THE COMPLEX EVOLUTION OF DUNCAN FOLEY AS A …



THE COMPLEX EVOLUTION OF DUNCAN K. FOLEY AS A COMPLEXITY ECONOMIST

J. Barkley Rosser, Jr.

James Madison University

rosserjb@jmu.edu

August, 2011

Introduction

Duncan K. Foley has always dealt with the larger and deeper questions of economics since the day he walked into Herbert Scarf’s class on mathematical economics at Yale University in the 1960s, with Scarf becoming his major professor. Scarf (1960, 1973) was the leading figure in the study of how to compute general equilibria and the problems of uniqueness and stability of same, problems that Foley (2010a) has pursued to the present day in the form of the related issues of convergence to general equilibria from non-equilibrium states. The pursuit of these issues would lead him to the problem of microfoundations of macroeconomics, which he saw as involving money and studied with the late Miguel Sidrauski (Foley and Sidrauski, 1971). Considering the problem of money would lead him to pursue the study of Marxian economics as possibly resolving this problem (Foley, 1982), a pursuit that would lead to his experiencing professional problems in the 1970s. But, he has continued to pursue the broader issue of the role of money in the economy, even while feeling some frustration along the way at trying to fill this “one of the big lacunae of economic theory.”[1]

While these topics as discussed so far are not obviously parts of what is called complexity economics, the pursuit of these matters would indeed lead him to pursue various forms of complexity economics. The initial link would be his reviving the model of Goodwin (1951) in a paper on the role of liquidity in business cycles, interested in the endogenous appearance of limit cycles, although these are at most on the edge of complexity models (Foley, 1986). This laid the groundwork for him to pursue ideas and approaches he had been previously interested in regarding statistical mechanics and also the role of the computer in economic decisionmaking and a more algorithmic and engineering approach to economics.[2] Meeting the late Peter Albin in the late 1980s would lead to collaboration with him (Albin and Foley, 1992; Albin with Foley, 1998) in which he pursued a more computational approach to complexity, albeit with some discussions of dynamic issues as well. During the same period he also took up the problem of applying statistical mechanics to general equilibrium theory (Foley, 1994), following on the work of Föllmer (1974), with this leading to a more specifically econophysics approach that linked more fully to considerations of the law of entropy (Foley and Smith, 2008) as he became associated with the Santa Fe Institute and moved more clearly into the complexity approach to modeling economics.

The rest of this essay honoring Duncan will focus on how his work in this area of complexity economics has related to ongoing controversies regarding the nature of complexity, and particularly which approach is most useful for economics. In particular, there has been an ongoing debate between those who focus more on computational complexity (Velupillai, 2009), in contrast with those who concentrate on dynamic complexity (Rosser, 2009). It can be broadly said that while Foley has tended to pursue and advocate more the computational complexity approach based on his work with Albin, he has also studied and articulated views about the dynamic approach and has suggested possible ways that they can be reconciled.

A Brief Overview of Competing Approaches to Complexity

What is complexity is a topic that has been beaten more or less to death in the existing literature, so we shall keep our discussion short here. Over a decade ago, Seth Lloyd of MIT famously gathered 45 definitions of complexity (Horgan, 1997, Chap. 11), with his not being a comprehensive list. This list can be viewed as comprising meta-complexity, the full set of definitions of complexity. Many of these 45 can be aggregated into sub-categories, with the sub-category arguably containing more of these definitions than any other being computational complexity, or variants thereof. Many of these definitions are variations on a theme of the minimum length of a computer program required to solve a problem, but there are indeed many variations on this. Furthermore, a hardline group argues that the only truly computationally complex systems are those that are undecidable due to halting problems in their programs (Blum, Cucker, Shub, and Smale, 1998). In any case, an argument made by those advocating the superiority of the computational approach to complexity is that it may be more precisely measurable, assuming one can agree on what is the appropriate measure, with well-defined hierarchies involved as well, although the hierarchy favored by Foley is not necessarily the same as that emphasized by some others.

Among economists almost certainly the main rival is dynamic complexity, defined by Rosser (1999), following Day (1994) as being systems that do not endogenously move to a point, a limit cycle, or a smooth implosion or smooth explosion.[3] Such systems inevitably involve either nonlinear dynamics or coupled linear systems that can be reduced to nonlinear systems (Goodwin, 1947). There are sub-categories of this form of complexity, with the “4 C’s” described by Horgan (op. cit.) being the main ones: cybernetics, catastrophe theory, chaos theory, and heterogeneous agent complexity. This latter has increasingly become what many mean by the term “complexity” in economics, with Arthur, Durlauf, and Lane (1997) laying out crucial characteristics of this approach, favored at the Santa Fe Institute, and often involving computer simulations. Given the interest of Foley in models using computer simulations of the sorts used at the Santa Fe Institute, such as cellular automata, it is not surprising that he is interested in these forms of complexity, although often emphasizing more the computability side of their use over the dynamics side of them. As laid out by Arthur, Durlauf, and Lane (1997), this approach is very much an antithesis of a general equilibrium perspective, with agents interacting locally with each other without ever necessarily being in any overall or general equilibrium and not ever necessarily arriving at one either. Many econophysics models arguably follow this pattern, although it can be debated whether or not Foley’s own forays into econophysics fully fit this model.

Foley on Computational Complexity

Foley’s most important direct work on questions related to computational complexity arose from his collaborations with the late Peter Albin. Most significantly, when Albin fell ill while working on a book covering his approach to this, Foley stepped in and edited a volume consisting mostly of previously published papers by Albin (Albin with Foley, 1998). As part of that, he wrote a substantial introductory chapter (pp. 3-72) in which he laid out his own perspective on such matters as nonlinear dynamical systems, the nature of complexity, various approaches to computational complexity, and other related issues. As the introductory chapter to this book, mostly by Albin, it followed very much Albin’s views, although it would seem that at least at that time, Foley’s views were fairly much in synch with those of Albin. This becomes important in that Albin was arguably the first economist to study computational complexity as it relates to economic systems (Albin, 1982, reprinted as Chap. 2 of Albin with Foley, 1998).

In particular, Albin emphasized computability problems associated with the halting problem, and the limits (or “barriers and bounds”) this imposed on full rationality by economic agents. Drawing on the literature deriving from Gödel’s Incompleteness Theorem as connected to computer science by Alan Turing (1936-37), he emphasized especially the role of self-referencing in bringing about these failures to halt, or infinite do-loops in programs. Such self-referencing can lead to such problems as the Cretan Liar paradox, “Is a Cretan who says ‘All Cretans are liars’ telling the truth?” Such questions can lead to an endless going back and forth between saying “yes” and “no,” thus becoming undecidable as the program fails to halt at a solution. Albin argued that economies are full of such self-referencing that implies such non-decidabilities from a computability perspective, a swarm of unstoppable infinite regresses, and Foley agreed.[4]

Albin was strongly influenced by the ideas of Stephen Wolfram (1986), who in turn has been strongly influenced by those of Noam Chomsky (1959). In his discussion of complexity in this introductory chapter, Foley follows through on these lines, discussing in succession computational complexity, linguistic complexity, and machine complexity. His initial discussion of computational complexity makes it clear that he is going to embed it within the four-level hierarchy developed by Chomsky and show how undecidability is a form of computational complexity that undermines full rationality. This is followed by a discusson of how computational complexity and dynamic complexity relate, which in turn is followed by a more detailed discussion of cellular automata and the forms of complexity in them.

Central to this entire discussion is the four-level hierarchy of languages, initially due to Chomsky (1959). He defines a formal language as consisting of finite sets of an alphabet of T terminal symbols (words), intermediate variables V, and a distinguished variable S that serves to initiate productions in the language. These take the form of P ( Q, where P is a string composed of one or more variables with zero or more terminals and Q is a string composed of any combination of variables and terminals.

The lowest of these is the category of regular languages, generated by grammars that take the form P ( T or P ( TQ, where P and Q are variables and T is a string of terminal symbols that make up the regular language. The next level up are the context-free languages, which include the regular languages. The grammars generating these take the form of P ( Q, where P is a string of variables and Q a string composed of terminal symbols and variables, without contextual restrictions. Above this is the level of context-sensitive languages that are generated by grammars such that P ( Q have the length of Q at least as long as the length of P, this group likewise containing the one lower down. For these, if Q is a non-empty string and P1PP2 ( P1QP2, then Q and P can be substituted for each other in the context of P1 and P2, but not necessarily in other contexts. The highest level of these grammars generates unrestricted languages, which only need to follow the most general rules described above and will include the most complex formal languages. Each of the higher levels contains within it the level below it in an embedded or nested form.

This hierarchy is then seen to translate directly to a four-level hierarchy of machine complexity. At the lowest level is the finite automaton that that has a finite set of states and reads simple inputs off a tape to generate simple outputs, with a pocket calculator an example. This corresponds to the regular languages. By adding an unbounded pushdown stack on which the automaton can store and retrieve symbols on a first-in and last-out basis, one can recognize context-free languages. Having two bounded pushdowns moves one to the context-sensitive level. Having two unbounded pushdowns generates the level of unrestricted languages and becomes equivalent to an abstract Turing machine. Foley interprets all of these as representing informational and computational problems for economic decisionmakers.[5]

Foley then pulls what is the neat hat trick of applying all this to the analysis of dynamic complexity, which he sees as also having such a hierarchy, although in this he follows Wolfram. The dynamic equivalent of regular languages and finite automata are simple linear dynamical systems that converge on point attractors. The next stage up equivalent to context-free languages and one pushdown automata consists of nonlinear systems that endogenously generate periodic cycles as their attractors.

Context-sensitive languages and the two-bounded pushdown automata are equivalent to nonlinear dynamical systems that can go to a chaotic attractor. The break between this level and the previous one is equivalent to the line between the non-dynamically complex and the dynamically complex systems according to the Day-Rosser definition. Foley notes that at this level there can be long-period relations such as a parenthesis that opens and is closed much later. He identifies this with such economic phenomena as a loan being made that must then be paid back at some later time. He also notes that for this level, computing costs may rise so sharply that it may become impractical to actually solve problems, even if they are decidable in principle. This is somewhat equivalent to the break between P and NP systems more generally in the computational complexity literature, although it remains unproven that these really are distinct levels.

Finally, the equivalent of the unrestricted languages and two-unbounded pushdown automata equivalent to Turing machines may be undecidable. Monotonicities holding at the other levels break down at this level.

Foley then discusses Wolfram’s more specific adaptation of these categories to cellular automata. Type 1 evolve to uniform states. Type 2 evolve to periodic patterns from arbitrary initial conditions. Type 3 evolve to irregular patterns. And Type 4 generate a much broader range of possibilities, including non-monotonic ones in which a simple outcome may be the result of an enormously complicated set of generated structures. Foley links this to “edge of chaos” ideas of self-organizing systems and sees this level as that where new structures can emerge. In Albin’s discussion of this he ties this to the original von Neumann (1966) formulation for cellular automata that can self-reproduce, with von Neumann likening the jump from one level to another as being marked by complexity thresholds.

This is a stunning way of integrating the supposedly competing approaches to complexity, and this observer finds it impressive. However, this observer also questions whether this fully captures all the dynamic phenomena one observes in dynamically complex systems. In particular is the matter of emergence, argued by many to be at the very heart of dynamic complexity in systems ranging from evolution to urban development (Rosser, 2011). Two comebacks to this are that it may be captured by that highest level of Chomsky-Wolfram complexity. Another is to say that the supposed inability of computational systems to deal with this is a sign that such emergence is really not a proper complexity concept, with computational systems that provide genuine novelty (Moore, 1990) being the more useful concept.[6]

The Dynamic Complexity of Foley’s Econophysics

In the wake of his work with Peter Albin, Duncan Foley was also moving in another direction into the world of complexity economics, that of reconsidering the role of statistical mechanics as a foundation for general equilibrium theory, or more accurately, an alternative version of that theory. Mirowski (1989) has documented that the seminal developer of statistical mechanics theory, J. Willard Gibbs (1902),[7] was a profound influence on efforts in the US to mathematize neoclassical economic theory, both through Irving Fisher and later through Paul Samuelson (1947), who always looked to Gibbs as a source and inspiration thanks to the direct influence of Gibbs’s student, Edwin Bidwell Wilson at the University of Chicago. However, after this time the ideas of Gibbs faded from influencing many in economics directly. In any case, Samuelson was more influenced by other parts of Gibbs’s work than that on statistical mechanics.

The idea that an equilibrium might not be just a vector of prices (an idea certainly consistent with Gibbsian vector analysis), but a probability distribution of prices was what Foley would pursue.[8] His important paper of 1994 in the Journal of Economic Theory would do just that and would fit in with the emerging new confluence of economics and physics ideas that was happening at the Santa Fe Institute with which he would become affiliated, and which would eventually form as econophysics shortly thereafter, with H. Eugene Stanley coining this term in 1995, a year after Foley’s article appeared (Rosser, 2008).

While Foley has avoided these debates, econophysics has become quite controversial. Several economists generally sympathetic to it have criticized it fairly recently (Gallegati, Keen, Lux, and Ormerod, 2006), with some physicists replying sharply (Ball, 2006; McCauley, 2006). The critics argue on four points: 1) that many econophysicists make exaggerated claims of originality due to not knowing relevant economics literature, 2) that they often fail to use a sufficiently rigorous statistical methodology or techniques, 3) that they make exaggerated claims regarding the universality of some of their empirical findings, and 4) that they often lack any theoretical foundation or explanation for what they study. Interestingly, Foley’s work on this topic may help out on this last point. It is a bit ironic that he is an economist using a physics-based approach that may provide a theoretical foundation for certain portions of current econophysics.

Central to Foley’s approach was the idea that an equilibrium state should be associated with a maximization of entropy, a deeply Gibbsian idea, as Gibbs was also a founder of chemical thermodynamics. This was an aspect that Samuelson did not approve of so much. He made this clear at a symposium honoring Gibbs (Samuelson, 1990, p. 263):

“I have come over the years to have some impatience and boredom to find those who try to find an analogue of the entropy of Clausius or Boltzman or Shannon[9] to put into economic theory. It is the mathematical structure of classical (phenomenological, macroscopic, nonstochastic) thermodynamics that has isomorphisms with theoretical economics.” [italics in original]

Not for Samuelson would there be stochastic processes or probability distributions of price outcomes. His Gibbs was the “nonstochastic” Gibbs.

Foley followed the later Gibbs of statistical mechanics, the student of the foundation of temperature in the stochastic processes of molecular dynamics. It was in this nonlinear world that phase transitions became important in studying such phenomena as the freezing or melting or boiling or condensation of water at critical temperatures. Foley’s interest in nonlinear dynamics had long been more along the lines of understanding bifurcations of dynamical systems than of the precise natures of erratic trajectories, whether chaotic or not. Statistical mechanics offered an approach to understanding such bifurcations within the purview of his old topic of study, general equilibrium theory.

In any case, he pursued the idea of a thermodynamic equilibrium in which entropy is maximized.[10] A difference between this sort of equilibrium and the more standard Walrasian one is that this statistical mechanics one lacks the welfare economics implications that the Walrasian one has. There would be no Pareto optimality for such a stochastic equilibrium, an equilibrium in which a good is being exchanged for many different prices across a set of heterogeneous traders in discrete bilateral transactions, with the set of those prices conforming to the distribution given by the maximization of entropy, this occurring across all goods and services.

One limit of the traditional Gibbsian model is that it is a conservative system, which usually means for a model of economic exchange that one is dealing with a pure exchange model. While much of general equilibrium theory has been developed with such an assumption, Foley was able to expand this to include both production and exchange. The key strong assumption he made was that all possible transactions in the economy have an equal probability. A curious outcome of this model is the possibility of negative prices coinciding with positive ones for the same good. While conventional theory rejects such ideas, indeed even the idea of a negative price for something at all,[11] such outcomes have been observed in real life, as in the Babylonian bride markets as reported by Herodotus, wherein the most beautiful brides command positive prices, while the ugliest command negative ones, even in the same auction. This phenomenon is coming to be known as the “Herodotus paradox,” (Baye, Kovenock, and de Vries, 2011), and it is to Foley’s credit that he allows for such outcomes in contrast with long traditions of general equilibrium theory that insist on strictly non-negative prices for equilibrium solutions..

In his model there are m commodities, n agents of r types, k with offer set Ak. The proportion of agents of type k who achieve a particular transaction x out of mn such possible ones is given by hk[x]. The multiplicity of an assignment is the number of ways n agents can be assigned to S actions, with ns being the number of agents assigned to action s, which is given by

W[{ns}] = n!/(n1!...ns!...nS!). (1)

A measure of this multiplicity is the informational entropy of the distribution, which is given by

H{hk[x]} = -Σk=1rWkΣx=1mnhk[x]lnhk[x]. (2)

This is then maimized subject to a pair of feasibility constraints given by

ΣxεAkhk[x] = 1, (3)

Σk=1rWkΣxεAkhk[x]x = 0, (4)

resulting in a unique solution if the feasibility set in non-empty of the canonical Gibbs form given by

hk[x] = exp[-πx]/Σxexp[-πx], (5)

where the vectors π ε Rm are the entropy shadow prices.

Foley has continued to follow along this line of inquiry (Foley and Smith, 2008), and this is more in line with developments in econophysics that have followed from this time forward. Foley has not pursued the issue of whether or not these distributions will obey power laws or not, a major concern of econophysicists, although he does accept their argument that financial market returns do tend to follow such laws. He has expressed the view that these newer ideas from physics can help move economics into a broader social science perspective along more realistic lines than has been seen in the past (Colander, Holt, and Rosser, p. 212):

“The study of economic data surely has a future, but the question is whether it will be recognizable as economics in today’s terms and whether it will exhibit any unity of subject matter and method.”

Conclusion

Duncan Foley’s intellectual career has emphasized fundamental issues of economic theory, particularly general equilibrium theory. This initially concerned the loose ends of uniqueness and stability and how the role of money might allow for a proper microfoundation for macroeconomics. However, as he pursued these questions through various approaches and over time, he eventually moved towards the two leading strands of complexity within economics, the computational and the dynamic. He became involved in the former through his collaboration with the late Peter Albin, the first economist to seriously deal with problems of computational complexity. They followed the route of using the four-hierarchy level model of Chomsky as adumbrated by Wolfram. A particular emphasis of their interests were in the problems of non-decidability that can arise in such systems and how this limits full rationality by agents.

He would also move towards the sorts of dynamics associated with the econophysics movement by using the statistical mechanics methods developed initially by J. Willard Gibbs, although as applied to information theory by such figures as Shannon and Jaynes. This allowed for him to establish a maximum entropy equilibrium of a distribution of price outcomes that could allow for both positive and negative prices for specific goods and that also lacked any welfare implications.

A particular outcome of the earlier work was a possible integration of the computational and dynamic approaches. This involved classifying patterns of dynamic paths arising from nonlinear economic systems into the four-level hierarchy developed by Chomsky and Wolfram. One may disagree that this is the best way to reconcile these mostly competing approaches to economic complexity. However, it must be granted that this is an ingenious intellectual achievement, not the only one that Duncan Foley has achieved in his distinguished intellectual career.

References

Albin, Peter S. 1982. “The metalogic of economic predictions, calculations, and propositions.” Mathematical Social Sciences 3, 329-358.

Albin, Peter and Duncan K. Foley. 1992. “Decentralized, dispersed exchange without an auctioneer: A simulation study.” Journal of Economic Behavior and Organization 18, 27-51.

Albin, Peter S. with Duncan K. Foley. 1998. Barriers and Bounds to Rationality: Essays on Economic Complexity and Dynamics in Interactive Systems. Princeton: Princeton University Press.

Arthur, W. Brian, Steven N. Durlauf, and David A. Lane. 1997. “Introduction.” In: Arthur, W.B, Durlauf, S.N., and Lane, D.A. (eds.). The Economy as an Evolving Complex System II. Reading: Addison-Wesley, pp. 1-14.

Ball, Philip. 2006. “Culture clash.” Nature 441, 686-688.

Baye, Michael A., Don Kovenock, and Caspar de Vries. 2011. “The Herodotus paradox.” Games and Economic Behavior, forthcoming.

Blum, Lenore, Felipe Cucker, Michael Shub, and Steve Smale. 1998. Complexity and Real Computation. New York: Springer.

Chomsky, Noam. 1959. “On certain properties of grammars.” Information and Control 2, 137-167.

Colander, David, Richard P.F. Holt, and J. Barkley Rosser, Jr. 2004. The Changing Face of Economics: Conversations with Cutting Edge Economists. Ann Arbor: University of Michigan Press.

Davidson, Julius. 1919. “One of the physical foundations of economics.” Quarterly Journal of Economics 33, 717-724.

Day, Richard H. 1994. Complex Economic Dynamics: An Introduction to Dynamical Systems and Market Mechanisms, Volume I. Cambridge: MIT Press.

Foley, Duncan K. 1982. “Realization and accumulation in a Marxian model of the circuit of capital.” Journal of Economic Theory 28, 300-319.

Foley, Duncan K. 1986. “Liquidity-profit rate cycles in a capitalist economy.” Journal of Economic Behavior and Organization 8, 363-376.

Foley, Duncan K. “A statistical equilibrium theory of markets.” Journal of Economic Theory 62, 321-345.

Foley, Duncan K. 2010a. “What’s wrong with the fundamental existence and welfare theorems?” Journal of Economic Behavior and Organization 75, 115-131.

Foley, Duncan K. 2010b. “Model description length priors in the urn problem.” In: Zambell, S. (ed.). Computable, Constructive and Behavioural Economic Dynamics: Essays in honour of Kumaraswamy (Vela) Velupillai. Milton Park: Routledge, pp. 109-121.

Foley, Duncan K. and Miguel Sidrauski. 1971. Monetary and Fiscal Policy in a Growing Economy. New York: Macmillan.

Foley, Duncan K. and Eric Smith. 2008. “Classical thermodynamics and general equilibrium theory.” Journal of Economic Dynamics and Control 32, 7-65.

Föllmer, Hans. 1974. “Random economies with many interacting agents.” Journal of Mathematical Economics 1, 51-62.

Gallegati, Mauro, Steve Keen, Thomas Lux, and Paul Ormerod. 2006. “Worrying trends in econophysics.” Physica A 370, 1-6.

Georgescu-Roegen, Nicholas. 1971. The Entropy Law and the Economic Process. Cambridge: Harvard University Press.

Gibbs, J. Willard. 1902. Elementary Principles of Statistical Mechanics. New Haven: Yale University Press.

Goodwin, Richard M. 1947. “Dynamical coupling with especial reference to markets having production lags.” Econometrica 15, 181-204.

Goodwin, Richard M. 1951. “The nonlinear accelerator and the persistence of business cycles.” Econometrica 19, 1-17.

Horgan, John. 1997. The End of Science: Facing the Limits of Knowledge in the Twilight of the Scientific Age. New York: Broadway Books.

Jaynes, E.T. 1957. “Information theory and statistical mechanics.” Physical Review 106, 620-638; 108, 171-190.

McCauley, Joseph L. 2006. “Response to ‘Worrying trends in econophysics’.” Physica A 371, 601-609.

Mirowski, Philip. 1989. More Heat than Light: Economics as Social Physics, Physics as Nature’s Economics. Cambridge: Cambridge University Press.

Mirowski, Philip. 2007. “Markets come to bits: Evolution, computation, and makomata in economic science.” Journal of Economic Behavior and Organization 63, 209-242.

Moore, Christopher. 1990. “Undecidability and unpredictability in dynamical systems.” Physical Review Letters 64, 2354-2357.

Rosser, J. Barkley, Jr. 1999. “On the complexities of complex economic dynamics.” Journal of Economic Perspectives 13(4), 169-192.

Rosser, J. Barkley, Jr. 2008. “Debating the role of econophysics.” Nonlinear Dynamics, Psychology, and Life Sciences 12, 311-323.

Rosser, J. Barkley, Jr. 2009. “Computational and dynamic complexity in economics.” In: Rosser, J.B., Jr. (ed.). Handbook of Complexity Research. Cheltenham: Edward Elgar, pp. 22-35.

Rosser, J. Barkley, Jr. 2011. Complex Evolutionary Dynamics in Urban-Regional and Ecologic-Economic Systems. New York: Springer.

Samuelson, Paul A. 1947. The Foundations of Economic Analysis. Cambridge: Harvard University Press.

Samuelson, Paul A. 1990. “Gibbs in economics.” In: Caldi, G. and Mostow, G.D. (eds.). Proceedings of the Gibbs Symposium. Providence: American Mathematical Society, pp. 255-267.

Scarf, Herbert.E. 1960. “Some examples of global instability of competitive equilibrium.” International Economic Review 1, 157-172.

Scarf, Herbert E. 1973. The Computation of Economic Equilibria. New Haven: Yale University Press.

Shannon, Claude E. 1951. “Prediction and entropy of printed English.” The Bell System Technical Journal 30, 50-64.

Stutzer, Michael J. 1994. “The statistical mechanics of asset prices.” In: Elworthy, K.D., Everitt, W.N., and Lee, E.B. (eds.). Differential Equations, Dynamical Systems, and Control Science: A Festschrift in Honor of Leonard Markus. New York: Marcel Dekker, pp. 321-342.

Tesfatsion, Leigh and Kenneth L. Judd (eds.). 2004. Handbook of Computational Economics, Volume 2: Agent-Based Computatonal Economics. Amsterdam: Elsevier.

Turing, Alan M. 1936-37. “On computable numbers with an application to the Entscheidungsproblem.” Proceedings of the London Mathematical Society, Series 2 43, 544-546.

Velupillai, K. Vela. 2000. Computable Economics. Oxford: Oxford University Press.

Velupillai, K. Vela. 2009. “A computable economist’s perspective on computational complexity.” In: Rosser, J.B., Jr. (ed.). Handbook of Complexity Research. Cheltenham: Edward Elgar, pp. 36-83.

von Neumann, John, completed by Arthur W. Burks. 1966. Theory of Self Reproducing Automata.Urbana: University of Illinois Press.

Weidlich, Wolfgang. 2000. Sociodynamics: A Systematic Approach to Mathematical Modelling in the Social Sciences. Amsterdam: Harwood.

Wilson, Alan G. 1970. Entropy in Urban and Regional Modelling. London: Pion.

Wolfram, Stephen (ed.).. 1986. Theory and Applications of Cellular Automata. Singapore: World Scientific.

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[1] Colander, Holt, and Roseser (2004, p. 196). This is from an interview with Foley in this book, and in connection with that remark he compared studying money to an odyssey like that of Ulysses trying to reach his home in Ithaca, finally confessing when asked “So, you haven’t reached Ithaca yet?” that “No, and I probably never will” (ibid.)

[2] He worked as a computer programmer in an instrument and control company for three summers when young (Colander, Holt, and Rosser, 2004, p. 186).

[3] Ironically, this definition is not among the 45 that Lloyd listed earlier in the 1990s, although a few of those he lists have some similarities to it.

[4] Velupillai (2000) has emphasized the distinction between “computable economics” and “computational economics,” with the former being more concerned with these matters of whether a program halts or not as Albin and Foley see it, whereas more conventional computational economics is concerned more with such issues as trying to figure out the most efficient way to program a system or problem (Judd and Tesfatsion, 2006). Curiously enough, the issue of computational complexity is involved with the concerns of computational economics as the most efficient program may also be the shortest in length, which is tied to many of the more widely used measures of computational complexity.

[5] Mirowski (2007) uses this four-level hierarchy to extend this analysis directly to markets, arguing that they are algorithms fundamentally, and that market forms themselves evolve in a natural selection manner, with the equivalent of finite automata being spot markets, with futures markets the next level up, options markets above and embedding them, and so forth.

[6] Foley (2010b) has continued to deal with problems of computational complexity at a theoretical level.

[7] Gibbs also invented vector analysis independently of Oliver Heaviside.

[8]Hans Föllmer (1974) would pursue this idea, only to have no one until Foley follow up on it with a more general approach.

[9] It was Shannon (1951) who would link entropy with information theory. A major influence on Foley’s thinking came through this link as discussed by Jaynes (1957).

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