The Virtues and Vices of Equilibrium and the Future of Financial Economics

The use of equilibrium models in economics springs from the desire for parsimonious models of economic phenomena that take human reasoning into account. This approach has been the cornerstone of modern economic theory.We explain why this is so, extolling the virtues of equilibrium theory; then we present a critique and describe why this approach is inherently limited, and why economics needs to move in new directions if it is to continue to make progress. We stress that this shouldn’t be a question of dogma, and should be resolved empirically. There are situations where equilibrium models provide useful predictions and there are situations where they can never provide useful predictions. There are also many situations where the jury is still out,i.e.,where so far they fail to provide a good description of the world,but where proper extensions might change this. Our goal is to convince the skeptics that equilibrium models can be useful, but also to make traditional economists more aware of the limitations of equilibrium models.We sketch some alternative approaches and discuss why they should play an important role in future research in economics. © 2008 Wiley Periodicals, Inc. Complexity 14: 11–38, 2009 J. DOYNE FARMER AND JOHN GEANAKOPLOS Key Words: equilibrium; rational expectations; efficiency; arbitrage; bounded ratio- nality; power laws; disequilibrium; zero intelligence; market ecology; agent-based modeling J. Doyne Farmer is affiliated with Santa Fe Institute, Santa Fe, NM 87501 and LUISS Guido Carli, 00198, Roma, Italy 1. INTRODUCTION (e-mail: [email protected]) he concept of equilibrium has dominated economics and finance for at least John Geanakoplos is affiliated with Yale 50 years. Motivated by the sound desire to find a parsimonious description University, New Haven, CT 06520 and T of the world, equilibrium theory has had an enormous impact on the way Santa Fe Institute, Santa Fe, NM 87501. that economists think, and indeed in many respects defines the way economists think. Nonetheless, its empirical validity and the extent of its scope still remains a matter of debate. Its proponents argue that it has been enormously successful, and that it at least qualitatively explains many aspects of real economic phenomena. Its detractors argue that when one probes to the bottom, most of its predictions are essentially unfalsifiable and therefore are not in fact testable scientific theories. We attempt to give some clarity to this debate. Our view is that while equilibrium theory is useful, there are inherent limitations to what it can ever achieve. Economists

This is an invited article for the Special Issue—Econophysics, Guest Editors: Martin Shubik and Eric Smith.

© 2008 Wiley Periodicals, Inc., Vol. 14, No. 3 COMPLEXITY 11 DOI 10.1002/cplx.20261 Published online 10 November 2008 in Wiley InterScience (www.interscience.wiley.com) must expand the scope of equilibrium to motivate corresponding models in how much happiness she gets out of con- theory, explore entirely new approaches, economics. In Section 8 we review sev- suming the goods.The agent does not get and combine equilibrium methods with eral alternatives approaches, including pleasure out of each good separately, but new approaches. now established approaches such as only in the context of all the goods she This article is the outcome of an eight- behavioral and experimental economics. consumes taken together. year conversation between an economist We also describe some less established The goal is to model the decisions and a physicist. Both of us have been approaches to dealing with bounded of agents incentivized solely by their involved in developing trading strategies rationality, specialization and heteroge- own selfish goals, and to deduce the for hedge funds, giving us a deep appre- neous agents. We make a distinction consequences for the production, con- ciation of the difference between theo- between the structural properties of a sumption, and pricing of each good. The ries that are empirically useful and those model (those depend on the structure model of Arrow and Debrieu is based on that are merely aesthetically pleasing. of institutions and interactions) vs. the four assumptions: Our hedge funds use completely differ- strategic properties (those that depend ent strategies. The strategy the econo- on the strategies of agents) and discuss 1. Perfect competition (price taking). mist developed uses equilibrium meth- problems in which structure dominates Each agent is small enough to have ods with behavioral modifications to (so equilibrium may not be very impor- a negligible effect on prices. Agents trade mortgage-backed securities; the tant). We also discuss how ideas from “take” the prices as given, in the sense strategy the physicist developed uses biology may be useful for understanding that they are offered prices at which time-series methods based on histori- markets. Finally in Section 9 we present they can buy or sell as much as they cal patterns to trade stocks, an approach some conclusions. want, but the agents cannot do any- that is in some sense the antithesis of thing to change the prices. the equilibrium method. Both strategies 2. Agent optimization of utility. At the have been highly successful. We initially given prices each agent indepen- came at the concept of equilibrium from 2. WHAT IS AN EQUILIBRIUM THEORY? dently chooses what to buy, what and very different points of view, one very In this section we explain the basic idea how to produce, and what to sell so supportive and the other very skepti- and the motivation for the neoclassi- as to maximize her utility. There is no cal. We have had many arguments over cal concept of economic equilibrium, free lunch—the cost of all purchases the last eight years. Surprisingly, we have and discuss some variations on the basic must be financed by sales. An agent more or less come to agree on the advan- theme. can only buy corn, for example, by tages and disadvantages of the equi- Market equilibrium models are selling an equal value of her labor, or librium approach and the need to go designed to describe the market interac- apples, or some other good from her beyond it. The view presented here is the tions of rational agents in an economy. endowment. result of this dialogue. The prototype is the general equilibrium 3. Market clearing. The aggregate This article is organized as follows: theory of Arrow and Debreu [1, 2]. The demand, i.e. the total quantity that In Section 2 we present a short qual- economy consists of a set of goods, such people wish to buy, must be equal itative review of what equilibrium the- as seed, corn, apples, and labor time, to the aggregate supply, i.e. the total ory is and what it is not, and in Section and a set of agents (households) who quantity that people wish to sell. 3 we discuss the related idea of mar- decide what to buy and sell, how and 4. Rational expectations. Agents make ket efficiency and its importance in what to produce, and how much to con- rational decisions based on perfect finance. In Section 4 we present what sume of each good. Agents are character- information. They have perfect mod- we think are the accomplishments and ized by their endowments, technologies, els of the world. They know the prices strengths of equilibrium theory, and in and utilities. The endowment of an agent of every good before they decide how Section 5 we discuss its inherent lim- is the set of all goods she inherits: for much to buy or sell of any good. itations from a theoretical perspective. example, her ability to labor, the apples In Section 6 we review the empirical on the trees in her backyard and so on. To actually use this theory one must evidence for and against equilibrium The technology of an agent is her col- of course make it concrete by specify- models. Then in Section 7 we develop lection of recipes for transforming goods ing the set of goods in the economy, how some further motivation for nonequilib- into others, like seed into corn. She might production depends on labor and other rium models, illustrated by a few prob- buy the seed and hire the labor to do the inputs, and a functional form for the lems that we think are inherently out job, or use her own seed and labor. Some utilities. General equilibrium theory was of equilibrium. We also compare equi- recipes are better than others, and not revolutionary because it provided clo- librium in physics and economics and all agents have access to the same tech- sure, giving a framework connecting the use nonequilibrium models in physics nologies.The utility of an agent describes different components of the economy to

12 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx each other and providing a minimum set there be an equilibrium? This is called in French) toward equilibrium. Unfortu- of assumptions necessary to get a solu- the existence problem. nately, after many years of analysis, it tion. The behavior of markets is an extra- In 1954 Arrow, Debreu, and McKen- became clear that in general this taton- ordinarily complicated and subtle phe- zie simultaneously showed that there nement can cycle without ever getting nomenon, but neoclassical economists always is an equilibrium, no matter close to equilibium. believe that most of it can be explained what the endowments and technologies on the basis of equilibrium. and utilities, provided that each utility 2.2.2. Omniscience The advent of general equilibrium displays diminishing marginal utility of An alternative is that the agents know the theory marked a major transition in the consumption and each technology dis- characteristics of their competitors and discipline of economics. It gave hope for plays diminishing marginal product. The know that they are rational and will not a quantitative explanation of the prop- margin is a very famous concept in eco- sell their goods at less than they could erties of the economy in one grand the- nomics that was discovered by Jevons, get elsewhere; if it is common knowl- ory, causing a sea change in the way Menger, andWalras in the so-called mar- edge that everybody is rational, and if economics is done and in the kind of ginal revolution of 1871. Apples display the characteristics are common knowl- people that do it. Mainstream econom- diminishing marginal utility if the more edge, then each agent will himself solve ics shifted from a largely qualitative pur- apples an agent consumes, the less addi- for equilibrium and the prices will simply suit, often called “political economy,” to tional happiness he achieves from one emerge. a highly mathematical field in which more apple. Diminishing marginal prod- This explanation for the emergence of articles are often published in theorem- uct means that the more a good like prices has an analogue in Nash equilib- proof format. labor is used in production, the less extra rium in game theory. A game is a con- We do not have the space to output comes from one additional unit test in which each player makes a move describe the many applications and con- (hour) of input. and receives a payoff that depends on sequences of equilibrium in any detail. The method of proof used by all this move as well as the moves of the The most important consequence for our these authors was first to guess a vec- other players. The question is how does a purposes is efficiency, which we shall tor of prices, one for each good, then player know what move to make without come to shortly. In the remainder of this to compute demand and supply at these knowing what moves his opponents will section we discuss some logical ques- prices, and then to increase the prices make? And how can he guess what they tions one might raise about whether for the goods with excess demand and to will do without knowing what they think equilibrium exists and how and whether decrease the prices of goods with excess he will do? This circular conundrum was it is attainable. Then we give a few supply. This defines a map from price resolved by Nash in 1951 who basically examples of how equilibrium models are vectors to price vectors. A price vector showed that there is always at least one being continuously extended to include generates an equilibrium if and only if self-consistent fixed point in the space more phenomena, and develop the con- it is mapped into itself by this map (for of strategies. To be more specific, a Nash cept of financial equilibrium, our central then there is no excess demand or excess equilibrium is an assignment of moves topic here. supply). By a clever use of Brouwer’s to each player such that if each player fixed point theorem, it was shown that thought the others would follow their there must always be a fixed point and assignments, then he would want to fol- 2.1. Existence of Equilibrium and therefore an equilibrium. low his. In game theory the question also Fixed Points arises: who announces this assignment? In equilibrium everybody’s plans are ful- 2.2. Getting to Equilibrium If the players are all perfectly logical, and filled. An agent who plans to buy three are aware of all their characteristics, they 2.2.1. Tatonnement ears of corn by selling two apples at the should deduce the assignment. given prices will find agents who want Who sets the equilibrium prices? How to sell him the three ears of corn and does the economy get to equilibrium? others who want to buy his two apples. Just because there is an equilibrium does 2.3. Financial Equilibrium Models A mother who goes to the store to buy not mean it is easy to find. One sug- 2.3.1. Time, Uncertainty, and her children bread will find as much as gestion of the marginalist Leon Walras Securities she wants to buy at the going prices on was that the equilibrium map could be One of the main limitations of general the shelves; yet at the end of the day iterated over and over. Starting from a equilibrium theory (as described so far) the store will not be left with extra bread guess, one could keep increasing and is that it is cast in a nontemporal set- that goes to waste. One might well won- decreasing prices depending on whether ting. There is only one time period, so der how this could happen without any there was too much demand or sup- agents don’t need to worry about the central coordinator. In short, why should ply. He called this groping (tatonnement future. They don’t look ahead and they

© 2008 Wiley Periodicals, Inc. COMPLEXITY 13 DOI 10.1002/cplx don’t speculate. This simplifies matters Financial models also extend the some good in some node in the finan- enormously but comes at the heavy cost notion of equilibrium by generalizing cial model must evaluate her utility in that the theory is powerless to deal with the notion of what can be traded to light of the decrease in the holding of temporal change. Another limitation to include securities, which are promises to some financial security that might then the theory so far is that agents only deliver certain goods at certain points in be required for her to balance her bud- trade goods. In the real world they trade time. Such promises can be contingent get, including the consequent loss of stocks and bonds and other financial on future states of nature. Examples are consumption in future states where that securities. Financial equilibrium extends stocks (which promise future dividends security would have paid dividends. general equilibrium to allow for time, that change depending on the profits 2.3.2. Financial Equilibrium uncertainty, and financial securities. of the firm), bonds (which promise the Uncertainty about the future is mod- same amount each period over a fixed Once the model is extended to allow for the tree of states of nature, and the trad- eled in terms of states of nature that maturity) and so on. The existence of ing of securities, we need to extend the unfold as the nodes in a tree [2, 3]. Each such securities is important because it notion of equilibrium. This is done by state represents the condition of things allows agents to decrease their risk. It following principles (1–4) above. Prices like the weather or political events whose also allows them to speculate in ways must now be given at every node of the future behavior is unknown, but exoge- that they might not otherwise be able tree. When agents optimize they must nous to the economy. States are given, to. The implication of such securities for choose an entire action plan contem- and while they can affect the economy, social welfare has proved to be highly plating what they will buy and sell at each the economy does not affect them. A controversial. node in the tree; part of their contem- change in the state of nature is often In the general equilibrium model plated trades will involve securities, and called a shock. The agents in the econ- each agent is aware of her total con- the other part commodities.2 Any pur- omy do not know the future states of sumption of every good, and makes her chase at any node must be financed by a nature, but they do know the tree from decisions in that context. That is why sale of equal value at the same node. The which the future states of nature are every decision is a marginal decision: sale could be of other goods, or it could drawn and the probability for reaching an increase in her consumption of one be of securities. The market for securi- each node. For example, suppose we good must be evaluated in light of the ties, as well as for goods, must clear at assume that the dividends of an agri- decrease in her consumption of some cultural company are made once a year every node in the tree. Finally, the ratio- other good that might be required to bal- and take on only one of two possible val- nal expectations hypothesis becomes ance her budget. In the financial equilib- ues, “high” or “low,” depending on the much more demanding: Agents are not rium extension we are describing, each weather that year. Then we can orga- only aware of the prices of all goods and agent is assumed to be aware of what nize these as a binary tree in which the securities today, but are also aware of her consumption will turn out to be at nth level with 2n nodes corresponds to how all the prices depend on each node every node, and to compute her utility the nth year in the future, and a path in the future. on that basis. One popular way to make from the beginning to the end of the tree that calculation is to evaluate the util- corresponds to a particular sequence of 2.4. Rational Expectations and Utility ity of consumption at any given node high and low dividends.1 If the situa- Maximization exactly as agents evaluated their utility tion was more complicated so that we Maximizing a utility function already of consumption in the general equilib- had to separate the weather in the west- embeds a large dose of rationality. The rium model (which is basically a spe- ern hemisphere from the eastern hemi- utility function is defined over the entire cial case of the financial model with sphere, then each node would have four consumption plan. The agents are also one node). One could then multiply the branches instead of two and there would presumed to know the tree of states of utility of consumption at a given node be 4n paths. The agents assign proba- nature, and the probabilities of every by the probability of ever reaching the bilities to each branch of the tree, giv- branch, and to correctly forecast the node. Finally, one could discount that ing a probability for reaching each node number again (i.e., multiply it by a fac- and thus a probability measure on the tor less than 1), depending on how far 2 terminal states in the tree. As the path of future states unfolds the away from the origin of the tree the agents will carry out the trades they antic- node is, to account for the impatience of ipated making on that path. Of course consumers. Summing over all the nodes only one path will eventually material- 1Financial models can also be formu- gives the expected discounted utility of ize, and the agents will never get a chance lated with continuous states and/or in consumption. An agent who contem- to implement the plans they had made for continuous time. plates increasing her consumption of unrealized branches.

14 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx prices that will emerge at each node if model to an infinite number of time peri- as productive as possible, or Pareto effi- they get there. So when an agent stops ods, calling it the overlapping genera- cient, i.e. that there is no change in pro- to buy an apple today, she does so only tions model, in order to study intergener- duction choices and trades that would after considering what other fruit she will ational issues like social security. In 2001 make everyone better off. In equilibrium likely have for dinner, how likely it is she Akerlof, Spence and Stiglitz received the everyone acts in his own selfish interest, will pass the same shop tomorrow, what Nobel prize for their work on asymmetric yet the decisions they make are in the else she could spend the money on the information, i.e. for studying situations common good in the sense that not even day after tomorrow, and so on. such as buying a used car, or negotia- a benevolent and wise dictator who told Not surprisingly, this rational utility tions between labor and management, everyone what to produce and what to maximization has received a great deal in which the information sets of agents trade could make all the agents better off. of attention and has been attacked from differ [12–14]. Another significant exten- In financial equilibrium the situation many different directions. In 2002 Kah- sion is to allow agents who sell secu- is much more complicated and interest- neman received a Nobel prize for show- rities to default on delivering the divi- ing. An important assumption underpin- ing that real people do not have simple dends those securities promise [15, 16]. ning much of the theory of financial eco- utility functions such as those that are This raises interesting problems such as nomics is that of complete markets. The typically assumed in standard economic how the market determines the amount markets are complete if at any node in theory, but instead have preferences that of collateral that must be put up for the tree it is possible to offset any future depend on context and even on fram- loans [17]. risk, defined as a particular future state, ing, i.e. on the way things are presented In view of all these extensions the by purchasing a security that pays if and [4]. The field of behavioral economics, meaning of the word “equilibrium” is not only if that state occurs. When all such which Kahneman exemplifies, investi- always clear. One definition would be securities exist so that markets are com- gates psychologically motivated modi- any model that could be interpreted as plete, financial equilibrium is necessarily fications of the rationality and utility satisfying hypotheses (1–4) as described Pareto efficient.3 assumptions. There are a range of levels in the introduction to this Section. But In contrast, when markets are at which one can do this. At the opposite we also wish to allow for some bound- not complete, financial equilibrium is extreme from rational expectations, one edly rational agents. Following the prac- almost never Pareto efficient. Worse still, can simply assume that agents have fixed tice of most economists, we will define the markets that do exist are not used beliefs, which might or might not corre- an equilibrium model as one in which properly. When markets are incomplete, spond to reality [5–7]. At an in-between at least some agents maximize prefer- a benevolent and wise dictator can level one can use a so-called noise trader ences and incorporate expectations in almost always make everyone better off model in which some agents have fixed a self-consistent manner. So, for exam- simply by taxing and subsidizing the beliefs while others are perfectly ratio- ple, a noise trader model is an equilib- existing security trades [21]. What form nal [8, 9] (This can actually make the rium model as long as some of the agents such government interventions should task of computing the equilibrium even are rational, but models in which all the take is a matter for economic policy. harder, since the rational agents have to agents act according to fixed beliefs are Is it a good approximation to assume have perfect models of the noise traders not equilibrium models. that real markets are complete? If they as well as of each other). At a higher are not complete, does the government level one can assume that agents are not have enough information to improve the 3. EFFICIENCY given the probabilities of states of nature a priori, but need to learn them [10, 11]. In equilibrium everyone is acting in his own selfish interest, guided only by the market prices, without any direction 3This was proved by Arrow [18] and from a central planner. One wonders if Debreu [19] under conditions (10-94), 2.5. Extensions of Equilibrium anything coherent can come from this including the rationality hypothesis on The rational expectations equilibrium decentralization of control. One might agents. Brock, Hommes and Wagener model that we have described so far is expect coordination failures and other have shown that when one deviates highly idealized, and in the next sections problems to arise. If activities were orga- from rational expectations, for example we will present a critique outlining its nized by rational cooperation, surely by assuming agents use reinforcement problems. The main agenda of cur- people could be made better off. Yet in learning, adding additional securities rent economic theory is to extend it by the general equilibrium model of Arrow can destabilize prices [20]. This suggests modifying or generalizing the assump- and Debreu, equilibrium allocations are that there are situations where mar- tions. For example, in 1958 Samuel- always allocatively efficient. Allocative ket completeness can actually decrease son extended the finite tree equilibrium efficiency means that the economy is utility.

© 2008 Wiley Periodicals, Inc. COMPLEXITY 15 DOI 10.1002/cplx situation? These questions have been a lived) asset must be equal to its funda- individual actions. Equilibrium theory is matter of considerable debate. mental value (also called present value if an agent based approach which does not Since allocative efficiency typically there is no uncertainty). The fundamen- admit any variables except those that fails for financial economies, economists tal value of an asset is the discounted can be explained in terms of individual turned to properties of security prices expected dividend payments of the asset choices. that must always hold. Informational over its entire life, where the expectation The advantages of the agent based efficiency is the property that secu- is calculated according to the martin- approach can be seen in the macroeco- rity prices are in some sense unpre- gale probabilities, and the discount is the nomic correlation between inflation and dictable. In its strongest form, informa- risk free rate (the interest rate on secu- output. At its face, such a correlation sug- tional efficiency is the simple statement rities such as treasury bonds for which gests that the monetary authority can that prices are a martingale [22, 23], i.e. the chance of default can be taken to stimulate increased output by engineer- that the current price is the best predic- be zero). This is important for investors ing higher inflation through printing tion of future prices.4 This is important who might feel too ignorant to buy money. According to equilibrium theory, because it suggests that all the informa- stocks, because it suggests that on aver- if inflation causes higher output, there tion held in the economy is revealed by age everybody will get what he pays for. must be an agent-based explanation. For the prices. If weather forecasts cannot From the point of view of finance, example, it may be that workers see their improve on today’s price in predicting arbitrage efficiency and informational wages going up and are tempted to work future orange prices, then one can say efficiency are particularly useful because harder, not realizing that the prices of the that today’s orange price already incor- they can be described without any goods they will want to buy are also going porates those forecasts, even if many explicit assumptions about utility. Many up so that there is no real incentive to traders are unaware of the forecasts.5 results in finance can be derived directly work harder. But if that is the explana- Arbitrage efficiency means that through from arbitrage efficiency. For exam- tion, then it becomes obvious that pol- buying and selling securities it is not ple, the Black-Scholes model for pric- icy makers will not be able to rely on possible for any trader to make profits ing options is perhaps the most famous the Phillips curve to stimulate output by in some state without taking any risk model in finance. The option price can printing more money. Agents will even- (i.e. without losing money in some other be derived from only two assumptions, tually catch on that when their wages are state). Again this is important because it namely that the price of the underlying rising, so are general prices.6 That skepti- suggests that there are no traders who asset is a random walk and that neither cism was validated during the stagflation can always beat the market by taking the person issuing the option nor the of the 1970s, when there was higher infla- advantage of less informed traders. Thus person buying it can make risk-free prof- tion and lower output. If printing money the uninformed should not feel afraid its. Because it does not rely on utility, sev- sometimes causes higher output, it must that they are not getting a fair deal. Infor- eral people have suggested that arbitrage be through a different channel, con- mational efficiency and arbitrage effi- efficiency forms a better basis for finan- nected say to interest rates and liquidity ciency are intimately related; in some cial economics than equilibrium; see for constraints. Finding what these are at contexts they are equivalent. example Ross [25]. As we will argue later, the agent level deepens the analysis and A consequence of the martingale because of all the problems with defining brings it closer to reality. property is that the price of every (finitely utility, this is a highly desirable feature. 4.2. The Need to Take Human Reasoning into Account 4More precisely, if no security pays a divi- 4. THE VIRTUES OF EQUILIBRIUM The fundamental difference between dend in period t +1, then there is a proba- economics and physics is that atoms bility measure on the branches out of any 4.1. Agent-Based Modeling don’t think. People are capable of rea- node in period t such that the expected Equilibrium theory focuses on individ- + soning and of making strategic plans price in period t 1 of any security (rela- ual actions and individual choices. By that take each other’s ability to reason tive to a fixed security), given its (relative) contrast Marx emphasized class strug- into account. Ultimately any economic price in period t, is just the period t price. gles, without asking whether each indi- model has to address this problem. By 5 vidual in a class would have the incentive Roll found in some Florida coun- going to a logical extreme, equilibrium to carry on the struggle. Similarly Key- ties that today’s orange prices predicted models provide a way to incorporate rea- nesian macroeconomics often posited tomorrow’s orange prices better than soning without having to confront the today’s weather reports, and also that reduced form relationships, such as a today’s orange prices predicted tomor- positive correlation between unemploy- row’s weather better than today’s weather ment and inflation (called the Phillips 6This has come to be called the Lucas forecasts [24]. curve), without deriving them from critique [26].

16 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx messiness of real human behavior. Even having to invent a new method to attack them from utility. Such a model is either though this approach is obviously unre- each problem, it provides a standardized silent on policy questions, or requires a alistic, the hope is that it may nonethe- approach, and a standardized language separate ad hoc notion of desireability in less be good enough for many purposes. in which to explain each conclusion. order to recommend one institution over A standardized model does not rule another. 4.3. Parsimony out new theories. New equilibrium theo- The rationality hypothesis allows us ries are of two forms. First, one can spe- to define equilibrium succinctly. But as 4.3.1. Rationality cialize the class of utilities, technologies, a byproduct it shapes the economist’s The rationality hypothesis is a par- and endowments to try to draw a sharp evaluation of the institutions. For exam- simonious description of the world and interesting conclusion that does not ple, one of the basic questions in eco- that makes strong predictions. Rational hold in general. If either the premise nomics is whether free markets orga- expectations equilibrium models have or conclusion can be empirically vali- nize efficient sharing of risks, leading to the advantage that agent expectations dated, one has found a law of economics. sensible production decisions. The equi- are derived from a single, simple and Second, one can extend the definition librium model is often used to show self-consistent assumption. Without this of equilibrium (adding say asymmetric that under certain conditions, free mar- assumption one needs to confront the information or default).The second form ket incentives lead selfish individuals to hard task of determining how agents facilitates the discovery of many the- make wise social decisions. We called actually think, and how they think about ories of the first form. Agreement on this allocative efficiency or Pareto effi- what others think. This requires formu- a unifying framework, such as equilib- ciency. But of course the rationality of lating a model of cognition or learn- rium and hypotheses (1–4), makes it eas- all the individuals is an indispensable ing, and thereby introducing additional ier to evaluate new economic theories, hypothesis in the proof. If agents are assumptions that are usually compli- and to make progress on applications. misinformed about future production cated and/or ad hoc. Without rationality But of course it stultifies radically new possibilities, or act whimsically, the free one needs either to introduce a set of approaches. market economy will obviously make behavioral rules of thumb or to intro- bad decisions. duce a learning model. While perfect Even if rational expectations doesn’t rationality defines a unique or nearly 4.4. The Normative Purpose of Economics provide a good model of the present, it unique model of the world, there are One of the principal differences between may sometimes provide a good model of an infinite number of boundedly ratio- physical and social science is that the the future. The introduction of an equi- nal models. To paraphrase Christopher laws of the physical world are fixed, librium model can change the future Sims, once we depart from perfect ratio- whereas the laws of society are mal- for the simple reason that once people nality, there are so many possible models leable. As human beings we have the better understand their optimal strate- it is easy to become lost in the wilderness capacity to change the world we live in. gies they may alter their behavior. A of bounded rationality [27]. Economics thus has a normative as well good example of this is the Black-Scholes Perfect rationality is an impossible as a descriptive purpose. A descriptive model; as people began using it to buy standard for any individual to attain, model describes the world as it is, while and sell mispriced options, the prices of but the hope of the economist in mak- a normative model describes the world options more closely matched its pre- ing a rational expectations model is that as it might be under a change in social dictions. Another example is the capi- in aggregate, people behave “as if” they institutions. The equilibrium model is tal asset pricing model. In that model were rational. This maybe true in some meant to allow us to describe both. For the optimal strategy for every investor situations, and it may not be true in oth- each arrangement of social institutions, is to hold a giant mutual fund of all ers. In any case, rational expectations such as those that prevail today, we com- stocks. Nowadays every investor who has models can provide a useful benchmark pute the equilibrium.The representation learned a little finance does indeed think for understanding whether or not peo- of agents by utilities (as well as endow- first of holding exactly such an index ple are actually rational, which can serve ments) enables us to evaluate the bene- (though few investors hold only that). as a starting point for more complicated fits to each person of the resulting equi- Unfortunately, many economists models that take bounded rationality librium, and thereby implicitly the utility have used the normative purpose of eco- into account. of the underlying institutions. Equilib- nomics as an excuse to construct eco- rium theory then recommends the insti- nomic theories that are so far removed 4.3.2. Succinctness tutions that lead to the highest utilities. from reality that they are useless. For an Equilibrium theory provides a unifying By contrast, consider a purely phenom- economic theory to have useful norma- framework from which many different enological model that assumes behav- tive value it must be sufficiently close conclusions can be drawn. Rather than ioral rules for the agents without deriving to reality to inspire confidence that its

© 2008 Wiley Periodicals, Inc. COMPLEXITY 17 DOI 10.1002/cplx conclusions give useful advice. Unless a a tree of possible future states. Wall trader might worry about future inter- theory can give some approximation of Street traders are trained in business est rates of all maturities, including the the world as it is, it is hard to have confi- schools and economics departments overnight (i.e one day) rate, the yearly dence in its predictions of how the world across the country to appreciate equi- rate, the ten year rate, and so on. If there might be. librium models. Nowadays so many Wall are 20 such rates, and each rate can take Street traders make conditional forecasts on 100 values, then each node in the 4.5. Why Does Wall Street Like the using concrete trees of future possibil- tree will require 10020 branches! Using Equilibrium Model? ities that this assumption has become the overnight rate alone requires a much Many Wall Street professionals try to more plausible as a descriptive model of smaller tree with only 100 branches from beat the market by actively looking the world. each node. The crucial point is that if for arbitrages. Fundamental analysts, One of the virtues of making con- the trader assumes there will never be for example, believe that they can find ditional forecasts is that it stimulates any arbitrage among interest rate securi- stocks whose prices differ substantially traders to find strategies (called riskless ties of different maturities in the future, from their fundamental values. Statis- arbitrages) that will work no matter what then the smaller tree will be sufficient to tical arbitrageurs believe they can find the future holds. If one is lucky enough to recover all the information in the larger information in previous stock move- find a riskless arbitrage, success is inde- tree even for a trader who cares about all ments that will help them predict the pendent of the probabilities of the future the rates. Given the smaller tree and the relative direction of future stock move- states of nature. In the real world risk- probabilities of each branch, the trader ments. The activities of these Wall Street less arbitrages are rare (as equilibrium can deduce all the other future long professionals seems proof that they theory says they should be). Most real rates, conditional on the future overnight think the equilibrium model is flawed. arbitrageurs actually make their money rates, without adding any new nodes to If they believe the market is in equilib- with risky arbitrages, and in this case her tree. (Today’s two day discount, for rium, why are they pursuing strategies the probabilities for branches of the example, can be deduced as the product that cannot succeed in equilibrium? tree become important. The estimation of today’s one day discount and tomor- Ironically, in fixed income and deriva- errors that are inherent in estimating row’s one day discount, averaged over all tive asset markets the arbitrageurs them- these probabilities compound the risk. possible values of the one day discount selves use the apparatus of equilibrium Not only might the future bring a state in tomorrow). Furthermore, if the trader models to find arbitrages. Many people which money will be lost, but the trader assumes there is no arbitrage between make good profits by betting that when might also underestimate the probabil- current long maturity interest rate secu- prices make large deviations from equi- ity of this state. But at least she can see rities and future short interest rate secu- librium these deviations will eventually the scenarios that constitute the model rities and current interest rate options, die out and return to equilibrium. explicitly, and can compute the sensitiv- then she can deduce the probabilities There are many waysWall Street prac- ity of her expected profits to variations in of each branch in the smaller overnight titioners use equilibrium models. We the probabilities of these scenarios. interest rate tree. review some of them below. The homage arbitrage traders pay to 4.5.2. The Power of the arbitrage efficiency is the most com- 4.5.1. Conditional Forecasting Is No-Arbitrage Hypothesis pelling evidence that in some respects it Good Discipline Tree building is most useful when the is true. Again we see the irony that the Many pundits, and even some pro- future possibilities are easy to imagine, road to arbitrage comes from assuming fessional economists (and especially when they are not too numerous to com- no arbitrage.7 macroeconomists), make unconditional pute, and when there is a reliable guide to forecasts. They say growth next quarter their probabilities. The hypothesis of no- will be slow, but by the fourth quar- arbitrage drastically reduces the num- ter it should pick up and unemploy- ber of states, making the computation 7The mortgage trader assumes no arbi- ment should start to decline. They do feasible, and often even determines the trage in the class of interest rate securi- not often bother to say, for example, probabilities. ties alone. By contrast, she does assume that their predictions might change if One reason that financial practition- there is an arbitrage involving interest there is a messy war in Iraq. In the ers like the no arbitrage assumption rate securities and mortgage securities. A equilibrium model agents do not know is because it makes things much sim- mortgage derivative trader might assume what state will prevail tomorrow, but pler. Arbitrage efficiency requires a con- there is no arbitrage involving interest they do know what prices will prevail sistency between prices that drastically rate securities and mortgage securities, conditional on tomorrow’s state. They reduces the size of the hypothetical but that there is an arbitrage once mort- make conditional forecasts, based on tree. For example, a mortgage derivative gage derivatives are added to the mix.

18 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx 4.5.3. Risk Reduction 5.1. Falsifiability? theory. All these factors make equilib- Hedging refers to the process of reduc- Empirical laws in economics are much rium theories difficult to test, and mean ing specific financial risks. To hedge the harder to find than in physics or the that many of the predictions are not as value of a particular security one cre- other natural sciences. Realistic experi- sharp as they seem at first. In many ates a combination of other securities, ments for the economy as a whole are situations, whether or not the predic- called a portfolio, that mimics the secu- nearly impossible to conduct (though for tions of equilibrium theory agree with rity that one wishes to hedge. By selling some small scale phenomena there is reality remains controversial, even after this combination of securities one can a thriving experimental research effort decades of debate. cancel or partially cancel the risk of buy- underway) and many causal variables Not only are there few sharp eco- ing the original security. Statistical arbi- are impossible to measure or even nomic predictions to test, but the trageurs, for example, try to make bets on to observe. Economists usually do not hypotheses of economic equilibrium the relative movements of stocks without dream of finding “constants of behav- also seem questionable, or hard to making bets on the overall movement of ior,” analogous to the constants of nature observe. A long-standing dispute is the stock market. The market risk can found in physics; when they do, they do whether utility is a reasonable founda- be hedged by making sure that the total not expect more than a couple of signifi- tion for economic theory.The concept of 8 position is market neutral, i.e. that under cant digits. We shall argue later that this utility as it is normally used in economic a movement of the market as a whole the may be a mistake, and that it is possible theory is purely qualitative. The func- value of the position is unaffected. This to find finely calibrated relationships in tional form of utility is generally cho- is typically done by maintaining con- economic data. sen for convenience, without any empir- straints on the overall position, and can Perhaps because of the difficulty of ical justification for choosing one form also be done by selling a security, such as empirical work, economic theory has over another. No one takes the functional a futures contract on a market index. emphasized understanding over pre- form and the parameters of utility func- To properly hedge it is necessary to dictability. For example, to economists, tions literally. This creates a vagueness have a set of scenarios for the future. equilibrium theory itself is most impor- in economic theory that remains in its Such scenarios can be represented by tant not for any empirical predictions, predictions. means of a tree, exactly as we described but for the paradoxical understanding Psychologists have generally con- for equilibrium models. Thus, the same it provides for markets: Without any cluded that utility is not a good way to structure we defined in order to discuss centralized coordination, all individual describe preferences, and have proposed equilibrium is central to hedging risks. plans can perfectly mesh; though every- alternatives, such as prospect theory [4]. Wall Street professionals are very con- one is perfectly selfish, their actions Some economists have taken this seri- cerned about risks, and for this reason promote the common good; though ously, as evidenced by the recent Nobel like the financial framework for under- everybody is spending lots of time hag- prize awarded to Kahneman. However, standing equilibrium. gling and negotiating over prices, their most theory is still built around con- Hedging and arbitrage are two sides behavior can be understood as if every- ventional utility functions, and so far of the same coin. In both hedging and body took all the prices as given and the alternatives are not well-integrated arbitrage, the trader looks for a portfolio immutable. into the mainstream. Attempts that have of securities that will pay off exactly what Equilibrium theory has not been been made to develop economic mod- the risky asset does. By buying the asset built with an eye exclusively focused els based on prospect theory still do not and selling the portfolio, the trader is on testability or on finding exact func- determine the parameters a priori (and completely hedged. If the trader discov- tional forms. Most equilibrium models in fact they have more free parameters). ers that the cost of the hedging portfolio are highly simplified at the sacrifice of Thus so far it is still not clear whether is more than the security, then hedging features such as temporal dynamics or more general notions of preferences can becomes arbitrage: The arbitrageur can the complexities of institutional struc- be used to make sharper and more quan- 9 buy the security and sell the portfolio ture. When it comes time to test the titative economic predictions. and lock in a riskless profit. model, it is usually necessary to make auxiliary assumptions that put in addi- 9 tional features that are outside of the One approach that has gained some 5. DIFFICULTIES AND LIMITATIONS OF attention is hyperbolic discounting [28]. EQUILIBRIUM Hyperbolic agents care a lot more about While equilibrium should be an impor- 8Okun’s Law for example, which states today than tomorrow, yet they act today tant component of the economist’s tool that every 1% increase in unemployment as if they will never care about the differ- kit, its dangers and limitations need to be is accompanied by a 3% decrease in GNP, ence between 30 days from today and 31 understood (and all too often are not). has one significant digit. days from today.

© 2008 Wiley Periodicals, Inc. COMPLEXITY 19 DOI 10.1002/cplx The other big problem that is intrinsic 5.3. Realism of the Rationality knowledge of all the other agents’ utili- to equilibrium theories concerns expec- Hypothesis? ties and endowments? These compu- tations about the probabilities of future It seems completely obvious that peo- tational problems can be intractable states. It is difficult to measure expecta- ple are not rational.The economic model even if all agents are fully rational, and tions. In practice we only observe a single of rationality not only requires them they can become even worse if some path through the tree of future states.The to be super-smart, it requires them to agents are rational and others aren’t. particular path that is observed histor- have God-like powers of omniscience. • Behavioral anomalies. There is perva- ically may be atypical, and may not be This should make anyone skeptical that sive evidence of irrationality in both a good indication of what agents really such models can ever describe the real psychological experiments and real believed when they made their deci- world. Rational expectations (i.e., know- world economic behavior. Even in sim- sions. Thus, even when we have a good ing the probabilities of each branch in ple situations where people should be historical record there is plenty of room the tree and knowing what the prices able to deal with the difficulties of to debate the conclusions. will be at each node) is often justified by computing an equilibrium it seems All these problems occur in testing the argument that people behave “as if” many do not do so [29, 30]. the assertion that markets are arbitrage they were rational. Skeptics do not find • Modeling cognitive cost and het- efficient, as we discuss later. Real arbi- such arguments convincing, particularly erogeneity. Real agents have highly trages are rarely risk free. Instead, they without supporting empirical evidence. diverse and context dependent involve risk, and determining whether There are many reasons to be suspicious notions of rationality. Because models one arbitrage is better than another of equilibrium models, and many situa- are expensive to create real agents involves measuring risk. Future risks tions where the cognitive tasks involved take shortcuts and ignore lots of infor- may not match historical risks. If a skill- are sufficiently complicated that the a mation. They use specialized strate- ful trader produces excess returns (above priori expectation that an equilibrium gies. The resulting set of decisions may and beyond the market as a whole) theory should work is not high. be far from the rationality supposed with a high level of statistical signifi- The conceptual problems with per- in equilibrium, even when taken in cance, a champion of market efficiency fect rationality can be broken down into aggregate. can always argue that this was possible several categories: only because the asset had “unobserved 5.4. Limited Scope risk factors.” What seems like a power- There are many interesting problems in ful scientific prediction from one point • Omniscience. To take each others’ expectations into account agents must financial economics that the equilibrium of view can look suspiciously like an framework was never intended to solve. unfalsifiable belief system from another. have an accurate model of each other, including the cognitive abilities, util- Under conditions that are rapidly evolv- ity, and information sets of all agents. ing, e.g. where there is insufficient time All agents construct and solve the same for agents to learn good models of the 5.2. Parsimony and Tractability Are Not tree. They must also agree on the prob- situation they are placed in, there is lit- the Same abilities of the branches. More real- tle reason to believe that the equilib- While we have argued that equilibrium istically one must allow for errors in rium framework describes the real world. models are parsimonious this does not model building, so that not all agents Some of the problems include necessarily mean that they are easy have the same estimates or even the to use. The mathematical machinery same tree of possibilities. • Evolution of knowledge. In a real mar- required to set up and solve an equilib- • Excessive cognitive demands. The cog- ket setting the framework from which rium model can be cumbersome. Find- nitive demands the equilibrium model we view the world is constantly and ing fixed points is much more diffi- places on its agents can be prepos- unpredictably evolving. The models cult than iterating dynamic maps. Thus terous in the sense that the calcu- used by traders to evaluate mortgage incorporating equilibrium into a model lations the agents need to make are securities in the 1980s were vastly more comes at a large cost, and may force extremely time consuming to perform. primitive than the models used to eval- one to neglect other factors, such as the Even given the tree and the prob- uate the same securities today. Peo- real structure of market institutions. As abilities and the conditional prices, ple have a difficult time imagining the a result of this complexity most equi- it may be difficult for an agent to possible ways in which knowledge will librium models end up being qualitative compute her optimal plan. But how evolve, much less assigning probabili- toy models, formulated over one or two does the agent know the conditional ties to all of its states. time periods, whose predictions are too prices unless she herself computes • Lack of tatonnement. The fact that an qualitative to be testable. the entire equilibrium based on her equilibrium exists does not necessarily

20 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx imply that it is stable. For stability it is this seems implausible—almost every- for the most part these laws bear a necessary to show that when a system one asks such questions. rather distant relation to empirical phe- starts out of equilibrium it will neces- nomena, and usually imply only qual- sarily move toward equilibrium. This 6. EMPIRICAL EVIDENCE FOR AND itative relations among observables .... requires the construction of a model AGAINST Thus, we know a great deal about the for price formation out of equilibrium, Equilibrium theory teaches us that direction of movement of one variable a process that is called tatonnement. expectations are important, and this is with the movement of another, a little Such models suggest that there are no less true in evaluating the success about the magnitudes of such move- many situations where prices will fail or failure of equilibrium theory itself. ment, and almost nothing about the to converge to equilibrium [31]. The booster can point to significant suc- functional forms of the underlying rela- • Inability to model deviations from cesses and the detractor can point to tions” [32]. Since they said this in 1977 itself. Equilibrium can’t model devia- significant failures. the situation has improved, but only a lit- tions from itself. It provides no way The booster can assert that equilib- tle. Apart from options and prime mort- to pose questions such as “How inef- rium theory has made many practical gage pricing (but not subprime), there ficient is the economy?”. One would suggestions that have been followed and are very few examples of economic the- like to be able to understand questions have been borne out as correct. Com- ories that cleanly fit the data and also such as the timescale for violations of munism did not produce as much out- unambiguously exclude nonequilibrium arbitrage efficiency to disappear, but put as the free markets of capitalism. alternatives. While some of the predic- such questions are inherently outside Diversifying your investments really is tions of equilibrium models are in quali- of the equilibrium framework. (See the a good idea, and many index funds tative agreement with the data, there are discussion in Section 7). have been created to accommodate this few that can be quantitatively verified in desire. A hedge fund that can show a really convincing manner. its returns are independent of much One of the conclusions of equilibrum While the financial implications of of the rest of the economy can attract theory is that relative prices should fol- equilibrium theory are on one hand investors even if it promises a lower low a martingale. It is well documented extraordinarily powerful, on the other rate of return than competitors whose that over the long-term risky securi- hand, their very power makes them returns are highly correlated with the ties like stocks have had higher returns almost empty.Economic theory says that market.10 Arbitrages are indeed hard to than safe securities like short term gov- there is very little to know about markets: find, and even harder to exploit. Stock ernment bonds, at least in the United An asset’s price is the best possible mea- 11 option prices are explained to a high States. This would seem to contradict sure of its fundamental value, and the degree of accuracy by the Black-Scholes the martingale pricing theorem. But the best predictor of future prices. If that is formula. Prepayment behavior for prime martingale pricing theorem says only true, there is no need to answer the kinds mortgages can be explained by maximiz- that asset prices follow a martingale with of questions that economists are asked ing the utility of individual households respect to some probability measure, not all the time, such as “Are stocks going up (with some allowances for inattention necessarily with respect to the objective or down?,” or“Is this a good time to invest and other irrational behaviors). probabilities. The skeptic naturally com- in real estate?.” Only by going outside the The skeptic counters that, with a few plains that this is giving the theory too equilibrium model can one raise many exceptions, most of the predictions of much freedom, making it close to tauto- pressing financial questions. equilibrium theory are qualitative and logical. Economists have responded by It is worth noting that the mere have not been strongly borne out empir- making more assumptions on the under- fact that people so persistently ask ically in an unambiguous manner. To lying utilities in the equilibrium model to such questions puts equilibrium theory quote Ijiri and Simon: “To be sure, eco- limit how the set of allowable martingale into doubt. There are three possibilities: nomics has evolved a highly sophisti- Either the people that do this are irra- cated body of mathematical laws, but tional, but if they are, then how can we 11Brown, Goetzmann and Ross point out expect equilibrium theory to describe that averaged over all western countries, them? Or they are rational, but if so, then 10Though diversification is often sighted stocks have not outperformed bonds [33]. their questions reflect that their under- as evidence for equilibrium theory, the An economic collapse, say during the standing is deeper than that of the equi- skeptic will point out that this only Russian revolution, usually crushes stock librium theorist. The only other expla- depends on portfolio theory, which is a prices more than bond prices. So the nation is that the people who ask such simple result from variational calculus American experience may not be repre- questions do not invest an economi- concerning statistical estimation and has sentative. Of course this illustrates the cally significant amount of money. But nothing to do with equilibrium theory. difficulty of testing the prediction.

© 2008 Wiley Periodicals, Inc. COMPLEXITY 21 DOI 10.1002/cplx probabilities can deviate from objective be disputed on the grounds that funda- While news arrival and market move- probabilities, making the theory falsifi- mental values are not easily measurable, ments are clearly correlated, the correla- able. The most famous model of that and perhaps none of the measures they tion is not nearly as strong as one would type is the so-called capital asset pricing use are correct (though they tried sev- expect based on rational expectations. As model (CAPM) developed by Markowitz, eral alternatives). But whether the theory asserted in the previous section, markets Tobin, Linter, and Sharpe [34–37]. In is wrong or whether it is merely unfal- appear to make their own news. One of one version of CAPM all utilities are sifiable, the failure to produce a good the problems in answering this question quadratic.12 Under this assumption the match to fundamental values represents empirically is that it is difficult to mea- martingale probabilities are not arbi- a major flaw in either case. sure information arrival—“news” con- trary, but must lie in a one dimensional Perhaps even worse, there are situa- tains a judgement about what is impor- set, making the theory more straight- tions where equilibrium makes predic- tant or not important, and is difficult forwardly testable. CAPM does explain tions that appear to be false. For exam- to objectively reduce to a quantitative why riskier securities should have higher ple, people seem to trade much more measure.14 returns than safe securities, and also pro- than they should in equilibrium [39]. vides a rigorous definition and quantifi- Global trading in financial markets is on 7. MOTIVATION FOR NONEQUILIBRIUM cation of the risk of a security. Data from the order of a hundred times as large MODELS: A FEW EXAMPLES the 1930s through the 1960s seemed to as global production [40, 41]. If people We begin by describing some empir- confirm the CAPM theory of pricing. are properly taking each others’ expecta- ical regularities in financial data that But little by little the empirical valid- tions into account why should they need seem salient yet which equilibrium the- ity of CAPM has unraveled. Of course to trade so much? One of the problems ory does not seem able to explain. Of there is always the possibility that a is that the theoretical predictions of how particular importance is price volatil- different specialization will give rise to much people should trade are not very ity. Not only do many people think that another model that matches the data sharp, due at least in part to the prob- prices change more than they should more closely. Or perhaps an extension of lems discussed in the previous section. under equilibrium, but also volatility the equilibrium model involving default Nonetheless, it seems that even under displays an interesting temporal corre- and collateral will be able to match the the most favorable set of assumptions lation structure that seems to be bet- data, without creating so many free para- people trade far more than can be ratio- ter explained by nonequilibrium mod- meters as to make the theory unfalsifi- nalized by an equilibrium model [42]. els. This correlation is an example of a 13 able. As already discussed in the previous power law, a functional form that seems A similar story can be told about section, one of the key principles of ratio- to underly many of the regularities in the conclusion that asset prices should nal expectations is that investors should financial economics. In physics power match fundamental values. Attempts correctly process information, and thus laws can’t be generated at equilibrium to independently compute fundamen- price movements should occur only in and are a signature of nonequilibrium tal values based on dividend series do response to new information. Studies behavior. Economic and physical equi- not match prices very well. For exam- that attempt to correlate news arrival librium are quite different, however, as ple, for the U.S. stock market Campbell with large market moves do not support and Shiller [38] show deviations between this. For example, Cutler et al. exam- 14 prices and fundamental values of more Engle and Rangel [45] demonstrate that ined the largest 100 daily price move- than a factor of two over periods as there is a positive correlation between ments in the S&P index during a 40 long as decades. This conclusion can low frequency price volatility and macro- year period and showed that most of economic fundamentals. Because their the largest movements occur on days results involve both a longtitudinal and 12 This is an example of the first kind where there is no discernable news, a cross-sectional regression involving of progress equilibrium theory makes, and conversely, days that are partic- both developed and undeveloped coun- in which specialized assumptions give ularly newsworthy are not particularly tries the interpretation of the correla- rise to more precise conclusions, as we likely to have large price movements tions that they observe is not obvious. discussed in Section 4.3.2. [43]. Other studies have shown that price No one disputes that prices respond to 13If successful, this would be an example volatility when markets are closed, even information—the question is,“What frac- of the second kind of progress equilib- on non-holidays, is much lower than tion of price movements can be explained rium theory makes, when the model is when the market is open [44]. The evi- by information?.” Timescale is clearly extended to incorporate new concepts,but dence seems to suggest that a substantial critical, i.e. one expects more correla- retains methodological premises 1-4, as fraction of price changes are driven by tion to information arrival at lower we discussed in Section 4.3.2. factors unrelated to information arrival. frequencies.

22 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx we try to make clear. Statistical testing for say that there is nothing to explain. If 7.2. Power Laws power laws is difficult and the existence the standard equilibrium theory is cor- As we already mentioned, clustered of power laws and their relevance for rect, changes in price are caused by the volatility and many other phenomena in economics have generated a great deal receipt of new information, so clustered economics are widely believed to display of debate, which we briefly review. volatility is simply a reflection of non- power laws. Loosely speaking, a power Next we observe that one of the most uniform information arrival. The rate of law is a relationship of the form y = frequently cited pieces of evidence for information arrival has to do with mete- kxα, where k and α are constants, that equilibrium is that arbitrage opportuni- orology, sociology, political science, etc. holds for asymptotic values of a vari- ties tend to disappear. There is substan- It is exogenous to financial econom- able x, such as x →∞[52, 53]. The tial evidence that real markets are effi- ics and so it is someone else’s job to first observation of a power law (in any cient at some level of approximation. But explain it, if such a question is even field) was made by Pareto, who claimed we present anecdotal evidence that effi- interesting. that this functional form fit the distri- ciencies disappear surprisingly slowly. An alternative point of view is that, bution of the incomes of the wealthi- We would like a theory of the transi- due to a lack of a perfect rationality, est people in many different countries tion to equilibrium, or of tatonnement the market is not in perfect equilibrium [54]. Power laws are reported to occur as it has sometimes been called. We end and thus prices do not simply passively in many aspects of financial economics, this section with a brief discussion of reflect new information. Under this view such as the distribution of large price one of the most remarkable episodes in the market acts as a nonlinear dynam- returns16 [47, 55–66], the size of trans- financial markets of the last ten years, ical system: agents process information actions [67, 68], the autocorrelation of which illustrates both the value and the via decision-making rules that respond volatility [69–74] the prices where orders limitations of the equilibrium model. to prices and other inputs, and prices are placed relative to the best prices are formed as a result of agent deci- [75–77], the autocorrelation of signs of 7.1. Some Economically Important sions. Since this information processing trading orders [78–81], the growth rate of Phenomena May Not Depend on Equilibrium is imperfect, the resulting feedback loop companies [82–84], and the scaling of the There are many empirically observed amplifies noise. As a result a significant price impact of trading with market cap- properties of markets that have so far component of volatility is generated by italization [85]. The empirical evidence not been explained under the equilib- the market itself. This point of view is for power laws in economics and their rium framework. One famous exam- supported by the fact that there are now relevance for economic theory remains ple is called clustered volatility. This many examples of nonequilibrium mod- controversial [86]. But there is a large lit- refers to the fact that there are sub- els that generate clustered volatility, and erature claiming that power laws exist stantial and strongly temporally corre- in some cases there is a good match to in economics and unless one is will- lated changes in the size of price move- empirical data. For many of these mod- ing to cast aside this entire literature, it ments at different points in time [46]. els it is possible to include equilibrium as seems a serious problem that there is no If prices move a lot on a Tuesday, they a special case; when this is done the over- explanation based on equilibrium. will probably move a lot the Wednesday all volatility level drops and clustered This problem might be resolved in one of several ways: Either as in physics, after, and probably (but not as likely) volatility disappears (see Section 8.3). 15 (1) power laws represent nonequilibrium move a lot on Thursday. The hard Another relevant point is that volatility is phenomena from an economic point of core rational expectations booster would widely believed to exhibit long-memory, view and thus provide a motivation for which we discuss shortly in the section nonequilibrium theory, or (2) power laws 15 on power laws. Mandelbrot and Clark suggest that it are consistent with economic equilib- Since the discussion of properties of is possible to describe clustered volatitlity rium, but still need to be explained by markets that are not currently explained as if time gradually speeds up and slows some other means, or (3) equilibrium is by equilibrium makes more sense in down, according to some random process a key concept in understanding power that is independent of the direction of the context of alternative models, we prices [47–49]. It has also been suggested will return to review this in more depth that the correct notion of trading time when we discuss nonequilibrium het- 16Under time varying volatility the dis- can be measured by either transaction erogeneous agent models in Section 8.3. tribution of returns can be interpreted as volume or frequency of transactions [50]; As we discuss in the next section, the a mixture of normal distributions with more careful analysis reveals that this questions of when equilibrium theory varying standard deviations. This gener- is not correct, and that fluctuations of is irrelevant or simply wrong can blur ically fattens the tails, and for the right volume are dominated by fluctuations in together. distribution of standard deviations can liquidity [51]. produce a power law.

© 2008 Wiley Periodicals, Inc. COMPLEXITY 23 DOI 10.1002/cplx laws but we just don’t know how to possible slowly varying functions; with 7.3. The Progression Toward Market do this yet.17 Or perhaps the exis- a finite amount of data it is only possi- Efficiency tence of power laws is just an illusion. ble to probe to a finite value of x and The theory of market (arbitrage) effi- But it seems that power laws fit the the slowly varying function may con- ciency is justified by the assertion that data to at least a reasonable degree of verge slowly. In practice one tends to if there were profit-making opportuni- approximation. Furthermore, there are look for a threshhold x beyond which ties in markets, they would be quickly many nonequilibrium models that nat- the points (log x, log y) almost lie on a found and exploited, and the resulting urally generate power laws, and for such straight line. (This can be made pre- trading activity would change prices in models it is easy to test that the phenom- cise using maximum likelihood estima- ways that would remove them. But is ena they generate are indeed real power tion, including the estimation of the this really true? Anecdotal empirical evi- laws. threshold x [90]). dence suggests otherwise, as illustrated It may seem strange to single out In the last 20 years or so the phe- in Figure 1, where we plot the per- a particular functional form for special nomenon of power laws has received a formance of two proprietary predictive attention. There are several reasons for great deal of attention in physics. Power trading signals developed by Prediction viewing power laws as important. One laws are special because they are self- Company (where one of the authors was of them comes from extreme value the- similar, i.e. under a scale change of the employed). The performance of the sig- ory [89], which says that in a certain dependent variable x = cx, treating K nals is measured in terms of their correla- sense there are only four possible con- as a constant, the function f (x) = cαf (x) tion to future price movements on a two vergent forms for the tail of a probability remains a power law with the same expo- week time scale over a twenty three year 18 19 distribution, and power laws are one nent α but a modified scale K  = cαK . period. In the case of signal 1 we see of them. The main motivation for look- a gradual degradation in performance One of the places where power laws are ing at tail distributions is that they pro- over the twenty three year period. The widespread is at phase transitions, such vide a clue to underlying mechanisms: correlation to future prices is variable but as the point where a liquid changes into a different mechanisms or behaviors tend the overall trend is clear. At the begin- solid. Self-similarity proved to be a pow- to generate different classes of tail ning of the period in 1975 it averages erful clue to understanding the physics distributions. about 14% and by 1998 it has declined of phase transitions and led to the devel- Statistically classifying a tail distribu- to roughly 4%. Nonetheless, even in 1998 opment of the renormalization group, tion is inherently difficult because one the signal remains strong enough to be which made it possible to compute the does not know a priori where the tail profitably tradeable. properties of phase transitions precisely. begins. In extreme value theory a power Does this evidence support market The main assumption of the renormal- law is more precisely stated as a rela- efficiency? On one hand the answer is ization group is that the physics is invari- tion of the form y = K (x)xα, where yes: The usefulness of the signal declines ant across different scales, i.e. the rea- K (x) is a slowly varying function. A with time. On the other hand it is no: It son the observed phenomena are power slowly varying function K (x) satisfies takes at least 23 years to do this, and at laws is because their underlying physics lim →∞ K (tx)/K (x) = 1, for every pos- the end of the period the opportunity for x is self-similar. This is an important clue itive constant t. There is a great variety of profits remains present, even if greatly about mechanism. diminished. The performance of signal 2 In physics power laws are associ- is even more surprising, and clearly does ated with nonequilibrium behavior and not support market efficiency. Due to a 17See articles by Calvet and Fisher for are viewed as a possible signature of structural change in the market, signal 2 examples where power laws seem to be nonequilibrium dynamics [52, 53, 91]. It begins in 1983 and subsequently grows consistent with equilibrium [87, 88]. was shown more than 100 years ago by in predictive power over the next decade 18 Boltzmann and Gibbs that in physical The four possible tail behaviors of and a half. This is the opposite of what probability distributions correspond to equilibrium, energies obey an exponen- distributions with finite support; thin tial distribution; thus a power law indi- tailed distributions such as Gaussians, cates nonequilibrium behavior. As dis- 19The signals were developed based on exponentials or log-normals; distribu- cussed in Section 7.4, equilibrium in data from 1991 to 1998. Afterward data tions where there is a critical cutoff physics is quite different from equilib- from 1975 to 1990 was acquired and the above which moments don’t exist (namely rium in economics, but it seems telling model was tested on this data without power laws); and distributions that lack that the current set of financial models in alteration. Thus the earlier data is com- any regular tail behavior at all. Almost all which power laws appear are (economic) pletely out of sample and the enhanced commonly used distributions are in one of nonequilibrium models, as discussed in performance prior to 1991 cannot be due the first three categories. Section 8.3. to overfitting.

24 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx FIGURE 1 normally a prerequisite to raising capital, takes another four years. We thus expect that the typical time scale before such strategies become profitable should be at least eight years.21 One also has to address the question of how prices are altered and what is required to make the inefficiency disappear. As investors start to recognize the opportunity and exploit it they will make it shrink (say by bid- ding up the price of the mortgage backed security) and thus make it harder for oth- ers to see, extending the time until its superior profitability is completely extin- guished [92]. In any case this argument at least indicates that we should not be surprised that inefficiencies disappear The inefficiency of the market with respect to two financial strategies over a 23-year period.The y slowly. axis plots the historical performance for a backtest of two trading signals developed by Prediction As originally pointed out by Milton Company. Performance is measured in terms of the correlation of each signal to the movement of Friedman, the idea of a fully efficient stock prices two weeks in the future. market is inherently contradictory. To remove market inefficiencies we must have traders who are motivated to exploit them. But if the market is perfectly effi- cient there is no possibility to make modeling. In this case one can approxi- standard dogma about efficient markets excess profits. While efficiency might be mate the rate at which the strategy can would suggest. true at first order, it cannot be true at sec- be discovered and exploited based on The need for a nonequilibrium the- ond order; there must be on-going vio- its Sharpe ratio S, which is defined as ory is apparent. Equilibrium theory pre- lations of efficiency that are sufficiently the ratio of the expected excess return dicts that markets are perfectly efficient, large to keep traders motivated. Devel- of the strategy compared to its risk as and thus violations of efficiency can- oping a theory capable of quantitatively measured by the standard deviation of not be addressed without going outside explaining this is a major challenge for its profits. Under the optimistic assump- it. The time scale for the degradation economics, and a significant motivation tion that excess returns are normally dis- of a profitable strategy is an inherently for developing a nonequilibrium theory. tributed, the statistical significance with disequilibium phenomenon. But what about signal 2 in Figure 1? In which an inefficiency with Sharpe ratio S An estimate of the time scale for √ this case, the inefficiency actually grows can be detected is roughly S t, where the progression toward efficiency can be for a period of more than a decade. We t is the time the inefficiency has been made by taking into account what is believe this is an example of a structural in existence. On an annual time scale needed to learn a good model and to change. In fact the information available a trading strategy with a Sharpe ratio acquire the capital to exploit it. Assume for constructing the signal was not even of one is considered highly desirable.20 that there is a structural change in the available prior to 1983, and only became market, suddenly creating a new ineffi- Thus to detect the existence of an inef- ciency. For example, suppose that lots ficiency at two standard deviations of of mortgages are pooled together and significance requires four years. Accu- 21There is always the possibility of con- securitized for the first time. On account mulating a statistically significant track ducting more sophisticated research to of lack of historical data or irrational record based on live trading, which is speed up the learning time. Instead of fear (say of adverse selection), the secu- observing the returns of the securitized rity might sell for a very low price. How MBS, one could find historical records of long will this inefficiency take to dis- 20The historical Sharpe ratio of the S&P individual loan performance from before appear? Assume also that the only pos- index is about 0.4, and the Sharpe ratio of the securitization. But this data is hard sible strategies for removing the inef- many famous investors, such as Warren to get, and the partial data that can be ficiency are econometric, i.e. they are Buffet or George Soros, is significantly less obtained might not be selected in the based purely on statistical time series than one. same way the securitized loans were.

© 2008 Wiley Periodicals, Inc. COMPLEXITY 25 DOI 10.1002/cplx available after a rule change in report- similarities. Both involve the simplify- detailed models it is possible to estimate ing. There are many examples of sud- ing assumption that the basic set up of the parameters of the diffusion process, den structural changes in markets, for the underlying situation is not chang- so that we can roughly estimate when example the introduction of a new deriv- ing in time. In economic equilibrium this the beer will become too warm, even if ative such as a mortgage-backed secu- is implicit in the sense that in order to we have no previous experience drink- rity, a market for a new type of good imagine that rationality is a reasonable ing beer in warm rooms.22 Thus we see such as an internet company, or the assumption all agents must have time to that in physics, equilibrium theory only introduction of a new technology such make good models of the world. In prac- explains part of what we need to know— as computers that enables new finan- tice good models cannot be built with- to fully understand what is going on we cial strategies such as program trading. out a learning period, in a setting that need a more detailed nonequilibrium When the ongoing introduction of finan- is reasonably constant. In physics, equi- theory that can explain the dynamics cial innovations is combined with the librium explicitly requires time indepen- of the transitions between equilibrium slow time scale for discovering such sig- dence. Equilibrium in economics allows states. One important advantage of a nals, perhaps the persistent existence of explicit time dependences, but requires nonequilibrium theory is that it makes it inefficiencies is not so surprising. that agents have a full understanding of possible to know when the equilibrium them. theory is valid. To illustrate how nonequilibrium The fact that we have no similar dise- models are used in physics, suppose you 7.4. Equilibrium and Nonequilibrium in quilibrium theory in economics is clearly put a can of cold beer in a warm room. Physics and Economics a serious problem. If there were such a Heat initially flows from the room into theory, we would know when equilib- To understand the motivations for non- the beer—the system is out of equilib- rium applies and when it doesn’t. There equilibrium models, and to give some rium. With time the beer gets warmer, have been many attempts to create such insight into how this might be use- and eventually to a good approximation a theory. Examples can be found in ful in economics, we make a compari- the beer comes to room temperature and Franklin Fisher’s book, Disequilibrium son with physics. The very name “non- the system is at equilibrium. foundations of equilibrium economics equilibrium” indicates something that is Equilibrium thermodynamics tells us [31]. This book reviews a research pro- defined in terms of its opposite, so before some useful things, such as the change gram involving many prominent econ- we can discuss nonequilibrium models in the pressure inside the can of beer omists that attempted to make models in physics we have to make it clear what between the time it is taken out of of the approach to equilibrium under equilibrium means and how it differs the refrigerator and when it comes to the Walrasian framework. They focused from the same concept in economics. room temperature. There are, however, on the problem of tatonnement, i.e. the As in economics, the word equi- many things it doesn’t tell us, such as process by which prices arrive at their librium in physics has several mean- how long will it take for the beer to equilibrium values. This body of theory ings. In mechanics the word “equilib- get warm (The analogy to the prob- was not very successful. In retrospect the rium” sometimes denotes the fairly triv- lem of inefficiencies in the previous difficulty was that the models required ial notion that forces are balanced, so strong and detailed assumptions about there is no acceleration. The more com- section should now be clear). From an the market mechanism and the conclu- mon use of the term is in thermodynam- empirical point of view, anyone who sions that could be derived were not very ics or statistical mechanics, where the drinks beer knows that on a hot day it general. This serves as a cautionary tale. word “thermal equilibrium” means that will become unpleasantly warm in less The hope for a new attempt is that we two systems in thermal contact reach the than an hour. A more quantitative, first have new tools (the computer), much same temperature and cease to exchange principles prediction is possible using better data, and perhaps a few new ideas. energy by heat. A system in equilibrium Fourier’s law, which is a nonequilibrium Another often-used point of com- will experience no change when isolated principle stating that the rate of change parison between equilibrium in physics from its surroundings, and exists at the in temperature is proportional to the and economics concerns power laws. minimum of a thermodynamic poten- temperature difference between the beer tial. This is typically a good approxima- and the room. This implies that the tem- tion only when a system has been left perature of the beer converges to that 22This is not at all trivial. The rate of heat undisturbed for a sufficiently long period of the room exponentially. This can be transfer depends on the heat conductiv- of time. derived from the heat equation (which is ity of the can, on the convection prop- While the notion of equilibrium in a standard diffusion equation and would erties of air, and several other factors. It physics is fundamentally different from look very familiar to financial econo- is nonetheless possible to make estimates that in economics, it does have some mists). Furthermore, by making more based on first principles.

26 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx As already mentioned, physical equilib- • A drop in housing prices made it clear This crisis involves some phenomena rium does not yield power laws. This that many sub-prime borrowers would that are inherently out of equilibrium. At can be explained fairly trivially: Under eventually have to default. the same time it shows the relevance of the Jaynes formulation of statistical • This stressed capital providers and the equilibrium model. mechanics, deducing the correct prob- caused them to tighten credit, for The crisis was originally precipitated ability distribution of states is just a mat- example by demanding more collat- by levels of credit that are difficult to ter of maximizing entropy subject to any eral for the same loans. One effect was justify as rational. Homeowners got sev- imposed physical constraints, such as that subprime borrowers themselves eral hundred thousand dollar loans with those that come from conservation laws. found it harder to refinance, which no money down (i.e. the loan was equal So, for example, for a closed system with paradoxically created more defaults to the appraisal value of the house), a fixed number of particles, conserva- and higher losses. and representations about income were tion of energy is just a constraint on the • Another effect is that the prices of com- taken at face value (i.e. the income was mean energy, and maximizing entropy panies that depend on credit to do not verified, but only stated by the home- subject to this constraint yields an expo- business were depressed. owner himself). While housing prices nential distribution of energies, which is • Institutions (including the largest were rising, this caused no trouble. If not a power law. One can also work this banks in the United States) hold- the homeowner did not pay, foreclo- out in more general settings, e.g. when ing AAA rated securities called CDOs sure could bring enough money to make the energy or the number of particles is backed by bonds which in turn were the loan good, or more common, its not conserved, and a similar answer is backed by sub-prime mortgages lost threat could induce the homeowner to obtained. It is not clear that this pertains something like $100 or $200 bil- refinance into another loan. As housing in economics. For example, doing the lion. Curiously these banks announced prices started to decline, homeowners same calculation under a constraint on their losses piecemeal. One week a who in fact had no income could not the mean logarithm yields a power law; if bank would announce $10 billion of pay, and foreclosures are not expected there were economic systems where this losses, the next week they would add to yield nearly enough cash to repay the constraint were natural (e.g., because of another $10 billion, then two weeks loans. the use of logarithmic utility), power laws later another $10 billion. As of this writing, the crisis is might emerge naturally. And as we have • In mid August 2007, in the midst of the based entirely on expectations, includ- already stressed, in any case the notions sub-prime mortgage crisis, there was ing expections about future home prices. of equilibrium in economics and physics a sudden meltdown of many statisti- The actual realized losses from foreclo- are quite different. cal arbitrage hedge funds trading equi- sures so far have only been around 1%. ties. Stocks held by these hedge funds The $250 billion loss that the banks and 7.5. The Sub-Prime Mortgage Crisis made unusual but systematic relative hedge funds have been forced to take is of 2007 movements—stocks owned by these based on the expectation that eventually, As we are writing this in the late sum- hedge funds dropped in value, while in several years, 50% of the homeowners mer of 2007, a financial crisis is unfolding those shorted by these funds rose in (2.5 million families) will be thrown out 23 that provides a good illustration of the value. of their houses with losses of 50 cents on One popular explanation is that the 24 The value of equilibrium models, at the same • the dollar for each foreclosed loan. institutions losing money in sub- equilibrium model is thus right to put time that it illustrates the need for non- prime mortgages sold positions in so much emphasis on the importance of equilibrium models. The crisis began in other markets, including stocks held by expectations. We have yet to see whether sub-prime mortgages and appears to stat arb hedge funds. This caused the these expectations are rational. be spreading to seemiingly unrelated stat arb hedge funds to take losses and One important contributing factor is sectors. This will take some time to to liquidate more stocks, causing more that collateral levels have changed dra- understand completely, but in cartoon losses and more liquidations. matically. Before the crisis, one could form the crisis unfolded something like The President announced that the cri- get a sub-prime loan with almost no this: • sis might be leading the country into a money down. As the crisis developed, the recession. required downpayment jumped to 25%. • Low interest rates, a rising housing Since most sub-prime borrowers have market, and confidence in the risk- reducing power of mortgage-backed securities led to a proliferation of sub- 23This is based on anecdotal conversa- 24There is $1 trillion in outstanding sub- prime mortgages, often under ques- tions with hedge fund managers. See also prime loans. Losses of 50% × 50% × tionable terms. [93]. $1trillion = $250 billion.

© 2008 Wiley Periodicals, Inc. COMPLEXITY 27 DOI 10.1002/cplx little free cash, this meant that effec- loans were sold and repackaged into loans, equilibrium determines the col- tively the sub-prime mortgage market bonds that were in turn sold to hedge lateral levels that will be used for dried up overnight. In the past, subprime funds and other investors. Since the bro- promises (equivalently the leverage bor- borrowers who made 36 straight pay- kers knew they would not bear the losses rowers will choose).The kind of bad news ments, demonstrating they deserved a from defaulted loans, they evidently did that increases uncertainty also increases better credit rating, would refinance into not have enough incentive to check the equilibrium collateral levels. In normal cheaper prime mortgages with nothing truthfulness of the borrowers. And the times these collateral levels are too lax, down. The refinancing rate was over 70% investors apparently did not monitor the and in crisis times they are too tight (see cumulatively for these people. This is brokers carefully enough, perhaps lulled refs. [17, 94]). important because the subprime inter- by the good behavior of the loans dur- est rates typically reset to higher rates ing the period of housing appreciation. 8. IF NOT EQUILIBRIUM, THEN WHAT? after 36 months. Now these same peo- There was a breakdown of rationality. Ken Arrow began a lecture he gave at the ple are being asked to put 25% down The next incredible blunder was that Santa Fe Institute a few years ago by say- to refinance into a new mortgage. Many the big banks bought so many collateral- ing that “economics is in chaos.” In say- of these homeowners will not be able ized debt obligations (CDOs). To be sure, ingthishedrewacontrasttothesituation to refinance, and they will eventually they only bought those securities in the in say, 1970, when rational expectations- default, which will force further foreclo- CDOs that were rated AAA by the rating based equilibrium theories were produc- sures, further depressing housing prices. agencies. But when they do a deal the ing many new results and it looked as The collateral tightening is spread- rating agencies are paid a commission though they might be able to solve many ing throughout the mortgage market. for rating the securities and if they do of the major problems in economics. At Multibillion dollar hedge funds now have not approve enough AAA securities the the time the path for a young theoret- to put up twice as much cash to buy deal does not get done and they get no ical economist was clear. Now, in con- mortgage derivatives as they had three money. The rating agencies do not own trast, this is up in the air.While the major- months before (that is, instead of bor- the securities themselves, and so they do ity of economists still use rational expec- rowing 90% of the price using a mortgage not have the best incentive to get their tations as the foundation for what they derivative as collateral, they can only ratings right. It is amazing that the banks do, even the mainstream is investigat- borrow 80%). did not take this into account and naively ing perturbations of this foundation, and A striking feature of the crisis is bought the AAA securities thinking they some are seeking an entirely new foun- that the situation evolved rapidly were truly safe. dation. In this chapter we will sketch and appeared to be driven partly by Though many aspects of this cri- some possible directions. Many of these emotion—the word “fear,” which is not sis involve out-of-equilibrium phenom- approaches are overlapping and we try to an equilibrium concept, appeared in ena like irrational optimism, equilibrium stress the connections between them. almost every newspaper article covering theory can still be very insightful in We are not suggesting that equilib- these events. understanding what happened. In fact, rium theory should be thrown out: as The crisis underlined the role of spe- it partially predicted the crisis. emphasized earlier, there are many sit- cialized players: banks, mortgage backed In the first place the brokers who uations where it is extremely useful. derivatives experts, stock funds using winked at bad loans while collecting What we do argue is that it should be valuation models, and quants build- their commisions, and the rating agen- one among many tools in the econo- mist’s toolbox. During the rise of the ing statistical arbitrage strategies. Key cies who gave high grades to bad bonds neoclassical paradigm over the last fifty aspects are the heterogeneous nature while collecting their commisions, were years the emphasis on equilibrium the- of the strategies and their linkages to both responding to their incentives, just ory has been so strong that most theo- each other through institutional group- as equilibrium theory would predict, and retical economists have been trained in ings and cross-market exposure. as some economists had predicted. little else.25 As a result economics has Is the crisis a simple consequence But more importantly, the swift tran- suffered. of irrationality, completely orthogonal to sition from lax collateral levels permit- the equilibrium model? It is not pos- ting too many risky loans to overly tight sible that lenders were so foolish that collateral levels strangling the sub-prime 25The obvious exception is econometrics, they did not imagine the possibility market is typical of the so-called lever- which is an essential part of economics, that housing prices might fall. But they age cycle in equilibrium theory. Once the and should remain so. But econometric apparently underestimated the amount equilibrium model is extended to allow modelsarenotfoundedonapriorimodels of fraud in the loans. The brokers arrang- for default on promises, and to allow of agent behavior, and are not theories in ing the loans did not own them. The for the posting of collateral to guarantee the sense that we are using the word here.

28 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx Disequilibrium models are not new in a good job of documenting the many its structure. In economics this occurs economics. Keynes, for example, made ways in which real investors are not ratio- when equilibrium plays a role that is models that were essentially dynami- nal, and has shown that this has impor- minor compared to other factors, such as cal, without imposing equilibrium con- tant financial consequences. Although interaction dynamics or budgetary con- ditions. These models fell out of favor there has been some progress in under- straints. We give several examples below. for the good reason that many of them standing the implications of these facts The substantial effort needed to capture failed. In the 1950s and 1960s Keynesians for classical problems such as saving and the strategic aspects of a problem in an accepted the inflation-output trade-off asset pricing [28, 97], the jury is still out equilibrium model can often cause the we mentioned earlier, but government concerning whether this can be done in structural aspects of a problem to be efforts to exploit that relationship col- a fully quantitative manner. short-changed. In some cases the impor- lapsed completely during the stagflation One must also worry that behav- tance of structural factors may domi- of the 1970s (when there was both high ioral finance might be the victim of nate over strategic considerations, and inflation and low output). These failures its own success. If we discover new models that place too much emphasis were driving forces in the modern equi- empirical rules for the irrational com- on strategic interactions without proper librium approach to macroeconomics, ponent of human behavior, might not emphasis on structural properties will led by Lucas. It is clear that for many these rules be violated as people become simply fail to capture the essence of the problems in economics the ability to aware of them? For example, it has been problem. Of course in general one must incorporate human reasoning is neces- widely shown that people are strongly take both into account. sary. However, there may be other ways prone to overconfidence. If this knowl- to accomplish this goal, and there may edge becomes widespread, will savvy 8.2.1. Examples Where Structure also be other important economic prob- investors learn to compensate? In finan- Dominates Strategy lems where human reasoning is not the cial markets there is a lot of money on the As an example of what we mean, con- central issue. table and the motivation for overcoming sider traffic. Cars are driven by people, irrational behavior is large. Of course, if who anticipate the decisions of them- this were to occur that would already be 8.1. Behavioral and Experimental Finance selves and others. Yet an examination of a major achievement for the field. the literature on traffic and traffic con- As already stressed in Section 5.3, one trol suggests that game theory and equi- of the principal problems with rational librium play only a minor role. Instead expectations equilibrium is the lack of 8.2. Structure vs. Strategy the dominant theories look more like realism of the agent model. The impor- In comparing theoretical economists to physics [99, 100]. Traffic is modeled by tance of work in this direction was theoretical biologists, Paul Krugman has analogy with fluid flow. The road is a acknowledged by the mainstream with commented that the two are not so dif- structural constraint and cars are inter- the Nobel prizes of Daniel Kahneman, ferent as it might seem: both make exten- acting particles. Appropriately modified who is one of the pioneers behavioral sive use of game theoretic models [98]. versions of the same ideas that are used economics, and Vernon Smith, who is This observation is true, but it obscures to understand the phases of matter in one of the pioneers of experimental eco- a very important matter of degree. In physics are used to understand traffic: A nomics. Because these important fields biology game theory models are one of road with only a few cars is like a gas, a are already well-covered in many review many approaches used by theoreticians. crowded road where traffic is still flowing articles [29, 30, 95, 96] we will not review In economics, game theory models and is like a liquid, and a traffic jam is like a them here, but rather only make a few their corollary form, equilibrium mod- solid. In the liquid phase perturbations comments. els, are almost the only approach. From can propagate and induce traffic jams; We think that a proper characteriza- a certain point of view this is natural— the equations that are used to under- tion of human behavior is an essential the need to take into account strategic stand this are familiar to fluid dynami- part of the future of economics. This is interactions in economics is larger than cists.26 This is not because equilibrium not going to be easy; people are compli- it is in biology.The problem is that strate- cated and their behavior is malleable and gic interactions are not the only impor- context dependent. Experimental eco- tant factor in economic models. Other nomics offers hope for categorizing the factors can also be important, such as 26In the medium density range, when spectrum of human behavior and pre- the nature of economic institutions, and drivers have delayed reactions or over- dicting how people will behave in given how the interactions of agents aggre- reactions to random variations in the situations, but the results are difficult gate to generate economic phenomena traffic flow, a breakdown in traffic flow to reduce to quantitative mathematical at higher levels.We will call the aspects of is caused by a dynamic instability of the form. So far behavioral finance has done a problem that do not depend on strategy stationary and homogeneous solution.

© 2008 Wiley Periodicals, Inc. COMPLEXITY 29 DOI 10.1002/cplx is not relevant, but rather because dri- The crowd dynamics amplify small per- 8.2.3. Zero Intelligence Models ving is tightly constrained by roads, traf- turbations and create pressure fluctua- of Auctions fic control, driving habits, and response tions that caused people to be crushed. The continuous double auction, which is times. The strategic considerations are The insights that were obtained by the perhaps the most commonly used price fairly simple and their importance is study of the Hajj data led to several mod- formation mechanism in modern finan- dominated by the mechanics of particles ifications in the methods of crowd con- cial markets, provides a nice example of (automobiles) moving under structural trol, and to a safe Hajj pilgrimage the how in some cases it can be very use- constraints. following year. ful to focus attention on structural rather The same holds true for crowd The skeptical economist might than strategic properties. In a continu- dynamics, where game theoretic notions respond that these are not economic ous double auction buyers and sellers are of equilibrium would also seem to be problems, and just because physical allowed to place or cancel trading orders important. In moving through a crowd models such as fluid dynamics apply whenever they like, at the prices of their one tries to avoid collisions with one’s to crowd behavior does not imply that choosing. If an order to buy crosses a pre- neighbors, while trying to reach a given they should be useful in economics. We existing order to sell, or vice versa, there goal, such as a subway entrance. One’s included the examples above because is a transaction. The way in which orders neighbors are thinking about the same they provide clear examples of situations are placed is obviously closely related thing. There are lots of strategic interac- that involve people, where despite the to properties of prices, such as volatility tions and it would seem that game theory presence of strategic interactions, their and the spread (the price gap between should be the key modeling principle. In ability to reason is not a dominant point the best standing sell order and the best fact this does not seem to be the most in understanding their aggregate behav- standing buy order). There have been important consideration for modeling ior. We now discuss a few problems in many attempts to model the continuous the movement of a crowd. economics where similar considerations double auction using equilibrium the- The problem of crowd dynam- apply. ory. One famous example is the Glosten ics becomes particularly important in model [114], which predicts the expected extreme situations when the density of equilibrium distribution of orders under people is high. For example, on Janu- 8.2.2. Distribution of Wealth a single period model.Testsof this model ary 12, 2006 roughly three million Mus- and Firm Size show that it does not match reality very lims participated in the Hajj to perform well [115]. As originally observed by Pareto, the dis- the stoning ritual. There was a panic In contrast, an alternative approach tribution of income displays robust reg- and several hundred people died as a to this problem assumes a zero intelli- ularities that are persistent across dif- result of either being crushed or tram- gence model.28 This approach was orig- ferent countries and through time [54]. pled to death. In an effort to under- inally pioneered by Becker [116], who For low to medium income it has a stand why this occured the output of showed that some aspects of supply and functional form that has been variously video cameras was processed in order demand curves could be understood described as exponential or log-normal to reduce the crowd dynamics to a set without any reliance on strategic think- [102, 103], but for very high incomes of quantitative measurements. Analysis ing. The notion that a zero intelligence it is better approximated by a power by Helbing et al. [101] showed that high model might be useful for understand- law. Since the early efforts of Cham- densities of people facilitate the forma- ing economic phenomena was further pernowne, Simon, and others, the most tion of density waves, similar to those developed by Gode and Sunder [117], 27 successful theories for explaining this that occur in driven granular media. who popularized this phrase and argued have been random process models for that observations of efficiency in class- the acquisition and transfer of wealth room experiments on auctions could [102,104–108]. If these theories are right, be explained without relying on agent then the distribution of wealth, which 27During a New Year’s Eve celebration is one of the most remarkable and per- at Trafalgar Square in London in 1980 sistent properties of the economy, has 28“Zero intelligence”loosely refers to mod- one of the authors (jdf) personally expe- little to do with the principles of equilib- els in which the agents have minimal rienced the propagation of density waves rium theory, and indeed little to do with cognitive capacity. It should more accu- through a large crowd. The density waves human cognition. Other problems with rately be properly called “ intelligence”, propagate remarkably slowly. Their prop- a similar flavor include the distribution since often agents do have some cognitive agation speed can be crudely estimated of the sizes of firms, the size of cities, and ability, e.g. the capacity to draw from a by regarding each person as an inverted the frequency with which words are used known distribution or to enforce a budget pendulum. [32, 105, 109–113]. constraint.

30 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx intelligence. Zero intelligence models of These models suggest that for many pur- A few common examples are fundamen- the continuous double auction make poses the structure of the institution tal valuation, technical trading (inter- the assumption that people place orders used to make trades plays an important preting patterns in historical price move- more or less at random. We say “more role. ments), many forms of derivative pric- or less” because there is more than The zero intelligence approach is sim- ing, statistical arbitrage, market making, one way to characterize their behavior, ilar to the equilibrium model in the index arbitrage, and term structure mod- and one can argue that some of these sense that both approaches are parsi- els. The champions of heterogeneous involve at least some strategic think- monious but sometimes highly unreal- models believe that to understand asset ing, and so are not strictly speaking istic. By making the simple assumption pricing as it manifests itself in the real zero intelligence. The development of that people do not think at all, one can world, it is necessary to understand the zero intelligence models of the contin- focus on aspects of a problem that would behaviors of at least some of these types uous double auction has a long history be missed by an equilibrium analysis. of agents and their inter-relationships [75, 77, 118–130]. Equilibrium and zero intelligence mod- and interactions. In some cases zero intelligence mod- els provide ways to enter the wilderness During the last couple of decades els make successful predictions. For of bounded rationality without becom- a new school of boundedly rational example, by assuming that order flow ing totally lost; the two approaches are heterogeneous agent (BRHA) models follows simple Poisson processes, the complementary because they enter this has emerged in finance. The primary model of Daniels et al. [127, 129] derives wilderness from opposite sides. motivation of this work has been to equations of state relating statistical Of course, many problems, including explain phenomena such as bubbles properties of order flow such as the rates the double auction, require an under- and crashes, clustered volatility, excess of trading order placement and cancel- standing of both structure and strategy, volatility, excess trading volume and the lation to statistical properties of prices and the art of good modeling is to find heavy tails of price returns. As discussed such as volatility and the bid-ask spread. the right compromise in emphasizing in Sections 6 and 7.1, these phenomena these two facets of economics. A good Some of these predictions are borne out have so far not been explained by equi- recent example is the theory of Wyart by empirical data [130]. The develop- librium theory. In contrast there are now et al. who show that a mixture of mar- ment of such zero intelligence models many BRHA models that produce these ket efficiency and structural arguments has guided more empirically grounded phenomena. The typical BRHA mod- can be used to explain the relationship “low intelligence models,” which refine els that have been constructed so far between the spread, price impact and the statistical processes for order place- study a single asset and assume that volatility in transaction time [131]. ment and cancellation. For certain cat- exogenous information enters through egories of stocks this results in accu- a stochastic process corresponding to rate quantitative prediction of the dis- 8.3. Bounded Rationality, Specialization, the dividends paid by that asset. The tribution of volatility and also of the and Heterogeneous Agents most common technique is to simulate distribution of the spread [77]. As originally pointed out by Adam Smith, the resulting market behavior on a com- These models can be criticized the cognitive limitations of real agents puter, though some studies also derive because they fail in an important respect: cause them to specialize.29 In finance, analytic results. This style of work often the resulting price series are not effi- agents use specialized trading strategies. goes under the name of agent-based cient. This is a failing, but it illustrates modeling [134–136]. an important point: the fact that a good Roughly speaking BRHA models can prediction of the distribution of volatility 29Equilibrium models often begin with be divided into those that identify can be made from a model that does not asymmetric information a priori. See for classes of agents a priori and those that satisfy efficiency suggests that the distri- example the classic paper of Grossman use learning to generate heterogeneous bution of volatility may not depend on and Stiglitz [132], which can be viewed agents de novo. Examples of models that efficiency. Instead, the structural prop- as an example of a rational treatment assume heterogeneous agents a priori erties of the continuous double auction of heterogeneous agents emerging from are found in references [5, 6, 92, 137– appear to be more important. In this case idiosyncratic information. Noise trader 149]. The most commonly studied cate- by “structural properties” we mean those models provide another example [8]. For gories of agents are value investors, who that come directly from the structure of a modern treatment of agents who are price stocks according to their funda- the auction, e.g., the rules under which a priori the same but choose to special- mentals, and trend followers, who buy transactions take place, the dynamics of ize and learn different things on account stocks that have recently increased in removal and deposition of orders, and of information processing constraints see price and sell stocks that have recently their interactions with price formation. [133].

© 2008 Wiley Periodicals, Inc. COMPLEXITY 31 DOI 10.1002/cplx decreased in price. The learning mod- more profitable agents gain capital, models apper to confirm the physics els assume heterogeneous learning algo- giving them a greater effect on price for- lesson that certain kinds of empirical rithms (or starting points) a priori and mation. After sufficient simulation time regularities (power laws) are the conse- then let the strategies evolve as learning these models reach a state in which the quences of nonequilibrium behavior of takes place, thereby generating the het- remaining strategies are all equally prof- the market. erogeneous strategies de novo. Examples itable (and are in this sense arbitrage of this approach are given in references efficient). When this occurs the autocor- 8.4. Finance Through the Lens of Biology [134, 150–156]. This approach lets strate- relations in prices also disappear. This The influence of biology on econom- gies co-evolve in response to the condi- is not complete efficiency in the sense ics dates back at least to Alfred Mar- tions created by each other, as well as in that it is possible (at least in principle) shall [92, 159–165]. Both biology and response to exogenously changing fun- to introduce new strategies that are not economics involve specialized agents damentals. The typical result produces within the generated set, that can exploit evolving through time under strong a highly heterogeneous population of nonlinear price patterns and temporar- selection pressure, and it is not surpris- trading strategies. ily make profits. (For some examples see ing that they share many features, such Simulations of such systems show [92].) After enough time for capital to as extensive use of game theory. In this that they naturally generate the appar- reallocate the advantage of those strate- section we consider other aspects of biol- ently nonequilibrium phenomena gies also disappears. This shows that that ogy that are not captured by game the- the prediction of efficiency by equilib- mentioned earlier, including clustered ory and ask whether financial economics rium models is not as striking as it might volatility, heavy tails, bubbles and might not benefit by taking some lessons seem—the simulation models discussed crashes, and excess trading. In most of from biology. above generate results that are difficult these models the primary cause of clus- Note that the biological point of view to distinguish from true efficiency. Thus, tered volatility is changes in the pop- is particularly complementary with that these models provide a means, albeit ulations of different kinds of trading of BRHA modeling. Biology is after all a crude, to take reasoning into account, strategies. The population can shift due study of specialized organisms and their and satisfy many of the desiderata for to exogenous conditions, but most of interactions with each other, and thus modeling agent behavior discussed in the shifting appears to be largely ran- provides a lens for viewing many aspects Section 4. dom, due to context-dependent fluctu- of financial markets. ations in the profitability of one group A great deal remains to be done in this vs. another. Changes in the exogenously area. Many of the the models mentioned 8.4.1. Taxonomy of Financial provided fundamentals are amplified above illustrate the principle (by gener- Strategies with feedback. In some cases it is ating clustered volatility), but they do not Taxonomy is the study of classification. possible to show that in an analogous make quantitative predictions about the By sorting complex phenomena such equilibrium model none of these phe- real economy. Most of them are not very as organisms into categories we com- nomena would arise [150, 154, 155]. Fur- parsimonious. There are still major open press an enormous amount of informa- thermore, in some cases it is possible questions about the necessary and suffi- tion into a comprehensible scheme. Tax- to calibrate these BRHA models against cient conditions required for generating onomy has historically been one of the real data and get a good fit for properties the phenomena of interest. For instance, principal activities of biologists. Perhaps such as the distribution of returns and it is not absolutely clear what features of the most famous taxonomist is Carl Lin- the autocorrelation function of volatility the models that generate the phenom- naeus, who in the middle of the eigh- [152, 153, 156–158].Thus the more devel- ena are incompatible with equilibrium. teenth century introduced the hierarchi- oped models in this class make testable It is even possible that some equilibrium cal classification of organisms into cate- predictions. Note that such studies do model might also generate the same phe- gories at varying levels of differentiation, not claim that information arrival does nomena, but equilibrium models are so proposed a list of characteristics for clas- not have an effect, but only that a sub- hard to compute that examples have sification, and created the standardized stantial component of volatility, and in not yet been discovered. Further valida- nomenclature that is still largely used particular its interesting statistical prop- tion and testing and simplification are today. The taxonomic classification of erties, are internally generated. needed before we can have a high degree organisms has benefited from the par- Most of these BRHA models pro- of confidence that the detailed explana- ticipation of hundreds of thousands or duce uncorrelated prices and arbi- tions of these BRHA models are correct. perhaps millions of individuals for more trage efficiency between the simulated Still more work is required to apply the than three hundred years and contin- strategies. In the more sophisticated lessons of BRHA models to asset pricing ues to be a field of active research and models, capital is reallocated so that and market design. Nonetheless, these sometimes heated debates in biology.

32 COMPLEXITY © 2008 Wiley Periodicals, Inc. DOI 10.1002/cplx Although taxonomy might seem to be a Under the analogy to biology a finan- invested in strategy j, one can define a very limited end in and of itself, it has cial strategy can be viewed as a species, gain matrix Gij = ∂ρi/∂Cj.IfGij < 0 and proved to be an important prerequisite and the capital invested in it can be Gji < 0 strategies i and j have a compet- for the development and testing of the viewed as its population. If strategies are itive relationship; if Gij > 0 and Gji < 0 theory of evolution in biology. successful they accumulate more cap- then they have a predator-prey relation-

An economist, Adam Smith, was one ital, which causes them to have more ship, in which i preys on j, and if Gij > 0 of the first to point out the importance of influence in setting prices. The price and Gji > 0 they have a mutualistic (or specialization, and in some branches of provides the principal channel through symbiotic) relationship. The ecological economics, such as industrial econom- which different strategies interact—the view asserts that the set of such rela- ics, this is a field of active research. Not movements of prices influence the buy- tionships in the market is critical to its so in finance. To our knowledge there are ing and selling activity of strategies, function, and that the proper diversity only two articles that gather empirical which in turn affects prices. On a longer of financial strategies may be critical for data on the nature of real trading strate- timescale the movements of prices also market stability. This view also provides gies in financial markets (and both of affects profitability, which alters the cap- an evolutionary framework in which to these papers only address the frequency ital invested in strategies and is the main understand the path toward efficiency. of usage of two preassigned categories driver of the selection process underly- Until recently this ecological view of of strategies, technical trading, and value ing market evolution. markets was essentially hypothetical and investing) [166, 167]. The diversity and In most BRHA models, price setting it was not clear that the resulting mod- specialized functions performed by the is done using a standard method, such els were testable. Recently, however, the trading desks of major financial institu- as market clearing, but even when this acquisition of data sets in which there tions suggest that there is an enormous is the case, it can be very useful to view is information about brokerages or indi- amount waiting to be done if we are to the interactions of strategies in terms of vidual investors has opened up the pos- understand the taxonomy of real trad- their market impact. The market impact sibility of calibrating such models and ing strategies. The BRHA models men- is the price movement corresponding testing their predictions against real data tioned above suggest that such diver- to the initiation of a trade of a given [168, 169]. It will be interesting to deter- sity is important and plays a key role size, and is equivalent (up to a con- mine whether one can link properties in price formation. If this endeavor is to stant of proportionality) to the demand of financial ecologies to properties of enter the realm of hard science it needs a elasticity of price originally introduced price formation. For example, do alter- grounding in empirical data. We believe by Marshall [Farmer, et al. unpublished ations in financial ecologies influence that market taxonomy should become a research]. Market impact is an increas- price volatility? major field of study by financial econ- ing function of trading volume. As the 8.4.3. Evolutionary Finance omists. Electronic data storage presents capital invested in a strategy grows, The three essential elements of evolu- the possibility of gathering data with market impact lowers realized returns tionary theory are descent, variation, information about the identity of agents, and ultimately limits the trading capi- and selection. Financial ecologies are which can potentially form the basis tal that any given strategy can support. formed by these precisely these forces: for such taxonomies. Preliminary results With a proper understanding of market impact it is possible to derive differen- in this direction have already yielded tial equations for the dynamics of capital Descent. Financial strategies are interesting results [168, 169]. • flows between strategies, analogous to passed down through time as traders the generalized Lotka-Volterra models of change and start new firms. 8.4.2. Market Ecology population biology [92]. • Variation. Old strategies are modified For biological systems ecology is the Market impact also provides a way to and improved to create new strategies. study of organisms and their relation- understand the strength of the interac- • Selection. Successful strategies prolif- ship to their environment, including tions between strategies and to classify erate and unsuccessful strategies dis- each other. Simulations of BRHA mod- them in ecological terms. In particular, it appear. Success is substantially based els make it clear that understanding the is possible to make a pairwise classifica- on profitability, but is also influenced relationships between financial strate- tion of strategies according to their influ- by other factors such as marketing and gies is also important in financial eco- ence on each other’s profits by under- psychological appeal. nomics. Market ecology is an attempt standing how a variation in the capital to develop a theory that gives a deeper associated with strategy j influences the The view that profit selection is and more universal understanding of profits of strategy i. Letting ρi be the important is not new, having been cham- this [92]. return of strategy i and Cj be the capital pioned as a force for creating efficient

© 2008 Wiley Periodicals, Inc. COMPLEXITY 33 DOI 10.1002/cplx markets by central figures in financial too simple—to understand real finan- tractable, rational expectations equilib- economics such as Milton Friedman and cial economies we need to do the hard rium is likely to provide a good explana- Eugene Fama [22, 170]. The theory of work of understanding who the agents tion. Some examples where this is true market efficiency amounts to the asser- are, what factors cause them to change, include option pricing, hedging, or the tion that it is possible to short-circuit the and how they interact with each other. pricing of mortgage-backed securities. evolutionary process and go directly to To do this we need better taxonomic In other cases where the cognitive task its endpoints, in which evolution has run studies of real markets, better simulation is extraordinarily complex, such as the its course and profits of well-developed models, and better theory. To the exter- pricing of a new firm, or where esti- strategies are reduced to low levels. The nal observer, current research in finan- mation problems are severe, such as suggestion that the timescales for this cial economics has so far largely failed portfolio formation, human models may process are not short, as presented in to capture the richness of real markets, diverge significantly from rational mod- Section 7.3, motivate more detailed stud- which provide some of the best examples els, and the equilibrium framework may ies of the transient properties of the evo- of complex systems. More robust con- be a poor approximation. For good sci- lutionary process and its dynamics. This tact between financial economics and ence one must choose the right tool for view implicitly underlies all the BRHA the emerging field of complex systems the job, and in this case the good sci- models we have discussed so far, and is could help remedy this situation. entist must use an assortment of dif- also now being given a more mathemat- ferent tools. Close-mindedness in either ical underpinning [171, 172]. 9. CONCLUSION direction is not likely to be productive. What is still largely lacking is an We have written this article with a dual As we have stressed, equilibrium the- empirical foundation on which to build purpose. On one hand, we worry that ory is an elegant attempt to find a par- an evolutionary theory. This ultimately physicists often misunderstand the equi- simonious model of human behavior in depends on developing a taxonomy of librium framework in economics, and economic settings. It can be criticized, real financial strategies together with a fail to appreciate the very good reasons though, as a quick and dirty method, database of studies of financial ecologies for its emergence. On the other hand, a heroic attempt to simplify a com- as they change through time. the majority of economists have become plex problem. Now that we have begun so conditioned to explain everything in to understand its limitations, we must 8.5. The Complex Systems Viewpoint terms of equilibrium that they do not begin the hard work of laying new foun- Complex systems refers to the idea that appreciate that there are many circum- dations that can potentially go beyond it. systems composed of simple compo- stances in which this is unlikely to be nents interacting via simple rules can appropriate. We hope that physicists will ACKNOWLEDGEMENTS give rise to complex emergent behaviors, begin to incorporate equilibrium into We would like to thank Legg-Mason and in which in a certain sense the whole is their models when appropriate, and that Bill Miller for supporting the conference greater than the sum of its parts. With economists will become more aware of “Beyond Equilibrium and Efficiency,” his introduction of the concept of the analogies from other fields and begin to held at the Santa Fe Institute in 2000, that “invisible hand” Adam Smith was one explore the possibilities of alternatives to originally stimulated this article, and of the earliest to articulate this point of the standard equilibrium framework. for supporting our subsequent research. view. The general equilibrium theory of Our own belief is that one must choose This work was also supported by Bar- Arrow and Debreu can be regarded as modeling methods based on the con- clays Bank and NSF grant HSD-0624351. an attempt to cut through the complex- text of the problem. In situations where Any opinions, findings and conclusions ity of individual interactions and reduce the cognitive task to be solved is rela- or recommendations expressed in this the invisible hand to a tractable math- tively simple, where there is good infor- material are those of the authors and do ematical form. It appears, though, that mation available for model formation, not necessarily reflect the views of the for many purposes this approach is just and where the estimation problems are National Science Foundation.

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