**IMPORTANT POINT:** Agent-based Computational Economics (ACE) is not dependent on the existence of power laws, punctuated equilibrium, increasing returns, or any other particular feature discussed below that researchers have hypothesized might characterize complex adaptive systems in general or economic processes in particular. Rather, as will be seen in Part I.B of the course, ACE is a methodology that can be used to carry out controlled replicable experiments to test such hypotheses.
Physics is not as lawful as it appears -- physical reality is observer created. Doubts about the existence of a unique, observer-independent reality. (pp. 2-3)
LT NOTE: Andy Clark explicitly refuses to push his comprehensive conception of the mind to this limit, which some have argued is the logical limit of his theory.
Many physicists now agree that many fundamental processes shaping our natural world are stochastic and irreversible. Physics is becoming more historical and generative. (p. 4)
But unlike physics, economics has hardly changed at all...its central dogma still revolves around stable equilibrium principles. Students of economics are taught to believe that prices will converge to a level where supply equals demand much as water, flowing between two containers, finally comes to rest at a common level. More markets/agents = more tanks of water connected together. In physics, this kind of treatment is referred to as "mean field approximation." Mean field theories do not work well for systems that are subject to diversity and change. (pp. 4-5)
"The point of departure for this book, in fact, is that our economic world is heterogeneous and dynamic, not homogeneous and static. It is full of pattern and process. Development unfolds..." (p. 7)
"Our world is pluralistic because two "strange bedfellows" are at work together: chance and necessity." (p. 8)
LT NOTE: I would say that the world is pluralistic because three "strange bedfellows" have worked together in intricate combination over time: chance, necessity, and design. That is, human beings engage in deliberate attempts to design aspects of their external environments. They are not purely subject to the whims of chance and necessity.
"The interesting thing is that seemingly simple interactions between individual agents can accumulate to a critical level, precipitating unexpected change. What's even more surprising is that some of this change can produce patterns displaying impressive order." (p. 9)
Batten uses Per Bak's sand pile model to illustrate the difference between weakly and strongly interactive systems, and to motivate the idea of "self-organized criticality" (SOC).
Batten's discussion is too terse to provide a clear understanding of this interesting (and controversial) model. Nathan Winslow (Geological Sciences, University of Michigan) provides a clearer discussion of the SOC concept in general, and Per Bak's sand pile model in particular, in a relatively nontechnical 1997 paper titled Introduction to Self-Organized Criticality and Earthquakes. The Winslow paper is recommended for those wishing to pursue this topic further. The brief description, below, relies heavily on Winslow's paper.
Bak's conception of sand piles as systems exhibiting SOC can be intuitively explained as follows. When you first start building a sand pile on a tabletop of finite size, the system is weakly interactive. Sand grains drizzled from above onto the center of the sand pile have little effect on sand grains toward the edges. However, as you keep drizzling sand grains onto the center, a small number at a time, eventually the slope of the sand pile "self organizes" to a critical state where breakdowns of all different sizes are possible in response to further drizzlings of sand grains and the sand pile cannot grow any larger in a sustainable way. Bak refers to this critical state as a state of self-organized criticality (SOC), since the sand grains on the surface of the sand pile have self-organized to a point where they are just barely stable.
What does it mean to say that "breakdowns of all different sizes" can happen at the SOC state?
Starting in this SOC state, the addition of one more grain can result in an "avalanche" or "sand slide," i.e., a cascade of sand down the edges of the sand pile and (possibly) off the edges of the table. The size of this avalanche can range from one grain to catastrophic collapses involving large portions of the sand pile. The size distribution of these avalanches follows a power law over any specified period of time T. That is, the frequency of a given size of avalanche is inversely proportional to some power of its size, so that big avalanches are rare and small avalanches are frequent. For example, over 24 hours you might observe one avalanche involving 1000 sand grains, 10 avalanches involving 100 sand grains, and 100 avalanches involving 10 sand grains. This is consistent with a power law having form
|(*)||N = KC-s = K/Cs|
where N = number of avalanches, K = 1000, C = number of sand grains involved in the avalanche, and s = 1. (See the Glossary of Basic Concepts at the end of these notes for a more detailed discussion of power laws.)
At the SOC state, then, the sand grains at the center must somehow be capable of transmitting disturbances to sand grains at the edges, implying that the system has become strongly interactive. The dynamics of the sand pile thus transit from being purely local to being global in nature as more and more grains of sand are added to the sand pile. Batten refers to this phenomenon as "emergent dynamics."
Winslow (1997) describes how Bak's conception of sand pile dynamics can be approximated on a computer. A sand pile on a tabletop can be modelled as a two-dimensional "cellular automaton" (checker-board grid) in which each cell (checker-board square) keeps numerical track of the "average slope (gradient)" G of the sand pile in that cell as successive sand grains are added to the sand pile. Starting from some distribution of G values across the entire automaton (e.g., all G values set to 0), a cell is initially chosen at random and its G value is increased by one. If the resulting G value exceeds some critical value (user-specified to be greater than or equal to 3), then this value of G is decreased by 4 and the values of G in the north, south, east, and west neighboring cells of this cell are each increased by 1. This process captures in stylized fashion the way in which the slope of a sand pile may become too steep at some point, causing sand grains to roll down to nearby points. If this redistribution of G values results in a G value in a neighboring cell that exceeds its critical value, then another redistribution occurs. Otherwise, another cell is chosen at random, its G value is increased by 1, and the process repeats. Winslow shows (Figures 2 and 3) that a log-log plot of the avalanche size C versus the frequency of occurrence N of avalanches of size C obeys a power law distribution having form (*) above, where C is the number of cells whose G values are changed as a result of the avalanche.
On the other hand, Winslow (1997) also discusses the difficulties that experimenters have had in trying to get actual sand piles to behave in the idealized way captured in Per Bak's theory and implemented through simple computer models. Winslow cites interesting attempts by Nagel (1992) and Bretz (1992) to conduct experiments with real sand piles. These researchers were unable to obtain SOC results with actual sand piles unless the experimental conditions were rather delicately tuned, leading Winslow to question whether actual sand piles can legitimately be said to have self-organizing critical states even when critical slope values are found. The micro-level interactions among sand grains apparently involve effects not considered in Bak's idealized macro sand pile model, effects that -- in the absence of suitable controls -- can prevent the appearance of avalanches obeying a power law distribution.
For the purposes of our course, the bottom-line question that Batten raises in Chapter 1 (and takes up again in Chapter 7) regarding SOC and power law distributions is whether these ideas are useful conceptions for economics. As noted above, the ACE methodology is entirely independent of any particular conception of agent interactions, and most certainly does not rely on the existence of SOC/power laws in economic systems. The important point is that a number of economists (e.g., Rob Axtell of the Brookings Institution and Paul Krugman of MIT) are currently exploring the power law characteristics of various economic processes (e.g., firm formation, urban growth), and ACE provides a way to test their power law claims through controlled experiments.
Schelling's famous city segregation model (also sometimes referred to as a "tipping model") illustrates how a highly integrated city can rapidly shift to being highly segregated in response to a local disturbance even when everyone is fairly tolerant regarding their type of neighbors.
In Schelling's city segregation model there are two classes of agents. The agents live in a two-dimensional square "chessboard" city consisting of sixty-four squares, to be interpreted as a symmetrical grid of house locations. Each agent cares about the class of his immediate neighbors, i.e., the occupants of the abutting squares of the chessboard. Each agent has a maximum of eight possible neighbors, the exact number depending on the agent's position on the chessboard (straight edge, corner, or interior). Each agent has a "happiness rule" determining whether he is happy or not at his current house location. If unhappy, he either seeks an open square where his happiness rule can be satisfied or he exits the city altogether.
Example of a Happiness Rule: An agent with only one neighbor will try to move if the neighbor is of a different class than his own; an agent with two neighbors will try to move unless at least one neighbor is of the same class as his own; an agent with from three to five neighbors will try to move unless two neighbors are of the same class as his own; and an agent with from six to eight neighbors will try to move unless at least three neighbors are of the same class as his own.
The exact degree of segregation that emerges in the city depends strongly on the specification of the agents' happiness rules. Batten notes that, under some rule specifications, Schelling's city can transit from a highly integrated state to a highly segregated state in response to a small local disturbance. For example, if some agent decides to exit the city altogether, his neighbors might become discontented with their current locations and try to move. These efforts can lead to a chain reaction in which increasing numbers of agents become discontented and attempt to move. Thus, subject to some small initial disturbance (e.g., the exit of a few agents from the city), a city initially in a highly integrated state can "tip" to a highly segregated state.
TECHNICAL NOTE: Schelling's model, like Bak's sand pile model, can be implemented on the computer as a two-dimensional "cellular automaton." Cellular automata as a form of interaction model will be more carefully discussed in a later part of the course.
Batten's discussion here is very informal. He suggests (without theoretical proof or constructive demonstration) that the size distribution of avalanches in Per Bak's sand pile model and the size distribution of chain reactions in Schelling's segregated city model display power law properties. In Figure 1.7 (p. 21), Batten provides a graph of what a power law distribution for Bak's sand pile model might look like, with the "size class" of avalanches appearing as the independent variable c on the horizontal axis and the "number of avalanches of size class c" appearing as the dependent variable n on the vertical axis, both variables measured in logs.
Batten goes on to state (p. 22): "Although it's too early to say for sure, it's likely that many dynamic phenomena discussed in this book obey power laws." He takes up scale invariance and power laws again in chapters 2 and 5.
Power laws are now a controversial topic. So many researchers have claimed to find power law properties in so many different systems that one wonders what the substantive content of such claims really is. This could be an interesting (challenging) topic for a student project.
"Punctuated equilibrium" is another term that tends to be loosely applied to many complex adaptive systems. It would be good to have a more precise definition to avoid ambiguities, preferably one that does not make use of the term "equilibrium."
For example, in my ACE labor market experiments, and in my experiments with iterated prisoner's dilemma with choice and refusal of partners, I have witnessed numerous runs in which the average fitness levels of the agent types undergo long periods of stasis interrupted by periods of sudden dramatic change. On the other hand, other aspects of these agents (e.g., their strategies, their interaction partners) are continually evolving -- there is no stasis. In what sense, then, can the system be said to be in an "equilibrium state" in the supposed "stasis" periods in which average fitness levels are approximately constant? The answer clearly depends on the precise definition given for a system's "state."
In these pages, Batten notes that a punctuated equilibrium pattern appears to characterize a wide variety of systems, including: Per Bak's sand pile model; Tom Ray's Tierra (a continuously evolving computer world of digital hosts and parasites); the real-world process of scientific advance (as described by Thomas Kuhn), real-world economies (as described, for example, by the Austrian economist Joseph Schumpeter); and some of the most beloved pieces of classical music. This interesting hypothesis warrants a much more careful scrutiny.
Batten questions the "efficient markets hypothesis," according to which stock market traders process new information so efficiently that stock share prices instantly change to reflect all new information. Consequently, the current price of a stock share is the best possible predictor of its true ("fundamental") value. If the efficient markets hypothesis were true, technical trading (trying to use patterns in the historical record of past stock market prices to predict future stock market prices) would be a complete waste of time.
The puzzle is that many real-world stock market traders swear by technical trading. They believe that patterns in past stock market prices can be important predictors of future price trends. Many also believe that "nonfundamental" phenomena such as market psychology, fads, and herd (bandwagon) effects -- that is, phenomena having nothing to do with the actual financial soundness and profitability (hence dividend payouts) of firms -- can significantly influence the movement of stock share prices.
Batten notes, correctly, that the evidence regarding the efficiency of stock markets is mixed. Stock markets seem to be reasonably efficient. On average, traders have a hard time beating the return rate on a large diversified "stock market portfolio" such as the Standard and Poor 500 Index. [The latter index incorporates 500 stocks drawn from various industries, and accounts for more than 80 percent of the market value of all stocks listed on the New York Stock Exchange.]
On the other hand, trade volume and volatility data appear to be at odds with the purely "fundamentalist" explanation of stock share pricing implied by the efficient markets hypothesis: namely, that the current price of a stock share should equal the present value of its future expected dividend stream. Moreover, some researchers have concluded that financial price data exhibit definite types of regularities, including patterns of self-similarity on different time scales and conformity to power laws.
Attempting to understand these puzzling "stylized facts" about financial markets is now a hot topic for ACE researchers. For pointers to some of this work, see ACE Research Area: Financial Market Issues.
One important conclusion that Batten draws from his discussion in this section is that complex patterns in financial data, and in economic data in general, are created by long periods of evolution. He cautions (p. 28) that such patterns cannot be understood "by studying economic change within a time frame that is short compared with the economy's overall evolution."
Another important lesson he draws (p. 29) is that "wherever contingency is pervasive, detailed long-term prediction becomes impossible. For example, many kinds of economic changes are unpredictable. But that very fact doesn't mean that they're also unexplainable." He concludes that "the main problem with understanding our economic world is that we have no reliable benchmarks with which to compare it." By this he appears to mean that, for the real world, we can't re-run the tape.
Batten takes up these issues again in Chapter 7 of his book, a chapter which focuses on coevolving markets.
Batten notes, correctly, that conventional economic theory tends to focus on economic models in which only negative feedback loops are operative and diminishing returns to scale prevails. Such models tend to have well defined equilibria that are stable, predictable, and resistant to change. Thus, such models predict stasis as the ultimate economic outcome.
Batten argues, to the contrary, that real world economies are characterized by positive feedback loops and increasing returns to scale. An important point he stresses is that increasing returns prevent an economy from returning to its original state. Thus, economies governed by increasing returns undergo path-dependent structural change, which Batten refers to as morphogenesis. They can also exhibit "lock in" effects, whereby firms gain an advantage over their rivals simply by being the first to produce a certain good or service.
Although economists generally recognize that feedback loops in real-world economies can be positive and that increasing returns are possible, models incorporating increasing returns tend to be analytically intractable, hence difficult to work with. Moreover, some economists disagree with Batten regarding the empirical importance of increasing returns and lock-in effects. The latter issue will be taken up more carefully in later Batten chapters. It has also recently gained attention in the popular press due to the prominent role it has played in the Microsoft Anti-Trust case. For pointers to resources related to this issue, see ACE Research Area: Networks, Path-dependence, and Lock-In Effects.
Based on work by famous researchers such as Ilya Prigogine and Joseph Schumpeter, Batten formulates the following important conjectures (pp. 40-41): (1) "Morphogenesis and disequilibrium are more influential states in an evolving economy than stasis and equilibrium;" (2) "(S)elf-organizing human systems possess an evolutionary drive that selects for populations with an ability to learn, rather than for populations exhibiting optimal behavior;" and (3) "Learning isn't just evolutionary; it's coevolutionary." Much of the rest of his book is devoted to the justification of these claims.
|(1)||f(x(t)) = - x(t)|
Then dx(t)/dt = 0 for x(t) = 0, hence 0 is a rest point (equilibrium state) for system (1).
|(2)||f(x(t)) = x(t)2|
Then dx(t)/dt = 0 for x(t) = 0, hence 0 is a rest point (equilibrium state) for system (2).
|(3)||N = KC-s|
|(4)||n = k - sc|
Example: Government unemployment benefit programs (i.e., government programs that distribute payments to unemployed workers) constitute a negative feedback loop. In a recession, the income that workers lose due to job loss is partially offset by a rise in unemployment benefits. The reverse occurs during the expansion phase of a business cycle.
Example: Bank panics. Suppose you have a deposit account at First National Bank. If you see other people hurrying to withdraw their money from First National Bank because (for whatever reason) they fear the bank is going to go under, you might also begin to fear for your money and hurry to withdraw your deposit account funds. If this "chain reaction" continues, lots of people will end up seeking to withdraw their funds from First National Bank, all within a relatively short period of time. The end result will then be the bankruptcy of First National Bank even though, prior to the bank panic, the bank was not actually in any financial difficulty.