Multidimensional Uncertainty and Herd Behavior in Financial Markets
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Peter Zemsky, Christopher Avery
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Title:
Multidimensional Uncertainty and Herd Behavior in Financial Markets
Author:
Peter Zemsky, Christopher Avery
Year:
1998
Publication:
The American Economic Review
Volume:
88
Issue:
4
Start Page:
724
End Page:
748
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Language:
English
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Updated: August 23rd, 2012
Abstract:
Multidimensional Uncertainty and Herd Behavbsr in Financial Markets
By CHRISTOPHERAVERYAND PETERZEMSMY
*
We study the relationship between asset prices and herd behavior, which occurs when truders follow the trend in past trades. When traders fmve ~rivnte informlion on only a single dimension of uncertainty (the effect of a shock to the asset value), price adjustments prevent herd behavior. Herding arises when there are two dimen sions of uncertainty (the existence and effect of a shock), but it need not distort prices because the market discounts the informativeness of trades during herding. With a third dimension of uncertuinty (the quali~y of traders' infommtion), herd behavior can lend to a signgcant, shortrun mispricing. (JEL 612, C14, D83, D84)
In standard models of general equilibrium, tional cascade occurs in finite time with prob the simultaneous execution of a large number ability 1. That is, social learning completely of trades produces efficient outcomes, presum breaks down as all consumers from some time ing that the Walrasian auctioneer has set prices forward make the same choice and reveal no correctly. As a challenge to these results and new information. Because that choice is wrong the associated view that decentralized markets with strictly positive probability, the equilib tend to be efficient, an explosion of papers in rium of these sequential market games is inthe last several years argue that imitative or efficient, even in the long run. herdlike behavior can impede the flow of in The herding literature recalls a once formation in an economy when consumers act prominent view of asset markets as driven by sequentially rather than concurrently. [Abhijit "animal spirits," where investors behave like Banerjee (1992), Sushi1 Bikhchandani et al. imitative lemmings. While the rational actor ( 1992), Christophe Chamley and Douglas approach has largely driven this view from Gale ( 1992), Andrew Caplin and John Leahy mainstream research in financial economics, it (1993, 1994), and Jeremy Bulow and Paul is far from gone. Both market participants and Klemperer ( 1994)l. With sequential actions, financial economists reportedly still believe the earliest decisions can have a dispropor that imitative behavior is widespread in finan tionate effect over longrun outcomes in the cial markets (Pnndrea BlBevenow and Ivo economy. A slight preponderance of pub Welch, 1996). 'ffhis has led some researchers lic information is sufficient to induce all to assert that market participants engage in agents to follow the lead of the market, com nonrational herd behavior (e.g., Andrei pletely ignoring their private information. Shleifer and 1,awrencs H. Summers [I9901; Bikhchandani et al. (BHW) describe that sit Alan Kirman [I 9931). uation as an "informational cascade." In We investigate the relationship between ra BHW and Banerjee9s models, an informa tional herd behavior and asset prices. Past
work on rational herding is not well suited to address this relationshit) because. in almost all cases, herding models kix the price for taking
" Avery: Kennedy School of Government, Harvard an action ex ante, retaining that price ~nflexibly
University, Cambridge, MA 02138; Zemsky: INSEAD,
under all circumstances.' Wc address the fol
Boulevard de Constance, 77305 Fontainebleau, Cedex, France. We are grateful for comments and advice from Anat Admati, Darrell Duffie, Glenn Ellison, Drew Fudenberg, Jim Nines, Allan Kleidon, Paul Pfleiderer, ' An exception is Bulow and Kiemperer's model, hut Lones Smith, Peter Sorensen, Xavier Vives, Richard it still fixes the price after each purchase for a sufficient Zeckhauser, and especially Paul Milgrom. period to produce herciing.
724
lowing questions: Can there be informational cascades in financial markets? Can herd be havior lead to the longrun mispricing of as sets? Does it produce bubbles and crashes? Might it offer an explanation for excess vola tility? We begin our analysis with an example which motivates a final question.
I. A Simple Example
Our model retains the basic features of the simplest model considered by BHW (p. 996), with the notable addition of a mice mecha nism. It is useful to review that model and to consider what happens when prices are allowed to vary over time in response to agents' actions2 In BHW, agents face a choice of whether or not to adopt a new technology, and the cost of adoption is fixed at c .= 'I,. The value of the new technology, denoted V, is ei ther 1 or 0. Each agent gets an independent, imperfect signal about V,denoted x E ( 0, 1} , where P (x = V) = p > '1,. Agents act se quentially and observe H,, the history of ac tions up until time t. Let n:= P (V = 11Ift). The choice made by an agent depends on whether the expected value of adopting is greater than c. Consider the expected value of an agent with bad news (the value for an agent with good news is similar):
Assuming all prior agents have acted in accord with their signal, n:increases with the differ ence between the number of prior agents who adopted and those who did not. Indeed, when ever there are two more adopters than non adopters, it is the case that V' (x = 1 ) > Vt (x = 0) > '1,. Then agents at time t adopt regardless of their signal and an informational cascade begins.
Now suppose that the agents are traders in a financial market and that their choice is
We use notation consistent with the rest of this paper. In comparing our results to those in BHW, our reference is to the simple model described here.
whether to buy or sell a unit of an asset where the lrue value of the asset is given by I/. Further. suppose that the financial market is in formationally efficient in that the cost of a unit of the asset reflects all publicly available information:
'The key observation from this simple exercise is that
The asset price adjusts precisely so that there is no herding and agents always trade in accord with their signal! Reflecting on Adam Smith's invisible hand, it is not too surprising that an arbitrary fixed price leads to herd behavior and the persistence of inefficient decisions in an economy. We conclude thal whether or not herd behavior affects asset prices, asset prices can certainly affect herd behavior. In this ex ample, they completely eliminate it. Given the reported prevalence of herd behavior in finan cial markets, this raises the irnportant question of whether herd behavior is consistent with a market composed of rational traders.
PI. Overview of the Paper
In Section I11 we describe a general model and define terms. Of particular importance, we define herd behavior as a trade by an informed agent which follows the trend in past trades even though that trend is counter to his initial infom~ation about the asset value.
In Section IV, we show that there are limits to the distortions that can arise in a financial market where informed traders are rational ac tors and prices incorporate all publicly avail able information. We show that informational cascades are impossible: at any point in time there is always the possibility that new infor mation reaches the market. Consistent with this steady flow of information, prices always converge to the true value. Hence, herd be havior can cause no longrun mispricing of as sets. We show that the ex ante expected volatility in prices Is determined by funda mentals, which means that herd behavior can not be the source of excess volatility. Finally, we generalize the example from Section I by stating a monotonicity condition on private signals such that herding is impossible. With monotonic signals, there is only a single di mension of uncertainty confronting the mar ket, which we term value uncertainty.
In Section V we exhibit a plausible infor mation structure in which herding does occur, by adding event uncertainty to value uncer tainty. With event uncertainty, the market is uncertain as to whether the value of the asset has changed from its initial expected value, We show that any amount of event uncertainty produces the possibility of herd behavior. As event uncertainty becomes extreme (i.e., the probability that the asset has not changed value goes to 1 ), there is an arbitrarily long period of herd behavior when the asset value changes. This herd behavior is similar to the informational cascade of BHW in that the mar ket does not learn about whether the asset value is high or low as all informed traders either buy or sell. Surprisingly, this extreme herd behavior has little effect on asset prices. We show that the movement in the asset price is bounded and that this bound can be small. Finally, we argue that herd behavior is not clearly at odds with optimal social learning in financial markets.
Given the above results, one might expect that we find no connection between herd be havior and market crashes. However, this is not the case. In Section VT we investigate the combination of event uncertainty with what we term composition uncertainty, which means that there is uncertainty as to the aver age accuracy of traders' information. We are then able to identify certain (highly unlikely) states of the world in which herd behavior can lead to a price bubble and crash. In these states, market participants have a mistaken, but rational, belief that most traders possess very accurate information. Then, market par ticipants have trouble differentiating between a market composed of wellinformed traders and one with poorly informed traders who are herding :in each case, there is a preponderance of activity on one side of the market. The re sulting confusion allows uninformative herd behavior to have dramatic effects on prices. Our theory of price bubbles resembles the ex planation advanced by Sanford J. Grossman (1988) and Charles Y. Jacklin et al. (1992) for the stock market crash of 1987: traders under estimated the prevalence of noninfornlative computerbased insurance trading.
Based on these results, we conclude that de spite the significant constraints imposed by a rational financial market, herd behavior is robust to the operation of the price mechanism. In particular, as the number of dimensions of uncertainty with which the price mechanism must contend increase, herding becomes prev alent and extreme effects based on herd be havior occur in identifiable (but unlikely) states of the world.
In Section VII we consider the converse of herding. 'contrarian behavior," where agents ignore their private information about value uncertainty to trade against the trend in past trades. We show that composition uncertainty can give rise to such behavior. The existence of herd and contrarian behavior rationalizes the observed practice of price charting. Sec tion VlII concludes.
111. The General Model
A. Description
We begin by specifying a general model; we will add further assumptions in later sections, The market is for a single asset with true value V,which is restricted to be in 10,11. Prices are set by a competitive market maker who inter. acts with an infinite sequence of individuals chosen from a continuum of traders. Each trader is risk neutral and has the o~tion to buy or sell one unit of stock or to refrain from trad ing. The sequence of traders is indexed by t = 0, 1, 2, . .. . We denote by H, the publicly observable history of trades up until time t.
There are two broad classes of traders. In formed traders receive private information and maximize expected profit at the market maker's expense, while noise traders act for exogenous motives and without regard for ex pected profit.' Let p < 1 be the probability sf an informed trader arriving in any given pe
'Without the presence of noise traders, the notrade theorem of Paul Milgrom and Nancy Stokey (1982) applies and the market breaks down.
riod; 1 p is the probability of a noise trader arriving. For convenience, we assume that noise traders buy, sell, and do not trade with equal probability y = ( 1 p)/3.
Informed traders receive private informa tion X, E [0, l], where x, is drawn from the distributionfH(x, I V) and 0 is a trader's type.4 We assume a finite number of possible types and we denote the probability that a trader of type 0 arrives by p, > 0. The expected value of an informed trader is denoted V',(x) = E[V I HI, xo = x] . The market maker's expected value for the asset given public infor mation is denoted Vb, = E[V I H,] , which we shall sometimes refer to as the price.5
We assume that there is always a minimal amount of "useful" information in the mar ket. That is, as long as past trading does not identify the value perfectly, then there is strictly positive probability that some trader has an assessed value that differs from the market maker's (by a nontrivial amount). More precisely, we assume that if there does not exist a v such that P (V = v 1 H,) = 1, then there exists at least one 0 and set of signal realizations R C [0, 11 with P(x~ ER I H,) > 0 such that V&(xo) f VI, for X,E R. Moreover, if I VI, V ( = 6 > 0 then for some ~(6)> 0, (VQ(x8) VI,( > ~(6).
The market maker allows for adverse selec tion by setting a (bidask) spread between the prices at which he will sell and buy a unit of stock. Perfect competition among market mak ers restricts the market maker to zero profits at both the bid and ask prices. That is, the trader who arrives in period t faces a bid, B', and an ask, A', which satisfy:
Thus, a trader potentially has two pieces of private information to trade onthe value of xH E [O, 11 and his type 0. A trader's type constitutes private infonnation if there is uncertainty about the composition of the market. See Section VI for details.
We do this when we want to abstract from the exis tence of the bidask spread in interpreting our results.
where h, is the action taken by the trader who arrives in period t, with h, = B indicating a buy, h, = S indicating a sell, and h, = NT indicating no trade. Finally, we define the market maker's assessed distribution for the possible values as TI, = P (V = v (HI). By Bayes' theorem, these priors respond to trade as follows:
P(h,(V == u, H,)
(1) 7r;+'=T:,
P(h', HI)
where P(h,, Hz) = 2, nl B(h,(V= u, H,).
Our model is a special! case of the model developed by Lawrence Glosten and Milgrom (1985) with the notable simplifi cation that our noise traders have completely inelastic demand. Because our noise traders are willing to absorb any amount of losses, the market never breaks down due to adverse selection and zero profit equilibriuln prices always exist.
PROPOSITION 1: In each period t there ex ist unique bid and ask prices which satisfy B' 5 VI, 5A'. VI, and TI, are martingales with respecl to ff,.
PROOF: See Appendix.
The market maker accounts for the infor mation which is contained in buy and sell or ders in setting prices. Thus, A' > VI, and B' I V:, .V:,, and ni are expectations based on all of the information contained in the prior history of trade, HI. Therefore, they are martingales with respect to HI; if this were not the case, then the market maker's assessment of Vl, and 7rt would be systematically mistaken in a man ner which should be predictable to him.
B. The Definition of Herd Behavior
We differentiate between an informational cascade and herd behavior. In the example of Section I, herd behavior always implies an in formational cascade. With the simple infor mation structure used there, no information reaches the market when traders with bad sig nals (x = 0) and those with good signals (x = 1) are taking the same action. However, in a more general model, imitative behavior need not imply an informational cascade.
Dejinitioil I: An informational cascade occurs in period t when
In an informational cascade, no new infor mation reaches the market because the dis tribution over the observable actions is independent of the state of the world. In par ticular, this happens when the actions of all informed traders are independent of their pri vate information, such as when they are all buying (in all states of the world).
De$nition 2: A trader with private infoma tion x, engages in herd belzavior at time t if he buys when Vj(xH) <: Vf,< V:,,or if he sells when Vi(x,) > V!,> V:,,;and buying (or sell ing) is strictly preferred to other actions.
Herd behavior by a trader satisfies three properties, which we discuss for the case of herd buying. First, it must be that initially (be fore the start of trade) a trader's information leads him to be pessimistic about the value of the asset so that he is inclined to sell: V;(x,> < V:,. Second. the history of trading must be positive: Vg, <V:,,.Finally, the trader must want to buy given this positive history and his signal, which implies that V:, 5 A' < Vi(x~).These three properties demonstrate the extreme nature of herd behavior. Initially, the trader's signal constitutes negative informa. tion, causing him to reduce his assessment of the asset's value. Yet, after observing the trad ing history, the signal constitutes positive information, causing him to increase his as sessment of the asset's value from V:,.
In our definition. herd behavior occurs when agents imitate the prior actions (buying or sell ing) of others. An alternative approach is to define herding as a socially inefficient reliance on public information (see Xavier Vives, 1994) .' In contrast, we start with a behavioral
"n settings where agents learn from the actions of oth ers, an informational externality naturally arises in that future agents benefit when earlier agents take actions that reveal their private information. Hence the connection be
definition of herding and then study the extent to which such behavior leads to distortions and inefficiencies.
ilV, Bounds on the Effect of Herd Behavior
Asset prices have a profound effect on herd behavior. As suggested by our earlier example, the price mechanism eliminates the possibility of infonnational cascades.
PROPOSITION 2: An informational cascade never occurs in market ey~ilibriurn.~
PROOF: See Appendix
Our assumption of minimal useful infor mation implies that there is always private in formation in the economy. As long as private information exists, some traders must base their trading strategy on that information, but this assures that the distribution over observed actions is not independent of the state. Hence, an informational cascade is impossible. Like several of our results, Proposition 2 relies on a basic intuition about our model: informed trade is driven by information asymrnetndes be tween traders and the market maker.
Proposition 2 requires limited frictions in the market. Otherwise, trade and the flow of information can stop. In Ho Lee ( 1995) shows that informational cascades arise if there are transaction costs. Over time, the expected profit of informed traders declines to zero as the asset price becomes more accurate. If there are transaction costs totrading, informed trad ers will (almost surely) stop trading at some point. Then, no new information reaches the market. Similarly, in the original Glosten and
tween efficjency and herd behavior, which can obscure private infor1n;ition. There are two drawbacks to using an cfficicncy basi:d dcfinition of herd behavior for studying financial markets. Firsi, it requires a welfare benchmark, which is generally lacking in asymmetric information models of asset markets. Second, traders can place a very high weight on public information without exhibiting the sort of strongly imitative behavior studied here (see the work of Vivcs, 1995, 1997).
? We are very grateful to an anonymous referee for sug gesting this result.
Milgrom paper, informational cascades arise if the market breaks down due to adverse
selection.
Consider the following restriction on the private information in the economy.
Dejinition 3: A signal x, is monotonic if there exists a function u(x#) such that VA(x,) is al ways (weakly) between u(x,,) and V:,,for all trading histories H,.
Monotonic signals are particularly well be haved. Given any public information, they al ways move a trader's expected value towards some fixed valuation, u(xti). Monotonic sig nals are pervasive in the literature on asym metric information in financial markets. For instance, the signals in the example of Section I, which are often used in GlostenMilgrom style models, are monotonic because V(x) E [x, Vk,].In addition, noisy rational expecta tions models (e.g. Grossman and Joseph E. Stiglitz, 1980)require monotonic signals for tractability. We now show that it is the ubiq uitous assumption of monotonic signals that explains the absence of herd behavior in the received literature on the microstructure of financial markets.
PROPOSITION 3: A trader with a mono tonic signal never engages in herd behavior.
PROOF:
Suppose a trader with a monotonic signal xU engages in herd buying at time t. Then Vk (no) >A' 2 V:,,. Since the signal is mono tonic, this implies that ~(x,) > a/:, . But then Vj(x,) > Vjl, and the trader was not originally pessimistic. This is a contradiction. Similarly, herd selling never occurs.
With monotonic signals, a trader who wants to buy when the price has risen must also want to buy initially, which assures that any buying is not herding.' If we abstract from the exis
Vives (1995) develops a dynamic noisy rational ex pectations model which complements our analysis. Con sistent with Proposition 3 and his use of monotonic signals, traders in Vives' model never engage in herd be
tence of a bidask spread (as when p is small), agents with monotonic sagnals have particu larly simple trading strategies. They buy if u(x,,) is above the price, V:,,,and sell if it is below that. Then, traders need not concern themselves with the trading history at all! This rules out herd behavior, since it leaves no room for the trend in the trading history to in fluence trading. When traders have monotonic signals, we say that there is only a single di mension of uncertainty in the market. Our mo tivation is that a scalar, VL,,can summarize for traders all the information they need to extract from the trading history. We label this single dimension of uncertainty as value uncertainty, as it relates directly to the underlying value of the asset.'
In Section V we show that there exist plau sible nonmonotonic signals which produce herd behavior. However, we now show that the effect of herd behavior on prices must be limited. The impossibility of inforniational cascades implies that each period of trade re veals some information even if there is herd behavior. Since there is a continual flow of information, it is natural that the trading price must converge to the true asset value.
PROPOSITION 4: The bid and ask prices converge almost surely to the true value V.
PROOF: See Appendix.
Glosten and Milgrom (1985) show that the bid and the ask prices must converge together as long as the market does not break down. Hence, all private information becomes public

havior as defined here. They always buy if the value of their signal is above the price and sell if it is below. The amount that they buy or sell does change over time as public information accumuiates.
we do not make precise our notion of "dimensions" of uncertainty, but leave it as an intuitive construct that we find useful for interpreting our results. We shall speak of the asset price as having a single dimension in our model, even though technically thrre is both a bid and an ask price. We think of multidimensional prices as arising when there are derivative securities (such as options) that are traded.
over time. Convergence is then a direct con sequence of our assumption that a nontrivial amount of private information always exists so long as the true value is not yet identified by the trading history.
While this convergence result is not new, it has significant implications for the applicabil ity of results from the recent herding literature to financial markets. Price convergence di rectly rules out the sort of longrun inefficien cies found in earlier herding papers. Further, when coupled with the martingale property of prices, convergence provides a bound on the volatility of prices. We denote the change in the market maker's expectations from one pe riod to the next as AV:,, = V:,,Vi; l.
COROLLARY 1: The variance of price paths is bounded as follows.'"
Additionally, for a $.xed t,
Var (V VI:,) = Var (V) Var (V::,) .
PROOF: See Appendix.
The first part of Corollary 1 states that the expected volatility is bounded by the funda. mental uncertainty over V. Hence, it is not possible to explain volatility in excess of fun damentals in our general model, whether or not there is herd behavior. In addition, as time passes, the "remaining variance" in the price change process diminishes, so that V:,, must be more and more accurate over time, as implied by the second part of the corollary. That is a natural property with an important implica tion: any set of volatile price paths must either converge quickly to the true value or (as the next corollary emphasizes) they can only oc cur with small probability.
COROLLARY 2: Consider some a < V:,,. Then P(V < alH,) 5 (1 V:,,)/(l a).
'"We are grateful to Paul Milgroin for suggesting this result.
PROOF: See Appendix.
This result implies that high prices occur only rarely when asset values are low. Consider a market where V E ( 0, 1 ) as in BHBV. Corollary 2 implies that P (a/ 0)= 1 a/:,(i.e., as the piice V:,,goes to one, tile probability that V =: 0 goes to zero). This result limits the probability of a pxke bubble, which is a situation where the asset value movesfax away from its true value. In general, there is an inverse relationship be. tween the magnitude of a price bubble and the probability with which it occurs. In particular, extreme price bubbles, where the asset incor rectly attains its n~aximurn possible value, are zero probability events.
V. Event Uncertainty and Herd Behavior
A. Existerace Herd Behavior
Proposition 3 poses a puzzle. How do we reconcile the reported prevalence of herd be havior with its absence in a rational financial market with monotonic signals? Aclosely re lated puzzle is the existence of price "chart ing," where traders use detailed charts of price histories in their trading strategies (David Brown and Robert Jennings,1989). Charting is puzzling because the trading history plays at most a limited role in a trader's strategy when he lras a monotonic signal. For any his. tory, the set of poteritial buyers (traders who are more optimistic then the market maker) is given by the condition v(x,) > A', while the set of potential sellers is given by the conditionv(x~,)< W'."
While it is certainly not difficult to specify nonmonotonic signals, the more interesting question is whether such signals are likely to be common in fillancia1 markets. Consider
" Hence, the only role of the trading history is in help ing traders to assess whether their private information is suffi~iei~tlystrong to justify trading given the bidask spread. We do not find this weak history dependence to be a satisfactory theory of price charting. We seek a rationalization based on strong history dependence, where the trading history drives a trader from buying to selling. as occurs under our definitions of herd and contrarian behavior.
then, that many shocks to an asset's value are not publicly known, at least initially. For ex ample, a trader might learn from a contact who works at a company that there will be a change in management, that a new product has been developed, or that a merger is being consid eredall before a public announcement. Then, the trader has private information about two "dimensions" of uncertainty. In addition to information related to value uncertainty
is it a good or a bad merger?the trader has private information that there has been a shock to the underlying value of the asset. We follow the finance literature and refer to this second dimension as event uncertain0 j David Easley and Maureen O'Hara. 1992). We offer the fol lowing formalization of event uncertainty.
De$nition 4: There is event uncertain@ when 1 > P (V = Vf,) > 0.12
We now extend the simple BEaW information structure used in our introductory example to incorporate event uncertainty. Traders are lnformed if there is an information event (i.e., V s Vf,) and if there is, then traders have signals as in BHW. Formdly, information structure I(IS I) is defined as follows. The true value of the asset is V E(0,'I,, 1 ) with initial priors satisfying T:,, > 0and T: = T: > 0.Then, V:, = 'I2and there is event uncertainty. There is a single type of informed trader with signal x,where
if V = 0,
P(x=O[V)=
1p ifV=1,
' > p > ",. If there were event uncertainty
"The event uncertainty that we study satisfies the ad ditional property that informed traders know whether or not an information event has occurred. That is, P (V =
V: 1 E 10, 11.
(i.e., T?,~= Q), then the above signals are monotonic. The addition of event uncertainty in this way makes signals nonmonotonic and herd behavior possible."
PROPOSITION 5: Under IS I, price paths with herd behavior occur with positive prob ability for p < 1. They do not occur for p =
1. Herd behavior is misdirected with positive probability. l4
PROOF: See Appendix.
In BHW, a preponderance of one action chosen by earlier agents leads others to believe that the action is a good one, regardless of their own private information. Similarly, a suffi cient excess of buys over sells in our model leads a trader to believe that the asset value is more likely to have gone up rather than down, regardless of his signal. However, with infor mationally efficient prices, rational individuals only act based on information asymmetries be tween themselves and the market maker. The history of trades can only be the source of asymmetric information if it is interpreted dif ferently by the market maker and the informed traders. With only value uncertainty, there is only a single interpretation of the history of trade and hence herd behavior is impossible.
With the addition of event uncertainty, in formed traders know that an information event has occurred, while the market maker does not. This information asyrnmetry gives the traders an advantage in interpreting the history of trades. They are quicker to adjust their valuation to the trend in past trades than the
"It is possible to have event uncertainty while pre serving the monotonicity of signals. For example, if V E
{0,'/,,1),P(x=V)=p,P(~=l/V=~/~)=P(x=
OlV = 'I2)= (I py2, P (x = II2/V= 1) = P (x = 'I2/ V = 0) = q,andp > q 2 ( 1 p)/2, then .u is mono tonic. Note that such a signal precludes informed traders from knowing with certainty whether or not an informa tion event has occurred, which we find to be a natural feature of event uncertainty,
I' Easley and O'Hara (1992) study IS I for the special case wherep = 1.Their focus is on how the market maker learns that an information event has occurred.
market maker, who must consider the possi bility that there has been no change in the un derlying value of the asset and the trend is due to noise traders. Thus, event uncertainty dulls price adjustment in the short run.
PROPOSITION 6: Conszder IS I and some tradzng history Hr that resulfi in priors IT: = P (V= vllP,). For ~r:= 71 j,, there is no herd behavior. For IT ', + n6,there exists a critical valuefir the precision of ttmden' signalsjf(p, .ir[,, .ir: > such that ~raders engage in herd be havior in period t if and only fp < jf. This p decreases with,u and increases with 7r:/, (holding n:/7rj, consfant). IT: > nil, then any herd behavior involves buying and ji increases with IT { /nj, (holding IT { /, constant). If x:< 716,then any herd behavior involves sell ing and p increases with IT h/n ;(holding n:/* constant).
PROOF: See Appendix.
Proposition 6 identifies the three forces that produce herd behavior. First, herd behavior re sults when the weight of infomation in the history of trade overwhelms an individual's private information about value uncertainty. A reduction in p reduces the information contained in a private signal about value uncer tainty, while an increase inIT: nb1 increases the amount of information contained in the history. Either change makes it easier for the trading history to overwhelm the in formation about value uncertainty in private signals and thus makes it easier for herding to arise. Second, herding occurs when prices be come sufficiently unresponsive to the trading history. As the probability of an information event decreases (LC., 7~:/, increases), prices respond less to the trading history and thus more of the information in the trading history is private.I5 Third, herding requires that the bidask spread not deter potential herders from
Note that the infolmation gleaned by informed trad ers from observing the trading history is fixed by .ir\lir:, regardless of the value of nl,?because they know whether or not an information event has occurred.
trading. A decrease in p reduces adverse se lection and leads to a tighter bidask spread.
B. Existence of Pronounced Herd Behavior
Proposition 5 shows that herding is possible for any p < 1 and IT,/^ > 0. We now show that as event uncertainty becomes extreme, herd behavior becomes pronounced, resembling the cascades of BHW.
PROPOSITION 7: Consider IS 1 with p < I and suppose that an information event occurs. In the limit as the probability ofan information event becomes arbitrarily small ( i.e., 7rYl2 4
1 ), the probability that there is some herd be havior in the trading history goes to 1. Moreover, the trading history almost surely bkes the following form: (i) a Jinite, initial period of trading during which herd behavior does not occur, (ii) an arbitrarily long period of herd behavior of one Qpe (i.e., always buy or always sell). This herd behavior is in the wrong direction with a strictly positive probability,
In the limit as p + 0, the probabiliiy qf herd behavior in the wrong direction goes to 1 p .
PROOF: See Appendix.
When an information event is very surpris ing (i.e., nil, close to I), the market maker discounts almost completely the infomative ness sf trading. The price remains fixed at the initial expected asset value of '1, for an arbi irarily long period of time. With a fixed price, our model almost recreates the BHW model and hence it is not surprising that cascadelike behavior arises. The only difference with BHW is the existence of noise traders.''
"The main effect of noise traders is to increase some what the probability of herding in the wrong direction. In BHW it takes two more adopters than nonadopters to start an informational cascade. Here the imbalance between
There is an important distinction between the herding of Proposition 7 and an informa tional cascade, which cannot occur in equilib rium. In an informational cascade, no new information reaches the market. Under the conditions of Proposition 7, behavior resern bles a cascade when there has been an infor mation event. For a very long interval, all informed traders act as buyers (or sellers). However, new information still reaches the market. Had there been no information event, informed traders would not be trading and the volume of trade and the imbalance in trade would both be less. Hence, the market maker is learning that there was an information event. After a sufficient period of time, the market maker learns enough about event uncertainty that prices adjust, which ends herding.
The addition of event uncertainty makes herd behavior possible and even extreme. However, herd behavior in IS I.no matter how extremedoes not distort the asset price. Herding keeps information about the new asset value from entering the market and rational actors (including the market maker) account for this.
PROPOSITION 8: Consider IS I. During any interval of trading in which there is herd behavior, the movement in the asset price is less than
In the limit as either y +0 or p + 'I2,A = 0.
buys and sells necessary to start herd behavior depends on the probability of a noise trader. The greater the probabil ity of noise traders, the greater the imbalance required. However, the key statistic for a cascadethe probability that it is in the wrong directionis very similar. In BHW, the probability of a wrong direction cascade is the prob ability that two agents have the wrong signal (1 p)Zl
(p2 + (1 p))'. Here X is between the probability that two traders have the wrong signal and the probability that just one trader has the wrong signal.
PROOF;: See Appendix
Herding is triggered when the information contained in a trade pushes the value of an informed trader past the bid or the ask. How ever, during the entire interval of herding that follows, the valuations of traders are fixed be cause they realize that no new information about value uncertainty is being revealed. Therefore, as soon as the price moves by more than the information contained in the last trade prior to herding, herd behavior stops. In some circumstances, the information contained in a single irade is small. In particular, for a small probability p of an informed trader arriving or for lowprecision signals, each trade conveys little information and the price movement dur ing herding is small. In conclusion, no one is fooled by herd behavior in IS I. Any price rise during periods with herding results only from information about the new value which was contained in trading prior to herding. All that is learned during herding is that an information event has occurred.
C. Information Aggregation an,d
Herd Behavior
We now explore the effect of herd behavior on the aggregation of dispersed private infor mation in a financial market. A useful bench mark is Vives (1997), who studies the effect of imitative behavior on the efficiency of in formation revelation in a fixedprice setting. He considers an economy where a sequence of agents make the same, irreversible tlecision and have monotonic private signals about the optimal decision. Vives proposes as a welfare benchmark a team solution which assigns agents decision rules that minimize the mean decision error. He finds that in the decenrral ized economy, agents put too much weight on the decisions of others relative to the welfare benchmark. Agents do not i~~ternalize
the neg ative externality on later agents caused by their imitative behavior, which obscures tlheir pri vate information.
Vives' approach docs not transfer well to financial markets. In an asset market, the nat ural team solution would be to maximize the profit:; of the informecl traders. However, trading profits based on asymmetric informa tion have no clear link to social welfare. What would seem more important is the information revealed in the trading historyespecially the information reflected in the asset price. It is such information which is likely to affect decisionmaking in the real economy. With event uncertainty, there are potentially two pieces of information to be revealed: first, whether an information event has occurred, and then if it bas, whether it is a positive or negative event.
PROPOSITION 9: Consider IS I. A period of herding reveals more information about the existence of an information event than a period in which agents trade based on their information about value uncertainly. Spec$ cally, herding is more effective at decreasing
E[T:;~'/ V f 'I2,H,] .I7
PROOF: See Appendix.
While herding is costly in that it obscures information about value uncertainty, it has a benefit. It is more effective at revealing the existence of an information event. By focusing the trade of the informed on a single action when there is an event, herding reduces the effect of noise trading. For example, with herd buying, a sell order muse come from a noise trader and a buy order becomes a strong signal that there has been an information event. We now consider how well a decentralized market trades off these costs and benefits of herd behavior.
PROPOSITION 10: Consider IS I and sup pose that an information event occurs. In the limit as p + 0, the choice of informed traders between herding and trading based on their information about value uncertain@ minimizes
"NotethatsinceE[.irj;,' IH,] =P(V:='12)E[xiT;~Z'
IV= 'I2, H,] + P (V * 'lz)E[.irj;z' /V # 'I2,H,], minimizing E[T',;,' I V # 'I,, H,] is equivalent to maximizing E[.iriT;TZ1V = 'I2,I$?].Hence Proposition 9 also establishes
/
that herding is more effective at revealing that an infor mation event has not occurred, should this be the case.
the deviation of the asset przte froin rts new value. That IS, the t~adzrzg stmtegq of the in formed upproaches the tradzng ~trafeg?]
vv'hich minimzzes E[/V:,,'' Vl 1 V + 'IT9Hr].
PROOF: See Appendix.
The decentralized economy can come arbi trarily close to maximizing the movement of the asset price towards its new vali~e.~"e conclude that the incentives of selfinterested traders with private information do not diverge from social interests as much as the fixedprice herding literature suggests. We now recon, sider the incentives for irrfomed traders to herd, as identified in Proposition 6, to see why they are consistent with rnaxi~alal information revelation (via the price). Traders herd when they have sufficiently lowprecis~on slgnalc but then the infom~ation lost from held behav lor is small They do not herd when n1 nX.but then ~nfonnrabion that an event hasoccurred has no impact on prices since there is no public information as to whether the event is good or bad. Conversely, traders herd when there is already good information about khc new asset value (i.e., lrri 7~; Barge), bur
1
this is when the additional inforrnaiion froin more trad~ag based on value uncertainty I\ small They herd when there is little a\j\iarenc,s oi an info~mat~on large), but tbcri
event (T ',,2 price does not respond strongly to new ~riloeination abov~t value wncertarnty
Paopssition 10Ir a fitting end to Sectnon~ bV and V, which sing the praises of intorrnatiolr aly efficient financial markets populated by Ia tionai traders
Vii, Herd Behavior and Price Bubbles
Herd behavior in a financial n~arlcet is of particular interest because of the possibility that it might offer an explanation for price hwkbles and excess volatility, Because price is a
l8 However, Proposition 10 only addresses information revelation in one period of trade. An analybis of optimal inforrnation revelation over longer time horizons is be yond the scope of this papcr.
martingale that converges to the true value, it is not possible to have volatility in excess of fundamentals in our general model (Corollary 1) ,nor can price bubbles be both likely to oc cur and extreme (Corollary 2). However, it still may be possible to identify (unlikely) cir cumstances which consistently produce highly volatile price paths. Here we investigate whether herd behavior can produce an unsus tainable runup in price that results in a crash. In the previous section, we saw that herd be havior need not distort prices at all. In IS I, herding produces an imbalance in trading, but market participants understand that this is due to herd behavior and hence prices and valua tions do not respond. We now consider whether herding is always likely to be so transparent.
A. Uncertainty About the Composition of the Market
When a trader learns of an information event, his assessment of its impact on the asset value is sometimes precise and sometimes im precise. For example, a trader may or may not be confident in his ability to predict the effect on profits of a change in a firm's product mix or of a merger decision, depending on whether he has complementary pieces of information. such as detailed information about the merger partner. For the market as a whole, some in formation events will have a high proportion of wellinformed traders, while others will have only a few. If the market is uncertain ex ante about the proportion of different types of traders, we have a third dimension of uncertainty.
Definition 5: There is composition uncer tninty when the probability of traders of dif ferent types, p~,
is not common knowledge.
Composition uncertainty complicates learn ing for market participants, especially in the presence of herd behavior. Note that trading patterns in a market with many poorly in formed traders and herding mimic the trading patterns in a market with wellinformed trad ers. In a poorly informed market, a sequence of buy orders is natural because of herding. In a wellinformed market, a sequence of buy or ders is also natural because the agents tend to have the same (very informative) private sig nal. Without knowledge of the composition of the market, it can then become difficult to dis tinguish whether a sequence of buy orders re veals a large amount of information about value uncertainty (because the market is well informed) or almost none a1 all (because the market is poorly informed and informed trad ers are herding). We specify a new informa tion structure in order to show that this confusion can lead to extreme shortrun price effects due to herding.
Information structure i'I (IS 11) adds com position uncertainty to IS I. The true value of the asset is still V E { 0, I/,, 1) . The signals of informed traders take the same form, but now there are two types of trader with 19E { H, L } . The difference between the two types of trad ers is the precision of their tnformation when there is an information event. In particular,
and p, = I while 1 > p, > 'I,.Hence H types are perfectly informed (i.e., E[V I xH] = V), while L types have noisy signals when the as set value changes.
The level of information, in the market is indexed by I E {W, P}. The difference be tween a wellinformed marltet (I = W) and a poorly informed market (I = P) is in the pro portions of each type of informed trader. Let pi be the probability of a type 0 trader in a type I market. For example, p$ is the proba bility of a highprecision trader in a wellinformed market. We assume that there is a fixed probability of an infomed trader (p; + pi = p) and that there are more H types in a wellinformed market than in a poorly in formed market (p: > ,&). The state of the world is given by the combination of the asset's underlying value and the amount of in formation available: (V, I). The market maker's assessed probabilities conditional on the trading history prior to time I are derroteci
7l'v ,.
We now show how prjcc bubbles csli arise in IS 11 by means of a sirnulation.'"Tbc initial priors for the sirriulation arc 7~:)~~0.9999,
=rrgM,/n:,, =. 99. The true skate is (V*/,) ('3. P) . These prior probabilities :atroragly suggest that the market is well informed and that :in inforniation evcrlt is unlikr,ly, but we assume that a negative inhrnaation event occurs (17 0) and that the market is poorly inforined. We focrrs on extreme event uncertainly as irk Section V, subsection B.The rcst of tile parame ters are: p, "95 1., y 0.25, =: 0.125, ~1;= 0.125, pL = 0, and ,u:' =0.25. With these parameters, a poorly informcd economy has no wellinfonned traders (p:; = 0) and hence there is very little information about value uncertainty in any trade (sine:: p, 0.51 f. Despite the lack of inforrnatiorr in this economy, typical sinnulated price paths, such as the: one shown in Figure I. are highly ~olallle.~!'
Fig~lrc: 2shows the 20perirsd moving average of the probability of a buy aid ar'no tradi: (the pobability of a sell is rlie resititrrri).
With ~:xtrcmc event uncertainty, thc price remains close to lor thc first 30 grf:riods. Durk~lgthis initial in$erv:~l,klo~~ie~~i:i.,
Figllrc 2 shows a large buildup of buy orders, which is dale to herding. Three of rhe first five traders buy and this is enough to prompt L iypes to engage in herd buying. IIcrdbuying lasts from period 5 to period 56. As in the case of event uncertainty alone, the mmkcr maker continu ally increases his prior on an irrformation cvexlt as buy orders continue to arrive at a high rate. However, unlike the case of event uncertainty
'"'The analysis of price paihs is a nontrivial exercise. The stochastic process that generates prices is especially complex with herding. In any period the history takes one of thrcc possiblc value? and, depending on the history up until that period, there can be any one of six diffcra:nt distributions over thos.: valucs. This is why we rcsort lo asirnulation at this point in the analysis.
''T'hc price path in i'igare I is typical in that most price paths take on extreme values and tl~en return to prices oi' around 'I,. It is app!.oximately equally likcly that the ex treme value is 0 or 1.
alone, the pice moves dramatically during the interval with herding (from 0.5 to 0.94) as the rnarket rnaker concludes that there has been an information event. For comparison, the maxi. rriuin price rise during an interval with herding if the composition of the market were known is A = 0.03, from Proposition 8.
Since it is impossible to distinguish be. iween wellinformed and poorly informed economies during herding, both individual traders and the market maker must rely on their initial assessments. Because the initial assessment.; are that a wellinformed econ omy Is relatively more likely than a poorly informed one, the marker n;aker increases the price and L types increase their valua tions throughout the pcriod of herding al rriost as if the market were \veil informed. Eveiitually, the rnarket rnaker ends herding byincreasing the price and the spread suffi a:iently in period 57. Figure 2 shows the el' ftxt of the end of herding. There is a fa11 in buying and a rise in no trade as the bid/ask spread forccs E types out of the market. This drop in trading volume signals (over time) that previous actions were due to herding rather than to trading by N types. Note the similarity in the two flai spots in the price path. In periods 140, the market maker takes time to learn that there has been a change in fundamentals, while in periods55100,he takes time to learn that the mar ket is poorly informed, In each case, the niarket maker is slow 60 rcspond because he has extreme beliefs,
Oxlce it becomes apparent that the market is poorly informed, the price naturally has to drop, for there simply is not as much infor ~x~ationin previous trading as had been assulned.This brings us back to the case of event ?mcetTainly alone: the price should only have adjusted according to the information content oi' one poorly informed signal rather than to that of many wellinformed trades. As a result, the price falls to near 'I2before any further infom~ed trading takes place. Around period 220, the probability of no trade declines as L types reenter the marketthis time trading on alreir information about the new asset value. The market is only learning about one dimen sion of ~lncestainty a? a time. At first, the mar ket learns that an information event has occurred. Then it learns that the market is poorly informed. Only when these first two di mensions are sufficiently resolved, does the market begin to aggregate information about value uncertainty. The price bubble arises because the market mistakenly thinks that it is learning about both event and value uncertainty.
C. Discussion and Connections to Prior Research
It is possible to formalize the above ex ample to show that price bubbles consis tently occur under certain identifiable, but unlikely, conditions. There are three key fea tures of the example. First, an information event is very unlikely, T:',~+1. This assures that the price is fixed for a long period of time, so that a substantial amount of herd behavior occurs. Second, it is very likely that the market is well informed, ~1),~/~1).~
I+ 1. This assures that at first the market maker completely discounts the possibility that the market is poorly informed and that the sub stantial imbalance in trade is due to herding. Third, all informed traders are of type L in a poorly informed market, ,ug = 0. Then a poorly informed market with herd behavior behaves exactly like a wellinformed market and nothing is learned about composition uncertainty if there is herding. These effects combine to create highly volatile prices
Time
when the market is poorly informed about an
information event. In particular, the price
tends arbitrarily close to am extreme value,
then returns to 'I2The cxtseme value can be
either 0 or 1 .''
The existence of price bubbles in IS 11 for
extreme parameters is consistent with a general
intuition. The combination of event and com
position uncertainty leads herd behavior to dis
tort asset prices. So long as the market maker
can not completely distinguish between a well
informed market and a poorly informed market
during periods of herding, the herd behavior that
arises from event uncertainty will distort prices.
The more the market maker 1s surpriseti that the
market is poorly informed, tlhe more prices will
respond to the herd behavior.
Our conclusion that rational herding can ex
plain price bubbles and crashes contrasts with
several papers which argue implicitly that
herding and crashes, specifically the stock
market crash of 1987, cannot be explained in
models of rational trading (Robert li. Shiller
[I9891 gives a collection of papers to this ef
fect; Allan W. Kleidon [1992] summarizes
and criticizes this line of thought). For ex
ample, several papers explain the failure of
markets to produce effective prices as the
21 We formalize this result in an earlier version of this paper (available from the authors), where we make some additional simplifying assumptions.
0,8 T no trade
result of unsophisticated strategies and sub optimal behavior by market parlicipants (e.g., Gerard Gennotte and Hayne Leland, 1990; Shleifer and Summers, 1990),
Of prior work, Jacklin et al. ( 1992) and David Romer (1993) come closest to providmg a ra tional actor theory of price bubbles. Both papers have two dimensions of uncertainty: value un certainty and composition uncertainty. We be lieve that our use of three dimensions of uncertainty (value, event, and composition uncertainty) provides a more complete theory of price bubbles. Romer studies a noisy rational ex pectations model with a form of composition un certainty very much like that in IS H.His theory explains how price corrections can occur without contemporaneous changes in fundamentals: when markets learn about the composition of the mar ket, they reevaluate the information contained in past trades. However, his theory relies on the exogenous mispricing of assets 22 and on the exogenous arrival of information about the
"In Romer's model, mispricing is driven by noise trad ing and an assumption that traders receive signals which are inaccurate even when perfectly aggregated. Similarly, Lee's (1995) theory of sudden market corrections does not explore the mechanisms by which asset prices become mispriced, beyond noise trading or the arrival of many misinformed traders.
composition of the market.21 In contrast, rnispric ing in our model arises endogenously through the interaction of herd behavior and composition un certainty and composition uncertainty is endoge nously resolved through the pattern of trade.
Jacklin et al. consider a market with a class sf insurance traders who buy stock when the price rises and sell when it declines. They show that such insurance trading creates a positive feed back loop which can produce bubbles and crashes when the market is surprised by the ex tent of insurance trading. Wle such insurance trading has some desirable properties when in vestors hold a diverse portfolio of stock, Jacklin et al. take the use of these strategies as exoge nously given. In contrast, the herding strategy that produces our bubble is endogenous.
VIH. Contrarian Behavior
In Section V, subsection A, we began to ad, dress the puzzle of price charting. We show
''There is a further limitation to Romer's theory. Our results in Section VII below suggest that in a sequential trading model, there is a countervailing force that opposes price bubbles when there is composition uncertainty but no event uncertainty. Composition uncertainty creates an incentive for poorly informed traders to trade against the trend in prices. This should limit the formation of price bubbles.
there that an agent's trading strategy can be strongly based on the history of past trades. In particular, we show that with event uncertainty a trader may ignore his private infomlation about value uncertainty in order to trade with the trend in past trades. However, such herd behavior is only one of two possible types of strongly historydependent behavior. The other possibil ity is trade which opposes the trend in past trades at the expense of private information about value uncertainty. We start with a formal definition of such contrarian behavior.
Dejinition 6: A trader with private information XO engages in contrarian behavior at time t if either he buys when V:(xs) < Vf, and Z(xli) < VL, < Vjj, or he sells when Vj(xi,) > V:, and v(x,) > V:,, > VR,, where Z(x,) is defined as follows:

v (x,) = lim E[V / n draws from the
n+x
distributionf,( I V) all have
the value xs]
The definition of contrarian behavior is the analogue of herd behavior with the additional requirement that the trend in past trades does not overshoot the "limit value" ?(x,) of the signal. To see why such an addition is neces sary, consider a trader who knows that V = '1, and who trades in a market where V:, = 'I,. Initially, the trader wants to buy. If the trend in past trades pushes the price above V4, he will sell. This is not historydependent be havior. The trading strategy depends only on the price and the signal value (e.g., buy if and only if the asking price is less than 'I4). In de fining contrarian behavior, we seek to exclude situations where the trader reverses his behav ior simply because the trend in past trades has become more positive (or negative) than the trader's information about value uncertainty. With IS 11, U(X,) = xL, SO that there is con trarian selling if a trader with x, :== 1 sells when the trend in trade is positive V:,, > V;,.
PROPOSITION 11: A trader with a mono tonic signal never engages in contrarian behavior.
PROOF:
Suppose a trader with monotonic signal xli engages in contrarian buying at time t. Then Vk (x~) 2A' 2 Vi. Since the signal is mono tonic, this implies that v(x,) 2 V:,,. But since

v (x,) == v (x,) ,this contradicts contrarian buy ing. Similarly, contrarian selling never occurs.
Monotonicity is sufficient to rule out contrar ian behavior. Thus, Propositions 3 and 11 dem onstrate that an assumption of monotonic signals is inconsistent with strongly historydependent behavior of both the herd and contrarian variety. While event uncertainty can. produce herd be havior, we now show that composition uncer tainty can produce contrarian behavior.
PROPOSITION 12: Consider IS I1 without event uncertainty (i.e., n?,2 = 0). A suficient condition for L types to engage in contrarian behavior with positive probability is
PROOF: See Appendix.
When there is composition uncertainty (and no event uncertainty), traders of type L place less weight on previous trades than does the market maker. Whv? Hv definition. an in
.
*
formed trader is more likely to get a low precision signal in a poorly informed market than in a wellinformed rnarket (i.e., j~f > py). Hence, an L type trader assigns a higher probability to I = P than does the market maker: "If this is such a wellinformed mar ket, why did I receive such poor informa tion?" In a poorly informed market, a given imbalance between buys and sells is less in formative than in a wellinformed market. Hence, with composition uncertainty, the mar ket maker adjusts his expected value more in response to past trading than does a trader of type L.24
l4 We suspect that composition uncertainty can also create herd behavior. Note that H types believe the market
The sufficient condition in Proposition 12 has three parts. The lefthand side is a measure of the amount of information L types have about the composition of the market. The term p, I( 1 p, )on the righthand side is a measure of the amount of information that L types have about the new asset value. The second right handside term results from the existence of the bidask spread: as p 40, the bidask spread goes to zero and (y + p)I(y + p;) 4 1~ Hence, contrarian behavior due to composition uncertainty is shown to be possible when the information of L's about the composition of the market is large relative to their information about value uncertainty and relative to the bid ask spread.
With event uncertainty, herd behavior is possible for any imperfect signal (see Propo sition 5),but here contrarian behavior only oc curs if signals are sufficiently imprecise. The difference arises because with event uncer tainty, informed traders know that some states are impossible (i.e., V + 'I2) while with corn position uncertainty, L's only believe that some states are less likely than the market maker. Note that with pr = 0, 1,types know for sure that the market is poorly informed and the sufficient condition is satisfied for y, < I, which parallels the result for event uncertainty and herd behavior. We draw the following general conclusion. The existence of history dependent behavior (in either its herd or con trarian form) requires (i) that there exist multiple dimensions of uncertainty, and (ii) that traders' asymmetric information about value uncertainty be sufficiently poor relative to their information about one of the other di mensions of uncertainty.
Multidimensional uncertainty provides a justification for the phenomena of price chart ing (Easley and O'Mara [I9921 and Lawrence Blume et al. [1994] reach similar conclu sions). A trader who wants to make optimal
is more likely well info~med than does the market maker. Hence, they put higher weight on the history of trade than does the market maker. Because we assume H types have a perfect signal about value uncertainty (pH= 1), their signal always drives their trading. However, if their signal were imperfect then their private information about market composition might lead them to engage in herd behavior.
use of all dimensions of his information needs to know more about the trading history than just the price.?j
VIII. Conclusion
We reexamined the role of the price mech. anism in the aggregation sf dispersed private information in an economy when trade is sequential rather than simultaneous. In our gen eral model, the price mechanism assures that longrun choices are efficient and with simple information structures, it assures that herd be.havior is impossible. However, we show char more complex information structures can lead to herd behavior and that a sufficiently corn plex information structure makes price bubbles possible. Price is a singledimensional instrument and it only assures that the econ omy learns about a single dimension of uncertainty at one time. As a result, multiple dimensions of uncertainty can "overwhelm" the price mechanism during some stretches of trading. Then, interesting shortrun behavior
such as herding, price bubbles, and contrarian
behavior become possible.
Our results are consistent with the litern ture on trading and connmon knowledge. Re. peated communication leads a11 agents to agree in their assessments of the true value: they cannot "agree to disagree" (John U. Geanakoplos and Heraklis M. Polemarchalas, 1982). In the simplest examples discussed by Geanakoplos (1992), a single round of conn rnunication causes agents to unite in their beliefs; a richer set of possible outcomes necessitates further rounds of communication before the agents agree in their assessments. Adding a new dimension of uncertainty in our model is analogous to enriching the set of oaxt comes in a common knowledge game. Our re. sults show that com~nunication need not
"Adding simple aggregate statistics such as the trading volume or the imbalance of the market maker's sales need not produce a sufficient statistic for the corn plete history. The meaning of a buy or sell at time r depends on the extent of herd and contrarian behavior at that time.
happen uniformly in a financial market. The market may only be learning about one di mension of uncertainty at a time and with a sufficient number of dimensions, this can lead to highly volatile price paths.
We have investigated whether herding might allow an arbitrageur to profitably ma nipulate the market's learning process. Con sider an arbitrageur who buys in hopes of generating herd buying so as to sell at a later period for a profit. We show elsewhere (Avery and Zemsky, 1998) that such simple trading strategies can not be profitable for a trader with the same information as the mar ket maker.
We close with some ideas for future work. First, we hypothesize that herding and bub bles are less pronounced when prices have multiple dimensions. A natural source of multidimensional prices is derivative secu rities such as options. Second, while excess volatility can not be explained in our general model, our results demonstrate that volatility concentrates in certain identifiable situa tions. Our identification of conditions under which price bubbles arise is but a first step in investigating the pooling of variance. Third, we have not fully explored the topic of multidimensional uncertainty. For exam ple, we look at a market where there is either one or no information events. In an economy in which information events arrive stochas tically, the market might be uncertain as to how many information events are unfolding at any given time. Empirically, it would be useful to know more about how traders use price history in their trading strategies.
PROOF OF PROPOSITION 1:
We prove existence and uniqueness of an equilibrium ask price. The proof is similar for bid prices. Let h, = B, the evcnt that there is a buy order at time t. An eqailib rium ask price satisfies E[V I HI, A', h,] A' = 0. The conditional expected value given a buy order is the weighted average of two terms: the weighted expected value of informed buyers whose assessments sat isfy V,(x,) ? A', and the assessment of a noise trader V:,.
AtA1= V:,, E[V(H,, A', h,] ?A'. AsA' increases from V:,, E [VI HI, A', h,] changes only when A' outstrips the assessment of some informed traders, at which point those traders stop buying. Consequently, E [V (HI,A', h,] is weakly increasing in A' whenever A' < E[V I H,, A', h,] ,and weakly decreasing in A' wheneverAr > EIVIHr,Ar, h,].
The implication is that E[V (HI, A', h,] A' is strictly decreasing once it reaches zero. If there is an equilibrium price, it is unique. E [V (If,,A', h,] A' is continuous in A'except at finitely many points, where these dis continuities never change the sign of E[V I H,, A', h,] A'. In addition, E[V(Hr, A', h,] A' is nonpositive at A' = V:,,
, where the market maker gains nothing from noise traders, and nonnegative at A ' = 1, where the market maker loses nothing to informed traders. Therefore, a zero profit price must exist, and we know from above that it is unique and that A' r VL.
The market maker's expected value for the asset and priors are martingales with respect to H, since H, contains all of the market maker's information.
PROOF OF PROPOSITION 2:
Suppose there is an informational cascade in period t. In an informational cascade, the market maker learns nothing from a trade and hence B' = A' = V:,. With noise trading, all histories occur with strictly positive probabil ity in all states of the world, Hence, there does not exists a v such that P (I/ = v 1 HI) = 1 and there is a nontrivial set of traders with useful information, [i.e., traders for whom VA(xlI) # V:,]. With B' = A' = VL, these traders must be buying or selling.
Suppose traders with signals x8 E RE are buying and P(X# E RB(H,)> 0. Since this is an informational cascade, P(xH E RE1 V, HI) = P(x, EREI HI)VV, which implies that E[V I xo E RE, H,] = V:,,. This contradicts all types x8 E RE buying. Similarly, no positive measure of informed traders can be selling. But this contradicts a nontrivial set of traders having useful information. We conclude that an informational cascade it; impossible.
PROOF OF PROPOSITION 4:
This result is a direct consequence of Proposition 4 of Glosten and Milgrom (1985), which states that the beliefs of in formed traders and the market maker con verge over time so long as trade on both sides of the market is bounded away from zero. Since we assume a stationary proba bility that noise traders buy and sell in each period, there is a positive probability for a buy order and a positive probability for a sell order in each period. Therefore, the Glosten and Milgrom result applies and the expec tations of the market maker and of all the informed traders converge over time. If ex pectations converge to the true value V, then the prices must do so as well.
Suppose that the market maker's expecta tion does not converge to V. Then for some 6 > 0, there is strictly positive probability in each period that the market maker's expecta tion differs from Vby at least 6. But then, there is a strictly positive probability (in each pe riod) that an informed trader's assessment dif fers from the market maker's assessment by at least 46)> 0. This contradicts the conver gence of these assessments.
PROOF OF COROLLARY 1:
Since Vh, is a martingale, E[ AV::, . AV:;] = 0 for each t, * t, . That is, Cov (AV:;, AVZ)
0. Since VA; = V:, + C : 1:AV:, , we can write the variance of V:, as the sum of variances of AV:,,: Var(V:,;) = C:Ii" Var(BV1,). As ti: 4 a,Viy converges almost surely to V, so Var(VA:) converges to Var(V) and the first part of the proposition follows.
For the second part, note that Var(V:; V:;,) = E:Z:;+, Var(AV:,,) = Var(V:,') Var(V:,). The result follows by taking the limit as t2grows barge.
PROOF OF COROLLARY 2:
Let m = P(V < n Iff,). Because V:, is a martingale converging to V,
Hence,
Since E[VJV 5. a, Pb,] > Vh, and E[VIV < a, N,]< a, an upper bound on m is given by settingE[VIVz a,Pb,] 1 andE[VIV<n, ff]=a.
PROOF OF PROPOSITION 5:
Suppose p = I. Then E[V 1 x, H,] = x and signals are monotonic. Hence there is no herd behavj or.
Suppose that y < 1 and V #= 'I,. Because of noise trading, any finite history occurs with positive probability. Suppose that there is probability 0 of herding in the first Ntrades for each finite N. Fix E > 0. Without herding in the first N trades, each buy order increases the expected value of the asset (with an upper limit of I ) . Choose n such that an informed trader who observes n I buy orders and a pignal x = 0 has expected value for the asset greater than '1, + c. Note that each no trade increases nj,, with an upper limit of 1. Consider a history of length t = m + n < N which consists of m no trades followed by n buy orders, where m is sufficiently large
that i7lit + 11 + 1
, ,, > 1 E. Under these conditions A"' ''I < '1, + t and an informed trader with x = 0 will buy at time m + n. Further, this his tory occurs with positive probability, contra dicting the assumption that there was no herding at time m + n < N.A similar argu ment establishes that herd selling also occurs with positive probability. Therefore, herding in the wrong direction occurs with positive probability.
PROOF OF PROPOSITION 6: There is herd buying if E[V I x = 0, H,] > A', which is equivalent to
Setting y = (1 p)/3 and^',,^ = 1 71.; 7r:,the above condition is equivalent to
We have A(1) < 0 and A('/,)= (i7fi(l 7r;) i7t(l 7rh))(1p), which is positive if and only if 7r; > i7;. Hence, when i7: > i7;, there exists a unique p E( 'I2,1) such that A(p3 =0. Since aA/ap < 0, there is herd buying for p < p(p, 7r6, T:), where the closedform expression forpcomes from solv ing A(p) = 0. It is then straightforward to show thatdpllap < 0. To show that p is in creasing in 7r;,,,take the expression for p and set i7; = CYTTT;,7r: = a7r; and i7;,, = 1 a(i7h + 7r:) and then note that dpllaa < 0. Finally, to show that p is increasing in i7',/7rl, set i7h = k i7: and note that dp/d 7r', > 0. The results extend to herd selling by symmetry.
PROOF OF PROPOSITION 7:
Suppose that in the first t periods there is no herding and there are b buy orders and s sell orders, where wlog b 2 s. In these t periods, informed traders with x = 1 submit buy orders while those with x = 0 submit sell orders. At time t + 1, the assessment of an informed trader with signal x = 0 is
and V(0) is simply a function G(b s). G(b s) > '1, whenever (b s) is equal to or greater than a critical value A. Define SI = G(A) 'I,, and 6, = '1, G(A 1). Generically, 6,> 0 as well.
Let S* = min(S, , 6,). Assessments of in formed traders always differ from '/, by at least S* when there is an imbalance in prior trades.
So long as prices remain in the range ('1, S*, '1, + S* ),the market behaves as if the price were fixed at '1,. We now observe that as 7rYl2 + 1, both the bid and ask prices remain in this range ('I2S*, iS*) for an arbi trarily long time.
If there is no herding through period t,then the market maker's assessed probability of an information event is bounded below by his assessment after a history with tconsecutive trades:
For 7rYl2 sufficiently close to 1, this implies that i7:7,' > 1 26*. As a result, the bid and ask prices must be in the range ('1, S*, '1, + S* ) in each period prior to t. Define t* as the last period such that any trading history pro duces bid and ask prices in the range ('/, S*, '1, i6") for each period through t*. As 7rYI2+ 1, t* + m.
Prior to time t*, informed traders with x = 0 sell while traders with x = 1 buy so long as the absolute difference in buy and sell orders is less than A. If the imbalance reaches A in period q < t*, herding begins and no new information about value uncer tainty reaches the market. Hence, V(x) = Vq(x)for t* 2 t 2 q and herding continues until at leastt*, when it becomes possible for the price to adjust enough to break the herd.
The imbalance between buy and sell or ders is a random walk with drift ?2pp prior to time t*. As t* grows large (i.e., 7rYl2 approaches 1), the law of large numbers im plies that the probability that the absolute imbalance reaches A prior to time t* increases to 1. Therefore, herding arises with probability 1 as T:,, + 1.
The probability of herding in the wrong di rection is It follows from G(Yi 1) I'I2 5 G(Yi),that
Y2
Y + .:/21
Hence,
As p + 0, Yi gets arbitrarily large, G(Yi)+ '1, andA+(l p).
PROOF OF PROPOSITION 8:
Suppose there is no herding in period t, h, = B, and herd buying begins in period t + 1 and continues through period T. Then, V1+' (X = 0) > B'+ > B' > V' (X = 0)and since57:' ' > T;+ for herd buying, we also have B' > '1,. Since V (x = 0) = V+ (x = 0)for t + I < s 5 T, an upper bound on the change in price from t to T, is given by
(Al) V+'(x = 0) max{V(x = O),:}.
Defining v = V(x = 0),we can write
Since 0 < dV+' (x = 0)ldv < 1 for 'I2
v and 0 < aVf (x = O)/av for v < 'I2, an
upper bound on price changes is given by ex
pression (Al)evaluated at V (x = 0) = '/,,
which simplifies to give A = 3p(p 
]/2)/(2+ p).
PROOF OF PROPOSITION 9:
Let 4, = E[n',;,]I V = 'I2, HI]when in. formed traders herd and let +, be the same quantity when traders trade based on their in formation about value uncertainty. With herd buying, all informed traders buy when x # 'I2 and either sell or refrain from trading when x = 'I2. Then, using equation ( 1) we have The expression for herd selling is identical. When traders trade with their information about value uncertainty, they buy when x = 1, sell when x = 0 and refrain from trading when x = 'I2. Then,
Y2
4v
= ..:I2
The difference between these two quantities takes the following form:
where f(x) = (~fiI2y2)l(y+ x), a = p(1 n:,,) and b + c = a. The result that herding is more effective at revealing information (i.e., +,, > 4,)follows from the convexity off.
PROOF OF PROPOSITION 10:
Let +h = E[(IVF1V1)IV f 'I2, HI]= T:E[~V::' IV= 11 + 7r;E[VL+'0lV= 0] when informed traders herd and let 4, be the same quantity when traders trade based on their information about value uncertainty. Wlog, suppose that herding involves buying. We can reduce +, and 4, to expressions in P(h,l V = v) and ?r: using equation (1) forT:+'(h,)and the following equations:
E[V+ IV = v, Pi,]
When traders engage in herd buying, Iyh, = SIV)=P(h,=NTIV+. 'I2)=P(h,=BIV= 'I,) = y, and P (h, = BI V # 'I2) = P (hi = NT I V = '1,) = y + p. When traders trade with their information about value uncertainty,
P (h, = S/V = Il2) = P (hi = BIV = '12) = P(h, = NTIv # '1,) = y, P(h, = NTIV= 'I2) = y + p, P (hi = BIV = 1) = P (hi = SIV=0)= y +pp,andP(h,=SIV= 1)= P (h, = BIV =0) = y + p(1 p). Hence, $, $, is a function of the exogenous param eters, p and p, and the current priors. In par ticular, one can show that +, $,, takes the form a2p2+alp3 + a4p4,where
and a, and a, are also independent of p. Hence, as p goes to zero, the sign of $, 4, is given by a,. As p t 0, A(p) t a,, where A(p)is as defined in the proof of Proposition
6. Hence, as p becomes small, herding occurs precisely when it serves to minimize
$v
4h.
PROOF OF PROPOSITION 12:
The proof shows the possibility of contrarian selling, the result for contrarian buy ing follows from symmetry. Note that V(xL)= x,, so that if a trader with x, = 1 sells whenVi, > Vi, then there is contrarian selling. The proof proceeds in three steps. The first shows that given (2),a sufficiently long pe riod of buys implies that all L types stop buy ing. Step 2 shows that a sufficiently long period of buying and (2)implies that L types all sell given a condition on the priors when the buying started. Step 3 shows that an ini tial history of trading exists such that this condition on the priors is satisfied. We say that L types trade with their signal in some period t if those with xL = 0 sell and those with xL = 1 buy.
Step 1: A sufficiently long sequence of buys leads L types with xL= 1 to stop buying given condition (2).
If L's are trading with their signal, then the ask at time t is
dr= [(y + p: + p!'pL.)~:,LV
 +
 (7 + p; + p;pL)..:,Pl
 +
 [(Y + pHW + pLwpL)r:,W i(Y + t p:pL)x:,P
+ (Y + PT(
PL))~~;,W
+ (7 + pf(1 PL))~,,PI.
L types with a signal x, == 1do not buy at time t if A' > Vi(1) , which is equivalent to
l,P
~b w
.+ 
(pLw(1P,)(Y + PC) YPLPLW)
..;,p

> 0.
In each period where h, = B and L's are trading with their signal, the marker maker's priors are updated as follows
7r6,, ( y + pLv( 1 p,)) 7rbjV' 0,W I
A
a, I 7r;,W (Y + p: + PL"PL)' niT;>l I,W
Note that ai E [0, 11 and a, > max { a:!, a3, a, } .
Consider some t < S. If L's are trading with their signal in periods t and t + 1, h, = B and h, , = S, then the market maker's updated priors satisfy
Hence, one can construct an initial series of trades which make T ;,In G,, arbitrarily close to zero.
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