Information and Learning in Markets: The Impact of Market Microstructure

Markets

Introduction and Lecture Guide

Information is at the heart of the economy today. Despite this there is still a lively debate over whether markets aggregate the dispersed information of agents in the economy or whether they are at the mercy of herd behavior and fads and in the hands of short-term speculators and insiders. The debate has old roots and goes back at least to the exchange between Hayek and Lange in the 1930s about the economic viability of socialism. While Lange (1936, 1937) argued that socialism was viable because a competitive allocation can be replicated by a central planner, Hayek contended that the superiority of markets was to be found in their ability to aggregate the dispersed information of agents in the economy (Hayek 1945, p. 526):

The mere fact that there is one price for any commodity—or rather that local prices are connected in a manner determined by the cost of transport, etc.—brings about the solution which (it is just conceptually possible) might have been arrived at by one single mind possessing all the information which is in fact dispersed among all the people involved in the process."

In contrast to Hayek, Keynes in his General Theory pioneered the view of the stock market as a beauty contest where investors try to guess average opinion, instead of fundamentals, and end up chasing the crowd (Keynes 1936, p. 136):

[P]rofessional investment may be likened to those newspaper competitions in which the competitors have to pick out the six prettiest faces from a hundred photographs, the prize being awarded to the competitor whose choice most nearly corresponds to the average preferences of the competitors as a whole; so that each competitor has to pick, not those faces which he himself finds prettiest, but those which he thinks likeliest to catch the fancy of the other competitors, all of whom are looking at the problem from the same point of view.

This view is consistent with people behaving like a colony of penguins that herd and follow a first individual that jumps of a cliff. Indeed, it has been claimed that markets are in the hands of short-term speculators, insiders, and manipulators who induce bubbles and crashes. To this it may be added that traders are not rational to start with and that market outcomes are heavily influenced by psychological biases, behavioral rules, and persistent mistakes made by decision makers.

The central issue is whether we can trust markets to aggregate information in a world where information is dispersed. Does the stock market reflect the underlying fundamentals of the traded firms? Or, to put the question in more concrete terms, does the price in the futures market for wheat aggregate the relevant information of producers and speculators?

The view of Hayek is optimistic in the sense that we can trust markets to provide the right signals for decision making. Indeed, Hayek thought that the price system is a marvel because of its power in aggregating information.

In this book we will analyze in depth the information aggregation properties of markets and the impact of specific trading arrangements (the market microstructure) on the information aggregation process and the quality or performance of the market.

Hayek’s ideas are the basis for the rational expectations models that explain how rational agents make optimal inferences from prices, and other public statistics, about the relevant parameters about which they are uncertain.¹ The statistics that aggregate the dispersed information in the economy are indicators of aggregate activity such as output, employment, investment, or trade volume. The result is that learning from market statistics makes up for the lack of knowledge of traders and producers.

The book will try to disentangle whether the two views of markets, informational efficiency and herding, are compatible. Do prices in the stock market reflect all available information in the hands of traders and fundamental values, or are prices in the hands of short-term speculators, insiders, manipulators, and gurus and subject to fads, herding behavior, and bubbles? We will explain how rational well-informed agents may disregard their private information and make persistent errors when acting. Indeed we will see that we do not need irrational traders with behavioral biases making persistent mistakes to explain, for instance, crises and bubbles, or financial market anomalies like excess volatility. The book therefore does not cover the otherwise very interesting contributions of the behavioral finance literature.²

Thinking a little bit more generally, we are asking in what circumstances social learning, that is, learning from the actions of other people, leads to information revelation and efficient outcomes. This cuts across more disciplines than markets and economics and extends to sociology and political science. Indeed, markets are not the only mechanism for the aggregation of information. Deliberation is an old method. Information technology and the Internet have provided a range of instruments to aggregate information, from blogs to wikis to the open source movement, that hint at an information aggregation revolution.³ Those instruments are based on the desire of people to build a reputation, attain status, or simply driven by altruism and conformation to a social norm. They have proved effective although they are vulnerable to manipulation and attack. Markets work based on another mechanism: people, each putting their money where their mouth is, prove to be, on average, tremendously effective at aggregating information. Prediction markets, betting on the outcome of an election for example, have recently emerged, combining the features of markets and information technology aggregating procedures.⁴ In markets agents learn from other agents with the intermediation of the price system or market aggregates like market shares. For example, a high market share of a brand may provide a signal of the quality of the product similarly to when we buy a best seller because we think that the book must be good. Sometimes people learn directly from the actions of other people. Depositors may run on a bank when they see others doing so, or tourists may decide not to patronize an empty restaurant when visiting a city.

All these information aggregation mechanisms share the Hayekian idea of the power of many minds in collecting information. For example, the blogosphere is seen by some analysts as the newest mechanism for society to pool knowledge.⁵ It is tempting to think that Hayek’s one single mind possessing all the information which is in fact dispersed among all the people is represented today by the World Wide Web. The main point, however, is that, very much in line with Hayek’s vision, the World Wide Web is not akin to a central planner but to a very decentralized institution with many minds at work. Furthermore, it is yet to be seen whether some of these new information aggregation methods can match the accuracy and economy of knowledge of the price system.

Approach of the Book

This book takes the perspective of explaining aggregation of information and learning in markets with rational agents who understand market conditions and make the most of them. In particular, our agents will be Bayesian and will learn accordingly. We will therefore leave aside bounded rationality and non-Bayesian methods of learning. We concentrate on Bayesian rational models because of the discipline they introduce into the analysis, because they deliver results, including the explanation of market anomalies, and because there is evidence that those models are useful in explaining the behavior of agents. Indeed, a Bayesian model requires making explicit the assumptions of the learning agent and the context of learning, and it makes life harder for the researcher when trying to explain a phenomenon. Furthermore, despite the challenges to the Bayesian model coming from the experimental literature emphasizing boundedly rational and error-prone heuristics to explain cognitive judgments under uncertainty, there is recent evidence of the Bayesian optimality of human cognition in realistic scenarios as well as in experimental settings.⁶ Market pressure may also imply aggregate behavior consistent with Bayesian prediction even allowing for substantial departures for individual agents.⁷

Our approach will bring together results from the rational expectations literature and recent analysis of herding phenomena in a coherent game-theoretic framework. This framework will be used when considering both competitive and strategic agents. The advantage of such an approach is that the details of the environment are modeled precisely and outcomes are the results of the actions of the agents and the information they have. The approach in the book is game-theoretic but this is not a book of learning in games⁸; rather, it emphasizes the consequences of market interaction and social learning for informational and economic efficiency. The book therefore brings together dynamic models of rational expectations with Bayesian learning models, including the herding variety.⁹

The strategy of the book is to analyze the different topics considering a basic workhorse asymmetric-information model in a linear-quadratic mean–variance environment. Most results will be obtained in the context of a model where optimal actions end up being a linear function of the information agents have. The model is developed in two basic versions: a partial equilibrium production market and a financial market. The book examines information aggregation mechanisms, progressing from simple to complex environments: from static to dynamic models, from competitive to strategic agents, from simple market strategies (such as noncontingent orders or quantities) to complex ones (such as price-contingent orders or demand schedules).

This is a book that focuses on theoretical models.¹⁰ A nonexhaustive outline of topics with some of the basic ideas covered in the book follows.

Outline of Selected Themes and Issues

Herding, Rational Expectations, and the Underlying Information Externality

If the actions of others strongly suggest a certain action, then a rational individual may ignore his own information and follow the crowd or herd. When this happens information is lost and it is not revealed to others. There is no further accumulation of information and an informational cascade obtains. The result is that the outcome may be very inefficient. This is an insight derived from the sequential prediction model considered in the herding literature¹¹ which has been suggested to apply to investors and decision makers in markets. In principle, the results would give support to the pessimistic view of markets and would seem to contradict directly the rational expectations literature with its emphasis, following Hayek’s ideas, on the informational properties of markets.

We provide an analytical framework that makes apparent that rational expectations and herding models have in common a basic information externality. Indeed, a trader or agent when acting does not take into account the informational benefits of his action on the other traders or agents. When the trader or agent cannot fine-tune his action to his information (e.g., with discrete actions) and receives signals of bounded strength, then herding on a wrong action occurs with positive probability. This cannot happen with a rich choice set, in which case information will end up being revealed. However, even then with noisy observations revelation will be slow.

Slow Learning from Others and Welfare

Traders learning about the fundamentals of the market will typically learn slowly over time from a noisy public statistic (e.g., the price). A rational (Bayesian) trader will respond to increased price informativeness with decreased weight given to his private information when formulating his trade. This will mean that less of his private information will be incorporated into the public statistic. The result will be that the increase in informativeness in the public statistic will be smaller the larger is his absolute level, slowing down information revelation. Learning from others has a self-correcting property.

What are the welfare consequences? Welfare analysis has typically been neglected in herding models. We introduce an appropriate welfare benchmark with private information. This is team efficiency, where agents use decentralized decision rules but have as a common objective the welfare of a representative agent. The information externality implies that the market outcome is inefficient with respect to this team benchmark, which internalizes the externality. Herd behavior and informational cascades are extreme manifestations of the self-correcting aspect of learning from others. With discrete action spaces and signals of bounded strength, public information may end up overwhelming the private signals of the agents, who may (optimally) choose not to act on their information. More generally, herding means that agents put too little weight on their private information with respect to the team benchmark and this results in under-accumulation of public information. An interesting possibility is that more public information may hurt. The reason is that a higher amount of public information may reduce the effort to collect private information. A similar analysis can be performed in rational expectations models. The bottom line is that the welfare analysis of rational expectations and herding models is not qualitatively different.

Information Externalities and Payoff Externalities

With social interaction there is typically both an informational externality and a payoff externality. In a pure prediction model there is only an information externality. How do the informational and the payoff externalities interact? In the first place we show in a static model that informational efficiency is not the same as economic efficiency. We will see that prices in a rational expectations equilibrium of a market with substitute products will tend to convey too little information when the informational role of prices prevails over its indexof-scarcity role and too much in the opposite situation. The market will be team-efficient only in exceptional circumstances (i.e., when the information externality vanishes).

Explaining Market Dynamics

Do prices converge to full-information values as trading periods accumulate? Is the market an effective price-discovery mechanism in the presence of asymmetric information? If so, how fast? How long does the adjustment process takes? How does the payoff externality modify the result of slow learning from others in prediction models?

The results obtained in prediction models generalize to market settings. Traders learn the unknown parameters and prices converge to full-information values with repeated interaction in a stationary market environment. However, learning an unknown parameter and converging to full-information equilibrium is not equivalent. For example, with serial correlation in period shocks it is possible to learn an unknown parameter at a lower speed than converge to a full-information equilibrium. In nonstationary environments learning and convergence are not equivalent.

We find that in market models where traders learn from prices about the information of other traders about a valuation parameter, as in a financial market learning may be fast because of the payoff externality induced by market makers. Market makers induce a deeper market as more information about the fundamental value is revealed by the trading process. With discrete actions, as in the simple prediction models, there is no herding; with a rich choice set and noise in public information, convergence to a full-information equilibrium is fast because risk-averse informed traders respond more intensely to their private signals as market makers induce a deeper market. The role of a competitive market-making sector proves crucial for price discovery. Its presence implies that current prices reflect all public information.

The Impact of Market Microstructure

The information aggregation properties of the market depend on the details of market organization or market microstructure. First of all, information aggregation depends on whether the market mechanism is smooth, like a Cournot market, or has the winner-takes-all feature, like auction or voting mechanisms. The latter aggregate information better than the former both in the sense that they aggregate information in more circumstances and that they do so more economically, that is, without the necessary coming together of a very large number of agents. Second, information aggregation depends on whether agents use noncontingent strategies, like market orders, or price-contingent ones, like limit orders or demand schedules in financial markets. The more complex contingent strategies do better at aggregating information. This also affects the incentives to acquire information. Third, it also matters whether the market is order or quote driven. In order-driven markets informed traders move first while in quote-driven markets uninformed market makers move first.¹²

Can Market Anomalies Be Explained?

Market anomalies, seemingly inconsistent with rational expectations models, can in fact be explained without recourse to the irrationality of traders. Excess volatility, where asset prices are more volatile than fundamental values, can be consistent with dynamic trading models where market makers are risk averse and do not fully accommodate shocks. The analysis of patterns of stock prices, or technical analysis, is valuable if the current price is not a sufficient statistic for public information about the fundamental value. This is typically so except where market makers are risk neutral. However, learning from past prices may be slow. The presence of traders with short horizons may reduce the informational efficiency of the market and explain incentives for investors to herd in information acquisition. For example, short-term traders may care more about the information that other short-term traders have than about the fundamentals, like in Keynes’s beauty contest, because of strategic complementarities in information acquisition. Similarly, discrepancies between the average expectations of investors and stock prices, which sometimes may look like a bubble, can be rationalized in the presence of short-term traders who induce multiple market equilibria. More generally, price dynamics in financial markets may have a Keynesian or a Hayekian flavor depending on parameters such as the degree of persistence of shocks or the amount of residual uncertainty on the liquidation value of the asset.

Crises and Crashes

Sudden and significant drops in asset prices, even in the absence of major news, may be explained with our models, for example, by an abrupt information revelation about the quality of information of investors brought by a small price movement; by a misinterpretation of a price drop due to underestimation of the extent of portfolio trading generating multiple equilibria; or by a liquidity shortage as a result of an underestimation by market timers of the degree of dynamic hedging activity. Crises that arise from a coordination problem of investors, such as exchange rate crises or bank runs, leading to multiple equilibria can be explained with the interaction of private and endogenously generated public information. Policy implications may be derived from the analysis. For example, it need not be true that more transparency is always good since it may coordinate the expectations of investors in a bad equilibrium.

The Impact of Large Traders

Large informed traders are more cautious when responding to their private signals because they are aware of the price impact of their trades. The result is that prices are less informative in their presence. Insiders, when forced to reveal their trades, will engage in dissimulation strategies. Otherwise, they may have incentives to manipulate the market and slow down information revelation. For example, they may do so by using a contrarian strategy to neutralize the potential revelation of information by the trades of competitive informed agents. The effects of insider trading are multiple. Insider trading generates adverse selection, inducing market makers to protect themselves by making the market thinner, and advances the resolution of uncertainty, increasing price informativeness. The first effect tends to be bad for investment and welfare. The impact of the second effect, under risk aversion, depends on what is the alternative to insider trading. If the alternative is that information is disclosed, then insider trading may be good because too much information disclosure destroys insurance opportunities. If the alternative is that no information is collected, then insider trading will typically be bad for welfare.

A relevant issue from the point of view of policy is whether market power or asymmetric information looms larger in accounting for welfare losses in relation to full-information competitive equilibria. The general result is that market power dissipates faster than asymmetric information as a market grows large and supports more traders, and therefore in moderately sized markets asymmetric information is likely to be a larger source of the welfare loss. This is the case in Cournot markets, for example.

Some Conclusions

Apparently contending theories, such as market informational efficiency and herding, in fact build on the same principles of Bayesian decision making. The upshot is that we do not need irrational agents to explain herding behavior, crises, and crashes. Traders may be rational but learn slowly or even on some occasions disregard private information and end up making persistent errors when acting; or they may be trapped in a coordination failure. However, informational and economic efficiency need not, and in general will not, coincide. In any case the impact of market microstructure on the informational efficiency of prices is crucial. The stock market conveys information about the fundamentals of firms although the misalignment of stock prices and fundamentals is possible but typically not extremely long-lived.

Outline of Chapters

In chapters 1–3 and 7 we consider mostly real partial equilibrium markets, with Cournot competition as the leading example, although we also study auctions, supply function competition, and its relation to rational expectations equilibria (in chapter 3). In chapters 4, 5, 8, and 9 we consider financial markets and spend time explaining how the market microstructure affects trading and the informational efficiency of prices. The equivalent of Cournot competition in a financial setting is when traders submit market orders, while the equivalent of supply function competition is when traders submit demand schedules to a centralized trading mechanism. Chapters 1, 3, 4, and 6 deal with competitive models and chapters 2, 5, and 9 (mostly) deal with models of strategic traders. Chapters 1–5 deal with static models and chapters 6–9 with dynamic models. Chapter 6 builds a bridge between stylized social learning models and dynamic rational expectations models.

Let us review in more detail the content of the chapters and the organization of the book.

Chapters 1 and 2 study market mechanisms, such as auctions or Cournot markets, in which traders do not have the opportunity to condition their actions on prices or other market statistics. This provides a benchmark in which traders can only use their private information to decide how much to trade. In this framework we study under what circumstances the market replicates the outcome of shared-information equilibrium. If it does, we say that the market aggregates information. Chapter 1 studies first in detail large Cournot markets with demand uncertainty and asymmetric information and uncovers when information aggregation obtains, providing a taxonomy and a welfare analysis of the value of information in a general setting. When information aggregation obtains in a large market a second issue of interest is how large a market is needed for the result. In any case, as a market grows large strategic behavior vanishes and price-taking obtains.

Chapter 2 analyzes the convergence properties of finite markets to price-taking equilibria as the market gets larger and the number of agents grows. When the limit is first-best efficient, with full-information aggregation, we compare the rates of information aggregation in different market mechanisms and disentangle the sources of the welfare loss at the market solution in terms of market power and private information. Both chapters 1 and 2 compare the information aggregation properties of smooth market mechanisms, like Cournot markets, with auctions.

Chapter 3 introduces the concept of a rational expectations equilibrium (REE), with its different variants—fully revealing REE, partially revealing REE, and noisy REE—in the context of a static partial equilibrium homogeneous product market with demand and cost uncertainty and a continuum of firms. It is shown that not all REE are implementable. That is, not every REE can be seen as the outcome of a well-specified game among market participants. A game form is then considered in which firms compete in supply functions and Bayesian equilibrium in supply functions is studied. The chapter goes on to perform a welfare analysis of REE introducing the appropriate team efficiency benchmark that internalizes information externalities. That is, the chapter checks the alignment of informational efficiency and allocative and productive efficiency. The information externality at the market equilibrium is characterized and related to economic efficiency. The chapter also considers the information aggregation properties of the double auction mechanism.

Chapters 4 and 5 look at the basic static financial market models with asymmetric information and review the impact of different market microstructures with competitive (in chapter 4) and strategic (in chapter 5) players. The standard competitive noisy rational expectations financial market is presented along with variations in which informed traders submit market orders to market makers. A model is presented that has as special cases virtually all the competitive models in the literature. The potential paradox of informationally efficient markets when information is costly to acquire is addressed. Chapter 5 goes on to study strategic behavior in different market structures with simultaneous and sequential order placement, uniform and discriminatory pricing, and distinguishing the cases in which informed traders move first and those in which uninformed traders move first.

Chapter 6 considers social learning models. First it presents the basic sequential decision herding model and variations. It highlights the assumptions that drive the results and develops a basic smooth model of learning from others that serves as a benchmark for the analysis and that is close to rational expectations models. It is found that the basic driving forces in herding and rational expectations models are the same. A self-correcting property of learning from others and its implications in terms of information revelation and its speed are at the center of the chapter. Furthermore, the information externality of social learning is characterized and optimal learning studied. Results are extended to the case where information acquisition is costly and the value of public information studied. Finally, a static version of the model of learning from others with a rational expectations flavor is presented and a welfare analysis is performed.

Chapter 7 deals with dynamic information aggregation models and studies their convergence properties as the number of trading periods grows without bound. The basic model considered is a repeated Cournot model with an uncertain demand parameter and period-specific shocks with firms receiving private signals about the unknown demand. The rate of convergence to full-information equilibrium is characterized in the cases of independent and correlated shocks to demand. Results for the latter case make clear that learning an unknown parameter is not equivalent to converging to full-information equilibrium. A variation of the model with an unknown cost parameter is considered and it is shown that learning and convergence to full-information equilibrium may be slow. The latter is a model of learning from others and extends the results of chapter 6 to an environment with payoff externalities.

Chapters 8 and 9 study dynamic competitive and strategic models of financial markets. They address the difficult issue of solving for and characterizing an equilibrium when risk-averse traders have private information and a long horizon. The resolution of the problem illuminates trading and price temporal patterns and serves as a benchmark to compare with the case in which informed speculators have short horizons. A detailed analysis of the impact of market microstructure on the dynamic properties of market quality parameters, such as price volatility and informativeness, market depth, and volume traded, is performed. These chapters provide explanations for technical analysis, trading with no news, excess volatility, and market crashes, as well as the impact of short-term traders. We also address the interaction between the potential coordination failure of investors, e.g., in an exchange rate crisis, and endogenously determined market signals to generate multiple equilibria. All this is done without introducing any irrationality on the part of traders. The properties of a price-discovery mechanism are studied in both competitive and strategic versions showing the importance of the market microstructure and the presence of strategic traders in the speed of price discovery. The trading strategies of large informed traders are analyzed, including the possibility of dissimulation of trades and market manipulation.

The technical appendix (chapter 10) presents in an accessible way the basic tools needed to follow comfortably the material in the book. These include information structures with particular attention to the Gaussian and related models, convergence properties of random processes, and game-theoretic concepts (such as Bayesian equilibrium). It includes in particular a concise self-contained development of the linear-Gaussian model.

Road Map and Lecture Guide

The book may be read according to different field itineraries:

• Real markets: chapters 1–3, 6, and 7.

• Financial markets (market microstructure): sections 1.1, 1.2, 3.1, chapters 4–6, 8, and 9.

Alternatively, it may be read according to the following model itineraries:

• Static models: chapters 1–5.

• Dynamic models: chapters 6–9.

Or:

• Competitive models: chapters 1, 3, 4, and 6–8, section 9.1.

• Strategic models: chapter 2, section 3.4, chapters 5 and 9.

Readers familiar with Bayesian updating techniques and the basics of rational expectations and interested only in financial markets may jump directly to chapter 4 after reading sections 1.1, 1.2, and 3.1. Readers who are familiar with static models may jump directly to chapter 6.

The level of difficulty of chapters is pretty uniform, with dynamic models being in general more demanding. Chapters 2 and 7 are more demanding and may be skipped in a first reading without major loss of continuity. The same applies to sections which are more advanced (marked with an asterisk (*)). Instructors who use the text for advanced undergraduates or masters students with little formal background in economics may want to introduce the major results with the examples which usually follow the major propositions, and present the general results later on. Each chapter is closed by a summary and an appendix with proof of results in the text, and worked-out exercises. The level of difficulty of problems varies from very challenging (marked with a double asterisk (**)) because of inherent difficulty or the amount of work they require, challenging (marked with an asterisk (*)), and easy (with no mark).

References

Ackert, L., B. Church, and M. Shehata. 1997. Market behavior in the presence of costly, imperfect information: experimental evidence. Journal of Economic Behavior and Organization 33:61–74.

Anderson, L., and Ch. Holt. 1997. Information cascades in the laboratory. American Economic Review 87:847–62.

Banerjee, A. V. 1992. A simple model of herd behavior. Quarterly Journal of Economics 107:797–817.

Berg, J., R. Forsythe, T. Nelson, and T. Rietz. 2005. What makes markets predict well? Evidence from the Iowa electronic markets. In Handbook of Experimental Economic Results (ed. C. R. Plott and V. Smith). Amsterdam: Elsevier.

Bikhchandani, S., D. Hirshleifer, and I. Welch. 1992. A theory of fads, fashion, custom, and cultural change as informational cascades. Journal of Political Economy 100:992–1026.

Brunnermeier, M. 2001. Asset Pricing under Asymmetric Information. Oxford University Press.

Camerer, C. 1995. Individual decision-making. In Handbook of Experimental Economics (ed. J. Kagel and A. Roth). Princeton University Press.

Chamley, C. 2004. Rational Herds. Cambridge University Press.

Chen, K. Y., and C. R. Plott. 2002. Information aggregation mechanisms: concept, design and implementation for a sales forecasting problem. California Institute of Technology, Social Science Working Paper 1131.

Cipriani, M., and A. Guarino. 2005. Herd behavior in a laboratory financial market. American Economic Review 95:1427–43.

Drehmann, M., J. Oechssler, and A. Roider. 2005. Herding and contrarian behavior in financial markets: an Internet experiment. American Economic Review 95:1403–26.

Evans, G. W., and S. Honkapohja. 2001. Learning and Expectations in Macroeconomics. Princeton University Press.

Forsythe, R., and R. Lundholm. 1990. Information aggregation in an experimental market. Econometrica 58:309–47.

Forsythe, R., F. Nelson, G. R. Neumann, and J. Wright. 1992. Anatomy of an experimental political stock market. American Economic Review 82:1142–61.

Fudenberg, D., and D. Levine. 1998. The Theory of Learning in Games. Cambridge, MA: MIT Press.

Garman, M. 1976. Market microstructure. Journal of Financial Economics 3:257–75.

Griffiths, T., and J. Tenenbaum. 2006. Optimal predictions in everyday cognition. Psychological Science 17:767–73.

Grossman, S. 1989. The Informational Role of Prices. Cambridge, MA: MIT Press.

Guesnerie, R. 2001. Assessing Rational Expectations: Sunspot Multiplicity and Economic Fluctuations. Cambridge, MA: MIT Press.

——. 2006. Assessing Expectations, volume 2: Eductive Stability in Economics. Cambridge, MA: MIT Press.

Hasbrouck, J. 2007. Empirical Market Microstructure. Oxford University Press.

Hayek, F. 1945. The use of knowledge in society. American Economic Review 35:519–30.

Hung, A., and C. R. Plott. 2001. Information cascades: replication and an extension to majority rule and conformity-rewarding institutions. American Economic Review 91:1508–20.

Jamal, K., and S. Sunder. 1996. Bayesian equilibrium in double auctions populated by biased heuristic traders. Journal of Economic Behavior and Organization 31:273–91.

Kahneman, D., P. Slovic, and A. Tversky. 1982. Judgment under Uncertainty: Heuristics and Biases. Cambridge University Press.

Keynes, J. 1936. The General Theory of Employment, Interest and Money. Cambridge University Press.

Lange, O. 1936. On the economic theory of socialism. Part One. Review of Economic Studies 4(1):53–71.

——. 1937. On the economic theory of socialism. Part Two. Review of Economic Studies 4(2):123–42.

Ljungqvist, L., and T. J. Sargent. 2000. Recursive Macroeconomic Theory. Cambridge, MA: MIT Press.

Plott, C. R., and S. Sunder. 1988. Rational expectations and the aggregation of diverse information in laboratory security markets. Econometrica 56:1085–118.

Rabin, M. 1998. Psychology and economics. Journal of Economic Literature 36:11–46.

Radner, R. 1979. Rational expectations equilibrium: generic existence and the information revealed by prices. Econometrica 47:655–78.

Sandroni, A. 2000. Do markets favor agents able to make accurate predictions? Econometrica 28:1303–41.

——. 2005. Efficient markets and Bayes’ rule. Economic Theory 26:741–64.

Shiller, R. J. 2005. Irrational Exuberance. Princeton University Press.

Shleifer, A. 2000. Clarendon Lectures: Inefficient Markets. Oxford University Press.

Sunstein, C. 2006. Infotopia. How Many Minds Produce Knowledge. Oxford University Press.

Thaler, R. 2005. Advances in Behavioral Finance. Princeton University Press.

Tversky, A., and D. Kahneman. 1971. Belief in the law of small numbers. Psychology Bulletin 76:105–10.

——. 1974. Judgement under uncertainty: heuristics and biases. Science 185:1124–31.

Wolfers, J., and E. Zitzewitz. 2004. Prediction markets. Journal of Economic Perspectives 18:107–26.

Some of the material presented in the book has been adapted from the following:

Bru, L., and X. Vives. 2002. Informational externalities, herding, and incentives. Journal of Institutional and Theoretical Economics 158:91–105 (chapter 6).

Burguet, R., and X. Vives. 2000. Social learning and costly information acquisition. Economic Theory 15:185–205 (chapter 6).

Jun, B., and X. Vives. 1996. Learning and convergence to a full-information equilibrium are not equivalent. Review of Economic Studies 63:653–74 (chapter 7).

Medrano, L., and X. Vives. 2001. Strategic behavior and price discovery. RAND Journal of Economics 32:221–48 (chapters 4, 5, and 9).

——. 2004. Regulating insider trading when investment matters. Review of Finance 8:199–277 (chapters 4 and 5).

Rochet, J. C., and X. Vives. 2004. Coordination failures and the lender of last resort: was Bagehot right after all? Journal of the European Economic Association 2:1116–47 (chapter 8).

Vives, X. 1988. Aggregation of information in large Cournot markets. Econometrica 56:851–76 (chapters 1 and 2).

——. 1990. Trade association disclosure rules, incentives to share information and welfare. RAND Journal of Economics 21:409–30 (chapter 1).

——. 1993. How fast do rational agents learn? Review of Economic Studies 60:329–47 (chapters 6 and 7).

——. 1995. Short term investment and the informational efficiency of the market. Review of Financial Studies 8:125–60 (chapters 4 and 8).

——. 1995. The speed of information revelation in a financial market mechanism. Journal of Economic Theory 67:178–204 (chapters 4 and 9).

——. 1996. Social learning and rational expectations. European Economic Review 40:589–601 (chapter 6).

——. 1997. Learning from others: a welfare analysis. Games and Economic Behavior 20:177–200 (chapter 6).

——. 2002. Private information, strategic behavior, and efficiency in Cournot markets. RAND Journal of Economics 33:361–76 (chapter 2).

——. 2005. Complementarities and games: new developments. Journal of Economic Literature 43:437–79 (chapter 8).

¹ See Radner (1979) and Grossman (1989).

² See Shleifer (2000), Shiller (2005), and Thaler (2005) for accounts of behavioral finance models.

³ Wikis are web pages that can be freely edited by any user with access to them. A leading example is Wikipedia, a free online encyclopedia that can be edited by anyone. In open source projects the source material (e.g., computer code) is available freely for other people to use and improve (typically under the condition that the outcome will also be made freely available). See Sunstein (2006).

⁴ See Plott and Sunder (1988) and Forsythe and Lundholm (1990) for experimental evidence on information aggregation by prices, and Wolfers and Zitzewitz (2004) for an introduction to prediction markets. Forsythe et al. (1992) and Berg et al. (2005) show that prices in the Iowa Electronic Market for presidential elections were typically closer to the actual vote shares than opinion polls. According to Chen and Plott (2002) even the price forecasts in the (relatively small) internal market set up by Hewlett-Packard were closer to the actual sales of the company than the official forecast.

⁵ From the introduction to the Becker–Posner blog at www.becker-posner-blog.com/archives/2004/12/: Blogging is a major new social, political, and economic phenomenon. It is a fresh and striking exemplification of Friedrich Hayek’s thesis that knowledge is widely distributed among people and that the challenge to society is to create mechanisms for pooling that knowledge. The powerful mechanism that was the focus of Hayek’s work, as of economists generally, is the price system (the market). The newest mechanism is the ‘blogosphere’.

⁶ See Tversky and Kahneman (1971, 1974), Kahneman et al. (1982), Camerer (1995), and Rabin (1998) for surveys on error-prone heuristics (such as making strong inferences from small samples—law of small numbers, confirmatory bias—where agents ignore information that contradicts beliefs held, and representativeness, where agents tend to neglect prior probabilities). Griffiths and Tenenbaum (2006) present recent evidence of the Bayesian optimality of human cognition in realistic scenarios, and Anderson and Holt (1997), Hung and Plott (2001), Cipriani and Guarino (2005), and Drehmann et al. (2005) evidence in experimental settings.

⁷ Sandroni (2000, 2005) shows that Bayesian agents drive behavioral non-Bayesian agents out of the market (see the discussion in section 7.1.1). See Jamal and Sunder (1996) and Ackert et al. (1997) for experimental evidence.

⁸ See Fudenberg and Levine (1998) for a treatment of this topic.

⁹ In the macroeconomics field, see Evans and Honkapohja (2001) for a treatment of adaptive learning models and Ljungqvist and Sargent (2000) for an exhaustive study of recursive methods. See Guesnerie (2001, 2006) for an assessment of rational expectations models.

¹⁰ The reader is referred to Hasbrouck (2007) for a recent survey of empirical market microstructure models.

¹¹ See Banerjee (1992), Bikhchandani et al. (1992), and Chamley (2004) for an assessment.

¹² The term market microstructure in finance was coined by Garman (1976) in a piece of work about market making. Brunnermeier (2001) provides a nice introduction to market microstructure models in finance.

1

Aggregation of Information in Simple Market Mechanisms: Large Markets

The aim of this chapter is to provide an introduction to the most basic models of information aggregation: static simple market mechanisms where traders do not observe any market statistic before making their decisions. Each agent moves only once, simultaneously with other agents, and can condition his action only on his private information. The issue is whether market outcomes replicate or are close to the situation where agents have symmetric information and share the information in the economy. Examples of such market mechanisms are one-shot auctions and quantity (Cournot) and price (Bertrand) competition markets. In chapters 3–5 we will consider market mechanisms in the rational expectations tradition in which agents can use more complex strategies, conditioning their actions on market statistics. For example, a firm may use a supply function as a strategy, conditioning its output on the market price (chapter 3), or a trader may submit a demand schedule to a centralized stock market mechanism (chapters 4 and 5).

The plan of the chapter is as follows. Section 1.1 introduces the topic of information aggregation, some modeling issues, and an overview of results. Section 1.2 analyzes a large Cournot market with demand uncertainty and asymmetric information. It studies a general model and two examples: linearnormal and isoelastic-lognormal. Section 1.3 examines the welfare properties of price-taking equilibria in Cournot markets with private information. Section 1.4 presents a general smooth market model to examine information aggregation and the value of information. Section 1.5 deals with the case of auctions and section 1.6 introduces endogenous information acquisition.

1.1 Introduction and Overview

1.1.1 Do Markets Aggregate Information Efficiently?

As we stated in the Introduction and Lecture Guide, this has been a contentious issue at least since the debate between Hayek and Lange about the economic viability of socialism.

Hayek’s basic idea (1945) is that each trader has some information that can be transmitted economically to others only through the price mechanism and trading. A planner cannot do as well without all that information. We say that the market aggregates information if it replicates the outcome that would be obtained if the agents in the economy shared their private information. The Hayekian hypothesis to check is whether a large (competitive) market aggregates information.

In this chapter we study, as a benchmark, market mechanisms that do not allow traders to condition their actions on prices or other market statistics. In this sense we are making things difficult for the market in terms of information aggregation. In chapter 3 we will allow more complex market mechanisms that allow traders to condition on current prices. Market mechanisms such as Cournot or auctions have the property that when a participant submits his or her trade it can condition only on his private information. For example, in a sealed-bid auction a bidder submits his bid with his private knowledge of the auctioned object but without observing the other bids; in a Cournot market firms put forward their outputs with some private estimate of demand conditions but without observing the market-clearing price or the outputs submitted by other producers.

A first question to ask is whether those simple market mechanisms aggregate information, at least when markets have many participants. We must realize, however, that a large market need not be competitive but rather can be monopolistically competitive. Indeed, firms may be small relative to the market but still retain some market power. If a large market is competitive and it aggregates information, then first-best efficiency follows according to the First Welfare Theorem. This means, in particular, that the full-information Walrasian model may be a good approximation to a large market with dispersed private information. In this case informational and economic efficiency go hand in hand. When a large market is not competitive, informational efficiency will not imply in general economic efficiency. We will discuss in detail the relationship between informational and economic efficiency in chapter 3.

Information aggregation does not obtain, in general, in market mechanisms in which outcomes depend continuously (smoothly) on the actions of the players, like quantity (Cournot) or price (Bertrand) competition markets with product differentiation. In fact, why should a market in which each trader conditions only on its private information be able to replicate the shared-information outcome? We will see that a large Cournot market in a homogeneous product world with a common shock to demand in which each producer receives a private signal about the uncertain demand does not aggregate information in general despite firms being approximately price takers. However, in the same context information may be aggregated if there are constant returns or if uncertainty is of the independent-values type, when the types of traders are independently distributed and, in fact, in the aggregate there is no uncertainty.

In contrast, winner-takes-all markets like auctions or voting mechanisms, in which outcomes do not depend continuously on the actions of players, tend to deliver aggregation of information more easily. Winner-takes-all markets force traders to condition effectively on more information when taking their decisions.

1.1.2 Methodological Issues and Welfare

In the following sections we use the continuum model of a large market. We will examine games with a continuum of agents in which no single one of them can affect the market outcomes. This methodology is in line with the literature on large Cournot markets with complete information (Novshek 1980), and with the view that the continuum model is the appropriate formalization of a competitive economy (Aumann 1964). The advantage of working with the continuum model is that it is very easy to understand the statistical reasons why a competitive market with incomplete information does not aggregate information efficiently in general, and to characterize the equilibrium and its second-best properties. One must check that the equilibria in the continuum economy are the limit of equilibria of finite economies and not artifacts of the continuum specification. This will be done in chapter 2.

The analysis of competitive equilibria in asymmetric-information economies is somewhat underdeveloped. To start with, the very notion of competitive equilibrium needs to be defined in an asymmetric-information environment.¹ If the market aggregates information, then the competitive equilibrium corresponds to the standard concept with full information. Otherwise, we may define the concept of Bayesian (price-taking) equilibrium. This is the situation, e.g., in a Cournot market, where firms’ strategies depend on their private information but a firm does not perceive to affect the market price. This will be justified if the firm is very small in relation to the market, that is, in a large market.

Whenever the outcome of a large market, in which agents are price takers, is not first-best efficient, the question arises about what welfare property, if any, it has. The answer is that a price-taking Bayesian equilibrium maximizes expected total surplus subject to the restriction that agents use decentralized strategies (that is, strategies which depend only on the private information of the agents). This welfare benchmark for economies with incomplete information is termed team efficiency since the allocation would be the outcome of the decision of a team with a common objective (total surplus) but decentralized strategies (Radner 1962). This means that the large market performs as well as possible, subject to the constraint of using decentralized mechanisms. We will see this result in section 1.3.

1.2 Large Cournot Markets

In this section we will consider a large homogeneous product market in which demand is affected by a random shock and each firm has a private estimate of the shock and sets an output. This is a very simple and standard framework in which to analyze information aggregation. Quantity setting corresponds to the Cournot model of an oligopolistic market. Here we will have many firms and each will be negligible in relation to the size of the market.

There is a continuum of firms indexed by i ∈ [0, 1].² Each firm has a convex, twice continuously differentiable, variable cost function C( · ) and no fixed costs. Inverse demand is smooth and downward sloping, and given by p = P(x, θ), where x is average (per capita) output (and also aggregate output since we have normalized the measure of firms to 1) and θ is a random parameter. Firms are quantity setters.

Firm i receives a private signal si, a noisy estimate of θ. The signals received by firms are independently and identically distributed (i.i.d.) given θ and, without loss of generality, are unbiased, i.e., E[si | θ] = θ.

We make the convention that the strong law of large numbers (SLLN) holds for a continuum of independent random variables with uniformly bounded variances. Suppose that (qi)i∈[0,1] is a process of independent random variables with means E[qi] and uniformly bounded variances var[qialmost surely (a.s.). This convention will be used, taking as given the usual linearity property of the integral.³ In particular, here we have that, given θ, the average signal equals E[si | θ] = θ (a.s.).

Section 1.2.1 characterizes the equilibrium with strictly convex costs, section 1.2.2 its welfare properties, section 1.2.3 deals with the constant marginal cost case, and section 1.2.4 provides some examples.

1.2.1 Bayesian Equilibrium

Suppose that C( · ) is strictly convex. A production strategy for firm i is a function Xi( · ) which associates an output to the signal received. A market equilibrium is a Bayes–Nash (or Bayesian) equilibrium of the game with a continuum of players where the payoff to player i is given by

and the information structure is as described above.⁴ At equilibrium, Xi(si) maximizes

E[πi | si] = xiE[P(x;θ) | si] − C(xi)

and the firm cannot influence the market price P(x; θ) because it has no influence on average output x. This is, therefore, a price-taking Bayesian equilibrium. Restricting our attention to strategies with bounded means and uniformly bounded variances across players (this would obviously hold, for example, with a bounded output space), the equilibrium must be symmetric. In particular, with E[Xi(si) | θ] < ∞ and var[Xi(si) | θ] uniformly bounded across players, the random variables Xi(si) are independent (given θ) and, according to our convention on a continuum of independent random variables, we have that

It follows that the equilibrium must be symmetric, Xi(si) = X(si) for all i, since the payoff is symmetric, the cost function is strictly convex and identical for all firms, and signals are i.i.d. (given θ, the best response of a player is unique and identical for all players: Xi(si) = X(si). Consequently, in equilibrium the random variables X(si) will be i.i.d. (given θ.

An interior (symmetric) equilibrium X ( · ), that is, one with positive production for almost all signals, is characterized by the equalization of the expected market price, conditional on receiving signal si, and marginal production costs:

E[P(x; θ) | si] = C′(X(si)).

This characterizes the price-taking Bayesian equilibrium. Firms condition their output on their estimates of demand but they do not perceive, correctly in our continuum economy, any effect of their action on the (expected) market price. Thus the price-taking Bayesian equilibrium coincides with the Bayesian Cournot equilibrium of our large market.

1.2.2 Welfare and Information Aggregation

If firms were to know θ, then a Walrasian (competitive) equilibrium would be attained and the outcome would be first-best efficient. How does the Bayesian market outcome compare with the full-information first-best outcome, where total surplus (per capita) is maximized contingent on the true value of θ?

Given θ and individual production for firm i at xi, total surplus (per capita) is

If all the firms produce the same quantity, xi = x for all i, we have

Given strict convexity of costs, first-best production X°(θ) is given by the unique x which solves P(x; θ) = C′(x). If firms were able to pool their private signals, they could condition their production to the average signal, which equals θ a.s., and attain the first-best by producing X°(θ). Will a price-taking Bayesian equilibrium, where each firm can condition its production only on its private information, replicate the first-best outcome?

A necessary condition for any (symmetric) production strategy X ( · ) to be first-best optimal is that, conditional on θ, identical firms produce at the same marginal cost, namely, C′(X(si))= C′(X°(θ)) (a.s.). However, with increasing marginal costs this can happen only if X(si)= X°(θ) (a.s.), which boils down to the perfect-information case. Therefore, we should expect a welfare loss at the price-taking Bayesian equilibrium with noisy signals. Proposition 1.1 states the result and a proof follows.

Proposition 1.1 (Vives 1988). In a large Cournot market where firms have (symmetric) strictly convex costs and receive private noisy signals about an uncertain demand parameter, there is a welfare loss with respect to the full-information first-best outcome.

Proof. We show that no symmetric production strategy X , then expected total surplus (per capita) contingent on θ is given by

We have that

The first inequality is true since X°(θ) is the first-best, and the second is true since

, and the signals are noisy (which means that, given θ, X(si

1.2.3 Constant Marginal Costs

A necessary condition for the market outcome to be first-best optimal is that marginal costs be constant. Firms will then necessarily produce at the same marginal cost. If the information structure is regular enough, first-best efficiency is achieved in the price-taking limit (Palfrey 1985; Li 1985). The intuition of the result is as follows. Suppose that marginal costs are zero (without loss of generality) and that inverse demand intersects the quantity axis. Firm i will maximize E[πi | si] = xiE[P(x; θ) | si], where, as before, x is average output. In this constant-returns-to-scale context a necessary condition for an interior equilibrium to exist is that E[P(x; θ) | si] = 0 for almost all si. Looking at symmetric equilibria, Xi(si) = X(si) for all igiven θ for almost all sifor almost all θ.

Palfrey (1985) shows that if the signal and parameter spaces are finite and the likelihood matrix is of full rank (and demand satisfies some mild regularity conditions), then symmetric interior price-taking equilibria are first-best optimal. The result is easily understood with a two-point support example: θ can be either θL or θH, 0 < θθH, with equal prior probability. Firm i may receive a low (sL) or a high (sH) signal about θ with likelihood P(sH | θH) = P(sL | θL) = q, the signal is uninformative; if q be, respectively, the equilibrium prices when demand is high (θH) and low (θL). Then

and the symmetric interior equilibrium is efficient.

Remark 1.1. For the result to hold, equilibria have to be interior. For example, suppose that p = θ − x and that θL = 0 and θH = 1, then at the price-taking Bayesian equilibrium X(sL) = 0 and X(sH) = q/(q² +(1 − q)²).