The Economy As An Evolving Complex System II (Santa Fe Institute Studies in the Sciences of Complexity Lecture Notes)

Citation preview

About the Santa Fe Institute The Santa Fe Institute (SFI) is a private, independent, multidiscipiinary research and education center, founded in 1984. Since its founding, SFI has devoted itself to creating a new kind of scientific research community, pursuing emerging science. Operating as a small, visiting institution, SFI seeks to catalyze new collaborative, multidiscipiinary projects that break down the barriers between the traditional disciplines, to spread its ideas and methodologies to other individuals, and to encourage the practical applications of its results.

All titles from the Santa Fe Institute Studies in the Sciences of Complexity series will carry this imprint which is based on a Mimbres pottery design (circa A.D. 950-1150), drawn by Betsy Jones. The design was selected because the radiating feathers are evocative of the outreach of the Santa Fe Institute Program to many disciplines and institutions.

This page intentionally left blank

Santa Fe Institute Series List Lecture Notes Volumes in the Santa Fe Institute Studies in the Sciences of Complexity Volume I: John Hertz, Anders Krogh, Richard G. Palmer, editors: Introduction to the Theory of Neural Networks Volume II: Gerard Weisbuch: Complex Systems Dynamics Volume III: Wilfred Stein and Francisco J.Varela, editors: Thinking About Biology Volume IV: Joshua M. Epstein: Nonlinear Dynamics, .Mathematical Biology, and Social Science Volume V: H.F. Nijhout, Lynn Nadel, and Daniel L. Stein, editors: Pattern Formation in the Physical and Biological Sciences Proceedings Volumes in the Santa Fe Institute

Studies in the Sciences of

Complexity

Volume I: David Pines, editor: Emerging Synthesis in Science Volume II: Alan S, Perelson, editor: Theoretical Immunology, Part One Volume III: Alan S. Perclson, editor: Theoretical immunology, Part Two Volume IV: Gary D. Doolen, senior editor: Lattice Gas Methods for Partial Differential Equations Volume V: Philip W. Anderson, Kenneth J. Arrow, and David Pines, editors:The Economy as an Evolving Complex System Volume VI: Christopher G. Langton, editor: Artificial file Volume VII: George I. Bell and Thomas G. Marr, editors: Computers and DNA Volume VI11: Wojciech H. Zurek, editor: Complexity, Entropy, and the Physics of Information Volume IX: Alan S. Perelson and Stuart A. Kauffman, editors: Molecular Evolution on Rugged Landscapes: Proteins, RNA, and the Immune System Volume X: Christopher G. Langton, Charles Taylor, J. Dovrie Farmer and Steen Rasmussen, editors: Artificial Life II Volume XI: John A. Hawkins and Murray Gell-Mann, editors: The Evolution of Human Languages

Volume XII: Martin Casdagli and Stephen Eubank, editors: Nonlinear Modeling and Forecasting Volume XIII: Jay E. Mitlenthal and Arthur B. Baskin, editors: The Principles of Organizations in Organisms Volume XIV: Daniel Friedman and John Rust, editors: The Double Auction Market: Institutions, Theories, and Evidence Volume XV: Andreas S. Weigend and Neil A. Gershenfeld, editors:Time Series Prediction: Forecasting the Future and Understanding the Past Volume XVI: George j . Gummcrman and Murray Gell-Mann, editors: Understanding Complexity in the Prehistoric Southwest Volume XVII: Christopher G. Langton, editor: Artificial Life HI Volume XVIII: Gregory Kramer, editor: Auditory Display: Bonification, Audilication, and Auditory Interlaces Volume XIX: George A. Cowan, David Pines, and David Meltzer, editors: Complexity: Metaphors, Models, and Reality Volume XX: David Wolpert, editor: The Mathematics of Generalization Volume XXI: P.E. Cladis and P. Palffy-Muhoray, editors: Spatio-Temporal Patterns in Nonequilibrium Complex Systems Volume XXII: Harold J. Morowitz and Jerome L. Singer, editors: The Mind, the Brain, and Complex Adaptive Systems Volume XXIII: Bela Jules-/. and llona Kovacs, editors: Maturational Windows and Adult Cortical PlasticityVolume XXIV: Joseph A. Fainter and Bonnie Bagley fainter, editors: Evolving Complexity and Environmental Risk in the Prehistoric Southwest Volume XXV: John B. Rundle, Donald L.Turcottc, and William Klein, editors: Reduction and Predictability of Natural Disasters Volume XXVI: Richard K. Belew and Melanie Mitchell, editors: Adaptive Individuals in Evolving Populations Volume XXVII: W. Brian Arthur, Steven N. Durlauf, and David A Lane, editors:The Economy as an Evolving Complex System II Volume XXVIII: Gerald Myers, editor: Viral Regulatory Structures and Their Degeneracy

Santa Fe Institute Editorial Board December 1996 Ronda K. Butler-Villa, Chair Director of Publications, Santa Fe Institute Prof. W. Brian Arthur Citibank Professor, Santa Fe Institute Dr. David K. Campbell Chair, Department of Physics, University of Illinois Dr. George A. Cowan Visiting Scientist, Santa Fe Institute and Senior Fellow Emeritus, Los Alamos National Laboratory Prof. Marcus W. Feldman Director, Institute for Population & Resource Studies, Stanford University Prof. Murray Gell-Mann Division of Physics & Astronomy, California Institute of Technology Dr. Ellen Goldberg President, Santa Fe Institute Prof. George J. Gumerman Center for Archaeological Investigations, Southern Illinois University Prof. John H. Holland Department of Psychology, University of Michigan Dr. Erica Jen Vice President for Academic Affairs, Santa Fe Institute Dr. Stuart A. Kauffman Professor, Santa Fe Institute Dr. Edward A. Knapp Visiting Scientist, Santa Fe Institute Prof. Harold Morowitz Robinson Professor, George Mason University Dr. Alan S. Perelson Theoretical Division, Los Alamos National Laboratory Prof. David Pines Department of Physics, University of Illinois Dr. L. Mike Simmons 700 New Hampshire Avenue, NW, Apartment 616, Washington DC 20037 Dr. Charles F. Stevens Molecular Neurobiology, The Salk Institute Prof. Harry L. Swinney Department of Physics, University of Texas

This page intentionally left blank

Contributors to This Volume Anderson, Philip W., Joseph Henry Laboratories of Physics, Badwin Hall, Princeton University, Princeton, NJ 08544 Arthur, W. B., Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 Blume, Lawrence E., Department of Economics, Uris Hall, Cornell University, Ithaca, NY 14853 Brock, William A.. Department of Economics, University of Wisconsin at Madison, Madison, WI 53706 Darley, V. M., Division of Applied Sciences. Harvard University, Cambridge, MA 02138 Durlauf, Steven, Department of Economics, University of Wisconsin at Madison, Madison, WI 53706 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501 Geanakoplos. John, Cowles Foundation. Yale University, 30 Hillhouse Avenue, New Haven, CT 06520 Holland, John H., Department of Computer Science and Engineering, University of Michigan, Ann Arbor. MI 48109 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 loannides, Yannis M., Department of Economics, Tufts University, Medford. MA 02155 Kauffman, Stuart A., Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 Kirman, Alan P , G.R.E.Q.A.M., E.H.E.S.S. and Universite d'Aix-Marseille III, Institut Universitaire de France, 2 Rue de la Charite, 13002 Marseille, FRANCE Kollman, Ken, Department of Political Science and Center for Political Studies, University of Michigan, Ann Arbor, MI 48109 Krugman, Paul, Department of Economics, Stanford University, Stanford, CA 95305 Lane, David, Department of Political Economy, University of Modena, ITALY LeBaron, Blake, Department of Economics, University of Wisconsin, Madison, WI 53706 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 Leijonhufvud, Axel, Center for Computable Economics, and Department of Economics, UCLA, 405 Hilgard Avenue, Los Angeles, CA 90095 Lindgren, Kristian, Institute of Physical Resource Theory, Chalmers University of Technology and Goteborg University, S-412 96 Goteborg, SWEDEN Manski, Charles F., Department of Economics, University of Wisconsin at Madison, Madison, WI 53706 Maxfield, Robert, Department of Engineering and Economic Svstems, Stanford University, Stanford, CA 95305

Miller, John H., Department of Social and Decision Sciences, Carnegie Mellon University, Pittsburgh, PA 15213 North, Douglass C Department of Economics,.Washington University, St. Louis, MO 63130-4899 Padgett, John F., Department of Political Science, University of Chicago. Chicago, IL 60637 Page, Scott, Division of Humanities and Social Sciences, California Institute of Technology 228-77, Pasadena, CA 91125 Palmer, Richard, Department of Physics, Duke University, Durham, NC 27706 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 Shubik, Martin, Yale University, Cowles Foundation for Research in Economics, Department of Economics, P.O. Box 208281, New Haven, CT 06520-8281 Tayler, Paul, Department of Computer Science, Brunei University, London, UK Tesfatsion, Leigh, Department of Economics, Heady Hall 260, Iowa State University, Ames, IA 50011-1070

Acknowledgment

The conference at which these papers were presented was sponsored by Legg Mason, whose support we gratefully acknowledge. Over the years the Santa Fe Institute's Economics Program has benefited from the generosity of Citicorp, Coopers & Lybrand, The John D. and Catherine T. MacArthur Foundation, McKinsey and Company, the Russell Sage Foundation, and SFFs core support. We thank Eric Beinhocker, Caren Grown, Win Farrell, Dick Foster, Henry Lichstein, Bill Miller, John Reed, and Eric Wanner, not only for their organizations' financial support but for the moral and intellectual support they have provided. Their many insights and suggestions over the years have greatly bolstered the program. We also thank the members of SFFs Business Network, and the many researchers who have taken part in the program. George Cowan took a chance early on that an economics program at the Institute would be a success. We thank him for his temerity. One of the pleasures of working at the Santa Fe Institute is the exemplary staff support. In particular we thank Ginger Richardson, the staff Director of Programs, and Andi Sutherland, who organized the conference this book is baaed on. We are very grateful to the very able publications people at SFI, especially Marylee Thomson and Delia Ulibarri. Philip Anderson and Kenneth Arrow have been guiding lights of the SFI Economics Program since its inception. Their intellectual and personal contributions are too long to enumerate. With respect and admiration, this book is dedicated to them.

W. Brian Arthur, Steven N. Durlauf, and David A. Lane

This page intentionally left blank

Contents

Introduction W. B. Arthur, S. N. Durlauf, and D. Lane

1

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

15

Natural Rationality V. M. Darky and S. A. Kauffman

45

Statistical Mechanics Approaches to Socioeconomic Behavior 5. N. Durlauf

81

Is What Is Good for Each Best for All? Learning From Others in the Information Contagion Model D. Lane

105

Evolution of Trading Structures Y. M. loannides

129

Foresight, Complexity, and Strategy D. Lane and R. Maxfield

169

The Emergence of Simple Ecologies of Skill J. F. Padgett

199

The Economy as an Evolving Complex System II, Eds. Arthur, Durlauf, and Lane SFI Studies in the Sciences of Complexity, Vol. XXVII, Addison-Wesley, 1997

XI

Xli

Contents

Some Fundamental Puzzles in Economic History/ Development D. C. North

223

How the Economy Organizes Itself in Space: A Survey of the New Economic Geography P. Krugman

239

Time and Money M. Shubik

263

Promises Promises J. Geanakoplos

285

Macroeconomics and Complexity: Inflation Theory A. Leijonhufvud

321

Evolutionary Dynamics in Game-Theoretic Models K. Lindgren

337

Identification of Anonymous Endogenous Interactions C. F. Manski

369

Asset Price Behavior in Complex Environments W. A. Brock

385

Population Games L. E. Blume

425

Computational Political Economy K. Kollman, J. H. Miller, and S. Page

461

The Economy as an Interactive System A. P. Kirman

491

How Economists Can Get ALife L. Tesjatsion

533

Some Thoughts About Distribution in Economics P. W. Anderson

565

Index

567

W. B. Arthur,* S. N. Durlauf,** and D. Lanet "Citibank Professor, Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 ""Department of Economics, University of Wisconsin at Madison, 53706 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM, 87501 fDepartment of Political Economy, University of Modena, ITALY

Introduction

PROCESS AND EMERGENCE IN THE ECONOMY In September 1987, twenty people came together at the Santa Fe Institute to talk about "the economy as an evolving, complex system." Ten were theoretical economists, invited by Kenneth J. Arrow, and ten were physicists, biologists, and computer scientists, invited by Philip W. Anderson. The meeting was motivated by the hope that new ideas bubbling in the natural sciences, loosely tied together under the rubric of "the sciences of complexity," might stimulate new ways of thinking about economic problems. For ten days, economists and natural scientists took turns talking about their respective worlds and methodologies. While physicists grappled with general equilibrium analysis and noncooperative game theory, economists tried to make sense of spin glass models, Boolean networks, and genetic algorithms. The meeting left two legacies. The first was a volume of essays, The Economy as an Evolving Complex System, edited by Arrow. Anderson, and David Pines. The

The Economy as an Evolving Complex System II, Eds. Arthur, Durlauf, and Lane SFI Studies in the Sciences of Complexity, Vol. XXVII, Addison-Wesley, 1997

2

W. B. Arthur, S. N. Durlauf, and D. Lane

other was the founding, in 1988, of the Economics Program at the Santa Fe Institute, the Institute's first resident research program. The Program's mission was to encourage the understanding of economic phenomena from a complexity perspective, which involved the development of theory as well as tools for modeling and for empirical analysis. To this end, since 1988, the Program has brought researchers to Santa Fe, sponsored research projects, held several workshops each year, and published several dozen working papers. And, since 1994, it has held an annual summer school for economics graduate students. This volume, The Economy as an Evolving Complex System II, represents the proceedings of an August 1996 workshop sponsored by the SFI Economics Program. The intention of this workshop was to take stock, to ask: What has the complexity perspective contributed to economics in the past decade? In contrast to the 1987 workshop, almost all of the presentations addressed economic problems, and most participants were economists by training. In addition, while some of the work presented was conceived or carried out at the Institute, some of the participants had no previous relation with SFI—research related to the complexity perspective is under active development now in a number of different institutes and university departments. But just what is the complexity perspective in economics? That is not an easy question to answer. Its meaning is still very much under construction, and, in fact, the present volume is intended to contribute to that construction process. Indeed, the authors of the essays in this volume by no means share a single, coherent vision of the meaning and significance of complexity in economics. What we will find instead is a family resemblance, based upon a set of interrelated themes that together constitute the current meaning of the complexity perspective in economics. Several of these themes, already active subjects of research by economists in the mid-1980s, are well described in the earlier The Economy as an Evolving Complex System: In particular, applications of nonlinear dynamics to economic theory and data analysis, surveyed in the 1987 meeting by Michele Boldrin and William Brock; and the theory of positive feedback and its associated phenomenology of path dependence and lock-in, discussed by W. Brian Arthur. Research related to both these themes has flourished since 1987, both in and outside the SFI Economics Program. While chaos has been displaced from its place in 1987 at center stage of the interest in nonlinear dynamics, in the last decade economists have made substantial progress in identifying patterns of nonlinearity in financial time series and in proposing models that both offer explanations for these patterns and help to analyze and, to some extent, predict the series in which they are displayed. Brock surveys both these developments in his chapter in this volume, while positive feedback plays a central role in the models analyzed by Lane (on information contagion), Durlauf (on inequality) and Krugman (on economic geography), and lurk just under the surface of the phenomena described by North (development) and Leijonhufvud (high inflation). Looking back over the developments in the past decade and the papers produced by the program, we believe that a coherent perspective—sometimes called

introduction

3

the "Santa Fe approach"—has emerged within economics. We will call this the complexity perspective, or Santa Fe perspective, or occasionally the process-andemergence perspective. Before we describe this, we first sketch the two conceptions of the economy that underlie standard, neoclassical economics (and indeed most of the presentations by economic theorists at the earlier 1987 meeting). We can call these conceptions the "equilibrium" and "dynamical systems" approaches. In the equilibrium approach, the problem of interest is to derive, from the rational choices of individual optimizers, aggregate-level "states of the economy" (prices in general equilibrium analysis, a set of strategy assignments in game theory with associated payoffs) that satisfy some aggregate-level consistency condition (market-clearing, Nash equilibrium), and to examine the properties of these aggregate-level states. In the dynamical systems approach, the state of the economy is represented by a set of variables, and a system of difference equations or differential equations describes how these variables change over time. The problem is to examine the resulting trajectories, mapped over the state space. However, the equilibrium approach does not describe the mechanism whereby the state of the economy changes over time—nor indeed how an equilibrium comes into being.I1! And the dynamical system approach generally fails to accommodate the distinction between agent- and aggregate-levels (except by obscuring it through the device of "representative agents"). Neither accounts for the emergence of new kinds of relevant state variables, much less new entities, new patterns, new structures.'2' To describe the complexity approach, we begin by pointing out six features of the economy that together present difficulties for the traditional mathematics used in economics: l3i DISPERSED INTERACTION. What happens in the economy is determined by the interaction of many dispersed, possibly heterogeneous, agents acting in parallel. The action of any given agent depends upon the anticipated actions of a limited number of other agents and on the aggregate state these agents cocreate.

"'Since an a priori intertemporal equilibrium hardly counts as a mechanism. '^Norman Packard's contribution to the 1987 meeting addresses just this problem with respect to the dynamical systems approach. As he points out, "if the set of relevant variables changes with time, then the state space is itself changing with time, which is not commensurate with a conventional dynamical systems model." Mjohn Holland's paper at the 1987 meeting beautifully—and presciently—frames these features. For an early description of the Santa Fe approach, see also the program's March 1989 newsletter, "Emergent Structures."

4

W. B. Arthur, S. N. Durlauf, and D. Lane

NO GLOBAL CONTROLLER. No global entity controls interactions. Instead, controls are provided by mechanisms of competition and coordination among agents. Economic actions are mediated by legal institutions, assigned roles, and shifting associations. Nor is there a universal competitor—a single agent that can exploit all opportunities in the economy. CROSS-CUTTING HIERARCHICAL ORGANIZATION. The economy has many levels of organization and interaction. Units at any given level—behaviors, actions, strategies, products—typically serve as "building blocks" for constructing units at the next higher level. The overall organization is more than hierarchical, with many sorts of tangled interactions (associations, channels of communication) across levels. CONTINUAL ADAPTATION . Behaviors, actions, strategies, and products are revised continually as the individual agents accumulate experience—the system constantly adapts. PERPETUAL NOVELTY, Niches are continually created by new markets, new technologies, new behaviors, new institutions. The very act of filling a niche may provide new niches. The result is ongoing, perpetual novelty. OUT-OF-EQUILIBRIUM DYNAMICS. Because new niches, new potentials, new possibilities, are continually created, the economy operates far from any optimum or global equilibrium. Improvements are always possible and indeed occur regularly. Systems with these properties have come to be called adaptive nonlinear networks (the term is John Holland's5). There are many such in nature and society: nervous systems, immune systems, ecologies, as well as economies. An essential element of adaptive nonlinear networks is that they do not act simply in terms of stimulus and response. Instead they anticipate. In particular, economic agents form expectations—they build up models of the economy and act on the basis of predictions generated by these models. These anticipative models need neither be explicit, nor coherent, nor even mutually consistent. Because of the difficulties outlined above, the mathematical tools economists customarily use, which exploit linearity, fixed points, and systems of differential equations, cannot provide a deep understanding of adaptive nonlinear networks. Instead, what is needed are new classes of combinatorial mathematics and populationlevel stochastic processes, in conjunction with computer modeling. These mathematical and computational techniques are in their infancy. But they emphasize the discovery of structure and the processes through which structure emerges across different levels of organization. This conception of the economy as an adaptive nonlinear network—as an evolving, complex system—has profound implications for the foundations of economic

Introduction

5

theory and for the way in which theoretical problems are cast and solved. We interpret these implications as follows: COGNITIVE FOUNDATIONS. Neoclassical economic theory has a unitary cognitive foundation: economic agents are rational optimizers. This means that (in the usual interpretation) agents evaluate uncertainty probabilistically, revise their evaluations in the light of new information via Bayesian updating, and choose the course of action that maximizes their expected utility. As glosses on this unitary foundation, agents are generally assumed to have common knowledge about each other and rational expectations about the world they inhabit (and of course cocreate). In contrast, the Santa Fe viewpoint is pluralistic. Following modern cognitive theory, we posit no single, dominant mode of cognitive processing. Rather, we see agents as having to cognitively structure the problems they face—as having to "make sense" of their problems—as much as solve them. And they have to do this with cognitive resources that are limited. To "make sense," to learn, and to adapt, agents use variety of distributed cognitive processes. The very categories agents use to convert information about the world into action emerge from experience, and these categories or cognitive props need not fit together coherently in order to generate effective actions. Agents therefore inhabit a world that they must cognitively interpret—one that is complicated by the presence and actions of other agents and that is ever changing. It follows that agents generally do not optimize in the standard sense, not because they are constrained by finite memory or processing capability, but because the very concept of an optimal course of action often cannot be defined. It further follows that the deductive rationality of neoclassical economic agents occupies at best a marginal position in guiding effective action in the world. And it follows that any "common knowledge" agents might have about one another must be attained from concrete, specified cognitive processes operating on experiences obtained through concrete interactions. Common knowledge cannot simply be assumed into existence. STRUCTURAL FOUNDATIONS. In general equilibrium analysis, agents do not interact with one another directly, but only through impersonal markets. By contrast, in game theory all players interact with all other players, with outcomes specified by the game's payoff matrix. So interaction structures are simple and often extreme— one-with-all or all-with-all. Moreover, the internal structure of the agents themselves is abstracted away. W In contrast, from a complexity perspective, structure matters. First, network4)ased structures become important. All economic action involves interactions among agents, so economic functionality is both constrained and carried by networks defined by recurring patterns of interaction among agents. These network structures are characterized by relatively sparse ties. Second, economic action is structured by emergent social roles and by socially supported procedures—that is, HI Except in principal-agent theory or transaction-costs economics, where a simple hierarchical structure is supposed to obtain.

6

W. B. Arthur, S. N. Durlauf, and D. Lane

by institutions. Third, economic entities have a recursive structure: they are themselves comprised of entities. The resulting "level" structure of entities and their associated action processes is not strictly hierarchical, in that component entities may be part of more than one higher-level entity, and entities at multiple levels of organization may interact. Thus, reciprocal causation operates between different levels of organization—while action processes at a given level of organization may sometimes by viewed as autonomous, they are nonetheless constrained by action patterns and entity structures at other levels. And they may even give rise to new patterns and entities at both higher and lower levels. From the Santa Fe perspective, the fundamental principle of organization is the idea that units at one level combine to produce units at the next higher level. i5l WHAT COUNTS AS A PROBLEM AND AS A SOLUTION. It should be clear by now that exclusively posing economic problems as multiagent optimization exercises makes little sense from the viewpoint we are outlining—a viewpoint that puts emphasis on process, not just outcome. In particular, it asks how new "things" arise in the world—cognitive things, like "internal models"; physical things, like "new technologies"; social things, like new kinds of economic "units." And it is clear that if we posit a world of perpetual novelty, then outcomes cannot correspond to steady-state equilibria, whether Walrasian, Nash, or dynamic-systems-theoretical. The only descriptions that can matter in such a world are about transient phenomena—about process and about emergent structures. What then can we know about the economy from a process-and-emergence viewpoint, and how can we come to know it? Studying process and emergence in the economy has spawned a growth industry in the production of what are now generally called "agent-based models." And what counts as a solution in an agent-based model is currently under negotiation. Many of the papers in this volume—including those by Arthur et al., Darley and Kauffman, Shubik, Lindgren, Kollman et al., Kirman, and Tesfatsion—address this issue, explicitly or implicitly. We can characterize these as seeking emergent structures arising in interaction processes, in which the interacting entities anticipate the future through cognitive procedures that themselves involve interactions taking place in multilevel structures. A description of an approach to economics, however, is not a research program. To build a research program around a process-and-emergence perspective, two things have to happen. First, concrete economic problems have to be identified for which the approach may provide new insights. A number of candidates are offered in this volume: artifact innovation (Lane and Maxfield), the evolution of trading networks (Ioannides, Kirman, and Tesfatsion), money (Shubik), the origin and spatial distribution of cities (Krugman), asset pricing (Arthur et al. and (51 We need not commit ourselves to what constitutes economic "units" and "leveis." This will vary from problem context to problem context.

Introduction

7

Brock), high inflation (Leijonhufvud) persistent differences in income between different neighborhoods or countries (Durlauf). Second, cognitive and structural foundations for modeling these problems have to be constructed and methods developed for relating theories based on these foundations to observable phenomena (Manski). Here, while substantial progress has been made since 1987, the program is far from complete. The essays in this volume describe a series of parallel explorations of the central themes of process and emergence in an interactive world—of how to study systems capable of generating perpetual novelty. These explorations do not form a coherent whole. They are sometimes complementary, sometimes even partially contradictory. But what could be more appropriate to the Santa Fe perspective, with its emphasis on distributed processes, emergence, and self-organization? Here are our interpretations of the research directions that seem to be emerging from this process: COGNITION. The central cognitive issues raised in this volume are ones of interpretation. As Shubik puts it, "the interpretation of data is critical. It is not what the numbers are, but what they mean." How do agents render their world comprehensible enough so that "information" has meaning? The two papers by Arthur, Holland, LeBaron, Palmer, and Tayler and by Darley and Kauffman consider this. They explore problems in which a group of agents take actions whose effects depend on what the other agents do. The agents base their actions on expectations they generate about how other agents will behave. Where do these expectations come from? Both papers reject common knowledge or common expectations as a starting point. Indeed, Arthur et al. argue that common beliefs cannot be deduced. Because agents must derive their expectations from an imagined future that is the aggregate result of other agents' expectations, there is a self-reference of expectations that leads to deductive indeterminacy. Rather, both papers suppose that each agent has access to a variety of "interpretative devices" that single out particular elements in the world as meaningful and suggest useful actions on the basis of the "information" these elements convey. Agents keep track of how useful these devices turn out to be, discarding ones that produce bad advice and tinkering to improve those that work. In this view, economic action arises from an evolving ecology of interpretive devices that interact with one another through the medium of the agents that use them to generate their expectations. Arthur et al. build a theory of asset pricing upon such a view. Agents— investors—act as market statisticians. They continually generate expectational models—interpretations of what moves prices in the market—and test these by trading. They discard and replace models if not successful. Expectations in the market therefore become endogenous—they continually change and adapt to a market that they create together. The Arthur et al. market settles into a rich psychology, in which speculative bubbles, technical trading, and persistence of volatility emerge. The homogeneous rational expectations of the standard literature become a special case—possible in theory but unlikely to emerge in practice. Brock presents

8

W. B. Arthur, S. N. Duriauf, and D. Lane

a variant of this approach, allowing agents to switch between a limited number of expectational models. His model is simpler than that of Arthur et al., but he achieves analytical results, which he relates to a variety of stylized facts about financial times series, many of which have been uncovered through the application of nonlinear analysis over the past decade. In the world of Darley and Kauffman, agents are arrayed on a lattice, and they try to predict the behavior of their lattice neighbors. They generate their predictions via an autoregressive model, and they can individually tune the number of parameters in the model and the length of the time series they use to estimate model parameters. Agents can change parameter number or history length by steps of length 1 each period, if by doing so they would have generated better predictions in the previous period. This induces a coevolutionary "interpretative dynamics," which does not settle down to a stable regime of precise, coordinated mutual expectations. In particular, when the system approaches a "stable rational-expectations state," it tends to break down into a disordered state. They use their results to argue against conventional notions of rationality, with infinite foresight horizons and unlimited deductive capability. In his paper on high inflation, Leijonhufvud poses the same problem as Darley and Kauffman: Where should we locate agent cognition, between the extremes of "infinite-horizon optimization" and "myopic adaptation"? Leijonhufvud argues that the answer to this question is context dependent. He claims that in situations of institutional break-down like high inflation, agent cognition shifts toward the "short memory/short foresight adaptive mode." The causative relation between institutional and cognitive shifts becomes reciprocal. With the shrinking of foresight horizons, markets for long-term loans (where long-term can mean over 15 days) disappear. And as inflation accelerates, units of accounting lose meaning. Budgets cannot be drawn in meaningful ways, the executive arm of government becomes no longer fiscally accountable to parliament, and local governments become unaccountable to national governments. Mechanisms of social and economic control erode. Ministers lose control over their bureaucracies, shareholders over corporate management. The idea that "interpretative devices" such as explicit forcasting models and technical-trading rules play a central role in agent cognition fits with a more general set of ideas in cognitive science, summarized in Clark.2 This work rejects the notion that cognition is all "in the head." Rather, interpretive aids such as autoregressive models, computers, languages, or even navigational tools (as in Hutchins6) and institutions provide a "scaffolding," an external structure on which much of task of interpreting the world is off-loaded. Clark2 argues that the distinctive hallmark of in-the-head cognition is "fast pattern completion," which bears little relation to the neoclassical economist's deductive rationality. In this volume, North takes up this theme, describing some of the ways in which institutions scaffold interpretations of what constitutes possible and appropriate action for economic agents.

Introduction

9

Lane and Maxfield consider the problem of interpretation from a different perspective. They are particularly interested in what they call attributions of functionality: interpretations about what an artifact does. They argue that new attributions of functionality arise in the context of particular kinds of agent relationships, where agents can differ in their interpretations. As a consequence, cognition has an unavoidable social dimension. What interpretations are possible depend on who interacts with whom, about what. They also argue that new functionality attributions cannot be foreseen outside the particular generative relationships in which they arise. This unforeseeability has profound consequences for what constitutes "rational" action in situations of rapid change in the structure of agent-artifact space. All the papers mentioned so far take as fundamental the importance of cognition for economic theory. But the opposite point of view can also be legitimately defended from a process-and-emergence perspective. According to this argument, overrating cognition is just another error deriving from methodological individualism, the very bedrock of standard economic theory. How individual agents decide what to do may not matter very much. What happens as a result of their actions may depend much more on the interaction structure through which they act—who interacts with whom, according to which rules. Blume makes this point in the introduction to his paper on population games, which, as he puts it, provide a class of models that shift attention "from the fine points of individual-level decision theory to dynamics of agent interaction." Padgett makes a similar claim, though for a different reason. He is interested in formulating a theory of the firm as a locus of transformative "work," and he argues that "work" may be represented by "an orchestrated sequence of actions and reactions, the sequence of which produces some collective result (intended or not)." Hence, studying the structure of coordinated action-reaction sequences may provide insight into the organization of economic activity, without bringing "cognition" into the story at all. Padgett's paper is inspired by recent work in chemistry and biology (by Eigen and Schuster3 and by Fontana and Buss,4 among others) that are considered exemplars of the complexity perspective in these fields. STRUCTURE. Most human interactions, even those taking place in "economic" contexts, have a primarily social character: talking with friends, asking advice from knowledgeable acquaintances, working together with colleagues, living next to neighbors. Recurring patterns of such social interactions bind agents together into networks.I61 According to standard economic theory, what agents do depends on their values and available information. But standard theory typically ignores where values and information come from. It treats agents' values and information as exogenous and autonomous. In reality, agents learn from each other, and their values may be influenced by others' values and actions. These processes of learning '^There is a voluminous sociological literature on interaction networks. Recent entry points include Noria and Eccles, 7 particularly the essay by Granovetter entitled "Problems of Explanation in Economic Sociology," and the methodological survey of Wasserman and Faust. 8

10

W. B. Arthur, S. N. Durlauf, and D. Lane

and influencing happen through the social interaction networks in which agents are embedded, and they may have important economic consequences. For example, one of the models presented in Durlauf's paper implies that value relationships among neighbors can induce persistent income inequalities between neighborhoods. Lane examines a model in which information flowing between agents in a network determines the market shares of two competing products. Kirman's paper reviews a number of models that derive economic consequences from interaction networks. Ioannides, Kirman, and Tesfatsion consider the problems of how networks emerge from initially random patterns of dyadic interaction and what kinds of structure the resulting networks exhibit. Ioannides studies mathematical models based on controlled random fields, while Tesfatsion works in the context of a particular agent-based model, in which the "agents" are strategies that play Prisoner's Dilemma with one another. Ioannides and Tesfatsion are both primarily interested in networks involving explicitly economic interactions, in particular trade. Their motivating idea, long recognized among sociologists (for example, Baker1), is that markets actually function by means of networks of traders, and what happens in markets may reflect the structure of these networks, which in turn may depend on how the networks emerge. Local interactions can give rise to large-scale spatial structures. This phenomenon is investigated by several of the papers in this volume. Lindgren's contribution is particularly interesting in this regard. Like Tesfatsion, he works with an agent-based model in which the agents code strategies for playing two-person games. In both Lindgren's and Tesfatsion's models, agents adapt their strategies over time in response to their past success in playing against other agents. Unlike Tesfatsion's agents, who meet randomly and decide whether or not to interact, Lindgren's agents only interact with neighbors in a prespecified interaction network. Lindgren studies the emergence of spatiotemporal structure in agent space—-metastable ecologies of strategies that maintain themselves for many agent-generations against "invasion" by new strategy types or "competing" ecologies at their spatial borders. In particular, he compares the structures that arise in a lattice network, in which each agent interacts with only a few other agents, with those that arise in a fully connected network, in which each agent interacts with all other agents. He finds that the former "give rise to a stable coexistence between strategies that would otherwise be outcompeted. These spatiotemporal structures may take the form of spiral waves, irregular waves, spatiotemporal chaos, frozen patchy patterns, and various geometrical configurations." Though Lindgren's model is not explicitly economic, the contrast he draws between an agent space in which interactions are structured by (relatively sparse) social networks and an agent space in which all interactions are possible (as is the case, at least in principle, with the impersonal markets featured in general equilibrium analysis) is suggestive. Padgett's paper offers a similar contrast, in a quite different context. Both Durlauf and Krugman explore the emergence of geographical segregation. In their models, agents may change location—that is, change their position in a social structure defined by neighbor ties. In these models (especially Durlauf's),

Introduction

11

there are many types of agents, and the question is under what circumstances, and through what mechanisms, do aggregate-level "neighborhoods" arise, each consisting predominantly (or even exclusively) of one agent type. Thus, agents' choices, conditioned by current network structure (the agent's neighbors and the neighbors at the sites to which the agent can move), change that structure; over time, from the changing local network structure, an aggregate-level pattern of segregated neighborhoods emerges. Kollman, Miller, and Page explore a related theme in their work on political platforms and institutions in multiple jurisdictions. In their agent-based model, agents may relocate between jurisdictions. They show that when there are more than three jurisdictions, two-party competition outperforms democratic referenda. The opposite is the case when there is only one jurisdiction and, hence, no agent mobility. They also find that two-party competition results in more agent moves than does democratic referenda. Manski reminds us that while theory is all very well, understanding of real phenomena is just as important. He distinguishes between three kinds of causal explanation for the often observed empirical fact that "persons belonging to the same group tend to behave similarly." One is the one we have been describing above: the behavioral similarities may arise through network interaction effects. But there are two other possible explanations: contextual, in which the behavior may depend on exogenous characteristics of the group (like socioeconomic composition); or correlated effects, in which the behavior may be due to similar individual characteristics of members of the group. Manski shows, among other results, that a researcher who uses the popular linear-in-means model to analyze his data and "observes equilibrium outcomes and the composition of reference groups cannot empirically distinguish" endogenous interactions from these alternative explanations. One moral is that nonlinear effects require nonlinear inferential techniques. In the essays of North, Shubik, and Leijonhufvud, the focus shifts to another kind of social structure, the institution. North's essay focuses on institutions and economic growth, Shubik's on financial institutions, and Leijonhufvud's on highinflation phenomenology. All three authors agree in defining institutions as "the rules of the game," without which economic action is unthinkable. They use the word "institution" in at least three senses: as the "rules" themselves (for example, bankruptcy laws); as the entities endowed with the social and political power to promulgate rules (for example, governments and courts); and as the socially legitimized constructions that instantiate rules and through which economic agents act (for example, fiat money and markets). In whichever sense institutions are construed, the three authors agree that they cannot be adequately understood from a purely economic, purely political, or purely social point of view. Economics, politics, and society are inextricably mixed in the processes whereby institutions come into being. And they change and determine economic, political, and social action. North also insists that institutions have a cognitive dimension through the aggregate-level "belief systems" that sustain them and determine the directions in which they change.

12

W. B. Arthur, S. N. Durlauf, and D. Lane

North takes up the question of the emergence of institutions from a functionalist perspective: institutions are brought into being "in order to reduce uncertainty," that is, to make agents' worlds predictable enough to afford recognizable opportunities for effective action. In particular, modern economies depend upon institutions that provide low transaction costs in impersonal markets. Shubik takes a different approach. His analysis starts from his notion of strategic market games. These are "fully defined process models" that specify actions "for all points in the set of feasible outcomes." He shows how, in the context of constructing a strategic market game for an exchange economy using fiat money, the full specification requirement leads to the logical necessity of certain kinds of rules that Shubik identifies with financial institutions. Geanakoplos' paper makes a similar point to Shubik's. Financial instruments represent promises, he argues. What happens if someone cannot or will not honor a promise? Shubik already introduced the logical necessity of one institution, bankruptcy law, to deal with defaults. Geanakoplos introduces another, collateral. He shows that, in equilibrium, collateral as an institution has institutional implications—missing markets. Finally, in his note concluding the volume, Philip Anderson provides a physicist's perspective on a point that Fernand Braudel argues is a central lesson from the history of long-term socioeconomic change. Averages and assumptions of agent homogeneity can be very deceptive in complex systems. And processes of change are generally driven by the inhabitants of the extreme tails of some relevant distribution. Hence, an interesting theoretical question from the Santa Fe perspective is: How do distributions with extreme tails arise, and why are they so ubiquitous and so important? WHAT COUNTS AS A PROBLEM AND AS A SOLUTION. While the papers here have much to say on cognition and structure, they contain much less discussion on what constitutes a problem and solution from this new viewpoint. Perhaps this is because it is premature to talk about methods for generating and assessing understanding when what is to be understood is still under discussion. While a few of the papers completely avoid mathematics, most of the papers do present mathematical models—whether based on statistical mechanics, strategic market games, random graphs, population games, stochastic dynamics, or agent-based computations. Yet sometimes the mathematical models the authors use leave important questions unanswered. For example, in what way do equilibrium calculations provide insight into emergence? This troublesome question is not addressed in any of the papers, even those in which models are presented from which equilibria are calculated—and insight into emergence is claimed to result. Blume raises two related issues in his discussion of population games: whether the asymptotic equilibrium selection theorems featured in the theory happen "soon enough" to be economically interesting; and whether the invariance of the "global environment" determined by the game and interaction model is compatible with an underlying economic reality in which rules of the game undergo endogenous change. It will not be easy to resolve the

Introduction

13

inherent tension between traditional mathematical tools and phenomena that may exhibit perpetual novelty. As we mentioned previously, several of the papers introduce less traditional, agent-based models. Kollman, Miller, and Page discuss both advantages and difficulties associated with this set of techniques. They end up expressing cautious optimism about their future usefulness. Tesfatsion casts her own paper as an illustration of what she calls "the alife approach for economics, as well as the hurdles that remain to be cleared," Perhaps the best recommendation we can make to the reader with respect to the epistemological problems associated with the processand-emergence perspective is simple. Read the papers, and see what you find convincing.

14

W. B. Arthur, S. N. Durlauf, and D. Lane

REFERENCES 1. Baker W. "The Social Structure of a National Securities Market." Amer. J. Social. 89 (1984): 775-811. 2. Clark, A. Being There: Putting Brain, Body, and World Together Again. Cambridge, MA: MIT Press, 1997. 3. Eigen, M., and P. Schuster. The Hypercycle. Berlin: Springer Veriag, 1979. 4. Fontana, W., and L. Buss. 'The Arrival of the Fittest: Toward a Theory of Biological Organization." Bull. Math. Biol. 56 (1994): 1-64 5. Holland, J. H. "The Global Economy as an Adaptive Process." In The Economy as an Evolving Complex System, edited by P. W. Anderson, K. J. Arrow, and D. Pines, 117-124. Santa Fe Institute Studies in the Sciences of Complexity, Proc. Vol. V. Redwood City, CA: Addison-Wesley, 1988. 6. Hutchins, E. Cognition in the Wild. Cambridge, MA: MIT Press, 1995. 7. Noria, N., and R. Eccles (Eds.) Networks and Organizations: Structure, Form, and Action. Cambridge, MA: Harvard Business School Press, 1992. 8. Wasserman, W., and K. Faust. Social Network Analysis: Methods and Applications. Cambridge, MA: Cambridge University Press, 1994.

W. Brian Arthur,! John H. Holland,}: Blake LeBaron,* Richard Palmer,* and Paul Tayler** •(•Citibank Professor, Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 |Professor of Computer Science and Engineering, University of Michigan, Ann Arbor, Ml 48109 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 'Associate Professor of Economics, University of Wisconsin, Madison, Wl 53706 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 * Professor of Physics, Duke University, Durham, NC 27706 and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501 "Department of Computer Science, Brunei University, London

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

We propose a theory of asset pricing based on heterogeneous agents who continually adapt their expectations to the market that these expectations aggregatively create. And we explore the implications of this theory computationally using our Santa Fe artificial stock market.M Asset markets, we argue, have a recursive nature in that agents' expectations are formed on the basis of their anticipations of other agents' expectations, which precludes expectations being formed by deductive means. Instead, traders continually hypothesize—continually explore—expectational models, buy or sell on the basis of those that perform best, and confirm or discard these according to their performance. Thus, individual beliefs or expectations become endogenous to the market, and constantly compete within an ecology of others' beliefs or expectations. The ecology of beliefs coevolves over time. Computer experiments with this endogenous-expectations market explain one of the more striking puzzles in finance: that market traders often believe in such concepts as technical trading, "market psychology," and bandwagon effects, while MFor a less formal discussion of the ideas in this paper see Arthur. 3

The Economy as an Evolving Complex System II, Eds. Arthur, Durlauf, and Lane SFI Studies in the Sciences of Complexity, Vol. XXVII, Addison-Wesley, 1997

15

16

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

academic theorists believe in market efficiency and a lack of speculative opportunities. Both views, we show, are correct, but within different regimes. Within a regime where investors explore alternative expectational models at a low rate, the market settles into the rational-expectations equilibrium of the efficient-market literature. Within a regime where the rate of exploration of alternative expectations is higher, the market self-organizes into a complex pattern. It acquires a rich psychology, technical trading emerges, temporary bubbles and crashes occur, and asset prices and trading volume show statistical features—in particular, GARCH behavior—characteristic of actual market data.

1. INTRODUCTION Academic theorists and market traders tend to view financial markets in strikingly different ways. Standard (efficient-market) financial theory assumes identical investors who share rational expectations of an asset's future price, and who instantaneously and rationally discount all market information into this price.!2! It follows that no opportunities are left open for consistent speculative profit, that technical trading (using patterns in past prices to forecast future ones) cannot be profitable except by luck, that temporary price overreaetions—bubbles and crashes—reflect rational changes in assets' valuations rather than sudden shifts in investor sentiment. It follows too that trading volume is low or zero, and that indices of trading volume and price volatility are not serially correlated in any way. The market, in this standard theoretical view, is rational, mechanistic, and efficient. Traders, by contrast, often see markets as offering speculative opportunities. Many believe that technical trading is profitable,!3! that something definable as a "market psychology" exists, and that herd effects unrelated to market news can cause bubbles and crashes. Some traders and financial writers even see the market itself as possessing its own moods and personality, sometimes describing the market as "nervous" or "sluggish" or "jittery." The market in this view is psychological, organic, and imperfectly efficient. From the academic viewpoint, traders with such beliefs— embarrassingly the very agents assumed rational by the theory—are irrational and superstitious. From the traders' viewpoint, the standard academic theory is unrealistic and not borne out by their own perceptions.M While few academics would be willing to assert that the market has a personality or experiences moods, the standard economic view has in recent years I2'For the classic statement see Lucas, 34 or Diba and Grossman. 16 I3! For evidence see Frankel and FVoot.!9 I4!To quote one of the most successful traders, George Soros 47 : "this [efficient market theory] interpretation of the way financial markets operate is severely distorted... .It may seem strange that a patently false theory should gain such widespread acceptance."

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

17

begun to change. The crash of 1987 damaged economists' beliefs that sudden price changes reflect rational adjustments to news in the market: several studies failed to find significant correlation between the crash and market information issued at the time (e.g., Cutler et al. 12 ). Trading volume and price volatility in real markets are large—not zero or small, respectively, as the standard theory would predict 32,44,45 — and both show significant autocorrelation.7,21 Stock returns also contain small, but significant serial correlations. 18,33,39,48 Certain technical-trading rules produce statistically significant, if modest, long-run profits.10 And it has long been known that when investors apply full rationality to the market, they lack incentives both to trade and to gather information.23,24,36 By now, enough statistical evidence has accumulated to question efficient-market theories and to show that the traders' viewpoint cannot be entirely dismissed. As a result, the modern finance literature has been searching for alternative theories that can explain these market realities. One promising modern alternative, the noise-trader approach, observes that when there are "noise traders" in the market—investors who possess expectations different from those of the rational-expectations traders—technical-trading strategies such as trend chasing may become rational. For example, if noise traders believe that an upswing in a stock's price will persist, rational traders can exploit this by buying into the uptrend, thereby exacerbating the trend. In this way positivefeedback trading strategies—and other technical-trading strategies—can be seen as rational, as long as there are nonrational traders in the market to prime these strategies. 13,14,15,46 This "behavioral" noise-trader literature moves some way toward justifying the traders' view. But it is built on two less-than-realistic assumptions: the existence of unintelligent noise traders who do not learn over time that their forecasts are erroneous; and the existence of rational players who possess, by some unspecified means, full knowledge of both the noise traders' expectations and their own class's. Neither assumption is likely to hold up in real markets. Suppose for a moment an actual market with minimally intelligent noise traders. Over time, in all likelihood, some would discover their errors and begin to formulate more intelligent (or at least different) expectations. This would change the market, which means that the perfectly intelligent players would need to readjust their expectations. But there is no reason these latter would know the new expectations of the noise-trader deviants; they would have to derive their expectations by some means such as guessing or observation of the market. As the rational players changed, the market would change again. And so the noise traders might again further deviate, forcing further readjustments for the rational traders. Actual noise-trader markets, assumed stationary in theory, would start to unravel; and the perfectly rational traders would be left at each turn guessing the changed expectations by observing the market. Thus, noise-trader theories, while they explain much, are not robust. But in questioning such theories we are led to an interesting sequence of thought. Suppose we were to assume "rational," but nonidentical, agents who do not find themselves in a market with rational expectations, or with publicly known expectations. Suppose we allowed each agent continually to observe the market with an eye to

18

W. B, Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

discovering profitable expectations. Suppose further we allowed each agent to adopt these when discovered and to discard the less profitable as time progressed. In this situation, agents' expectations would become endogenous—individually adapted to the current state of the market—and they would cocreate the market they were designed to exploit. How would such a market work? How would it act to price assets? Would it converge to a rational-expectations equilibrium—or would it uphold the traders' viewpoint? In this chapter we propose a theory of asset pricing that assumes fully heterogeneous agents whose expectations continually adapt to the market these expectations aggregatively create. We argue that under heterogeneity, expectations have a recursive character: agents have to form their expectations from their anticipations of other agents' expectations, and this self-reference precludes expectations being formed by deductive means. So, in the absence of being able to deduce expectations, agents—no matter how rational—are forced to hypothesize them. Agents, therefore, continually form individual, hypothetical, expectational models or "theories of the market," test these, and trade on the ones that predict best. From time to time they drop hypotheses that perform badly, and introduce new ones to test. Prices are driven endogenously by these induced expectations. Individuals' expectations, therefore, evolve and "compete" in a market formed by others' expectations. In other words, agents' expectations coevolve in a world they cocreate. The natural question is whether these heterogeneous expectations coevolve into homogeneous rational-expectations beliefs, upholding the efficient-market theory, or whether richer individual and collective behavior emerges, upholding the traders' viewpoint and explaining the empirical market phenomena mentioned above. We answer this not analytically—our model, with its fully heterogeneous expectations, is too complicated to allow analytical solutions—but computationally. To investigate price dynamics, investment strategies, and market statistics in our endogenous-expectations market, we perform carefully controlled experiments within a computer-based market we have constructed, the SFI Artificial Stock Market.!5] The picture of the market that results from our experiments, surprisingly, confirms both the efficient-market academic view and the traders' view. But each is valid under different circumstances—in different regimes. In both circumstances, we initiate our traders with heterogeneous beliefs clustered randomly in an interval near homogeneous rational expectations. We find that if our agents very slowly adapt their forecasts to new observations of the market's behavior, the market converges to a rational-expectations regime. Here "mutant" expectations cannot get a profitable footing; and technical trading, bubbles, crashes, and autocorrelative behavior do not emerge. Trading volume remains low. The efficient-market theory prevails. If, on the other hand, we allow the traders to adapt to new market observations at a more realistic rate, heterogeneous beliefs persist, and the market self-organizes f5lFor an earlier report on the SFI artificial stock market, see Palmer et al.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

19

into a complex regime. A rich "market psychology"—a rich set of expectations— becomes observable. Technical trading emerges as a profitable activity, and temporary bubbles and crashes occur from time to time. Trading volume is high, with times of quiescence alternating with times of intense market activity. The price time series shows persistence in volatility, the characteristic GARCH signature of price series from actual financial markets. And it shows persistence in trading volume. And over the period of our experiments, at least, individual behavior evolves continually and does not settle down. In this regime, the traders' view is upheld. In what follows, we discuss first the rationale for our endogenous-expectations approach to market behavior; and introduce the idea of collections of conditional expectational hypotheses or "predictors" to implement this. We next set up the computational model that will form the basic framework. We are then in a position to carry out and describe the computer experiments with the model. Two final sections discuss the results of the experiments, compare our findings with other modern approaches in the literature, and summarize our conclusions.

2. WHY INDUCTIVE REASONING? Before proceeding, we show that once we introduce heterogeneity of agents, deductive reasoning on the part of agents fails. We argue that in the absence of deductive reasoning, agents must resort to inductive reasoning, which is both natural and realistic in financial markets. A. FORMING EXPECTATIONS BY DEDUCTIVE REASONING: AN INDETERMINACY

We make our point about the indeterminacy of deductive logic on the part of agents using a simple arbitrage pricing model, avoiding technical details that will be spelled out later. (This pricing model is a special case of our model in section 3, assuming risk coefficient A arbitrarily close to 0, and gaussian expectational distributions.) Consider a market with a single security that provides a stochastic payoff or dividend sequence {dt}, with a risk-free outside asset that pays a constant r units per period. Each agent i may form individual expectations of next period's dividend and price, Ei[dl+i\It] and Ei{pt+i\It}, with conditional variance of these combined expectations, of t , given current market information It- Assuming perfect arbitrage, the market for the asset clears at the equilibrium price: pt = /3]T WjjiEjidt+ilIt] + Ej\pt+l\It\). i

(1)

In other words, the security's price pt is bid to a value that reflects the current (weighted) average of individuals' market expectations, discounted by the factor

20

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

(3 - 1/(1 + r), with weights wjyt = (l/ff£ t )/ £ f e l/ojt.t the relative "confidence" placed in agent j's forecast. Now, assuming intelligent investors, the key question is how the individual dividend and price expectations Ei[dt+i\It] and Ei\pt+x\It}, respectively, might be formed. The standard argument that such expectations can be formed rationally (i.e., using deductive logic) goes as follows. Assume homogeneous investors who (i) use the available information It identically in forming their dividend expectations, and (ii) know that others use the same expectations. Assume further that the agents (iii) are perfectly rational (can make arbitrarily difficult logical inferences), (iv) know that price each time will be formed by arbitrage as in Eq. (1), and (v) that (iii) and (iv) are common knowledge. Then, expectations of future dividends Ei[t(pt+i +dt+i). They then calculate their desired holdings and pass their demand parameters to the specialist who declares a price pt that clears the market. At the start of the next period the new dividend dt+i is revealed, and the accuracies of the predictors active at time t are updated. The sequence repeats. B. MODELING THE FORMATION OF EXPECTATIONS

At this point we have a simple, neoclassical, two-asset market. We now break from tradition by allowing our agents to form their expectations individually and inductively. One obvious way to do this would be to posit a set of individual-agent expectational models which share the same functional form, and whose parameters are updated differently by each agent (by least squares, say) over time, starting from different priors. We reject this in favor of a different approach that better reflects the process of induction outlined in section 2 above. We assume each agent, at any time, possesses a multiplicity of linear forecasting models—hypotheses about the direction of the market, or "theories of the market"—and uses those that are both best suited to the current state of the market and have recently proved most reliable. Agents then learn, not by updating parameters, but by discovering which of their hypotheses "prove out" best, and by developing new ones from time to time, via the genetic algorithm. This structure will offer several desirable properties: It will avoid biases introduced by a fixed, shared functional form. It will allow the individuality of expectations to emerge over time (rather than be built in only to a priori beliefs). And it will better mirror actual cognitive reasoning, in which different agents might well "cognize" different patterns and arrive at different forecasts from the same market data. In the expectational part of the model, at each period, the time series of current and past prices and dividends are summarized by an array or information set of J market descriptors. And agents' subjective expectational models are represented by sets of predictors. Each predictor is a condition/forecast rule (similar to a Holland classifier which is a condition/action rule) that contains both a market condition that may at times be fulfilled by the current state of the market and a forecasting formula for next period's price and dividend. Each agent possesses M such individual predictors—holds M hypotheses of the market in mind simultaneously—and uses the most accurate of those that are active (matched by the current state of the market). In this way, each agent has the ability to "recognize" different sets of states of the market, and bring to bear appropriate forecasts, given these market patterns.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

25

It may clarify matters to show briefly how we implement this expectational system on the computer. (Further details are in Appendix A.) Suppose we summarize the state of the market by J = 13 bits. The fifth bit might correspond to "the price has risen the last 3 periods," and the tenth bit to "the price is larger than 16 times dividend divided by r," with 1 signaling the occurrence of the described state, and 0 its absence or nonoccurrence. Now, the condition part of all predictors corresponds to these market descriptors, and thus, also consists of a 13-bit array, each position of which is filled with a 0, or 1, or # ("don't care"). A condition array matches or "recognizes" the current market state if all its 0's and l's match the corresponding bits for the market state with the #'s matching either a 1 or a 0. Thus, the condition ( # # # # 1 # # # # # # # # ) "recognizes" market states in which the price has risen in the last 3 periods. The condition ( # # # # # # # # # 0 # # # ) recognizes states where the current price is not larger than 16 times dividend divided by r. The forecasting part of each predictor is an array of parameters that triggers a corresponding forecasting expression. In our experiments, all forecasts use a linear combination of price and dividend, E(pt+i + dt+i) — a(pt + dt) + b. Each predictor then stores specific values of a and b. Therefore, the full predictor ( # # # # l # # # # 0 # # # ) / ( 0 . 9 6 , 0 ) can be interpreted as "i/ the price has risen in the last 3 periods, and if the price is not larger than 16 times dividend divided by r, then forecast next period's price plus dividend as 96% of this period's." This predictor would recognize—would be activated by—the market state (0110100100011) but would not respond to the state (0110111011001). Predictors that can recognize many states of the market have few l's and 0's. Those more particularized have more l's and 0's. In practice, we include for each agent a default predictor consisting of all #'s. The genetic algorithm creates new predictors by "mutating" the values in the predictor array, or by "recombination"— combining part of one predictor array with the complementary part of another. The expectational system then works at each time with each agent observing the current state of the market, and noticing which of his predictors match this state. He forecasts next period's price and dividend by combining statistically the linear forecast of the H most accurate of these active predictors, and given this expectation and its variance, uses Eq. (5) to calculate desired stock holdings and to generate an appropriate bid or offer. Once the market clears, the next period's price and dividend are revealed and the accuracies of the active predictors are updated. As noted above, learning in this expectational system takes place in two ways. It happens rapidly as agents learn which of their predictors are accurate and worth acting upon, and which should be ignored. And it happens on a slower time scale as the genetic algorithm from time to time discards nonperforming predictors and creates new ones. Of course these new, untested predictors do not create disruptions—they will be acted upon only if they prove accurate. This avoids brittleness and provides what machine-learning theorists call "gracefulness" in the learning process.

26

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

We can now discern several advantages of this multibit, multipredictor architecture. One is that this expectational architecture allows the market to have potentially different dynamics—a different character—under different states or circumstances. Because predictors are pattern-recognizing expectational models, and so can "recognize" these different states, agents can "remember" what happened before in given states and activate appropriate forecasts. This enables agents to make swift gestalt-like transitions in forecasting behavior should the market change. Second, the design avoids bias from the choice of a particular functional form for expectations. Although the forecasting part of our predictors is linear, the multiplicity of predictors conditioned upon the many combinations of market conditions yield collectively at any time and for any agent a nonlinear forecasting expression in the form of a piecewise linear, noncontinuous forecasting function whose domain is the market state space, and whose accuracy is tuned to different regions of this space. (Forecasting is, of course, limited by the choice of the binary descriptors that represent market conditions.) Third, learning is concentrated where it is needed. For example, J = 12 descriptors produces predictors that can distinguish more than four thousand different states of the market. Yet, only a handful of these states might occur often. Predictor conditions that recognize states that do not occur often will be used less often, their accuracy will be updated less often and, other things being equal, their precision will be lower. They are, therefore, less likely to survive in the competition among predictors. Predictors will, therefore, cluster in the more visited parts of the market state space, which is exactly what we want. Finally, the descriptor bits can be organized into classes or information sets which summarize fundamentals, such as price-dividend ratios or technical-trading indicators, such as price trend movements. The design allows us to track exactly which information—which descriptor bits—the agents are using or ignoring, something of crucial importance if we want to test for the "emergence" of technical trading. This organization of the information also allows the possibility setting up different agent "types" who have access to different information sets. (In this chapter, all agents see all market information equally.) A neural net could also supply several of these desirable qualities. However, it would be less transparent than our predictor system, which we can easily monitor to observe which information agents are individually and collectively using at each time.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

27

4. COMPUTER EXPERIMENTS: THE EMERGENCE OF TWO MARKET REGIMES A. EXPERIMENTAL DESIGN

We now explore computationally the behavior of our endogenous-expectations market in a series of experiments. We retain the same model parameters throughout these experiments, so that we can make comparisons of the market outcomes using the model under identical conditions with only controlled changes. Each experiment is run for 250,000 periods to allow asymptotic behavior to emerge if it is present; and it is run 25 times under different random seeds to collect cross-sectional statistics. We specialize the model described in the previous section by choosing parameter values, and, where necessary, functional forms. We use N = 25 agents, who each have M = 100 predictors, which are conditioned on J = 12 market descriptors. The dividend follows the AR(1) process in Eq. (4), with autoregressive parameter p set to 0.95, yielding a process close to a random walk, yet persistent. The 12 binary descriptors that summarize the state of the market are the following: 1-6 Current price x interest rate/dividend > 0.25,0.5,0.75,0.875,1.0,1.125 7-10 Current price > 5-period moving average of past prices (MA), 10-period MA, 100-period MA, 500-period MA 11 Always on (1) 12 Always off (0) The first six binary descriptors—the first six bits—reflect the current price in relation to current dividend, and thus, indicate whether the stock is above or below fundamental value at the current price. We will call these "fundamental" bits. Bits 7-10 are "technical-trading" bits that indicate whether a trend in the price is under way. They will be ignored if useless, and acted upon if technical-analysis trend following emerges. The final two bits, constrained to be 0 or 1 at all times, serve as experimental controls. They convey no useful market information, but can tell us the degree to which agents act upon useless information at any time. We say a bit is "set" if it is 0 or 1, and predictors are selected randomly for recombination, other things equal, with slightly lower probabilities the higher their specificity— that is, the more set bits they contain (see Appendix A). This introduces a weak drift toward the all-# configuration, and ensures that the information represented by a particular bit is used only if agents find it genuinely useful in prediction. This market information design allows us to speak of "emergence." For example, it can be said that technical trading has emerged if bits 7-10 become set significantly more often, statistically, than the control bits. We assume that forecasts are formed by each predictor j storing values for the parameters Q.J, bj, in the linear combination of price and dividend, Ej\pt+i +

28

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

dt+\\h] = aj(Pt + dt) + bj. Each predictor also stores a current estimate of its forecast variance. (See Appendix A.) Before we conduct experiments, we run two diagnostic tests on our computerbased version of the model. In the first, we test to see whether the model can replicate the rational-expectations equilibrium (r.e.e.) of standard theory. We do this by calculating analytically the homogeneous rational-expectations equilibrium (h.r.e.e.) values for the forecasting parameters a and b (see Appendix A), then running the computation with all predictors "clamped" to these calculated h.r.e.e. parameters. We find indeed that such predictions are upheld-—that the model indeed reproduces the h.r.e.e.—which assures us that the computerized model, with its expectations, demand functions, aggregation, market clearing, and timing sequence, is working correctly. In the second test, we show the agents a given dividend sequence and a calculated h.r.e.e. price series that corresponds to it, and test whether they individually learn the correct forecasting parameters. They do, though with some variation due to the agents' continual exploration of expectational space, which assures us that our agents are learning properly. B. THE EXPERIMENTS

We now run two sets of fundamental experiments with the computerized model, corresponding respectively to slow and medium rates of exploration by agents of alternative expectations. The two sets give rise to two different regimes—two different sets of characteristic behaviors of the market. In the slow-learning-rate experiments, the genetic algorithm is invoked every 1,000 periods on average, predictors are crossed over with probablity 0.3, and the predictors' accuracy-updating parameter 9 is set to 1/150. In the medium-exploration-rate experiments, the genetic algorithm is invoked every 250 periods on average, crossover occurs with probability 0.1, and the predictors' accuracy-updating parameter 6 is set to 1/75.110' Otherwise, we keep the model parameters the same in both sets of experiments, and in both we start the agents with expectational parameters selected randomly from a uniform distribution of values centered on the calculated homogeneous rationalexpectations ones. (See Appendix A.) In the slow-exploration-rate experiments, no non-r.e.e. expectations can get a footing: the market enters an evolutionary stable, rational-expectations regime. In the medium-exploration-rate experiments, we find that the market enters a complex regime in which psychological behavior emerges, there are significant deviations from the r.e.e. benchmark, and statistical "signatures" of real financial markets are observed. We now describe these two sets of experiments and the two regimes or phases of the market they induce.

t 10 lAt the time of writing, we have discovered that the two regimes emerge, and the results are materially the same, if we vary only the rate of invocation of the genetic algorithm.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

29

THE RATIONAL-EXPECTATIONS REGIME. As stated, in this set of experiments, agents continually explore in prediction space, but under low rates. The market price, in these experiments, converges rapidly to the homogeneous rationalexpectations value adjusted for risk, even though the agents start with nonrational expectations. In other words, homogeneous rational expectations are an attractor for a market with endogenous, inductive expectations.!11' This is not surprising. If some agents forecast differently than the h.r.e.e. value, then the fact that most other agents are using something close to the h.r.e.e. value, will return a marketclearing price that corrects these deviant expectations: There is a natural, if weak, attraction to h.r.e.e. The equilibrium within this regime differs in two ways from the standard, theoretical, rational-expectations equilibrium. First, the equilibrium is neither assumed nor arrived at by deductive means. Our agents instead arrive inductively at a homogeneity that overlaps that of the homogeneous, theoretical rational expectations. Second, the equilibrium is a stochastic one. Agents continually explore alternatives, albeit at low rates. This testing of alternative explorations, small as it is, induces some "thermal noise" into the system. As we would expect, in this regime, agents' holdings remain highly homogeneous, trading volume remains low (reflecting only variations in forecasts due to mutation and recombination) and bubbles, crashes, and technical trading do not emerge. We can say that in this regime the efficient-market theory and its implications are upheld. THE COMPLEX OR RICH PSYCHOLOGICAL REGIME. We now allow a more realistic level of exploration in belief space. In these experiments, as we see in Figure 1, the price series still appears to be nearly identical to the price in the rationalexpectations regime. (It is lower because of risk attributable to the higher variance caused by increased exploration.) On closer inspection of the results, however, we find that complex patterns have formed in the collection of beliefs, and that the market displays characteristics that differ materially from those in the rational-expectations regime. For example, when we magnify the difference between the two price series, we see systematic evidence of temporary price bubbles and crashes (Figure 2). We call this new set of market behaviors the rich-psychological, or complex, regime. This appearance of bubbles and crashes suggests that technical trading, in the form of buying or selling into trends, has emerged in the market. We can check this rigorously by examining the information the agents condition their forecasts upon. I 11 ' Within a simpler model, Blume and Easley 6 prove analytically the evolutionary stability of r.e.e.

30

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

FIGURE 1 Rational-expectations price vs. price in the rich psychological regime. The two price series are generated on the same random dividend series. The upper is the homogeneous r.e.e. price, the lower is the price in the complex regime. The higher variance in the latter case causes the lower price through risk aversion.

too

-T—

R.e.e. Price 80

: \/^^y'V^wA/^y

y

WV*A/\A

60 -

40

20

Price Difference

7^^^^ -20 253000

1 253050

1 253100

— i 253150

253200

Time FIGURE 2 Deviations of the price series in the complex regime from fundamental value. The bottom graph shows the difference between the two price series in Figure 1 {with the complex series rescaled to match the r.e.e. one and the difference between the two doubled for ease of observation). The upper series is the h.r.e.e. price.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

31

Figure 3 shows the number of technical-trading bits that are used (are l's or O's) in the population of predictors as it evolves over time. In both sets of experiments, technical-trading bits are initially seeded randomly in the predictor population. In the rational-expectations regime, however, technical-trading bits provide no useful information and fall off as useless predictors are discarded. But in the complex regime, they bootstrap in the population, reaching a steady-state value by 150,000 periods. Technical trading, once it emerges, remains.!12! Price statistics in the complex regime differ from those in the rationalexpectations regime, mainly in that kuitosis is evident in the complex case (Table 1) and that volume of shares traded (per 10,000 periods) is about 300% larger in the complex case, reflecting the degree to which the agents remain heterogeneous in their expectations as the market evolves. We note that fat tails and high volume are also characteristic of price data from actual financial markets.

Bits Used 600 T

0

H

0

—]

1

1

1

1

1

1—i

50000

1

1

1

1

1

1

100000

1

1

1

150000

1

1

r

200000

250000

Time FIGURE 3 Number of technical-trading bits that become set as the market evolves, (median over 25 experiments in the two regimes). ! 12 l\Vhen we run these experiments informally to 1,000,000 periods, we see no signs that technicaltrading bits disappear.

32

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

TABLE 1 Returns and volume statistics (medians) in the two regimes collected for 25 experiments after 250,000 periods,

R.e.e. Regime Complex Regime

Mean

Std. Dev.

Skewness

Kurtosis1

Vol. traded

0.000 0.000

2.1002 2.1007

0.0131 0.0204

0.0497 0.3429

2,460.9 7,783.8

Kurtosis numbers are excess kurtosis (i.e., kurtosis -3).

How does technical trading emerge in psychologically rich or complex regime? In this regime the "temperature" of exploration is high enough to offset, to some degree, expectations' natural attraction to the r.e.e. And so, subsets of non-r.e.e. beliefs need not disappear rapidly. Instead they can become mutually reinforcing. Suppose, for example, predictors appear early on that, by chance, condition an upward price forecast upon the markets showing a current rising trend. Then, agents who hold such predictors are more likely to buy into the market on an uptrend, raising the price over what it might otherwise be, causing a slight upward bias that might be sufficient to lend validation to such rules and retain them in the market. A similar story holds for predictors that forecast reversion to fundamental value. Such predictors need to appear in sufficient density to validate each other and remain in the population of predictors. The situation here is analogous to that in theories of the origin of life, where there needs to be a certain density of mutually reinforcing RNA units in the "soup" of monomers and polymers for such replicating units to gain a footing.17'26 Thus, technical analysis can emerge if trend-following (or mean-reversion) beliefs are, by chance, generated in the population, and if random perturbations in the dividend sequence activate them and subsequently validate them. Prom then on, they may take their place in the population of patterns recognized by the agents and become mutually sustainable. This emergence of structure from the mutual interaction of system subcomponents justifies our use of the label "complex" for this regime. What is critical to the appearance of subpopulations of mutually reinforcing forecasts, in fact, is the presence of market information to condition upon. Market states act as "sunspot-like" signals that allow predictors to coordinate upon a direction they associate with that signal. (Of course, these are not classic sunspots that convey no real information.) Such coordination or mutuality can remain in the market once it establishes itself by chance. We can say the ability of market states to act as signals primes the mutuality that causes complex behavior. There is no need to assume a separate class of noise traders for this purpose. We can

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

33

test this signaling conjecture in further experiments where we "turn off" the condition part of all predictors (by filling them with nonreplaceable #'s). Now forecasts cannot differentiate among states of the market, and market states cannot act as signals. We find, consistent with our conjecture that signaling drives the observed patterns, that the complex regime does not emerge. As a further test of the significance of technical-trading signals, we regress the current price on the previous periods plus the technical indicator (price > 500-period moving average). In the rational-expectations regime, the technical indicator is of course not significant. In the complex regime, the trend indicator is significant (with t-value of 5.1 for the mean of the sample of 25 experiments), showing that the indicator does indeed carry useful market information. The corresponding test on actual financial data shows a similar result.10 One of the striking characteristics of actual financial markets is that both their price volatility and trading volume show persistence or autocorrelation. And volatility and volume show significant cross-correlation. In other words, both volume and volatility remain high or low for periods of random length, and they are interrelated. Our inductive market also shows persistence in volatility or GARCH behavior in the complex regime (see Figure 4), with the Chi-square statistic in the Engle GARCH Test significant at the 95% level.!13! It also shows persistence in trading volume (see Figure 5), as well as significant cross-correlation between trading volume and volatility (see Figure 6). The figures include corresponding correlations for the often-used market standard, IBM stock. (Note that because our time period and actual market days do not necessarily match, we should expect no exact overlap. But qualitatively, persistence in our market and IBM's is similar.) These correlations are not explained by the standard model, where theoretically they are zero. Why financial markets—and our inductive market—show these empirical "signatures" remains an open question. We conjecture a simple evolutionary explanation. Both in real markets and in our artificial market, agents are constantly exploring and testing new expectations. Once in a while, randomly, more successful expectations will be discovered. Such expectations will change the market, and trigger further changes in expectations, so that small and large "avalanches" of change will cascade through the system. (Of course, on this very short time-lag scale, these avalanches occur not through the genetic algorithm, but by agents changing their active predictors.) Changes then manifest in the form of increased volatility and increased volume. One way to test this conjecture is to see whether autocorrelations increase as the predictor accuracy-updating parameter 6 in Eq. (7) in Appendix A is increased. The larger 9 is, the faster individual agents "switch" among their ! 13 lAutocorrelated volatility is often fitted with a Generalized Autoregresstve Conditional Heteroscedastic time series. Hence, the GARCH label. See Bolterslev et al. r and Goodhart and O'Hara. 21

34

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

Ree, Case Complex Case IBM Daily

FIGURE 4 Autocorrelation of volatility in rational-expectations and complex regimes, and in IBM daily returns.

VotACF Ree VolACF Complex VolACF IBM

FIGURE 5 Autocorrelation of trading volume in the rational-expectations and complex regimes, and in IBM daily returns.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

35

—o— Complex case —o— Bee. Case - * -

-.05

IBM Daily

I i i—i-1 i i i i i i i i i i i i i t i 1 i i i i t i i i i i i i i i i i i i i -

4

-

3

-

2

-

1

0

1

2

3

4

Lag FIGURE 6 Cross-correlation of trading volume with volatility, in the rationalexpectations and complex regimes, and in IBM daily returns.

predictors. Thus, the more such switches should cascade. Experiments confirm that autocorrelations indeed increase with 6. Such cascades of switching in time are absorbed by the market, and die away. Hence, our evolutionary market exhibits periods of turbulence followed by periods of quiescence, as do actual markets.!14!

5. DISCUSSION To what extent is the existence of the complex regime an artifact of design assumptions in our model? We find experimentally by varying both the model's parameters and the expectational-learning mechanism, that the complex regime and the qualitative phenomena associated with it are robust. These are not an artifact of some deficiency in the model.!15! i 14 'For a discussion of volatility clustering in a different model, see Youssefmir and Huberrnan 50 ; and also Grannan and Swindle. 22 ! 15 !One design choice might make a difference. We have evaluated the usefulness of expectational beliefs by their accuracy rather than by the profit they produce. In practice, these alternatives may produce different outcomes. For example, buying into a price rise on the basis of expectations may yield a different result if validated by profit instead of by accuracy of forecast when "slippage" is present, that is, when traders on the other side of the market are hard to find. We believe, but have not proved, that the two criteria lead to the same qualitative results.

36

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

It might be objected that if some agents could discover a superior means of forecasting to exploit the market, this might arbitrage complex patterns away, causing the market again to converge to rational expectations. We believe not. If a clever metaexpectational model was "out there" that might exploit others' expectations, such a model would, by aggregation of others' expectations, be a complicated nonlinear function of current market information. To the degree that the piecewise linear form we have assumed covers the space of nonlinear expectational models conditioned on current market information, agents would indeed, via the genetic algorithm, pick up on an approximate form of this superior metamodel. The complex regime owes its existence then not to limitations of forecasting, but rather to the fact that in our endogenous-expectations model market information can be used as signals, so that a much wider space of possibilities is open—in particular, the market can self-organize into mutually supporting subpopulations of predictors. (In fact, in a simpler, analytical model, with a small number of classes of trader whose beliefs adapt endogenously, Brock and Hommes11 find similar, rich, asset-price dynamics.) There is no reason these emergent subpopulations should be in stochastic equilibrium. Indeed, agents may mutually adapt their expectations forever, so that the market explores its way through this large space, and is nonstationary. In some early exploratory experiments, we "froze" successful agents' expectations, then reinjected these agents with their previously successful expectations much later. The reintroduced agents proved less successful than average, indicating that the market had evolved and was nonstationary. It might be also objected that by our use of condition bits in the predictors, we have built technical trading into our model. And so it is no surprise that it appears in the complex regime. But actually, only the possibility of technical trading is built in, not its use. The use of market descriptors is selected against in the model. Thus, market signals must be of value to be used, and technical trading emerges only because such market signals induce mutually supporting expectations that condition themselves on these market signals, If the market has a well-defined psychology in our model, does it also experience "moods"? Obviously not. But, notice we assume that agents entertain more than one market hypothesis. Thus, we can imagine circumstances of a prolonged "bull-market" uptrend to a level well above fundamental value in which the market state activates predictors that indicate the uptrend will continue, and simultaneously other predictors that predict a rapid downward correction. Such combinations, which occur easily in both our market and actual markets, could well be described as "nervous." What about trade, and the motivation to trade in our market? In the rationalexpectations literature, the deductively rational agents have no motivation to trade, even where they differ in beliefs. Assuming other agents have access to different information sets, each agent in a prebidding arrangement arrives at identical beliefs. Our inductively rational agents (who do not communicate directly), by contrast, do not necessarily converge in beliefs. They thus retain a motivation to trade, betting ultimately on their powers as market statisticians. It might appear that, because

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

37

our agents have equal abilities as statisticians, they are irrational to trade at all. But although their abilities are the same, their luck in finding good predictors diverges over time. And at each period, the accuracy of their predictors is fully accounted for in their allocations between the risk-free and risky asset. Given that agents can only act as market statisticians, their trading behavior is rational. Our endogenous-expectation theory fits with two other modern approaches. Our model generalizes the learning models of Bray and others 8,42 which also assume endogenous updating of expectations. But while the Bray models assume homogeneous updating from a shared nonrational forecast, our approach assumes heterogeneous agents who can discover expectations that might exploit any patterns present. Our evolutionary approach also has strong affinities with the evolutionary models of Blume and Easley.5'6 These assume populations of expectational (or more correctly, investment) rules that compete for survival in the market in a given population of rules, and that sometimes adapt. But the concern in this literature is the selective survival of different, competing, rule types, not the emergence of mutually supportive subpopulations that give rise to complex phenomena, nor the role of market signals in this emergence. Our inductively rational market, of course, leaves out many details of realism. In actual financial markets, investors do not perfectly optimize portfolios, nor is full market clearing achieved each period. Indeed, except for the formation of expectations, our market is simple and neoclassical. Our object, however, is not market realism. Rather it is to show that given the inevitable inductive nature of expectations when heterogeneity is present, rich psychological behavior emerges—even under neoclassical conditions. We need not, as in other studies,20'28 assume sharing of information nor sharing of expectations nor herd effects to elicit these phenomena. Nor do we need to invoke "behaviorism" or other forms of irrationality.49 Herding tendencies and quasi-rational behavior may be present in actual markets, but they are not necessary to our findings.

6. CONCLUSION In asset markets, agents' forecasts create the world that agents are trying to forecast. Thus, asset markets have a reflexive nature in that prices are generated by traders' expectations, but these expectations are formed on the basis of anticipations of others' expectations.!16! This reftexivity, or self-referential character of expectations, precludes expectations being formed by deductive means, so that perfect rationality ceases to be well defined. Thus, agents can only treat their expectations as hypotheses: they act inductively, generating individual expectational models that l 16 lThis point was also made by Soros 47 whose term reflexivity we adopt.

38

W. B. Arthur, J. H. Holland, B. LeBaron, R. Palmer, and P. Tayler

they constantly introduce, test, act upon, discard. The market becomes driven by expectations that adapt endogenously to the ecology these expectations cocreate. Experiments with a computerized version of this endogenous-expectations market explain one of the more striking puzzles in finance: Standard theory tends to see markets as efficient, with no rationale for herd effects, and no possibility of systematic speculative profit, whereas traders tend to view the market as exhibiting a "psychology," bandwagon effects, and opportunities for speculative profit. Recently the traders' view has been justified by invoking behavioral assumptions, such as the existence of noise traders. We show, without behavioral assumptions, that both views can be correct. A market of inductively rational traders can exist in two different regimes: Under a low enough rate of exploration of alternative forecasts, the market settles into a simple regime which corresponds to the rationalexpectations equilibrium of the efficient-market literature. Under a more realistic rate of exploration of alternative forecasts, the market self-organizes into a complex regime in which rich psychological behavior emerges. Technical trading appears, as do temporary bubbles and crashes. And prices show statistical features—in particular, GARCH behavior—characteristic of actual market data. These phenomena arise when individual expectations that involve trend following or mean reversion become mutually reinforcing in the population of expectations, and when market indicators become used as signaling devices that coordinate these sets of mutually reinforcing beliefs. Our endogenous-expectations market shows that heterogeneity of beliefs, deviations from fundamental trading, and persistence in time series can be maintained indefinitely in actual markets with inductively rational traders. We conjecture that actual financial markets lie within the complex regime.

Asset Pricing Under Endogenous Expectations in an Artificial Stock Market

39

APPENDICES APPENDIX A: DETAILS OF THE MARKET'S ARCHITECTURE

MODEL PARAMETERS. Throughout the experiments we set the interest rate r to 0.1, and agents' risk-aversion parameter A to 0.5. The parameters of the dividend process in Eq. (4) are p = 0.95, d — 10, r = 0.1, a\ = 0.0743. (This error variance value is selected to yield a combined price-plus-dividend variance of 4.0 in the h.r.e.e.) PREDICTOR ACCURACY. The accuracy, or precision, of agent I'S jth predictor is updated each time the predictor is active, and is recorded as the inverse of the moving average of squared forecast error: 1 A= 1 0< A< 1 A= 0 -1 < A < 0 A = —1 A < —1

unstable coordination game basic coordination game dominant Nash/coordination with preferences game zero game dominant substitution game exact substitution game unstable substitution game

We primarily study coordination games with 0 < A < 1, but we also address our results to substitution games. These utility functions imply assumptions of riskneutrality; a more general payoff function would include observed predictive errors

55

Natural Rationality

over some history, weighted by both their variance and mean. Hi A payoff function that heavily penalizes the variance in errors in preference to the mean is known as "risk averse." A risk-neutral payoff seeks only to minimize the mean. The payoff function we use is risk-neutral and blind to the past. 3.2 PREDICTIVE MODELS At date t, agent at is completely specified by its predictive model (tt(a) and its past time series bt(a), where the space of predictive models satisfies the following properties: DEFINITION. A predictive model ji € M defines a triplet T, c, and p, where: T = T(fj.) 6 N, the length of history to use as data for predictions; c = c(fi) € N, the complexity of the predictive model; p = p(/i) ; J3T(I+I-'V'I) —* B^\ a mapping from local historical observations to future predictions; so that ft allows an agent to form a prediction of the subsequent behavior of its neighbors, based upon their past behavior for the previous T consecutive timesteps. M Clearly both T and c constrain the choice of p. We use the predictor, a, as introduced earlier, except that it operates endogenously now (there is no global pt)- The basic model specifies how to calculate a to predict an expected behavior bt+i,e from its preceding c behaviors: bt, bt~u • • •, bt-c+i, via the following linear recurrence relation: c

&t+l,e = »S + Zl at'bt+\-t'

•

(9)

('=1

In order to calculate a*, we must minimize the total error over the history of length T. For computational reasons the standard technique is to minimize the least squares error over some past'6!;

error*(T,c)= f f

U -og - £ « £ * £ - 4

•

( 10 )

I4'In which case it may be necessary to separate the decision-making u from the "payoff" utility tt. I51 Other examples of the complexity could be the number of Fourier modes or the size of a correlation matrix used to fit the series. '"'However, we also consider random initialization of the a followed by random adjustment over time in which better predictive models are retained (this is known as "hill-climbing"), for which our experimental results are unchanged.

56

V. M. Parley and S. A. Kauffman

The sums here are given for the jth agent, a,j, with time series {b(}. There are a large number of choices to consider when extending such a predictive model to a population in which local neighbors are modeled. The basic choices concern whether we believe different time series are related; whether our neighbors should be modeled individually or together, and whether an agent should include its own time series in its model estimation. It transpires that such choices are usually not important, so we will fix on the case in which each agent has a single model with 1 + c\M\ coefficients, which are calculated with a simultaneous regression on all the neighbors' time series. This model is then used to predict the average (bt) directly. We now calculate the optimal a* = {p^o) U { ,a*c;j € N't}, so that the prediction is given by: c

C?

(11)

,t+i-

and the coefficients a* are given by a least-squares minimization over the set of neighboring lagged time series of length T. Further details on the more general cases in which agents may have separate models of each neighbor or use their own time series as regressors are given in Darley.9 These scenarios are portrayed in Figure 2. The techniques used to perform the minimizations are singular-value decomposition (details to be given9). Let us note, however, that numerical stability factors16 mean that it is actually better to solve the minimization problem rather than differentiating the above equations and solving the subsequent equalities exactly. Under any of these techniques, once neighboring predictions are calculated, the best-response rule/utility function dictate the agent's own behavior.

nzr (a) Just 6

iT (b) a and b

nar (c) All neighbours

m (d) a and all neighbours

FIGURE 2 For agent a to predict a given neighbor b using local information it can form a predictive model using any of the four data-sets shown. Furthermore, for (c) and (d) the agent may either use one or |jV| models.

Natural Rationality

57

3.3 DYNAMICAL UPDATE RULE The system's dynamics stem from the ability of the agents to tune their predictive models, based upon differences between observed and expected behaviors, in an attempt to optimize their own behavior with respect to the changing environment. We do not impose any exogenous shocks on our models. This procedure can be summarized by the following sequence of actions, carried out at every time-step, in parallel, by every agent a*: Prediction^', V) at time t: i.

Calculate private predictions frj+1>e € B giving the expected behavior of all other agents (j ^ i) at time t + 1.

ii. Find b* - arg maxb eB u(b, {bi+he}), agent a/s predicted optimal response. iii. Carry out action b*. iv. Observe actual behaviors bf+1 and, using g, calculate agent dj's utility u*=u(b*,{l4+l}). v. If 3/4 with T = T ± 1 or d = c ± 1 s.t. u(6*,, {b{+l}) > u* then pick the best such new model /ij. This is the model update rule. Step (v) is the "predictive model update rule," which dictates the main dynamical properties of our systems. Each agent compares its utility under its current model, /i, with the utility it would have had under a perturbed model, p!. If any of the alternative models would have been better, the best such model is selected. Under our perturbation scheme, movement in the space of predictive models, M, is local, by steps of length 1 in either the T or c direction. Hence, discrete changes to an agent's model may be considered as moves in the two-dimensional discrete lattice of history-complexity pairs, and agents can be thought of as performing local search in this space as they seek the best model of the world around them. As remarked earlier, the payoff function is blind to the past, and the only effect of a particularly poor prediction is presumably a change in one's predictive model to a better pair of parameters. The only level of evolution in the system is the survival of certain models (and, therefore, certain parameter values). As a consequence we do not directly investigate phenomena pertaining to the accumulation of utility (rule (v) only compares instantaneous utilities). Finally, we shall note two facts: first, the sequence of update steps (i)-(v) is completely deterministic, with no exogenous perturbations or shocks; second, the entire model is evolving in synchrony. This differs from the approach of Blume5 who considers "strategy revision opportunities" independently exponentially distributed, such that the probability of two or more agents updating simultaneously is zero.

58

V. M. Darley and S. A. Kauffman

3.3.1 HETEROGENEITY AND THE USE OF INFORMATION. Our agents are heterogenous because of two facts: that they may choose different models (or groups of models); and that the information upon which they base their predictions may be different. Given some slight variation in the world (originally brought about at least by nonuniform initial conditions), each agent will operate with u upon a slightly different set of behaviors giving rise to a variety of actions. We shall investigate whether this initial heterogeneity grows, diminishes, or even vanishes with the evolution of the system. It is worth pointing out that this research does not currently differentiate fully between the two sources of heterogeneity (models and information): at a first approximation the information used to model a given neighbor (its behavior time series) is common to all agents modeling that neighbor, so models are the only source of local heterogeneity. However, the manner in which correlations between agents are modeled is constrained to utilize slightly different neighborhood information sources. This is because we currently constrain agents to use local information only to make their predictions. This means the four agents who make models of a given common neighbor are all constrained to use different information sets to construct their models if they wish to model correlations. Further research will allow agents to use nonlocal information. These more sophisticated models will enable us to pinpoint more accurately the assumptions from which heterogeneous nonstationary worlds can be derived. 3.4 WHAT TO PREDICT?

Finally, given a class of predictive models, an economic strategy, and a set of neighborhood relations, we must decide what the agents should predict and the manner in which that determines their own behaviors. We shall consider two scenarios: 1. the forward predictive scenario in which each agent predicts what its neighbors will do tomorrow and uses the utility u to determine its own behavior. This is the natural, obvious predictive method. 2. the stabilized predictive scenario in which each agent ignores the just realized predictions. The dynamics are as follows: all agents make predictions, adjust their models based upon the success of those predictions, but then those predictions are forgotten and ignored. The agents generate a new set of predictions and repeat. Hence, the models are updated as above, but the behaviors and predictions undergo nothing more than a process of iterated refinement. The first scenario is the natural one. We also study the second because it is, effectively, a stabilized version of the first. Rather than predicting forward in time, the agent effectively re-predicts what it should have just done, and then carries out that action (the process could be considered one of perpetual refinement of action, not unlike the simple adaptive processes considered earlier).

Natural Rationality

59

We will find that the dynamics which arise from these different scenarios can be quite different in character. Intuitively scenario 1 may lead to excessively unstable dynamics, as the agents' forward predictions diverge from one another. A more sophisticated predictive model may be required to follow the dynamics. Scenario 2 on the other hand may be too stable, with predictions from one date to the next hardly varying at all. A third and fourth scenario, which we leave to future research, are the following: 3. each agent predicts using a discounted sum of expected future utility based upon neighbor predictions over a given planning horizon (and seeks to optimize the length of that horizon); 4. agents build a hierarchy of meta-models of each other—my model of your model of my model of,.. .and optimize the cut-off height of this hierarchy. We shall present our results for scenario 2 first, in section 4.1, which illustrates the coherently organized coupling between model-update events. Then in section 4.2 we give our results for scenario 1 in rather more detail.

4. OBSERVATION AND SIMULATION A system such as ours has a large number of possible rational expectations states. There are clearly an infinite number of possible Nash equilibria (using the coordination strategy at least), and many more periodic equilibrium states are possible. One important consideration when given multiple equilibria is to try and understand the problem of equilibrium selection. The parameters that characterize the choice of equilibrium index beliefs the agents have (individually and collectively) about their world. One common use of adaptive models is in equilibrium selection in just such a scenario. From our perspective we would like to understand more than just selection; we would like to know what happens when models are not forced to be stable. This implies that our agents do not have as a priori (and rather ad hoc) beliefs that the world is heading inexorably to a simple static equilibrium (this is implicit in Bray's model and our extensions to it). Our agents will naturally pick from a class of models (which includes a set implying stationarity) so as to maximize their immediate gain. So one question we must address is: Is a static (coordinated) equilibrium selected for? If not, then we will concern ourselves with understanding and explaining whatever nonstationary dynamics are observed. In particular we attempt to formulate a categorization of the natural rationality of agents whose mutual interactions form their world. We shall compare the results of our analysis and experimentation with the intuitive ideas introduced earlier. In order to initiate the simulations, agents are given an artificial randomized past time series, and history and complexity parameters. All the results are robust to

60

V. M. Darley and S. A. Kauffman

changes in the initialization technique (Gaussian, sinusoidal, and uniformly random initializations have all been tested.) We now present our experimental results for the two predictive scenarios. 4.1 OBSERVATIONS OF THE STABILIZED SCENARIO

The following results are all for predictive scenario 2. As remarked earlier, the stabilized scenario allows us to investigate feedback and propagation of information between models in a more restricted stable world. For this scenario, there is little difference in agent behavior dynamics between the small and large-population cases (the behaviors are simple and stable across the system). However, the large case shows interesting spatial order in the space of agent models, so we shall consider that case exclusively. We observe two regimes of behavior, which we label "coherent" and "random," each preceded by a short transitory phase. These two regimes are qualitatively different.inxxpredictive models, agent scenarios During the first, coherent regime predictive errors decrease exponentially fast, whilst variance in agent behavior (system heterogeneity) collapses exponentially fast onto the system mean. These are diffusive spatial dynamics in which any behavioral heterogeneity disperses rapidly. Such dynamics can be generated by a wide class of models in which agents try to imitate each other using simple adaptive models; such models are presented elsewhere.9 During the second, random regime predictive errors and system heterogeneity have reached a lower bound at which they remain. The occurrence of these regimes is explained below. Model update dynamics are more interesting: define an "avalanche" to be the number of consecutive time-steps an agent spends adjusting its predictive parameters c, T. Then in the coherent regime, we observe a power-law relationship between avalanche frequency and size: / «• c/lk, where k = 0.97 ± 0.07. Hence, the modelupdate process has organized model change into a critical state in which large, long-duration avalanches may occur. This requires coordination to build up endogenously across significant sub-groups of the population of agents, so we refer to this state as "self-organized." In the random regime the avalanche frequency distribution develops into an exponential fall-off: f = pi~l(l-p), where p = 0.323 ± 0.005. Furthermore, an examination of spatial (rather than temporal) avalanches in the lattice gives the same power-law then exponential result. In that sense the system is scale-invariant in both space and time. We should point out that there is a large practical difference between a powerlaw and an exponential fall off of avalanche size. For the exponential case, large avalanches effectively never occur, whereas in the coherent regime we have much data for disturbances of size nearing 100, for instance. This is an important distinction.

Natural Rationality

61

The reason for the dramatic change between regimes is as foliows: in the first regime behavior variance and predictive errors are both converging to zero exponentially fast (the former at least is expected for a diffusive dynamic). Once the differences between models' predictions are less than that discernible under the numerical representation used, numerical rounding errors contribute more than model difference. At that point model selection becomes a random process. This hypothesis has been confirmed using test experiments of varying floating point accuracy. The curve fit pl~l(l - p) in the random regime is just the expected number of consecutive moves in a random walk with probability p of moving. The random regime is, therefore, of lesser importance, and the self-organized critical behavior can be considered the predominant characteristic of the internal dynamics of the stabilized scenario. This critical behavior exists in a wide variety of observables, indicating that the system does truly self-tune to a boundary intermediate between order and disorder. Although coherence is eventually lost due to the overwhelming stability we have imposed, leading to a degenerate dynamic, it is clear that basic coevolutionary forces between the agents' models have profound influence upon the global dynamics, and the macroscopic behavior can be captured with a relatively simpler picture. An extension of the above observations can be found if each agent models each of its neighbors using a totally separate model. We still find the same two regimes (and avalanche characteristics) as before, but now we get exponential convergence to a nonuniform state. Each agent has a different behavior, but such behaviors are coordinated so the agents are still in a high-utility state. So, whereas the old system converged to a system-wide fixed mean, zero variance; the new system converges to a system-wide fixed mean, but nonzero variance. This raises a basic issue: persistent local diversity requires at least the capability of modeling that diversity—an agent which uses a single model for each of its neighbors (or, more generally, for the entire information set it observes) believes the system's equilibrium states are much simpler than the agent with multiple models who has no such presupposition. In systems such as ours in which the dynamics are endogenous, and agents' beliefs are reflected in those dynamics, it is important to take those beliefs into consideration. 4.2 OBSERVATIONS OF THE FORWARD PREDICTIVE SCENARIO The following results are all for predictive scenario 1. This is a more natural predictive situation, in which we can expect both models and behaviors to exhibit interesting dynamics. The most clear difference between this and the stabilized scenario is in observations of the agents' behaviors. They no longer always settle down over time. There are two generic cases:

62

V. M. Parley and S. A. Kauffman

SMALL SYSTEM OR STABILIZING MODELS—agents coordinate across the system on relatively simple strategies. The variance in behavior across the system is very low, and its mean follows a simple monotonic path. LARGE SYSTEM—an interesting interplay between between periods of coordinated behavior, and periods of disordered rapidly changing behaviors is observed. Figure 3 shows the basic behavior-space dynamics for large systems. Comparing this with Figure 6, we can see that the coordinated time-phases are precisely linked with periods of very low variance in agent behavior, whereas the uncoordinated periods show very high variance levels. Hence, the agents all converge to a particular selection, retain that behavior for some time, with only quite minor fluctuations, and finally the coordination breaks down and the agents pass through a disordered bubble of activity, selecting any of a great range of behaviors before settling upon a new metastable state. Consider the coordination and substitution games we introduced. We can summarize our results for these very succinctly: the "unstable" variants are indeed unstable because best-response dynamics drive the system away from equilibrium; dominant-Nash games are identical to coordination games (which are the specific case with no domination, A = 1) in the short term, but have more robust equilibria in the long term. Their short-term dynamics and the general dynamics of the coordination game exhibit interesting punctuated equilibria, cyclic oscillations,.. .which we shall analyze in detail below. We shall discuss the case with small populations first, before considering the large-population case.

5000

6500

FIGURE 3 Mean agent behavior in a 10 x 10 world. The maxima/minima represent almost completely homogenous behavior; in between behavior, is highly disordered.

63

Natural Rationality

4.2.1 SMALL POPULATIONS. The large-scale dynamics described above are sufficiently complex t h a t it is insightful in this case to consider small populations in which we can better see how the agents influence each other, and in which there is less room for variation, and fewer degrees of freedom to go unstable. The first thing to learn from Figure 4 is that the agents can learn to coordinate on particular behaviors, and that that coordination can take place in a relatively rapid, damped oscillatory form. Plots (b) and (c) show that coordination need not be on a static equilibrium, if the agents have a sufficiently large percentage of the total information available, i.e., if each agent is modeling a reasonably large fraction of the total number of agents, then global coordination on a i-dependent path may be achieved. Notice that, since the agents are constrained to the range [0,1000], this will cause a temporary instability on any path which hits the bounds.

(a) 2 agents, unit neighbourhood — the models become consistent

(b) 2 x 2 agents, M become consistent

(c) 4 x 4 agents, M — 4; again consistent models develop (very quickly in this case)

(d) 10 agents, unit neighbourhood. Meta-stability and punctuated equilibria are now observed generically.

4; the models

FIGURE 4 Agent behaviors in small populations. The horizontal axes are time; vertical is behavior. Each agent's behavior at each time step is plotted as a dot, which are connected in the first figure.

64

V. M. Darley and S. A. Kauffman

FIGURE 5 Behavior of a sample agent taken from a ring of 10 playing the exact substitution game.

The exact paths in Figures 4(a), (b), and (c) are completely history-dependent, although the range of possible qualitative dynamics is rather small. When we consider systems in which the proportion of agents modeled is low, a characteristic dynamic emerges. Equilibrium is only possible on a constant value, and those equilibria are metastable. The system will leap from one such equilibrium to the next via disordered transitions. This qualitative dynamic is very robust, and the only historical influence is in the actual equilibrium values. These results are all identical for the exact substitution game, for which we give a sample plot in Figure 5. The stabler variants of these games, with |A| < 1 damp down to the unique equilibrium after a A-dependent time, for both coordination and substitution games. For large populations this still occurs, but only after significant transient periods. 4.2.2 LARGE POPULATIONS. The only characteristic simple coordinated states for larger populations are static equilibria. These occur with great regularity, and are highly uniform across the entire system. Figure 6 shows how the variance of the agents' behaviors drops almost to zero at these times—a spatial plot of such a state is shown in Figure 7(a). Such states are only metastable, however, and hence, oscillatory dynamics are observed. 171 All the large systems we consider show a punctuated equilibrium dynamic. Successive equilibria are destabilized endogenously, leading to wild fluctuations before the agents settle to another equilibrium. Furthermore, these dynamics are not transient for the exact coordination or substitution game. However, interestingly enough, given some small level of dominance in the game (|A| = 0.9, say), the dynamics do become transient after sufficiently long periods of time. The agents eventually correlate their behaviors and models sufficiently that they reach of state of small fluctuations about a static equilibrium. The surprising dynamics of history and complexity by which this may be achieved are shown in Figure 9. Both the early punctuated equilibrium regime, and l ? lThe interested reader, with fast Internet access, should point their web browser at http://vwu.fas.harvard.edu/~darley/Vince-Thasis.htinl for some some movies of the twodimensional evolution of such a system.

65

Natural Rationality

the transition last for significant periods of time. Note, however, that external perturbations can destabilize that system, so that the natural dynamics may be either the punctuated or static equilibrium depending upon the natural frequency with which exogenous perturbations occur. As remarked in our introduction, it is always important to remember that distinction when analyzing systems with long transients, since that will dictate which of the possible behaviors is actually observed.

FIGURE 6 Variance in agent behavior—it is clear that when variance is low it is vanishingly small.

(a) A coordinated regime

(b) An uncoordinated regime

FIGURE 7 Spatial distribution of behaviors for the forward predictive scenario. This is for a 20 x 20 square lattice of agents, with smoothing applied for representational purposes.

66

V. M. Parley and S. A. Kauffman

FIGURE 8 Limit cycle behavior in (t,).

(13)

Two models (and, therefore, agents) are in the same equivalence class iff the models have the same sophistication. 2. Prom the observable behaviors in the world, we can derive a vector field, such that an agent may be considered to be in a local behavior field /3\ which is

70

V. M. Darley and S. A. Kauffman

some externally imposed behavior pattern (3 (assumed to be very small) plus a local field provided by the neighboring agents. p" = 0'([3, local agents).

(14)

3. There is a global difficulty of prediction, 7 € R + , derived from observations of the character and complexity of time series behavior observed in the predictive system. 7 = 7(Wo)}«eA). (15) 5.1.2 MEAN-FIELD ASSUMPTIONS. Given the above assumptions, it is reasonable to assume that the average effective model am in the field /?' will be given by the following law: behavior field p' difficulty of prediction " This states the average agent's model is proportional to the local behavior field and inversely proportional to the difficulty of prediction in the system as a whole. We assume that if prediction is hard the agents will tend to pick simpler models. The mean-field assumption is that the behavior field, due to neighboring agents, is a function of the average model am. We shall consider am to be small so we can expand p' in a power-series about p. Note that ft is the sole source of exogenous effects; other terms are purely endogenous. The first term is given by: P' = p + lam.

(17)

This relates the observed behavior field 0' to the character of the exogenous forcing and average local model. Note that it implicitly addresses the interactions from the point of view of a single individual. It can be extended to consider small clusters of individuals, but the qualitative consequences of the results are not changed. Now, writing 7 for the difficulty of prediction, c(/3+lam) ffm

—

7 e/3 cp 7 - Ic 7 - 7c ' where c, I are constants and 7C = Ic. This solution is only valid for 7 > 7C, otherwise am points in the opposite direction to 0, which is not meaningful. For 7 < 7C we must expand 0' to third order (second order terms are not present for reasons of symmetry) in 7C If the difficulty of prediction is high (the "high temperature" regime), in the absence of external forcing (0 = 0, as in the endogenously generated worlds we study) agents will: • •

Pick simple models (sophistication is proportional to the reciprocal of the difficulty). Since the average of all models is the "zero" model, there will be no preferred choice of a within any given equivalence class of sophistications. Agents' models of the world are not mutually consistent (the significance of this point will be expanded upon later).

If there were to be external forcing, signified by /?, then a preferred choice of model would exist. Currently this situation has no real interpretation in our predictive systems, V, as they stand. However, were we to impose a certain pattern of cyclical variation in the world (incorporating, for instance, an economic "sunspot" effect (Barnett et al. 4 )), we would expect the particular choice of agents' models to reflect the character of that forcing behavior. 2- 7 < 7c If the difficulty of prediction is low, even without any external forcing, the agents spontaneously break symmetry within their equivalence class of models, a, to pick out a preferred type of model. This occurs because the directional degree of freedom of the average model am is nonzero. That degree of freedom gives the preferred model, The interpretation here is that by selecting a preferred model, the symmetry breaking ensures that the agents' models are mutually consistent. The system naturally reaches a state exhibiting an important ingredient of what are normally the assumptions of rational expectations, Of course, this description and analysis only apply directly to a static equilibrium scenario, which we do not expect to arise in these systems. However, one expects a dynamic equilibrium situation to situate those dynamics around any

72

V. M. Parley and S. A. Kauffman

marginally unstable static equilibrium. Indeed the first step in taking the above analysis further is to consider the character of perturbations about the static equilibrium we have so far derived. There are two important points still to address: that of the assumed difficulty of prediction and that of interpreting the beliefs of agents' models as being mutually consistent or inconsistent. DIFFICULTY OF PREDICTION. Despite the form of assumption 3, in the analysis which followed, the difficulty 7 was treated as a tunable parameter, independent of the agents. In general, it is certainly not independent of the agents, as assumption 3 states, it is a function of the agents' behaviors: 7 = l({b(a)}aeA). Certainly if the agents behave in a reasonably varied fashion it will be easier to discern their underlying models and predict, than if all agents behave similarly (given some underlying level of noise and that we wish to consider relative ability of predictions). So the difficulty of prediction will be an emergent observable, tuned by the dynamics of the system. Its dynamics over time will depend upon the following forces: i.

the relative ability of a successful versus unsuccessful agent will be greater in a higher variance world;

ii. agents with fancy models will be more susceptible to subtle changes in the behavior of the world, i.e., a smaller discrepancy is capable of disproving their models; iii. if the systems are sufficiently dynamic in the sense that agents must adjust their models over time in order to do relatively well in the world, then nonstationarity is maintained. iv. if the world is too static or simple, an agent which encourages more varied behavior in its environment will lead to the potential for relative success to be possible in its neighborhood (following (i)), leading to destabilization of a small region. CONSISTENCY OF MODEL. The two regimes over 7, are characterized by. 7 > 7C =*• Agents have inconsistent models 7 < 7C => Agents have consistent models What does this actually mean? Consider an equivalence class of models with the same sophistication s(a). Any given choice of a within that class reflects a hypothesis of the character of the observed dynamics in the world. A simple analogy would be that a given equivalence class selects a fixed number, s(a), of fourier modes (or splines or...) to be used to model a given time series. The set of models within that equivalence class will be the set of all models which use exactly s independent fourier modes. Clearly the models within that class which

73

Natural Rationality

utilize different fourier modes to describe a series, reflect wildly differing hypotheses about the dynamics of the underlying process generating the given time series. So, for the former case, the dynamics of the world prevent information of agent's models from transferring through the subsidiary medium of generated behaviors. Agents will pick inconsistent models of the world. In the latter case, information transfer is achieved, and agents will pick consistent models. So, in the high-difficulty regime, agents' hypotheses about the underlying dynamics of the world are mutually inconsistent (the dynamics of the world prevent information of agent's models from transferring through the subsidiary medium of generated behaviors). In the low-difficulty regime agents will actually select, on average, a single model from a given class (information transfer is achieved). Thus the agents will have a common hypothesis for the world's dynamics. OBSERVATIONS. An estimate for the concept of difficulty of prediction, upon which the analysis is based, may be obtained from the observed predictive errors. Figure 10 shows how these vary over time, and a very clear correlation between time periods of high-difficulty and uncoordinated behavior can be discerned—compare with Figures 3 and 6. Although we have, as yet, no statistical estimates, the transitions between coordinated/uncoordinated and small/large errors are very sudden. This lends credence to the concept of an underlying phase transition.

I I ....

1

I p

ikkk 2000

(a) Mean error

WOO

GOOD

(b) Variance in error

FIGURE 10 Averaged predictive errors over a 20 x 20 world. These can be considered approximate measures of observed predictive difficulty. Errors are uniformly small or uniformly large.

74

V. M. Darley and S. A. Kauffman

5.2 STATICS AND DYNAMICS The difficulty of prediction 7 is analogous in this exposition to the temperature parameter in magnetic phase transitions, with one crucial difference. It is not a exogenous parameter—it is an endogenous observable, driven by the very dynamics it implicitly tunes. At this point we leave as an open theoretical question whether its emergent dynamics drive the system to the consistent or inconsistent regime, or to fluctuate between the two. The above analysis can only be considered a static stability analysis, and must be extended to derive dynamical results. Let us first note, however, if models are simple we would expect 7 to be small. Hence a large 7 encourages a small \a\ which in turn produces simpler dynamics and a smaller 7. Conversely, as 7 becomes smaller, a oc ^/jc - 7 increases and so we expect 7 to increase. These opposing forces will, therefore, push 7 away from extremal values. This is in agreement with the argument presented in the introduction and with empirical observations—we fluctuate between an approximate rational expectations state (in which agents' implicit expectations of the world are in almost perfect agreement), and a highly disordered state.

6. DISCUSSION AND CONCLUSIONS We shall first discuss a number of limitations and assumptions on which our work is based, and what might be done to alleviate some of the more artificial constraints. We then summarize our goals, progress, and results. 6,1 UTILITY CONSTRAINTS AND GENERALIZATIONS Throughout this study, we have confined ourselves to utility functions exhibiting a number of constraints (for the forward predictive scenario): HISTORICAL DEPENDENCY—first note that, given some level of random noise, all the results of our simulations are independent of the initial conditions. In particular, the initial distribution of history and complexity values has no influence upon the long-term dynamics. By independent we mean that the qualitative nature of the system's evolution is not dependent on initial conditions. In the absence of noise, one can find a few pathological initial conditions that are nongeneric (completely uniform starting values for history, complexity, for example). CONVERGENCE—the distinctions between coordination, substitution, and varying the A parameter have already been discussed.

Natural Rationality

75

CONTINUITY—all the games we use are smooth, such that a small difference in predictions causes only a small difference in behavior. If the space of behaviors (or those behaviors selected by the game) is discretized, then most of the observations and predictions we make are no longer valid. We have not investigated whether this can be alleviated via the choice of a predictive model more suited to discrete predictions. VARIATION—when making predictions one can use single models or multiple models; predict the average or the individuals. We have already discussed the trade-off between model stability and system size as it effects the system's dynamics. FREQUENCY—the trade-off between dynamicity of the system and the stabilizing forces of model selection and prediction can be altered by changing the frequency with which the agents can adjust their predictive models (i.e., only carrying out the predictive update step (v) every n th time-step). Our results are very robust against changes to this parameter: even if agents only adjust their model every, say, every 10 time steps, no difference is observed. This is further evidence (though no more than that) of the independence of our results of synchronous versus asynchronous modeling. TOPOLOGY—the neighborhood size affects the rate of information propagation, and we have noted that system stability increases with neighborhood size (for a fixed number of agents). The type of local neighborhood is not important. If each agent's neighbors are given by a fixed random assignment from the set of AT, the results are identical. This accords with our mean-field analysis. 6.2 COOPERATIVE VS. COMPETITIVE GAMES A significant limitation to our work is the following: the dynamics we have observed in these predictive systems are limited to games which are cooperative in the sense that the best response for a given agent is not excessively bad for its neighbors. Some counter-examples such as the "unstable substitution game" seem of little importance since we understand quite clearly why they are unstable. However, other nonlinear games such as the undercut game (in which each agent tries to pick as a behavior a number just a little bit smaller than its neighbors, unless they pick very small numbers in which case it is desirable to pick a very large one) seem to require a more sophisticated modeling and strategic approach than we provide. We have not investigated whether our results can be extended in these directions via a more competitive iterated system, in which multistep forward planning, is incorporated, and nonpure strategies are allowed. Procedures like this allow cooperation to become reasonably stable in the iterated Prisoners' Dilemma, for example.

76

V. M. Darley and S. A. Kauffman

A direct extrapolation of our results suggests the intuition that such a forwardplanning horizon would be evolutionarily bounded in the same manner in which history and complexity are in our current systems. 6.3 PREDICTIVE REGIME CHANGES Examining the evolution of behaviors through time, as in Figure 3, and in connection with the theoretical results, it is clear that no single, simple model should try to analyze and predict the coordinated and uncoordinated regimes together. A model that seeks to perform better should categorize the data from the past into blocks from each regime. It should use just the data from coordinated periods for predictions when the world is reasonably stable, and the remaining data for the uncoordinated periods. This requires the identification and selection of noncontiguous blocks of data, and hence, a much more sophisticated predictive approach. These kinds of value-judgments must be made when modeling real-world economic/market data—in a healthy economic climate, data affected by political turbulence, depressions, and market crashes will not be used for basic modeling. Our current research makes no attempt to address these issues. It is an interesting question whether that type of modeling would in fact stabilize our systems. 6.4 THE ACCUMULATION OF PAYOFFS Ordinarily one associates a payoff function with the gain of something of value (money, energy), which one would expect in any economic market or system of biological survival. However, these predictive systems operate solely via the use of a relative payoff, in which the only use of the payoff is in deciding which model should be used subsequently. The reason for this is that in our models there is no place for the accumulation of payoffs; a more complex class of model would allow the introduction of such a scheme. In particular, it will be possible to study the dynamics of price formation in market systems, an area in which economic theory is notoriously unsuccessful. Work in progress is beginning to achieve an understanding of how predictive agents interact to form prices. Other extensions to this work, upon which we do not report in detail in this paper, are the following: (i) agents use a probability distribution over an entire space of models rather than a single model, and update that distribution (using a Bayesian rule) according to model success, creating a probability flow over model space. Our results there show that all agents evolve to attach nonzero weight to only a small, finite fraction of the space. In this sense, model selection is also bounded; (ii) extensions to Bray's model in which single or multiple agents imitate, learn from, and model each other in simple environments; (iii) forward planning; (iv) hierarchies of models.

Natural Rationality

77

6.5 CONCLUSIONS It has long been observed (for a good discussion, see Anderson1) that qualitatively different emergent phenomena arise in large systems, usually accompanied by macroscopic events and correlations brought about by the accumulation of what are essentially simple, local microscopic dynamics. The spirit of this paper is that such collective behavior can be harnessed in a dynamic theory of economic systems. Furthermore, we believe that interesting global, dynamical phenomena can arise out of systems in which the agents are relatively simple, with homogenous behavior patterns based upon simple predictive rules. This is no simple case of "complexity begets complexity"; we do not demonstrate that any given complex phenomenon can be recreated with complex, heterogenous underlying rules, rather we wish to show that, given a reasonable, simple class of agents, they can generate a wide class of global, punctuated, metastable phenomena. So, as a first step in that direction, we selected a class of predictive systems, V, designed to relax rational expectations and stationarity assumptions toward a more natural unconstrained dynamic, in which agents' beliefs are allowed to evolve, and hence, select the most advantageous state. The systems were chosen to be as simple as possible, without exogenous shocks, but the games and rules were selected so as to bring about a large amount of feedback in the manner in which the interactions accumulate. It is because an individual's environment is no more than its neighbors, that changes in its behavior may cause changes in its neighbors' behaviors which will then both propagate and feedback, generating an interesting dynamic. 6.5.1 NATURAL RATIONALITY. We based our systems, and choice of agents, upon the need to investigate assumptions and questions of local interactions, coordination, optimal use of information, and the resulting dynamics of rationality. We gave heuristic arguments for why dynamics of this form will cause the system to drive itself toward interesting nonstationary dynamic equilibria, intermediate between conceivable ordered and disordered regimes in its character, separated by a critical point. A mean-field analysis of our systems shows very strong agreement with the qualitative nature of the simulated system's dynamics. There are two regimes: coordinated and uncoordinated, and the system is driven from one to another according to the level of difficulty of prediction in the world (although the mean-field model does not predict the fluctuations in detail), under a self-organizing, endogenous dynamic. Hence, the system cycles between selecting for a static rational expectations state and breaking down into a disordered state. We studied games with both an infinite number of Nash equilibria and a single equilibrium, expecting that a single equilibrium would stabilize the system so that persistent diversity would not be observed. However, this was shown to be wrong. Punctuated equilibrium may be observed for very long transient periods even in systems with such a single fixed

78

V. M. Darley and S. A. Kauffman

point. Under the influence of noise, such transients can become the predominant dynamic. The manner in which these dynamics pass from coordinated phases to disordered, uncoordinated bubbles of dynamics is very reminiscent of real-world market phenomena. In that sense, our results show that as systems of autonomous agents grow in size, periods of highly disordered activity (stock market crashes or bubbles?) are to be expected, and are not an anomaly. These results hold even in the absence of noise or exogenous perturbations.M There are some connections which can be drawn with business cycles and the exploitation of market niches. For example, if we consider the static state to represent a market in which all the firms have the same strategy, then a firm that offers a slightly different kind of product/service may well be able to exploit the rest of the system's unsophistication. However, our results suggest that such exploitation can destabilize the market, leading to a highly disordered regime (bankruptcies, etc.) from which a new standard emerges. It could be argued that the beginnings of such a process can be observed in the airline industry. On a rather different scale, the nature of the "wheel of retailing" (in which firms evolve from innovative, small operations through a more sophisticated growth phase and finally become a larger, stabler more conservative retailer, opening a niche for yet another low-cost, innovative operation to enter the market) could perhaps be analyzed via the same techniques (see McNair14 and Brown8 for elaboration on this "wheel"). In summary, our results demonstrate that agents evolve toward a state in which they use a limited selection of models of intermediate history and complexity, unless subjected to a dominant equilibrium when they learn to use simple models. It is not advantageous to pick models of ever increasing sophistication, nor to attach nonzero weight to all available information. As such, our agents in their mutual creation of an economic world, do not evolve toward a static rational expectations equilibrium, nor to what one would naively consider a perfectly rational state. Rather in their coevolutionary creation of a nonstationary world, they evolve to use a level of rationality which is strictly bounded, constrained within a finite bubble of complexity, information use, model selection. As they fluctuate within those constraints, these agents can still achieve system-wide coordination, but the coordinated states are metastable. The dynamics are driven by the need to create models of greater/lesser precision to match the world's observed dynamics. We interpret this state as an extension of rationality away from the simple static case to encompass the natural rationality of nonstationary worlds. Thus the natural (evolutionarily advantageous) rationality of these systems is a state in which information usage, model sophistication, and the selection of models are all bounded. It is in that sense that the natural rationality of our agents is bounded. Adaptive agents evolve to such a state, mutually creating a system which fluctuates between metastable coordinated and uncoordinated dynamics. This is the first step to understanding what agents must do in systems in which their mutual interactions i9lAt a very low level there is some noise caused by rounding and other numerical phenomena.

Natural Rationality

79

coevolve to create a nonstationary world, and is, therefore, the first step to an understanding of homo economicus in such systems.

7. ACKNOWLEDGMENTS This research was undertaken with support from Harvard University and the Santa Fe Institute (with core funding from the John D. and Catherine T. MacArthur Foundation, the National Science Foundation, and the U.S. Department of Energy). This research has benefited enormously from discussions and time at the Santa Fe Institute, in particular with Steve Durlauf, and also with Eric Maskin at Harvard University, and Maja Mataric at the Volen Center for Complex Systems, Brandeis University. Thanks are also due to Ann Bell, James Nicholson. Jan Rivkin, Dave Kane, Phil Auerswald, and Bill Macready.

80

V. M. Darley and S. A. Kauffman

REFERENCES 1. Anderson, P. W. "More Is Different." Science 177(4047) (1972): 393-396. 2. Arifovic, J. Learning by Genetic Algorithms in Economic Environments. Ph.D. Thesis, University of Chicago, 1991. 3. Bak, P., C. Tang, and K. Wiesenfeld. "Self-Organized Criticality: An Explanation of 1/f Noise." Phys. Rev. Lett. 59(4) (1987): 381-384. 4. Barnett, W. A., J. Geweke, and K. Shell, eds. International Symposium in Economic Theory and Econometrics (4th: 1987: Austin, TX.), International Symposia in Economic Theory and Econometrics. Cambridge, MA: Cambridge University Press, 1989. 5. Blume, L. 'The Statistical Mechanics of Strategic Interaction." Games & Econ. Behav. 5 (1993): 387-424. 6. Bray, M. M. "Learning, Estimation, and Stability of Rational Expectations." J. Econ. Theor. 26 (1982): 318-339. 7. Brock, W., and S. Durlauf. "Discrete Choice with Social Interactions I: Theory." Technical Report, University of Wisconsin, 1995. 8. Brown, S. "The Wheel of the Wheel of Retailing." Intl. J. Retail 3(1) (1988): 16-37. 9. Darley, Vincent M. "Towards a Theory of Optimising, Autonomous Agents." Ph.D. Thesis, (1997): forthcoming. 10. Ellison, G. "Learning, Local Interaction, and Coordination." Econometrica 61(5) (1993): 1047-1071. 11. Kauffman, Stuart A. The Origins of Order: Self-Organization and Selection in Evolution. Oxford: Oxford University Press, 1993. 12. Kreps, D. M. Game Theory and Economic Modelling. Oxford: Clarendon Press; New York: Oxford University Press, 1990. 13. Lane, D. "Artificial Worlds in Economics. Parts 1 and 2. J. Evol. Econ. 3 (1993): 89-108, 177-197. 14. McNair, M. P. "Significant Trends and Developments in the Post-War Period." In Competitive Distribution in a Free High Level Economy and Its Implications for the University, edited by A. B. Smith, 1-25. University of Pittsburgh Press, 1958. 15. Nash, J. Non-Cooperative Games. Ph.D. Thesis, Mathematics Department, Princeton University, 1950. 16. Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. F. Flannery. Numerical Recipes in C, 2nd ed. Cambridge: Cambridge Unviersity Press, 1992. 17. Sargent, T. Bounded Rationality in Macroeconomics. Oxford: Clarendon Press, 1993. 18. Schelling, T. "Dynamic Models of Segregation." J. Math. Sociol. 1 (1971): 143-186.

Steven N. Duriauf Department of Economics, University of Wisconsin at Madison, Madison, Wl 53706; Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501

Statistical Mechanics Approaches to Socioeconomic Behavior

This chapter provides a unified framework for interpreting a wide range of interactions models which have appeared in the economics literature. A formalization taken from the statistical mechanics literature is shown to encompass a number of socioeconomic phenomena ranging from outof-wedlock births to aggregate output to crime. The framework bears a close relationship to econometric models of discrete choice and, therefore, holds the potential for rendering interactions models estimable. A number of new applications of statistical mechanics to socioeconomic problems are suggested.

1. INTRODUCTION This chapter is designed to provide a unified discussion of the use of statistical mechanics methods'1' in the study of socioeconomic behavior. The use of these methods in the social sciences is still in its infancy. Nevertheless, a growing body of work has l l 'The statistical mechanics models I employ are also referred to as interacting particle system or random fields models.

The Economy as an Evolving Complex System II, Eds. Arthur, Duriauf, and Lane SFI Studies in the Sciences of Complexity, Vol. XXVII, Addison-Wesley, 1997

81

82

S. N. Durlauf

shown how statistical mechanics and related probability techniques may be used to study the evolution and steady state behavior of heterogeneous populations. Examples of the range of applications of statistical mechanics methods include social pathologies such as out-of-wedlock births and crime (Brock and Durlauf,20 Glaeser, Sacerdote, and Scheinkman34), asset price behavior (Brock17'18), expectation formation (Brock and Hommes21), business cycles (Bak et al.7 and Durlauf25,27), technology adoption (An and Kiefer1), and endogenous preferences (Bell11). In addition, Blume 15,16 has shown how these methods can provide insight into the structure of abstract game-theoretic environments. M These disparate phenomena are linked by the possibility that each is determined, at least partially, by direct interactions between economic actors. Put differently, each of these phenomena is a case where the decisions of each individual are influenced by the choices of others with whom he interacts. This interdependence leads to the possibility that polarized behavior can occur at an aggregate level solely due to the collective interdependence in decision-making. This explanation of polarized group behavior may be contrasted with explanations which rely on the presence of highly correlated characteristics among members of a group. Interactions between economic actors are commonplace in economic models. However, these interactions are typically mediated through markets. What distinguishes the bulk of the interactions in the recent inequality literature is the focus on interdependencies which are direct.13' Examples of such direct interactions are role model, social norm, and peer group effects. At first glance, statistical mechanics methods, which underlie the theory of condensed matter, would appear to have little to do with socioeconomic phenomena related to inequality. However, strong metaphorical similarities exist between the two fields of research. The canonical question in statistical mechanics concerns the determinants of magnetization in matter. As a magnetized piece of matter is one in which a substantial majority of the atoms share a common spin (which can either be up or down), magnetization would appear to be an extremely unlikely phenomenon, as it would require the coincidence of many atoms sharing a common property. However, if the probability that one atom has a particular spin is a function of the spins of surrounding atoms, the possibility of collective interdependence renders magnetism understandable. As techniques for the study of economic phenomena, statistical mechanics approaches have proven valuable for studying the ^'Alternative approaches to the modeling of complex interaction environments include Arthur,' 1 ' 5 loannides, 35 Kirman, 37 and Krugman. 39 '31 While markets may not exist to directly mediate these interactions, this does not imply that economic actors do not alter their behavior in order to account for them. For example, as discussed in Benabou 1 2 , 1 3 and Durlauf, 27,28 the presence of within-neighborhood interactions can play a primary role in determining the composition of neighborhoods.

Statistical Mechanics Approaches to Socioeconomic Behavior

83

aggregate behavior of populations facing interdependent binary choices. In particular, statistical mechanics methods hold the promise of providing a general framework for understanding how collective interdependence can lead to the emergence of interesting and rich aggregate behavior. M Of course, any metaphorical similarity between physical and social models of interdependent behavior is of little interest, unless the specific substantive models underlying each can be shown to have similar structures. An important goal of this chapter is to show how statistical mechanics structures naturally arise in a number of socioeconomic environments. Also, it is important to recognize that the potential for interesting aggregate behavior to emerge from individual decisions has been explored in previous economic contexts. Two prominent examples include Becker's8 work on aggregate demand with zero-intelligence agents and Schelling's46 analysis of racial segregation. 1*1 Statistical mechanics approaches should thus be regarded as complementary to disparate strands of previous work. The rest of this chapter is organized as follows. Section 2 outlines some general issues in modeling binary choices with interactions. Section 3 analyzes binary choice models with global interaction structures. Section 4 analyzes binary choice models with local interactions. Section 5 provides a discussion of limitations and outstanding questions which arise in statistical mechanics models of social behavior. Section 6 contains summary and conclusions.

2. GENERAL CONSIDERATIONS The statistical mechanics-inspired models of socioeconomic phenomena that typically have been studied focus on environments in which each individual of a group faces a binary choice. Many individual decisions which are relevant to understanding inequality are binary in nature. Standard examples include decisions to have a child out of wedlock, drop out of school, commit a crime. Not only are these decisions of interest in their own right, but they are well-known to affect a broad range of individual socioeconomic outcomes over large time horizons. While the importance of these binary decisions in cross-section and intertemporal inequality is beyond dispute, there is considerable controversy over the role of interactions, and so one purpose of the particular formulation I have chosen is to develop interaction-based models in a way which makes contact with the econometric literature on discrete choice. In addition, this approach provides a method for exploring the interconnections between different approaches to modeling interactions. Brock17 and Blume15 l4'See Crutchfield 24 for a discussion of the meaning of emergent phenomena. i5'Schelling's model possesses a structure that is quite similar to some of the statistical mechanics models which are discussed below.

84

S. N. Durlauf

originally recognized the connection between discrete choice and statistical mechanics models. The current development follows Brock and Durlauf.19 Binary decisions of this type may be formalized as follows. Consider a population of / individuals. Individual i chooses wi,% i) represents deterministic private utility, S(u)it % i,n\(Mi -i)) represents deterministic social utility, and e(wj) represents random private utility. The two private utility components are standard in the economics of discrete choice. Recent work is distinguished by the introduction of social utility considerations. Second, the form of the social utility and the form of the probability density characterizing random private utility are generally given particular functional forms. (It will become clear that restricting the form of the private utility term has no qualitative effect on the properties of the aggregate population.) The social utility component is formalized by exploiting the intuition that individuals seek

Statistical Mechanics Approaches to Socioeconomic Behavior

85

to conform in some way to the behavior of others in the population. Formally, a specification which subsumes many specific models may be written as

Ei(-) represents the conditional expectation operator associated with agent i's beliefs. The term Jj($ \& u...,

& ,, nt(ie _ ! ) , . . . , ftUki . / ) ) ~

i

The large sample behavior of the average choice level in this economy may be analyzed as follows. First, assume that all agents share a common expectation of the average choice level, i.e., m\ = meVi. (14) Second, let dF% denote the limit of the sequence of empirical probability density functions associated with individual characteristics % *, where the limit is taken with respect to the population size /. (I assume that such a limit exists.) The strong law of large numbers implies, for any common expected mean, that the sample mean fhj of population choices converges with a limit equal to lim m, = f tanh(/?(& )h{X) + p(X)j(X /=*-oo

J

)me)dFx

.

(15)

W

The model is closed by imposing self-consistency in the large economy limit, so that the limit of the sample mean corresponds to the common expected average choice level. A self-consistent equilibrium mean, m", is any root of

m* » J tanh(/?(£ )h{$) + j3{X )J(X )m*)dFx .

(16)

When all agents are identical, in that they are associated with the same X the mean-choice level in the economy will correspond to the roots of m* = tanh(0/i + /3Jm*).

j

(17)

In this special case, the model corresponds to the mean-field approximation of the Curie-Weiss model. Brock's paper17 is the first instance in which the CurieWeiss model was given an economic interpretation; the current formulation differs in emphasizing the equivalence between the mean-field approximation of the model and the assumption of a noncooperative interaction environment. The following theorem, taken from Brock and Durlauf,19 characterizes the number of self-consistent steady states.

88

S. N. Durlauf

THEOREM. 1. Existence of multiple average choice levels in equilibrium. i.

If {3J > 1 and h = 0, there exist three roots to Eq. (17). One of these roots is positive, one root is zero, and one root is negative.

ii. If 0J > 1 and h ^ 0, there exists a threshold H, (which depends on f3 and J) such that a.

for \0h\ < H there exist three roots to Eq. (17) one of which has the same sign as h, and the others possessing the opposite sign.

b. for \0h\ > H there exists a unique root to Eq. (17) with the same sign as h. Notice that the model exhibits nonlinear behavior with respect to both the parameters /3h and (3 J. This makes sense intuitively. Conditional on a given private utility difference between the choices 1 and —1, which equals 2ft, there is a level that the conformity effect 0 J must reach in order to produce multiple self-consistent mean-choice behavior. Recall that the random utility shocks are i.i.d., so that in absence of the conformity effects, there would be a unique mean whose sign is the same as ft. Conditional on a conformity effect (3J > 1, as /Jft increases in magnitude, any multiplicity will eventually be eliminated. This occurs because eventually the private utility differential between the choices will overcome any tendency for the conformity effect to produce a self-consistent mean with the opposite sign. This type of model illustrates the complementary nature of the roles of economic fundamentals and social norms in explaining the degree of social pathologies in different neighborhoods. For example, Theorem 1 states that high degrees of conformity can lead to mean-choice levels opposite to that dictated by private utility. To be concrete, even if economic fundamentals, as embodied in ft, imply that the average teenager should stay in school, conformity effects can produce an equilibrium in which most teenagers drop out. It is straightforward to show (Brock and Durlauf19) that the equilibrium in which the sign of ft is the same as the sign of the mean-choice level produces higher average utility than the equilibrium in which the signs are opposite. Hence, this model illustrates the potential for collectively undesirable behavior, such as high out-of-wedlock birth rates, which is individually optimal. This ranking of average utility provides the appropriate stochastic generalization to the Pareto rankings of equilibria studied in Cooper and John. 23

Statistical Mechanics Approaches to Socioeconomic Behavior

89

4. LOCAL INTERACTIONS An alternative approach in the study of interactions has focused on the aggregate implications of local interactions. Such models assume that each individual interacts with a strict subset of others in the population. For individual x, this subset is n< and is referred to as the individual's neighborhood. Hence, for agent i, Ji{Xi)=Qiij hc, the mean choice will converge to one of two nonzero values. The probability of converging to any one of these values will depend on # oWhile providing qualitatively similar features to the original Follmer formulation, the absence of any contemporaneous interactions does create some differences. In particular, the hc in the dynamic model does not equal the J c in the static model. This occurs because of the absence of any contemporaneous interactions. An exact equivalence between the static and dynamic models (with respect to parameter values and invariant measures) can be achieved through the following continuous time formulation. Suppose that agents adjust asynchronously in time. In particular, each agent is associated with a separate Poisson process such that at each arrival time for his process, the agent's choice is such that he takes the choices of the nearest neighbors as given. As the probability that any two agents choose at the same time is zero even in the infinite population limit. This model will generate multiple regimes, with Jc = rV'9' Follmer's model illustrates that it is difficult to provide a straightforward interpretation of contemporaneous local interactions in which the interdependence occurs with respect to choices. Alternatively, once can interpret this discussion as demonstrating the sensitivity of local interactions models to the assumptions placed on expectations. To see this, consider a local interaction formulation of social utility which preserves the Ising interaction structure, 5(«,,£«,tf(#-«)W

£

*j . (26) However, this assumption will not lead to the conditional probability structure (10) for individual choices. The reason is simple. Under perfect foresight, the choices of individuals i and j will be interdependent whenever \i—j\ — 1, and so each choice will be a function of both £ efc,,, then Prob^.oo = l|w _j B. If 0^f

x

- 1 ) = 1.

< 0^ „ then

i.

ProbKO0 = l|w_1 = - X ) < l .

ii.

Proh(moo = l\u_1

=

~l)^0.

The main result of the model is that when production decisions are interdependent, then a low level production trap is produced. In terms of the underlying probability structure, the model (and its associated properties) is a generalization of the Stavskaya-Shapiro model, described in Shnirman47 and Stavskaya and Pyatetskii-Shapiro.48 This model can be rewritten in the canonical discrete choice form as follows. For each industry i at t, reinterpret u(wi]t, •,•) as the expected discounted profits. which depends on current technique ehoice.lnl Following Eqs. (5) and (6), h(Z i,i) measures the relevant individual- and time-specific deterministic private payoff parameter associated with a technique choice. Further, assume that the only relevant characteristics determining the relative profitability of the two techniques for a given industry are the past technique choices of technologically similar industries, h(£

i,t) = h(uti-k,t-i,

• • • ,i+i,t-i) •

(37)

Restrict this function so that it is increasing in all arguments and positive when all previous technique choices were equal to 1, /i(l,...,l)>0. Finally, assume that 0$

(38)

^t) has the property that

0(& i,t) = 00 if Wi_/fc,i_i ss . . . as Wi+U-1

= 1.

(39)

Subject to these restrictions, there will exist a discrete choice specification which replicates the probabilities in the interacting industries model. This specification reveals an important feature of the particular specification studied in Durlauf26: a strong nonlinearity in the case where all influences are positive at t — 1 versus all other cases. This is not surprising given the unbounded support of the logistic density for finite 0 combined with the probability 1 assumption on technique choice ("'Recall that the structure of industries is such that technique choices do not reflect any consequences for future spillover effects, so that an industry always chooses a technique to maximize one-period ahead profits.

Statistical Mechanics Approaches to Socioeconomic Behavior

95

under the conditioning in Eq. (35). This suggests that the particular case studied in Durlauf26 is knife-edge in terms of parameter values under the general discrete choice framework. While this does not affect the theoretical interest of the model, it does suggest that it should be reparameterized if one is to use it in empirical work. Finally, it is worth noting that when interactions between decisions are all intertemporal, then the assumption of extreme-valued random utility increments can be dropped. The equilibrium properties of the dynamic models in this section can be recomputed under alternative probability densities such as probit which are popular in the discrete choice work. In fact, under the mean-field analysis of global interactions, alternative specifications incorporate probit or other densities as well. In both cases, the large-scale properties of models under alternative error distributions are largely unknown.

5. STOCHASTIC INTERACTION STRUCTURES A third area of work has focused on cases where the interaction environments are themselves stochastic. Three approaches to this have been taken. The first is to allow for heterogeneity within a group. The second allows for random group formation. The third treats group membership as a choice variable. I. MULTIPLE AGENT TYPES

Glaeser, Sacerdote, and Scheinkman34 provide a model of local interactions and crime that fits cleanly into the general discrete choice framework. They consider a model in which individuals are arrayed on a one-dimensional lattice. Each individual has one of three kinds of preferences with reference to the decision to commit a crime. The decision to commit is coded as w< = 1. Preference type 1 is such that the agent always decides to commit a crime. Preference type 2 is such that an agent will choose to commit a crime only if the agent to his left does as well. Preference type 3 is such that the agent never commits a crime. Representing the preferences of agents as Uj(u>i, Wj_i) where j denotes the preference type, i / 1 ( l , l ) > C / , ( - l , l ) ; Ui(l,-l)>Ui{-l,~l) U3(1,1)>U2(-1,1); U2(~l,-l)>U2(l,-l)

(40) (41)

%(-l,l)>D»(l,i)j £/3(~l,~l)>£/3(l,-l).

(42)

The distribution of preference types is i.i.d. across agents. The interaction structure described here is a variant of the so-called voting model in statistical mechanics; see Liggett40 for a detailed description.

96

S. N. Durlauf

From the discrete choice perspective, these preference assumptions can be thought of as doing the following. Each agent possesses a neighborhood consisting of the agent to his left, so that Jj$

i) = J($ i) if i = it — 1,0 otherwise.

(43)

Further, each agent is associated with a latent variable it with support {4>l, m,4>h}. The latent variable is the only personal characteristic which influences individual utility, so that the different preference types can be thought of as induced by l)m)=0, J ( 0 m ) > O h(0, J(h) = 0.

(44) (45) (46)

The joint probability measure for choices in this model is Prob(& | 0 l 5 . . . , 4>i) ~ I ] *MPh{4>M + 0J(i) • WiWi-i) • »

(47)

The unconditional probability measure can be computed once an assumption is made on the form of dF^ and thereby allow analysis of cross-sectional patterns. When 0 = oo, the model reduces to the deterministic choice structure of Glaeser, Sacerdote, and Scheinkman. A nice feature of this model is that by preserving nonoverlapping neighborhoods, the problems created by the contemporaneous determination of choices are avoided in the perfect foresight case. Further, notice that if the interactions are intertemporal and past behavior influences current behavior (as clearly seems natural in this context), then joint behavior will follow

Y^hi^iA Prob(J0 t )~fflcp[2>i, twM }

( 48 )

hi,t - A(u> i _i,t-i,w M _ 1 ,0 M ).

(49)

where In this case, the interaction structure is a special case of that studied in Durlauf.26

97

Statistical Mechanics Approaches to Socioeconomic Behavior

II. RANDOM COMMUNICATION STRUCTURES

Kirman, 37 Ioannides,35 and Durlauf,29 have studied economic environments in which the structure of bilateral interactions has important aggregate consequences. In their models, random communication links exist between any pair of agents i and j . Coalitions emerge across any grouping of agents such that a path of direct bilateral communication links can be formed between any pair of members in a coalition. As all agents within a coalition communicate and whereas members of different coalitions do not, this structure can illustrate the role of group membership in phenomena ranging from fluctuations in the prices of a particular good across trading regions to the role of the degree of specialization of labor in explaining business cycles. A rich set of results from random graph theory illustrate how the distribution of coalition sizes will depend sensitively on the probability of the bilateral links. In particular, when the bilateral links are conditionally i.i.d. (in the sense that the probability that any pair of agents is directly linked is independent of whether any other pair is linked), then as / => oo (1) if the probability of any link is less than l/I, then the largest coalition will be of order log I, (2) if the probability equals c/I for c > 1, then the largest coalition will be of order /, (3) if the probability is greater than clog 1/1 for c > 1, then all agents will be members of a common coalition. Previous papers employing random graph formulations have been interested in the size distribution of coalitions. However, for socioeconomic environments involving networks of contacts or friends, the approach can enrich the exogenous interaction structures which are typically assumed. The discrete choice structure can accommodate random interactions by assigning a probability measure to Jitj's such that Jije Jij

{0,1}, =

(50)

•?],*»

If JiM^hM ''" JU,i - 1> t n e n JiJ ~

(51) l

•

(52)

Any assumptions about the distribution of the bilateral interactions can be mapped in a straightforward fashion to the J;,j weights subject to these restrictions. This particular structure is related to the Mattis model in statistical mechanics, which is described in Fischer and Hertz.32 In the Mattis model, the individual weights are defined by Ji,3 - J£,i£,j (53) where & has support {1, -1} and is distributed i.i.d. across i. Using the transformation e

_ Uj + l

98

S. N. Durlauf

then the, Mattis model will produce a random graph structure and associated interactions weights with the formula Jij

-

J

a> I CO

c o -p 1-

o a. I O

o 10

20 n

30

40

FIGURE 1 The asymptotic proportion of adopters of the superior product B: d = -0.1; (i so ej3,

0-4

0.2

0.4 0.6 proportion of B-adopters

0.8

FIGURE 3 Distribution of the proportion of adopters of the superior product B after 100 adoptions', d = - 0 . 1 , pi - CB, O2 - 1, A = 1.6, r = 1, a — 2. For the density with mode at 1, n = 3; for the other density, n ~ 20.

The answer to "whether" is: of course, but not uniformly, in r and s. Figure 3 shows the distribution for the proportion of adopters of the superior product B, with the same values for d,fj, — CB, and a2 as in the Figure 1 example, A equal to 1.6, and two different values of n: 3 and 20. The asymptotic market-shares of the superior product in these two cases are 93.88% and 68.04%, respectively. In the example illustrated in Figure 3, the initial proportion of B-adopters is 2/3. The means of the two distributions are nearly identical: 0.712 and 0.707 for n = 3 and n = 20, respectively. Not surprisingly, the n = 3 distribution is much more spread

118

D.Lane

out than is the n = 20 distribution (sd's 0.24 and 0.13, respectively). The only sense in which the information inversion phenomenon is observable in Figure 3 is in the difference in the modes of the two distributions: 1 for n = 3 and about 2/3 for n = 20. For the n = 3 distribution, the superior product will have a marketshare in excess of 90% with probability 0.28, while for the n = 20 distribution, this probability is only 0.0915. The market-share distributions change slowly with the number of adopters, of course in the direction predicted by the asymptotic results. After 5000 adoptions, the mean of the n = 3 distribution has increased, but only to 0.75, while the mean of the n = 20 distribution has decreased to 0.697; the probability that the superior product will have a market-share over 90% for the n = 3 distribution has risen to 0.3, while that for the n = 20 distribution, has fallen to 0.0185. Information inversion, yes; but not nearly as striking as in the infinite agent case. And, if we start with more concentrated initial distributions, or ones more favorable to the inferior product, it takes much larger agent societies for the aggregatelevel advantage of less information to make itself evident. A NOTE ON RELATED WORK Two other "more information/less efficiency" results arise in social-learning models in the economic literature, but both differ in key respects from those I have reported here. In the model of Ellison and Fudenberg,9 agents choose between two competing products. Each agent in the uncountably infinite population uses one of the two products, and in each time period a fixed fraction of them consider switching on the basis of information obtained from other users. In contrast with the information contagion model, in the Ellison-Fudenberg model the actual product performance characteristics change over time, in response to exogenous shocks. The agents in this model use a decision rule similar to the diffuse ALBO rule with A = 0: they choose the product with the highest average performance in their sample, as long as they have information on both products; otherwise, they choose the product about which they have information. Ellison and Fudenberg find that when the number of informants per agent is small, the proportion adopting the superior product converges to 1, which does not happen when the number of informants is larger. This Ellison-Fudenberg version of information inversion is driven by their model's assumed "shock" to the products' performance characteristics: a currently unpopular product can be "resurrected" in a period in which its performance is particularly high only if many agents find out about this good performance, which happens with higher probability the more informants each agent has. Thus, more informants per agent guarantee that neither product is eliminated. More subtle relations between individual and aggregate level do not arise in the Ellison-Fudenberg model, because their assumption of an uncountable number of agents lets the strong law of large numbers sweep the delicate probabilistic effects of feedback under the rug.

Is What Is Good For Each Best For All?

119

In Banerjee,5 heterogeneous agents encounter an investment opportunity only by hearing about others who chose to invest. The yield of the investment depends on a state of the world unknown to the agents. For some of the agents, the investment is profitable no matter what the state of the world, but for others it is not. Agents update their state of the world distribution rationally when they learn that someone has invested, and then decide whether or not to invest themselves. The probability that an agent will learn about an investment in a particular time interval is an increasing function of the number of agents who have invested. The key to deciding whether to invest is the time at which one first learns about the investment. Banerjee's process begins when a certain fraction x of agents learn about the investment opportunity and the true state of the world; this is a one-time revelation, not repeated for later investors who must infer the state of the world as described above. Surprisingly, the aggregate-level efficiency (measured as the fraction of those who should invest that do, as a function of the true state of the world) does not increase with x, the amount of "hard information" injected into the system. To me, this result is interesting and counter-intuitive, but the phenomenon is different from what happens in the information contagion model, since in Banerjee's model the relevant "more information" comes in at the aggregate not the individual level. If we range further afield than social-learning models, we can encounter a number of results that bear a certain resemblance to what I found in the information contagion context. The general issue is: whether (and if so when) more information at the microlevel of a multilevel system result in degraded higher-level system performance? Many examples of this phenomenon come to mind. Think of stochastic Hopfield networks (see Hertz, Krogh, and Palmer,11 Chap. 2): their performance as associative memories is enhanced when each unit responds to its input signals from other units perturbed by noise. Here, I interpret "less noise" as "more information": thus, more information at the individual unit level, about the state of other units, can result in poorer aggregate- or system-level performance. i17l Or the "complexity catastrophe" that happens in the NK model, for sufficiently large values of the connectivity parameter K (Kauffman13). It would be interesting to speculate about whether there are some common underlying processes linking these different phenomena.

2. RATIONALITY, SUFFICIENCY, AND EFFICIENCY: MAX DOMINATES ALBO In section 1, I argued that ALBO agents were as rational as we have a right to expect agents to be in the information contagion context. In section 2,1 showed that i t 7 lln Hopfteld networks, too much noise always results in poor performance, while in the information contagion context very little—but not no—information at the individual level can in some circumstances result in very good aggregate-level results.

120

D.Lane

the aggregate-level "efficiency"—the proportion of agents that adopts the superior product—of ALBO rules depends in a complicated way on the model parameters. In particular, the market-share of the superior product can be substantially less than 100%, at least when the real difference between the products' performance characteristics is not too large.!18! In this section, I introduce another decision rule, which I call the Max rule. The Max rule has the remarkable property that it always leads to 100% market-share for the superior product, no matter how small the difference between the two products. Thus, from a social efficiency perspective, it dominates ALBO rules, however attractive the Bayesian updating and maximizing expected utility prescriptions may be at the individual level. Moreover, the Max rule violates one of the central canons of statistical inference and decision theory: it is based on a function of the data (X, Y) that is not sufficient for the unknown parameters CA and CB* The Max rule was "discovered" in the course of an experiment carried out by Narduzzo and Warglien.18 This experiment reproduced the context of the information contagion model, with Venetian business school students as agents. As part of their experiment, Narduzzo and Warglien conducted a protocol analysis, in which subjects explained why they preferred one product to the other. Not surprisingly, none of the subjects claimed to reach their decision by Bayesian optimization. Instead, they typically invoked one or another of four simple rules-of-thumb to account for their choices. One of these rules, by the way, corresponds to the Ellison-Fudenberg rule cited in the last section, the diffuse ALBO rule with A = 0. Another, in complete contrast to my rationality argument in section 1, uses only the information in X and completely ignores the information in Y: choose whichever product the majority of your informants chose.!19! l 18 llu fact, the proportion of ALBO agents that adopt the superior product can be substantially less than 1 even when the real difference between the products is arbitrarily large. For example, suppose that the agents are well-calibrated for the superior product B (that is, ft = CB) and, as usual, the public information does not distinguish between the two products, so the agents have the same prior distribution for CA as for eg. In such a situation, the market share for the inferior product A need not be essentially 0, no matter how large the actual difference between the two products. The reason is simple: unlike the usual single-agent Bayesian asymptotics, in the information contagion context the effect of the prior distribution does not go away as the number of adopters increases. When the proportion of adopters of the superior product B becomes sufficiently large, an agent with high probability samples only B adopters. Then, his posterior mean for CA is just the prior mean p = eg. Of course, the expected value of the average of his y-observations is also CB , so the probability that his posterior mean for CB is less than his posterior mean for CA is 1/2. Thus, if he is not very risk averse and the number of his informants is not too large, he will end up adopting A with appreciable probability. For example, if A = 0, n = 5, and a2 = 10, the inferior product will have a market-share of at least 18%, no matter how low its actual performance characteristic. i19!Not surprisingly, this "imitation" rule leads to complete market domination by one or the other product. If r = s, each product has a probability 1/2 of dominating. See Lane and Vescovini.15 Kirman 14 provides an interesting analysis of the effects of this rule when n = 1, the population of agents is finite, and agents can change their product choices.

121

Is What Is Good For Each Best For All?

The Max rule can be stated as follows: choose the product associated with the highest value observed in the sample. That is, denoting by max the index of y (n) = max(j/i,...,t/„), the rule is given by D({xi,yi),...,(xn,y„)) = xmax. Note that if all informants adopted the same product, then the Max rule opts for that product, regardless of the magnitude of the observations tfi,..., t/n-[20' The Warglien-Narduzzo experimental subjects who invoked the Max rule justified it with a curious argument. They claimed that they ought to be able to obtain product performance "as good as anyone else," and so they figured that the best guide to how a product would work for them was the best it had worked for the people in their sample who had used it. Narduzzo and Warglien18 describe this argument as an example of a "well-known bias in human decision-making," which they follow Langer16 in calling the "illusion of control." Of course, this justification completely fails to take into account sample-size effects on the distribution of the maximum of a set of i.i.d. random variables. Since the current market leader tends to be over-represented in agents' samples, its maximum observed value will tend to be higher than its competitors', at least as long as the true performance of the two products are about the same. Thus, it seems plausible that in these circumstances a market lead once attained ought to tend to increase. According to this intuition, the Max rule should generate information contagion and thus path-dependent market domination. This intuition turns out to be completely wrong. In fact, for any n > 2, the Max rule always leads to a maximally socially efficient outcome: PROPOSITION: Suppose n > 2. If the two products, in reality, have identical performance characteristics, the market-share allocation process exhibits path dependence: the limiting distribution for the share of product A is p(r, s). Suppose, however, that the two products are different. Then, the better product attains 100% limiting market-share with probability 1. A proof of this proposition, from Lane and Vescovini,15 is given in the appendix. Why ALBO should fail to lead to market domination by the superior product is no mystery. But it is not easy to understand why the Max rule does allow agents collectively to ferret out the superior product. The proof gives no insight into the reason, though it does make clear that the larger is n, the faster will be the convergence to market domination. In fact, I am not sure there is a real "reason" more accessible to intuition than the fact that drives the proof: with the Max rule, the l20)p(fc) can be calculated for the Max rule as follows: p(n) = 1 and p(0) = 0; for 0 < k < n, p(fc) is just the probability that the maximum of a sample of size fc from a N(CA,1) distribution will exceed the maximum of an independent sample of size (n - fc) from a N{CB, 1) distribution: p(fc) = / * f o - (Cfl -

where ip is the standard normal density function.

cA)}n-kk^[y}k-^[y}dy,

122

D.Lane

probability of adopting the inferior product given the current proportion of inferior product adopters is always smaller than that proportion. In fact, the superiority of the Max rule to ALBO rules is not particularly evident in "real time." The associated distributions for the superior product's market-share after a finite number of adoptions, tend to be both more spread-out and more dependent on initial conditions (r and s) than those associated with ALBO rules having the same number of adopters and number of informants per adoption. Nonetheless, these distributions do, sooner or later, inexorably converge to point mass at 100%.

3. SUMMARY This chapter has focused on a simple instance of a deep and general problem in social science, the problem of micro-macro coordination in a multilevel system. Behavior at the microlevel in the system gives rise to aggregate-level patterns and structures, which then constrain and condition the behavior back at the microlevel. In the information contagion model, the interesting microlevel behavior concerns how and what individual agents choose, and the macrolevel pattern that emerges is the stable structure of market-shares for the two products that results from the aggregation of agent choices. The connection between these two levels lies just in the agents' access to information about the products. The results described in sections 2 and 3 show that mechanism at the microlevel has a determining and surprising effect at the macrolevel. In particular, what seems good from a microlevel point of view can turn out to have bad effects at the macrolevel, which in turn can materially affect microlevel agents. The chapter highlighted two specific instances of this phenomenon. First, although individual decision-makers should always prefer additional costless information, performance at the aggregate level can decrease as more information is available at the microlevel. Second, despite the fact that individual decision-makers "ought" to maximize their expected utility, a very large population of Max rule followers will achieve a higher level of social efficiency than will a population of Bayesian optimizers.

123

Is What Is Good For Each Best For All?

APPENDIX: PROOFS 1. ALBO AGENTS EXPECT TO GAIN FROM MORE INFORMATION

Let Ii represent the information an agent obtains from his first i informants: Ii = {(Xi,Y\),(X2,Yi),. •. ,(Xi,Yi)}. Further, let A* and Bi represent the expected utility the agent calculates for products A and B, respectively, on the basis of Ii, and let Ui — max(Ai, Bi). Thus, Ui is the value to the agent of his currently preferred product. I now show that an agent always expects to increase the value of U by obtaining information from an additional informant. CLAIM 1: {Ai} and {B,} are martingales with respect to the cr-fields generated by/i. PROOF OF CLAIM: I show that {Ai} is a martingale. E(Ai+1\Ii)

= E(Ai+il{X^tmA}lIi) + E(Ai+il{Xt+imB}\Ii) = E(Ai+i\{Xi+i = A}Ji)P(Xi+1 = A\h)+ AtP(Xl+1 = B\I{).

Thus, 1 need to show that E(Ai+i\{Xi+i = A},h) = Af, Let n be the number of .A-adopters among the first i informants and let Zi,...,Zn denote the Y-values obtained from these informants. Let a denote { 0 , l } x . . . x { 0 , l } . n-1

Interactions across individuals are defined in terms of probability distributions for the state of individual t conditional on her environment, 7rt(wt|?jt). The collection of these distribution functions, V = { T T J ^ I is known as a local specification. We say that V is a nearest neighbor specification if it implies that the state of individual i depends only upon the state of her neighbors. A probability measure Jrt is said to define a Markov random field if its local specification for t depends only on knowledge of outcomes for the elements of v(i), the nearest neighbors of t. This definition of a Markov random field confers a spatial property to the underlying stochastic structure, which is more general than the Markov property. We normally ( 5 lThe development of Markov random fields originated in lattices, 32 but was later extended by Preston 97 to general graphs.

135

Evolution Of Trading Structures

impose the assumption that regardless of the state of an individual's neighbors her state is nontrivially random. Random fields are easier to study in a state of statistical equilibrium. While in most of our own applications we have a finite number of agents in mind, a countable infinity of agents is also possible.1,2,97 It is an important fact that for infinite graphs there may be more than one measure with the same local characteristics. When this happens the probabilities relating to a fixed finite set will be affected by the knowledge of outcomes arbitrarily far (or, just quite far, if infinity is construed as an approximation for large but finite sets) from that set. Existence of more than one measure with the same local characteristics is known as a phase transition J6' We say that a measure ft is a global phase for model (G, S, V) if fj, is compatible with V in the sense that n{u)L = s\