At Home, can become the locus of an economy, in which the sets of goods and services can expand over time and in which speciation and extinction events occur. In Origins and At Home, I built upon a suggestion of economist Paul Romer, and specified that each symbol string - or here, each Lego object - has some utility to a single consumer. The utility of each object to the consumer is subjected to exponential discounting over time. A Lego house today is worth more than a Lego house tomorrow and still more than a Lego house two days from now. And for simplicity’s sake, the total utility of a bundle of goods is the sum of their discounted utilities to the consumer.
Next, I invoked a “social planner?’ The task of the social planner is to plan a pattern of production activities over time that optimizes the discounted happiness of the consumer. A standard approach is to adopt a finite planning horizon. The social planner thinks ahead, say, ten periods, finds that pattern of construction activities over time that creates the sequence of symbol-string goods, or Lego objects, that maximizes the time-discounted happiness of the consumer. Then, the social planner initiates the first-period plan, making the first set of objects. Next, the planner considers a ten-period planning horizon from period 1 to period 11, deduces the optimal second-period plan, taking account of the newly considered eleventh period, and carries out the second-period plan.
Because the total utility to the consumer is a simple sum of the discounted utilities of all the possible goods in the economy, finding the optimal plan at any period is just a linear programming problem, and the results are a fixed ratio of construction activities of all the objects produced at that period. The fixed ratio of the activities is the mirror of price, relative to one good, taken as the arbitrary “currency’ or “numeraire.”
Over time, the model economy ticks forward. At each period, in general, only some of the possible goods and services are constructed. The others do not make the consumer happy enough. Over time, new goods and services come into existence, and old ones go out of existence in small and large avalanches of speciation and extinction events.
Thus, a grammar model, or a physical instantiation of a grammar model such as Lego World, is a toy world with a technology graph of objects and transformations. With the addition of utilities to the different objects for one consumer or a set of consumers and a social planner - or more generally, with a set of utilities for the objects that may differ among consumers and with different costs and scaling of costs with sizes of production runs of different Lego objects - a market economy
can be constructed. With defined start-up costs, costs of borrowing money, and bankruptcy rules, a model economy with an evolving set of goods and services can be created and studied.
In general, such economies will advance persistently into the adjacent possible. And because the number of unique Lego objects in each rank is larger than the number in the preceding rank, the diversity of opportunities and objects tends to increase as ever more complex objects are constructed.
More generally, we need to consider “complements” and “substitutes.” Screw and screwdriver are complements; screw and nail are substitutes. Complements must be used together to create value; substitutes replace one another. Rather obviously, the complements and substitutes of any good or service constitute the economic niche in which that good or service lives. New goods enter the economy, typically, as complements and substitutes for existing goods. There is just no point in inventing the channel changer before the television set is invented and television programming is developed.
An economic web is just the set of goods and services in an economy, linked by red lines between substitutes and green lines between complements.
As we have seen, over the past million years, and even the past hundred years, the diversity of the economic web has increased. Why? Because, as in Lego World, the more objects there are in the economy, the more complement and substitute relations exist among those objects, as well as potential new objects in the adjacent possible. If there are N objects, the number of potential complement or substitute relations scales at least as N squared since each object might be a complement or substitute of any object. Thus, as the diversity of the objects in the web increases, the diversity of prospective niches for new goods and services increases even more rapidly! The very diversity of the economic web is autocatalytic.
If this view is correct, then diversity of goods and services is a major driver of economic growth. Indeed, I believe that the role of diversity of goods and services is the major unrecognized factor driving economic growth. Jane Jacobs had made the same point in her thoughtful books about the relation between economic growth and economic diversity of cities and their hinterlands. Economist Jose Scheinkman, now chairman of economics at the University of Chicago, and his colleagues studied a number of cities, normalized for total capitalization, and found that economic growth correlated with economic diversity in the city. In a similar spirit, microfinancing of a linked diversity of cottage businesses in the third world and the first world seems to be achieving local economic growth where more massive efforts at education and infrastructure, Aswan dams and power grids, seem to fail.
Indeed, in the same way in an ecosystem, organisms create niches for other organisms. I suspect, therefore, that over the past 4.8 billion years, the growth of diversity of species is autocatalytic, for the number of possible niches increases more
rapidly than the number of species filling niches. And in the linking of spontaneous and nonspontaneous processes, the universe as a whole advances autocatalytically into its adjacent possible, driven by the very increase of diversity by which novel displacements from equilibrium come into existence, are detected, are coupled to, and come to drive the endergonic creation of novel kinds of molecules and other entities. Economic growth is part and parcel of the creativity of the universe as a whole.
Think of the Wright brothers’ airplane. It was a recombination between an airfoil, a light gasoline engine, bicycle wheels, and a propeller. The more objects an economy has, the more novel objects can be constructed. When we were working with rough stone in the Lower Paleolithic, we pretty much mastered everything that could be done until pressure flaking came along. Most forms of simple stone tools that could be made were made. Today, the adjacent possible of goods and services is so vast that the economy, stumbling and lunging into the future adjacent possible, will only construct an ever smaller subset of the technologically possible.
The economy is ever more historically contingent... As the biosphere is ever more historically contingent... As, I suspect, the universe is ever more historically contingent.
We are on a trajectory, given a classical 6N-dimensional phase space, where the dimensionality of the adjacent possible does seem to increase secularly and the universe is not about to repeat itself in its nonergodic flow.
A fourth law?
I now discuss an algorithmic model of the real economic web, the one outside in the bustling world of the shopping mall, of mergers and acquisitions. While powerful, however, no algorithmic model is complete, for neither the biosphere nor the econosphere is finitely prestatable. Indeed, the effort to design and construct an algorithmic model of the real economic web will simultaneously help us see the weakness of any finite description.
It all hangs on object-oriented programming.
A case in point is the recent advent of Java, an object-based language, which, as of February 1998 had a library of some eighty thousand Java objects. Goodness knows how fast this library of objects is growing. Among the Java objects are “carburetor” objects, “engine block” objects, and “piston” objects, and objects come with “functional” descriptors, such as “is a” “has a,” “does a” “needs a” “uses a?’
Both implicitly and, with modest work, explicitly, the “piston” object can discover that it fits into the cylinder hole in the “engine block” object to create a completed piston in a cylinder. The “carburetor” object can discover that it is to be located on top of the “engine block” object, connected to certain “gas line” objects in certain ways.
The physical engine block and piston, in reality, are complements, used together to create value. Thus, the representations of the engine block and piston as
algorithmic Java objects, together with algorithmic “search engines” to match the corresponding “is a,” “has a” “does a” functions of complements and even substitutes, can as a matter of principle - and practicality - create an image of the real complements and substitutes in the real economic web.
In a fundamental sense, an appropriate set of Java objects, together with search engines for complements and substitutes matching “is a,” “has a” “does a” constitutes a grammar of objects and linkings or transformations among objects. The grammar may be context independent or context sensitive or richer.
In short, properly carried out, Java objects and the proper search engines can create a technology graph of all the objects and transformed objects constructible from any founder set of objects. The Java objects are like Lego World. And Lego World, stated in terms of building simple combinatorial objects, is logically similar to a set of objectives that must be achieved by a military force to carry out its total objective. Entities and operations are deeply similar, as we will explore further below. Technology graphs concern objects and actions, things and objectives, products and processes in a single framework.
Therefore, in principle, we have begun to specify a means to characterize large patches of the global economic web. Let each of very many firms, at different levels of disaggregation, create Java objects proper to their activities, building materials, partially processed materials, partially achieved objectives, and objectives including products. Let these Java objects be characterized by appropriate “is a” “has a” “does a,” lists, with adequate search engines looking for complements and substitutes. The result is a distributed web of Java objects linked functionally in at least many or most of the ways that are actually in use as complements and substitutes creating the millions of different goods and services in the current economy.
Much is of interest about such data on the real economic web. Among other features, any such graph has graph-typical characteristics. Some goods are central to the web, the car, computer, and so forth. Others are peripheral, such as the hula hoop and pet rock. Presumably, location of its products in the web structure has a great deal to do with the strategic position of a firm.
But there is more, for the economic web states its own adjacent possible. Given the Queen Mary and an umbrella, the umbrella placed in the smoke stack of the Queen Mary is in the adjacent possible. Not much use. But what about a small umbrella on the back of a Cessna 172 that opens upon landing: Ah, an air brake is a possible new good in the adjacent possible a few steps from here.
And still more. What are the statistics of the transformation of an economic web over time as new goods and services arise in the niches afforded by existing goods and services, and drive old goods and services extinct? No one makes Roman siege engines these days. Cruise missiles do the job better.
I believe such object-based economic web tools will come into existence in the
near future. Indeed, Bios Group is involved in inventing and making them. And I believe that such tools will be very powerful means of coordinating activities within supply chains and within the larger economy when linked by automated markets.
But I do not believe any such algorithmic tool can be complete. Consider the case of the engineers discovering that the engine block is so rigid that the block itself can serve as the chassis for the tractor they are trying to invent. That exaptation seems a genuine discovery. Now imagine that we had had Java object models of all the parts that were to go into the tractor: engine block objects, carburetor objects, piston objects. If, among the properties of the engine block - the proud “is a” “has a” “does a” features - we had not listed ahead of time the very rigidity of the engine block or if that rigidity was not deducible from the other listed properties, then my vaunted economic web model with its algorithmically accessible adjacent possible could not have ever come up with the suggestion: Use the engine block, due to its rigidity, as the chassis.
You see again that unless there is a finite predescription of all the potentially relevant properties of a real physical object, our algorithmic approach can be powerful, but incomplete. Yet I cannot see how to construct such a finite predescription of all the potentially relevant properties of a real physical object in the real universe.
The world is richer than all our dreams, Horatio.
I must say to Arrow and Debreu, “Gentlemen, the set of goods and services is not finitely prestatable, so fixed-point theorems are of limited use”
And to my economist colleagues: Consider the economy as forever becoming, burgeoning with new ways of making a living, new ways of creating value and advantages of trade, while old ways go extinct. This too is the proper subject for your study, not just allocation of scarce resources and achievement of market-clearing prices. The economy, like the biosphere, is about persistent creativity in ways of making a living.
I find it intriguing to note certain parallels from our prior discussion of autonomous agents and propagating organization. At the level of molecular autonomous agents, I made the point repeatedly that work is the constrained release of energy and that autonomous agents do carry out work to construct the constraints on the release of energy such that the energy is released along specific channels and such that specific couplings of nonequilibrium energy sources to propagating organization arise. Think then of the role of laws and contracts, whose constraints enable the linked flow of economic activities down particular corridors of activities. The web of economic activities flows down channels whose constraints are largely legal in nature. The coming into existence of the enabling constraints of law is as central to economic development and growth as any other aspect of the bubbling activity.
My first purpose in investing in an entire box of Legos was to explore and define concepts of “robust constructibiity?’ We have succeeded, but run into fascinating problems of a general phase transition in problem solvability. In turn, this very phase transition suggests that in a coconstructing biosphere or econosphere rather specific restrictions arise and are respected by critters and firms, creatures and cognoscenti.
Recall the Lego founder set, and the rings, rank 1, rank 2,.., rank 11,983,... each containing the unique Lego objects first constructible from the founder set in a number of steps equal to the rank of that ring. Suppose a given Lego house is first constructible in twenty steps, hence, lies in rank 20. Now, it might be the case that there is a single construction pathway from the founder set to the Lego house in twenty steps. It might also be the case that there are thousands of construction pathways to the Lego house in twenty steps. In the latter case, intuitively, construction of the Lego house is robust. If one way is blocked, say because 1 x 3 blocks are temporarily used up, then a neighboring pathway will allow the Lego house to be constructed without delay, that is, in twenty steps, using other block sizes.
A related sense of robustly constructibiity concerns how the number of ways to construct the Lego house increases if we take more than the minimum twenty steps, say, twenty-one, twenty-two, twenty-three,... steps. The number of ways may not increase at all or very slowly or hyperexponentially. If the number of ways increases very rapidly, it might be worth using twenty-two steps to make the Lego house, for it would be virtually impossible to block construction even if several types of building blocks and machines were temporarily broken.
But recall Boeing’s question. They wanted to build a family of related objects. Hence, let us define still another related sense of robustly constructible.
Consider a family of Lego objects, a house, and a house with a chimney. Now consider each of the many ways from the founder set to build the Lego house. For each such way to build the Lego house, consider how to change construction minimally in order to build the Lego house with the chimney. Perhaps the chimney can just be added to the completed Lego house. More likely, it would be necessary to partially deconstruct that completed Lego house, then go on to construct the house with the chimney. So there is a last branch point during construction on the way both to the Lego house and the Lego house with the chimney.
The branch point object and/or operation that is simultaneously on the way to the house and the house with the chimney is an interesting intermediate-complexity object or operation because it is polyfunctional. It can be used in at least two further ways.
When we build a house, we all know that boards and nails are primitive objects and the completed house is the finished object. But some intermediate objects, say, framed windows and framed walls, are commonly used. Why? Because they are in-
termediate objects that are polyfunctional. The technology graph and its branch points are identifying the intermediate-complexity polyfunctional objects for us.
But things are more subtle. It might be the case that from the last branch point on a way to make both the house and the house with the chimney there is only a single pathway forward to the house and there is only a single pathway forward to the house with the chimney. Not robust. Stupid stopping spot. Either pathway can readily be blocked. Suppose instead we consider an intermediate object three steps prior to the last branch point on the way outward from the founder set. Ah, perhaps there are thousands of ways to complete the house and to complete the house with the chimney. Any single blockage or small set of blockages is readily overcome. The house, or the house with the chimney, can be built without delay if all 1 x 2 blocks are temporarily out of stock. Now this is a smart, robust, intermediate-complexity polyfunctional object-objective. And it may cost no more to stockpile such smart intermediate objects!
So, here is a new view of process design and inventory control.
Bios colleague Jim Herriot has made a delightful little Java computer model to show technology graphs in action. The program shows a “chair” object, a “seat” object, a “back” object, and a “leg” object. In addition, there are “foam” and “padding” objects, two “attachment” objects, a set of “screw” objects, “nail” objects, “wood” objects, a “saw” object, a “hammer” object, and a “screwdriver” object. Each object comes with its own characteristic set of “is a” “has a” “does a” features.
The program assembles coherent technology graphs and chair-assembly pathways as follows: An object tries a connection, shown by a black line, to another object. In effect, the chair object extends a line to the screw object as it says, “I need a lean-on! I need a lean-on!” The screw object responds, “I do twist holds, I do twist holds!” There is no match. After many random tries, the “chair” object extends a black line to the “back” object. “I need a lean-on, I need a lean-on’ says the “chair” object. “I do lean-ons, I do lean-ons,” cries the “back” object with cybernetic joy. The black line becomes a yellow line as a contract is signed between the “chair” and “back” objects. In a similar way, the “back” object needs “padding” and “wood” objects, and either the complementary pair “nail and hammer” objects or their substitutes, “screw and screwdriver” objects, to carry out an “attachment” operation.
In due course, the conditions are met to begin construction of partial objects on the way to the entire chair. Legs, then backs, begin to be assembled as screws and nails are used up. Eventually, a seat is constructed too, and the first chair triumphantly follows.
All goes well until all the screws are used up. The seat, having relied on screws and screwdrivers to attach padding, goes nuts and looks about frantically, asking the screwdriver to work on nails. No luck. Eventually, the seat tries nails and hammers jointly, and that complementary pair works. More chairs are constructed, then nails run out and failures propagate throughout the system.
Not a metaphor, the technology graph. Rather, a new tool to understand the general principles of robust constructibility, the structure of economic webs, a knowledge-management tool for a firm, a new hunk of basic science. Indeed, one of the interesting features of technology graphs is that they constitute the proper conceptual framework to consider process and product design simultaneously. As far as I can tell, we have not had such a conceptual framework before.
Nor is the technology graph limited to manufacturing. The same general principles apply, for example, in military or other logistic operations. Technology graphs, in these other contexts, become the sequential set of requirements needed to meet subobjectives that robustly culminate in the achievement of an overall objective. Taking hill 19 after diverting gracefully from orders to take hill 20 is logically related to making the house with the chimney after starting to make the house without the chimney.
Part of the basic science of technology graphs stems from generic phase transitions in problem solvability in many combinatorial optimization or satisficing problems in biology and economics. I turn now to discuss these generic phase transitions.
I begin with a metaphor. You are in the Alps. A yellow bromine fog is present. Anyone in the fog for more than a microsecond will die. There are three regimes: the “dead” the “living dead’ and the “survivable.”
The “dead”: The bromine fog is higher than Mont Blanc. Unfortunately, everyone dies.
The “living dead”: The bromine fog has drifted lower and Mont Blanc, the Eiger, and the Matterhorn jut into the sunlight. Hikers near these three peaks are alive.
But consider that even mountains are not fixed. Plate tectonics can deform the mountainous landscape. Or, in the terms of the last chapter, the mountainous fitness landscape of a species or a firm or an armed force can change and persistently deform due to coevolution when other species or firms or adversaries change strategies. If the mountainous landscape deforms, Mont Blanc, the Eiger, and the Matterhorn will eventually dip into the bromine fog. As this happens, perhaps new peaks jut into the sunshine. But those peaks will typically be far away from Mont Blanc, the Eiger, and the Matterhorn.
Alas, the hikers near those three initial peaks will die as they are dipped into the lethal fog and are too far from the newly emerged sun drenched peaks to reach them. This “isolated peaks regime” is the living dead regime.
But there is a third regime a phase transition away.
The survivable regime: Let the bromine fog drift lower. More and more peaks jut into the sunshine. At some point, some magical point, as more peaks emerge
into the sunshine, quite suddenly, a hiker can walk all the way across the Alps in the sunshine.
This is a phase transition from the isolated peaks regime. A connected web - mathematically, a percolating web - of connected “solutions” has emerged suddenly as the fog lowers. Now consider hikers striding across the Alps, knapsacks and hearts full. If plate tectonics rather slowly deforms the landscape, then whenever hikers are about to be dipped into the lethal bromine fog, they can take a sideways step in some direction and remain in the sunshine.
The percolating web of solutions regime is persistently survivable. In fitness landscape terms, if you are evolving on a fitness landscape that is deforming and are in your survivable regime, you can continue to exist by stepping somewhere from wherever you happen to find yourself.
This phase transition is generic to hard combinatorial optimization problems. A case in point is the well-known job shop problem. The job shop problem posits M machines and O objects. The idea is to build the O objects on the machines. Each object requires being on each machine in some set order for some fixed period of time. Perhaps object 1 must be on machine 13 for 20 minutes, then machine 3 for 11 minutes, then machine 4 for 31 minutes. In turn, object 2 must be on machine 1 for 11 minutes, then machine 22 for 10 minutes, and so on.
A schedule is an assignment of objects to machines such that all objects are constructed. The total length of time it takes to construct the set of objects is called the “makespan.” We may consider, for each schedule, a definition of neighboring schedules, such as swapping the order in which two objects are assigned to a given machine. Given the set of possible schedules, the neighborhood relation between schedules, and the makespan of each schedule, there is a makespan fitness landscape over the space of schedules of the job shop problem.
We want to minimize makespan. But to keep our mountainous landscape metaphor where high peaks are good, let us consider optimizing efficiency by minimizing makespan. So schedules with low makespan correspond to points of high efficiency on the job-shop fitness landscape. Clearly, short makespan makes the problem hard, long makespan makes the problem easy. So long makespan is like the bromine fog being low, while short makespan is like the bromine fog being high.
Does the phase transition occur in typical job shop problems as makespan is tuned from long to short? Figure 9.2 [HHC: not displayed] shows the results Vince Darley obtained. The figure plots makespan, short to long, on the x-axis and the number of schedules at a given makespan on the y-axis.
As you can see, for long enough makespan, as makespan decreases there are roughly a constant number of schedules at each makespan. But at a critically short makespan, the number of solutions starts to fall abruptly. The corner where the curve starts to turn is the phase transition between the survivable regime for longer makespans and the isolated peaks/living dead regime for shorter makespans. (I
cheat slightly, the sharpness of the corner increases as the size of the job shop problem increases in numbers of machines, M, and objects, 0.)
There are a number of direct tests for this phase transition. In the survivable regime, at longer makespans and lower efficiency than the phase transition makespan, start with a given schedule at a given makespan. N ow examine all “nearby” schedules and test if any is of an equal or better makespan. Continue to “walk” across the space of schedules via neighbors to test if there is a connected web of schedules with makespans at least as good as our initial schedule’s makespan. Either the web percolates across the space of solutions or it does not. If the web percolates, then the initial schedule was in the survivable regime. If only isolated regions of neighboring schedules of the same approximate makespan are found, then you are in the isolated peaks regime.
A second test looks at the “Hausdorf dimensionality” of the acceptable solutions at a given makespan or better. The Hausdorf dimension is computed by considering an initial schedule at a given makespan, then by considering from among all the 1-mutant neighbor schedules the number of them that are of the same or better makespan. as the initial schedule’s makespan, then doing the same among all the 2-mutant neighbor schedules. The Hausdorf dimension of the acceptable set of schedules at that makespan and point in the job shop space is the ratio of the loga-
Figure 9.2 [HHC: not displayed]
The transition from the isolated peaks regime to the survivable regime in a job shop problem as makespan increases. Robust survivable operations occur on the horizontal region of the curve near the phase transition region where the curve bends sharply downward as makespan decreases. Phase transition becomes sharper as size of job shop problem increases.
rithm of the 2-mutant acceptable schedules to the logarithm of the 1-mutant acceptable schedules. In effect, the Hausdorf dimension shows how rapidly - in how many dimensions of the job-shop schedule space - acceptable schedules of a given makespan or better are growing. In the survivable regime, the Hausdorf dimension, on average, is greater than 1.0. In the isolated peaks regime, averaged over the job shop space, the Hausdorf dimension is less than 1.0. At the phase transition, the dimensionality is 1.0.
The phase transition I have just noted is generic for many or most hard combinatorial optimization problems. It is not true for all fitness landscapes. For example, it would not hold on a conical Fujiyama landscape. But the Fuji landscape corresponds to a simple, typically linear, optimization problem. Hard combinatorial optimization problems are multipeaked due to conflicting constraints.
A further interesting connection relates the statistics of search on the job shop landscape to learning curves in economics. Learning curves, well known in arenas from airplane manufacture to diamond cutting to cigar manufacture, show that every time the total output of a plant is doubled, the cost per unit falls by a rough constant percentage, typically 5 to 10 percent. If the logarithm of the cost per unit is plotted on the y-axis and the logarithm of the cumulative number of units produced is plotted on the x-axis, one gets a typical straight-line power law that decreases downward to the right.
The fascinating thing is that this feature probably reflects the statistics of search for fitter variants on rugged, correlated fitness landscapes such as the job shop problem. Typically, in such problems, every time a fitter 1-mutant variant is found, the fraction of 1-mutant variants that are still fitter falls by a constant fraction, while the improvement achieved at each step is typically a constant fraction of the improvement achieved at the last step. These properties yield the learning curve. My colleagues Jose Lobo, Phil Auerswald, Karl Shell, Bill Macready, and I have published a number of papers on the application of the statistics of rugged landscapes and learning curves.
But there is another even more important point. We can control the statistical structure of the problem spaces we face such that the problem space is more readily solvable. We can, and do, tune the structure of the problems we solve. The capacity to tune landscape structure shows up in the job shop problem. In most cases, improving the structure of the problem space requires relaxing conflicting constraints to move the problem into the survivable regime at the makespan, or efficiency, you require. For a specific case, in my statement of the job shop problem, I asserted that the O objects must each have access to the M machines in some fixed order.
Now simple observation and experience tells you that you can put on your shirt and pants in either order, but you had better put on your socks before your shoes. In other words, some steps in life are permutable, others are not. Suppose in our
job shop problem, a fixed fraction, P, of the steps in the construction of each of the O objects were jointly permutable. As P increases and more steps are permutable, the conflicting constraints in the total problem are reduced. But this in turn means that the entire space of solutions improves, that is, the entire space shifts toward lower makespan, or higher efficiency.
And now the relation to the bromine fog metaphor can be stated clearly. As the number of permutable steps increases, the conflicting constraints are reduced. The entire makespan-efficiency landscape is lifted higher, the peaks are higher, and the landscape, with fewer conflicting constraints, is smoother. All in all, the result is that the percolating web of solutions where the hikers can walk all across the Alps occurs at a higher efficiency and shorter makespan.
In terms of Figure 9.2, [HHC: not displayed] the capacity to permute steps in the job shop problem shifts the phase transition point leftward, toward shorter makespan. Equivalently, if one wants to shift the curve in Figure 9.2 to the left, purchase a computable number of machines that are polyfunctional so that the same machine can be used for more than one job. That too reduces conflicting constraints.
There is another view of this generic phase transition between solvable and nonsolvable that again highlights the role of having alternative ways of doing things, robustly constructible strategies that are not easily blocked. In addition, the same simple model, the Ksat model discussed in the previous chapter, begins to account for at least the following anecdotal observation, which is apparently typical: Colleagues at Unilever noted to us that if they have a plant that manufactures different types of a product, say, toothpaste, then the plant does well when the diversity of products grows from three to four to ten to twenty to twenty-five, but at twenty-seven different toothpastes, the plant suddenly fails. So rather abruptly as product diversity in a given plant increases, the system fails.
Why? Presumably it is the same phase transition in problem solvability.
Consider again the Ksat problem, taken as a model for community assembly in the last chapter based on the work of Bruce Sawhill and Tim Keitt. Figure 8.13 repeats the Ksat problem and shows again the phase transition.
Recall that a Ksat problem consists in a logical statement with V variables, in C clauses, in normal disjunctive form: (A1 vA2) and (A3 vA4) and (not A2 vA4). As we discussed in the previous chapter, a normal disjunctive expression with C clauses, V variables in total, and K variables per clause is satisfiable if there is an assignment of true or false to each of the V variables, such that the expression as a whole is true.
The normal disjunctive form makes it clear that as there is an increase in the number of alternative ways, K, of carrying out a task, or making a clause true, it is easier to satisfy the combined expression. More generally, as noted in the last chapter and shown in Figure 8.13, there is a phase transition in the probability that a random Ksat expression with V variables, K per clause, and C clauses can be satis-
fied by some assignment of true or false to the V variables. The phase transition occurs in the horizontal axis, labeled C/V, which is the mean number of clauses in which any variable occurs. Obviously, as C/V increases, conflicting constraints increase. The phase transition from easily solvable to virtually impossible to solve occurs at a point on the C/V axis equal to log 2 x 2 raised to the K power, or 0.6 x 2K. Hence, as K increases, the phase transition shifts outward to greater C/V values.
But Figure 8.13 gives us an intuitive understanding of Unilever’s problem, in fact, the problem is far more general than Unilever’s assembly plants. Think of each clause, (A1 v A2), et cetera, as a way to make one of the toothpaste products, and think of the conjunction, (A1 v A2) and (A3 v A4) and..., as the way to make all twenty-seven toothpastes. Then as the number of clauses, hence toothpaste products, increases for a given plant with V variables to use in making these different products, all of a sudden the conjoint problem will have so many conflicting constraints that it will cross the phase transition from solvable to insolvable.
Moreover, for any given number of clauses and V variables, if a given assignment of true or false to the V variables satisfies the Ksat problem, we can ask if any of the 1-mutant neighbor assignments of true and false to the V variables that change the truth value assigned to one of the V variables also satisfy the Ksat problem. Thus, we can study the phase transition from a survivable percolating web of solutions regime when V/C is lower to an isolated peaks regime as V/C increases to virtual impossibility for high V/C.
Thus, just as in the job shop problem, as product diversity increases for a fixed plant, a phase transition from survivable to unsurvivable will occur because the conflicting constraints will increase with C/V. The resulting landscape becomes more rugged, the peaks lower, the yellow bromine fog rises from the survivable to the living dead to the dead regime, covering Mont Blanc and all hikers in the Alps (Figure 9.3).
But this takes us back to the technology graph and robust constructibility. Recall from above that the Lego house and Lego house with a chimney might be robustly constructible from wisely chosen intermediate-complexity objects on the pathway to both houses, with thousands of ways to get there. At no extra cost, we might choose to stockpile that intermediate-complexity polyfunctional object rather than another choice.
But intermediate-complexity polyfunctional objects are just what allows multiple pathways, multiple permutations of construction steps to our two final objects. Hence, these same smart intermediate objects reduce the conflicting constraints in the fitness landscape over the construction space to make our desired set of objects, or our set of toothpastes. Lowering the conflicting constraints makes the efficiency peaks of the fitness landscapes higher, hence, allows survivable operations at a higher level of product diversity.
Thus, by use of the technology graph to design both products and processes, we
FIGURE 9.3 [HHC: not displayed]
Transition from isolated peaks regime to survivable regime as the diversity of goods produced in a given facility decreases from high to low. Figure suggests a bound on diversity that can be manufactured in a given facility because conflicting constraints increase as diversity increases. Robust operations occur on the horizontal part of the curve just before the curve drops sharply as diversity of goods increases. The same concepts should apply to the complexity of a military campaign that can be waged in terms of the diversity of weapon systems – variables - and subobjectives - clauses. Robust operations should occur in the survivable regime.
can choose a family of products and construction pathways with highly redundant intermediate objects. That choice makes the problem space easy to solve rather than hard to solve. We have thereby tuned the statistical structure of our problem space into a survivable regime. Furthermore, we can test whether our choice of construction pathways to the house and/or house with a chimney is robustly survivable or in the living dead - isolated peaks regime. We need merely use the technology graph to test for percolating sets of 1-mutant neighboring pathways of construction of the same objects and the average Hausdorf dimension of such pathways.
No need to operate in the isolated peaks regime. Indeed, if you face loss of parts and machines, you had best locate back from the phase transition, deep enough into the survivable regime to survive. And if you are a military force fighting against an enemy whose strategy changes persistently deform your payoff landscape and whose efforts are to destroy your capacity to fight, you had best operate even further back from the phase transition in the survivable regime. Indeed, the normal disjunctive form in Figure 9.3 is a rough image of the complexity of a campaign you can fight - the number of clauses that must be jointly satisfied to meet
your objectives, where each clause is a subobjective and there are K alternative ways to meet that objective using V weapon systems.
Just as warfare and the economy as a whole have much in common, warfare and the biosphere have much in common. If you are a species coevolving with other species, you had best operate back from the phase transition well into the survivable regime.
There is a message: If you must make a living, for God’s sake, make your problem space survivable!
This brings us back to a point made in early chapters. Recall the no-free-lunch theorem proved by Bill Macready and David Wolpert. Given a family of all possible fitness landscapes, on average, no search algorithm outperforms any other search algorithm. Hill climbing is, on average, no better than random search in finding high peaks, when averaged over all possible fitness landscapes.
The no-free-lunch theorem led me to wonder about the following: We organisms use mutation, recombination, and selection in evolution, and we pay twofold fitness for sex and recombination to boot. But recombination is only a useful search procedure on smooth enough fitness landscapes where the high peaks snuggle rather near one another.
In turn, this led me to wonder where such nice fitness landscapes arise in evolution, for not all fitness landscapes are so blessedly smooth. Some are random. Some are anticorrelated.
In turn, this led me to think about and discuss natural games, or ways of making a living. Since ways of making a living evolve with the organisms making those livings, we got to the winning games are the games the winners play. Which led me to suggest that those ways of making a living that are well searched out and exploited by the search mechanisms organisms happen to use - mutation, recombination, and selection - will be ways of making a living that are well populated by organisms and similar species. Ways of making a living that cannot be well searched out by organisms and their mutation recombination search procedures will not be well populated.
So we came to the reasonable conclusion that a biosphere of autonomous agents is a self-consistently self-constructing whole, in which agents, ways of making a living, and ways of searching for how to make a living all work together to co-construct the biosphere. Happily, we are picking the problems we can manage to solve. Of course, if we could not solve our chosen ways to make livings, we would be dead.
And there is, I think, a molecular clue that the biosphere is persistently coconstructing itself in the survivable regime for a propagating set of lineages. We have just characterized the survivable percolating web regime where the fitness landscape of each creature deforms due to the adaptive moves of other creatures, but there are always neighboring ways of surviving. Genetically, those neighboring
ways are one or a few mutations or recombinations away from where the species population is right now. If the biosphere has coconstructed itself such that most species are in a survivable regime, then as coevolution occurs, most species will persist but may well transform, for example, to daughter species. One would guess that this mildly turbulent process is rather continuous, perhaps with some self-organized critical bursts on a power law scale.
The “molecular clock hypothesis” seems to fit these facts. If one compares hemoglobins from humans, chimps, horses, whales, and so on, in general, the longer ago we diverged from one another in the evolutionary record, the more amino acid mutations distinguish the hemoglobins of the two species involved. Our hemoglobin is very similar to chimp hemoglobin and quite different from the whale. So good is this correlation that, within given protein families, it is argued that mutations accumulate with timelike clockwork, hence the molecular clock hypothesis, in which a number of amino acid differences can be taken as a surrogate for time from the most common ancestor. Different protein family clocks seem to run at different rates.
There is evidence that the clock does not run quite smoothly. John Gillespie, a population biologist now at the University of California at Davis, showed some years ago that amino acid substitutions seemed to come in short bursts that accumulate over long periods of time to a rough molecular clock that “stutters.” Gillespie argued that fitness landscapes were episodically shifting and the bursts of amino acid substitutions were adaptive runs toward nearby newly formed peaks. I agree and suggest that the near accuracy of the molecular clock data over hundreds of millions of years and virtually all species strongly suggests that the biosphere has coconstructed itself such that species, even as they speciate and go extinct, are, as lineages, in the persistently survivable regime.
We as organisms have, in fact, constructed our ways of making a living such that those problem spaces are typically, but not always, solvable as coevolution proceeds. And, on average, the same thing holds for the econosphere. As old ways of making a living go extinct, new ones persistently enter. We too, it appears, have coconstructed our econosphere such that our ways of making a living, and discovering new ways of making a living, are manageable, probably in a self-organized critical manner, with small and large speciation and extinction events.
And there is a corollary: If you are lucky enough to be in the survivable regime, you can survive by being adaptable. What is required to be adaptable as an organism or organization? We discussed this in the last chapter. A good guess is that an organism or organization needs to be poised in an ordered regime, near the edge of chaos, where a power law distribution of small and large avalanches of change propagates through the system such that it optimizes the persistent balance between exploration and exploitation on ever-shifting, coevolving fitness landscapes.
Laws for any biosphere extend, presumably, to laws for any economy. Nor
should that be surprising. The economy is based on advantages of trade. But those advantages accrue no more to humans exchanging apples and oranges than to root nodules and fungi exchanging sugar and fixed nitrogen such that both make enhanced livings. Thus, economics must partake of the vast creativity of the universe. Molecules, species, and economic systems are advancing into an adjacent possible. In all cases, one senses a secular trend for diversity to increase, hence for the dimensionality of the adjacent possible to increase in our nonergodic journey.
Perhaps again we glimpse a fourth law.