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Trading panics and information acquisition : theory and experiments Kendall, Chad William 2014

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Trading Panics and InformationAcquisition: Theory and ExperimentsbyChad William KendallB.A.Sc., Simon Fraser University, 1996M.A., Simon Fraser University, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2014c© Chad William Kendall 2014AbstractThis dissertation studies the incentives of economic agents to acquire in-formation about financial assets when it must be acquired through time-consuming research. When information can not be obtained instantaneously,agents face a tradeoff between acting immediately and performing morethorough research. Better information allows for more informed tradingdecisions, but research is costly because other agents’ trades move pricesadversely as time passes.The first chapter develops a theoretical model in which agents sequen-tially trade a single financial asset. Each agent receives weak, private in-formation when they arrive to the market and may trade immediately, orinstead wait for additional information. Should they wait, other agentshave an opportunity to trade before the first agent receives their additionalinformation, which creates an endogenous cost to waiting. The analysisdetermines the conditions under which equilibrium behavior involves imme-diate trades (“panics”), and then studies the quantitative impacts of weaklyinformed trades on the ability of prices to aggregate information.The second chapter experimentally tests the theoretical model in a lab-oratory setting, in order to determine whether or not subjects understandthe tradeoff between better quality information and potential adverse pricemovements. Comparative static results establish that the theory broadlyexplains when panics, and the corresponding informational losses, occur.However, additional, “heuristic” panics are also frequently observed. Specif-ically, subjects exhibit a strong tendency to wait for more information whenhighly uncertain about asset values, but switch to trading as soon as possibleonce values become more certain.Motivated by the findings of the second chapter, the third chapter ex-iiAbstracttends both the theory of the first chapter and the experimental results of thesecond chapter to a second, richer environment. Different from the sequentialstructure of the first model, agents may trade simultaneously in the richermodel. Experimental results with the richer model produce trade cluster-ing and serial correlations in returns, as predicted by the heuristic behavioridentified in the second chapter. These phenomena are well-established fea-tures of real financial markets, suggesting that the heuristic subjects followin the laboratory may provide a novel explanation for these phenomena.iiiPrefaceThis dissertation consists of original, independent, unpublished work by theauthor, Chad Kendall. The experimental work discussed in Chapters 2 and3 is covered by UBC Behavioral Ethics Board Certificate number H13-01124.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Informational Losses in Rational Trading Panics . . . . . . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 The Expected Value Model . . . . . . . . . . . . . . . . . . . 112.3 Analysis of the Expected Value Model . . . . . . . . . . . . . 152.4 Effects of Panics . . . . . . . . . . . . . . . . . . . . . . . . 312.5 The Zero-Profit Model . . . . . . . . . . . . . . . . . . . . . 412.6 Empirical Implications . . . . . . . . . . . . . . . . . . . . . 453 Rational and Heuristic-Based Trading Panics in an Experi-mental Asset Market . . . . . . . . . . . . . . . . . . . . . . . 503.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.3 Theoretical Predictions . . . . . . . . . . . . . . . . . . . . . 533.4 Experimental Design . . . . . . . . . . . . . . . . . . . . . . 55vTable of Contents3.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644 Panics with Simultaneous Trading: Theoretical and Exper-imental Extensions . . . . . . . . . . . . . . . . . . . . . . . . . 674.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.2 Extended Model . . . . . . . . . . . . . . . . . . . . . . . . . 684.3 Theoretical Predictions . . . . . . . . . . . . . . . . . . . . . 704.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89AppendicesA Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 97A.1 Benefit Functions . . . . . . . . . . . . . . . . . . . . . . . . 97A.2 Stability of Equilibria . . . . . . . . . . . . . . . . . . . . . . 99A.3 Omitted Proofs . . . . . . . . . . . . . . . . . . . . . . . . . 100A.4 Formulas for the Zero-Profit Model . . . . . . . . . . . . . . 115A.5 Multiple Arrival (n > 1) . . . . . . . . . . . . . . . . . . . . . 116B Appendix to Chapter 3 . . . . . . . . . . . . . . . . . . . . . . 119B.1 Analysis Details . . . . . . . . . . . . . . . . . . . . . . . . . 119B.2 Rational Herding and the τ -Herding Heuristic . . . . . . . . 120B.3 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123B.4 Non-equilibrium Informational Losses . . . . . . . . . . . . . 124B.5 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 124C Appendix to Chapter 4 . . . . . . . . . . . . . . . . . . . . . . 129C.1 Analysis Details and Omitted Proofs . . . . . . . . . . . . . 129C.2 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134viTable of ContentsC.3 Non-equilibrium Informational Losses . . . . . . . . . . . . . 135C.4 Instructions . . . . . . . . . . . . . . . . . . . . . . . . . . . 135viiList of Tables3.1 Basic Model Treatments . . . . . . . . . . . . . . . . . . . . 533.2 Basic Environment Trading Results . . . . . . . . . . . . . . . 573.3 Basic Environment Timing Results . . . . . . . . . . . . . . . 583.4 Determinants of Rushed Trades in the Basic Treatments . . . 613.5 Frequency of Subjects of Each Type in the Basic Treatments 634.1 Extended Model Treatments . . . . . . . . . . . . . . . . . . 694.2 Extended Model Trading Results . . . . . . . . . . . . . . . . 714.3 Determinants of Rushed Trades in the Extended Wait Treat-ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.4 Correlation of Trading Returns in the Extended Wait Treatment 774.5 Signals and Rushed Trades in the Extended Wait Treatment 794.6 Frequency of Subjects of Each Type In Extended Wait . . . . 80viiiList of Figures2.1 Trade Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2 Example Benefit Functions . . . . . . . . . . . . . . . . . . . 202.3 Equilibrium Timing Strategies and Sample Price Chains . . 272.4 Sample Price Path . . . . . . . . . . . . . . . . . . . . . . . . 322.5 Heat Map of Simulated Percentage Slowdowns in Price Con-vergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.6 Simulation of Average Price Paths . . . . . . . . . . . . . . . 372.7 Simulation of Probability of Increase in Mispricing . . . . . . 394.1 Cumulative Distribution of Trades Occurring at t ≤ t′ . . . . 72B.1 Proportion of Rational Behavior as Trials Progress (Basic) . . 123B.2 CDFs of Difference Between Final Prices and Asset Values byTreatment (Basic) . . . . . . . . . . . . . . . . . . . . . . . . 125C.1 Proportion of Rational Behavior as Trials Progress (Extended)134C.2 CDFs of Difference Between Final Prices and Asset Values byTreatment (Extended) . . . . . . . . . . . . . . . . . . . . . . 136ixAcknowledgmentsI am extremely grateful to Francesco Trebbi and Ryan Oprea for the untoldnumber of hours they spent providing advice, encouragement, and construc-tive criticism. I would also especially like to thank Andrea Frazzini, Li Hao,and Patrick Francois for their invaluable contributions.I benefited from comments from many other faculty members withinEconomics and the Sauder School of Business, as well as numerous seminarparticipants.Most of all, thanks to Jenny and my family, who have always providedunwavering support and never once questioned my decision to embark onthis path.xChapter 1IntroductionInformation about financial assets fundamentally takes time to both acquireand to process into a trading decision. To uncover information that is notalready publicly available, and hence not already incorporated into prices,one must perform time-consuming research. The large literature on incen-tives to acquire information in financial markets, however, assumes that oneobtains private information by paying a monetary cost, so that informationis acquired instantaneously. In this dissertation, I relax this assumptionand ask what are the incentives to acquire information when it can only beacquired over time.When information takes time to acquire, agents face a tradeoff betweenacquiring relatively weak information quickly, or performing more researchto obtain better information. Better information allows one to make a moreinformed trading decision, but is costly if other agents may trade while oneis researching the asset. This cost of research means that it may be rationalto trade as soon as possible. If everyone rushes to trade immediately, atrading panic results. The existing literature on rational trading panics hasemphasized the role of panics in explaining price crashes while I insteadstudy how panics reduce incentives to perform time-consuming research,resulting in markets that aggregate only relatively weak information.In the first chapter, I develop a stylized model in which agents sequen-tially trade a single financial asset. When a trader arrives to the market,they receive relatively weak, private information. They may trade imme-diately, or wait to obtain better information and trade then. I adopt anoverlapping arrival structure so that, while one agent waits, the subsequentagent arrives to the market and has an opportunity to trade. The possibil-ity that the next agent may trade creates an endogenous cost of waiting,1Chapter 1. Introductionbecause others’ trades move prices adversely in expectation, reducing one’sprofits. Intuitively, others are more likely to buy the asset when you un-cover information that the asset is currently undervalued. I characterize theequilibria of the model and determine the conditions under which rationalpanics (immediate trades) occur. Interestingly, the model predicts that bet-ter information is acquired only when uncertainty about the fundamentalvalue of the asset is low, in stark contrast to existing models of informa-tion acquisition. I further quantify the effects of panics on the ability ofprices to aggregate information, showing that panics increase the time ittakes prices to converge to fundamental values. Finally, I develop severalempirical predictions, some of which are confirmed by existing studies.In the second chapter, I experimentally test the model of the first chapterin a laboratory setting. The purpose of the experiment is to determine theextent to which observed panics can be predicted by theory. Importantly,precise theoretical predictions allow me to categorize panics as either ratio-nal, equilibrium behavior, or the result of other trading heuristics. I findthat equilibrium behavior explains the majority of observed panics, but thatthe use of a particular heuristic results in additional panics. The identifiedheuristic is closely related to momentum trading in which subjects wait formore information when uncertain about the asset’s fundamental value, butrush to trade in the direction of past price movements once the uncertaintyis partially resolved.Motivated by the identification of heuristic-based panics, the third chap-ter extends both the theory and experimental work to a richer environmentin which agents can trade simultaneously. There, heuristic-based panics arepredicted to result in trade clustering and serial correlation in returns. Bothof these phenomena are found to occur in the experimental data, suggestingthat the identified heuristic may play a role in explaining these two commonfeatures of real-world financial markets.The theoretical work of the dissertation relates conceptually to two sep-arate strands of the literature on financial markets: that on informationacquisition and that on rational panics. The literature on information acqui-sition began with the work of Grossman and Stiglitz (1980). In their static2Chapter 1. Introductionmodel, agents are less likely to acquire information when others do so, dueto the fact that one can infer others’ information from prices. More recently,information acquisition has been studied in sequential models more closelyrelated to the one considered here. Chamley (2007) considers the situationin which traders trade for short-term profits, unwinding their investments inthe period after initial trade. He shows that, in this case, agents can be morelikely to acquire information when others do: there is strategic complemen-tarity in acquisition decisions, contrary to the strategic substitutability inGrossman and Stiglitz (1980). Strategic complementarity also arises in themodels considered here, although its nature is quite different. Nikandrova(2012) and Lew (2013) consider sequential trading models in which agentsare only present in the market for one period. In their models, informationis only acquired when uncertainty about the fundamental value of the as-set is high because it is at this time that information is most valuable. Inall of these models, agents acquire private information instantaneously bypaying a monetary cost. By considering the fact that information requirestime-consuming research, I obtain very different results. Information is for-gone during times of high uncertainty. Although it is most valuable at thistime, the endogenous cost due to others’ trades is also largest at this time.Intuitively, when uncertainty is high, prices move a lot, making the cost ofwaiting higher.The literature on rational panics, some of which shares the intuitionthat agents want to trade as soon as possible to avoid the price movementsof others’ trades, has focused on explaining price crashes. The most closelyrelated papers are Bulow and Klemperer (1994), Brunnermeier and Pedersen(2005), and Pedersen (2009).1 In Brunnermeier and Pedersen (2005) andPedersen (2009), rational panics occur when traders try to front-run eachother. In their model, an exogenous shock to a trader’s holdings forces herto liquidate her position. Other traders then try to sell before the distressedtrader pushes the price down, resulting in a price crash that may have havea ripple effect on other traders. A similar ripple effect occurs here, but it1For other papers on rational panics, see Romer (1993), Smith (1997), Lee (1998), andBarlevy and Veronesi (2003).3Chapter 1. Introductioncan happen in the absence of an exogenous shock. In Bulow and Klemperer(1994), a single seller sells K objects to K + L bidders through a series ofsequential auctions. Bidders trade off waiting for a lower price against theprobability that they will not receive a good. As a result, both clusteringof trades and price crashes arise. While the models considered here sharesome of the underlying intuition of these papers, the focus is on studyingthe effects of panics on the quality of information aggregated by prices, noton price crashes (which do not arise here).In terms of model structure, my work is related to the financial herd-ing literature that utilizes the sequential trading model first introduced byGlosten and Milgrom (1985) (and, more broadly, to the herding literaturethat originated with Banerjee (1992) and Bikhchandani et al. (1992)). Fora survey of the earlier financial herding literature, see Devenow and Welch(1996), and for more recent contributions, see Avery and Zemsky (1998),Park and Sabourian (2011), and Dasgupta and Prat (2008). In these pa-pers, traders may ignore their private information to follow (herd) or tradecontrary to their predecessors, which creates inefficiencies in informationaggregation. Herding and contrarianism do not arise here, but convergenceslows through another channel: not waiting to acquire information. Otherrelated theoretical papers include those that allow for trade timing, suchas Malinova and Park (2012), and Ostrovsky (2012), who extends the in-sider trading model of Kyle (1985) to multiple traders. In these papers,information is exogenously given, so information acquisition plays no role.The experimental work extends a relatively small experimental litera-ture on trade timing in financial markets. Importantly, the work extendsprevious studies by deriving precise theoretical predictions about when sub-jects should trade. Park and Sgroi (2012), in the most closely related paper,provide qualitative predictions guided by theory in an endogenous timingsetting. They focus on the possibility of herding with exogenously giveninformation. Shachat and Srinivasan (2011) study trading with sequentialarrival of information, but in an environment in which a no trade theoremapplies. Bloomfield et al. (2005) study the choice between market and limitorders in the absence of theoretical predictions. Several papers consider4Chapter 1. Introductiontiming decisions in environments in which prices are fixed and theoreticalpredictions are known. See, for example, Sgroi (2003) and Ziegelmeyer et al.(2005) who each implement the irreversible investment model of Chamleyand Gale (1994). In similar environments, see Ivanov et al. (2009), Ivanovet al. (2013), and C¸elen and Hyndman (2012). Incentives to wait are quitedifferent in these papers as observing others’ decisions can be beneficial whentheir information is not incorporated into prices. Finally, Brunnermeier andMorgan (2010) study timing decisions theoretically and experimentally in agame related to both preemption and war of attrition games.5Chapter 2Informational Losses inRational Trading Panics2.1 Introduction“Apple Stock Hits All-Time High As Investors Rush To Get InEarly Before iPhone 5” - August 17, 2012, Cult of Mac“Apple Stock Hit by Panic Selling: ’Someone Yelled Fire”’ -November 15, 2012, CNBCInformal commentary abounds with anecdotal evidence of“panic selling”and“panic buying” in financial markets. The two articles cited above claim toprovide explanations for both booms and crashes in Apple’s stock.2 In panicbuying, or “buying frenzies”, traders rush to buy a stock for fear of missingout on continued price increases. The first article quotes a fund manager assaying that “his biggest fear [...] is that he won’t get [the] chance to put allof his money into Apple before the share price skyrockets.” In the oppositedirection, panic selling refers to traders rushing to liquidate their positionsbefore prices fall further. The second article quotes David Greenberg ofGreenberg Capital as saying “Someone yelled fire in the theater [...] andas traders do, they will trample you trying to be first to get to the exit.”These quotations illustrate the common perception that panics are drivenby emotional or irrational investors who trade out of the fear of adverse pricemovements.2The first article can be found at: http://www.cultofmac.com/185279/apple-stock-hits-all-time-high-as-investors-rush-to-get-in-early-before-iphone-5/. The second is fromCNBC: http://www.cnbc.com/id/49842457.62.1. IntroductionThis chapter shows that this behavior is not necessarily irrational: per-fectly rational investors may optimally “panic”. Although the word “panic”in a financial market setting may have many different connotations, here Iassociate the term with the general meaning of the word: a sense of urgencyto act as soon as possible. I define panics in a market setting as tradingat the first opportunity, which does not necessarily imply irrational behav-ior or price movements in a single direction (which do not arise here). Ishow that panics are a natural consequence of unconditional correlation ofinformation. Under the standard assumption that private signals are in-dependent conditional on the asset’s true value, signals are unconditionallycorrelated: traders are likely to receive similar news. Consequently, others’trades are likely to move prices adversely, rationalizing the fear of adverseprice movements, and motivating one to preemptively trade. This rationalfear of adverse price movements appears in previous papers, specifically Bu-low and Klemperer (1994), Brunnermeier and Pedersen (2005), and Pedersen(2009). These papers have focused on explanations for price crashes, takinginformation as exogenously given. I instead consider the impact of panicson the informational content of trades, a topic that has not been previouslyaddressed. Intuitively, if traders panic, they forgo the opportunity to acquireadditional information through time-consuming research. This failure to“doone’s homework” can result in trades that are based on weak information,causing prices to more frequently deviate away from fundamental values andtake longer to converge. As other researchers have noted (Vives, 1993), con-vergence speeds are important because knowing that prices converge to aasset’s fundamental value is not practically useful if convergence is so slowthat, by the time it occurs, the value has changed.I build upon the classic trading model of Glosten and Milgrom (1985)in which privately-informed agents sequentially trade an asset with a mar-ket maker. In the period in which she arrives, a trader may buy or sellshort a single unit of an asset (with no restrictions on short sales). To thismodel I add an information acquisition decision in which traders can wait toobtain additional information before trading. Requiring traders to wait toobtain information reflects the fact that information is fundamentally gen-72.1. Introductionerated through time-consuming research and must be processed to generateone’s trading decision. It is this feature that sets the work here apart fromthe existing literature on information acquisition in financial markets, whichinstead assumes information can be acquired at a monetary cost. Equilib-rium predictions are substantially different when information takes time toacquire: waiting to obtain information cannot be equivalently modeled aspaying a monetary, opportunity cost.To focus on the simplest possible timing decision, traders may tradeonly once in one of two periods: the period in which they arrive or in thesubsequent period. Traders receive a private signal in the period in whichthey arrive and, should they choose to wait, an additional private signalin the subsequent period.3 In order to study preemptive trading, I adoptan overlapping timing structure such that if a trader waits, another tradermay front-run her. The option value of waiting therefore depends uponthe strategy of a trader’s successor. This payoff interdependence in a social-learning setting typically makes characterizing the set of equilibria difficult.4As in Glosten and Milgrom (1985), all trades are made with a marketmaker. In the first version of the model (the expected value model), I as-sume that the market maker sets a single price equal to the expected valueof the asset conditional on all available public information.5 In this model,I abstract from the bid and ask prices that arise from the adverse selec-tion problem in order to focus on the strategic interaction between traders.Under intuitive restrictions on off-equilibrium beliefs and prices, I obtain a3The substantive assumption is that, should one trade upon arrival, one cannot alsotrade after receiving the additional signal. This assumption is meant to capture, in atractable way, the idea that information arrives continuously, but trades are necessarilydiscrete (trading continuously, even if technically feasible, would be prohibitively costly).Therefore, a trader contemplating a trade at any point in time must decide how muchinformation to obtain before trading. For tractability, I focus on one such decision, whicheliminates incentives to manipulate prices or not reveal one’s private information (as inKyle (1985)).4For examples of papers with interdependent payoffs, see Callander (2007), Ali andKartik (1998), and Chamley (2007), which is discussed further below.5As in the financial herding literature, prices reflect all past information, so there is noincentive to wait to learn from others’ trades as there is in models of common investmentopportunities, such as Chamley and Gale (1994), Gul and Lundholm (1995), and Chariand Kehoe (2004).82.1. Introductioncomplete characterization of all equilibria. I then consider the zero-profitmodel, adopting the standard market microstructure setting in which com-petitive market makers post separate bid and ask prices, earning zero profitsin expectation. Here, a complete characterization of the equilibria is com-plicated by the additional strategic interaction between the traders and themarket maker. However, Section 2.5 demonstrates that the main insights ofthe expected value model extend to this case.Panics (in which all traders buy or sell immediately upon arrival) occurwhen prices reflect uncertain public beliefs. Interestingly, because infor-mation is most valuable when uncertainty is high, this result implies thatpanics cause information to be forgone precisely when it is most valuable.This counter-intuitive finding results from the fact that the cost of waitingis endogenously determined by the price impact of others’ trades, which islargest when uncertainty is high. This result goes against standard intuitionfrom the literature on information acquisition in financial markets. As firstnoted by Grossman and Stiglitz (1980), there is typically strategic substi-tutability between traders’ decisions to acquire information: when othersacquire information, prices have strong informational content so that onehas little incentive to do her own research.6 Conversely, here there is strate-gic complementarity: acquiring information is less costly when others aredoing so. Intuitively, if others are not in a rush to trade, there is no needfor you to rush either.Strategic complementarity in traders’ timing decisions leads to a multi-plicity of equilibria for some parameterizations. This multiplicity is reflectedin the effects of market commentary on real asset markets, where news ofpanic can lead to further panic. It may also suggest a role for market in-tervention. For example, if a trading halt (or circuit-breaker) interventionis sufficient to change expectations of panic, panics can in fact be soothed.7In a related finding, Chamley (2007)considers a Glosten-Milgrom setup in6See Barlevy and Veronesi (2010) for an interesting counterexample.7For an example of a successful trading halt,see http://www.ft.com/intl/cms/s/0/710240e6-1945-11e2-9b3e-00144feabdc0.html#axzz2NCmc7u66: “Google shares fell as much as 10 per cent to$676 before trading was halted. Their partial recovery to $695 when trading resumed ...”92.1. Introductionwhich market participants trade for short-term profit, unwinding their posi-tions in the period after they first trade. His setting also produces strategiccomplementarity in acquiring information and a multiplicity of equilibria.However, in his model, information is acquired at a monetary cost and tradetiming is exogenous, so the nature of the strategic complementarity is muchdifferent. He also does not explicitly study the effect of information acqui-sition on the informational content of prices, instead focusing on its abilityto produce discontinuous trader behavior and trading volumes.Because information is forgone when it is most valuable, price conver-gence is much different than in sequential trading models with monetarycosts of information, such as that of Nikandrova (2012) and Lew (2013). Intheir models, more information is acquired when uncertainty is high andinformation is most valuable. There, starting from a price reflecting highuncertainty, prices converge quickly to a price near the true asset value andthen convergence slows (or stops completely if there is a fixed cost com-ponent), whereas here, convergence slows down immediately. Simulationresults show that these slowdowns are quantitatively significant, as muchas doubling the time required to converge under certain realistic parame-terizations. Panics also cause prices to more frequently deviate away fromfundamental values.In a particularly counter-intuitive and perhaps troubling result, I showthat increasing the quality of the information immediately available can ac-tually slow convergence overall. This result can contribute to the debateregarding the impacts of high-frequency traders in the market (Biais et al.(2013) and Hoffmann (2013)). Technical innovations that make better qual-ity information available more quickly can actually reduce the amount ofinformation that is incorporated into asset prices.The model can capture several existing empirical findings, including thatthe estimated probability of informed trading (Easley et al., 1996) and per-sistent price impacts of trades (Hasbrouck, 1991) are lower when tradingvolume is higher. Section 2.6 develops several testable predictions and dis-cusses existing evidence that is suggestive of the novel predicted relationshipsbetween the informational content of trades and volume and uncertainty.102.2. The Expected Value Model2.2 The Expected Value ModelThe model is set in discrete time, t = 1, 2, . . . , T . A single asset of unknownvalue, V ∈ {0, 1}, may be traded in each period. Its value is realized at T .For convenience, I assume T → ∞ but the results do not depend on thisassumption.8 The prior belief that V = 1 is given by p1 ∈ (0, 1). In eachperiod, n new risk-neutral traders enter the market. In the remainder ofthe model description and the analysis of Section 2.2, I focus on the caseof n = 1. In Appendix A.5, I consider the case of n > 1 to establish acomparative static prediction with respect to volume.9Upon arrival, each trader receives a private, binary signal, st ∈ {0, 1},which matches V with probability q = Pr(st = 1|V = 1) = Pr(st = 0|V =0) ∈ (12 , 1). I identify a trader by the period in which she arrives, t. Givenher signal, trader t may choose to buy, sell, or not trade, at ∈ {B,S,NT}. Ifshe trades, she leaves the market. If she chooses not to trade, she receives anadditional, private, binary signal in period t+1, st+1 ∈ {0, 1}, which matchesV with probability q = Pr(st+1 = 1|V = 1) = Pr(st+1 = 0|V = 0) ∈ (12 , 1)and may then trade, at+1 ∈ {B,S,NT}. All signals are assumed to beindependent conditional on V . To allow for trades to occur during the timea trader waits to acquire the additional signal, each period is divided intotwo sub-periods with trader t + 1 trading prior to trader t if they trade inthe same period.10 The timing of possible trades is shown in Figure 2.1. Iabbreviate the time of trade as R for rush (at ∈ {B,S}) and W for wait(at = NT ). With this overlapping arrival structure, a trader t that choosesto acquire additional information may face up to two intervening trades:one from trader t − 1 (if she chose to wait) and one from trader t + 1 (ifshe chooses to rush). The complete history of trades, timing decisions, and8For finite T , the strategy of the final trader must be considered separately as nofurther traders arrive to the market.9The n > 1 results are also used in Section 2.4 to numerically assess the impacts ofpanics.10t is used to refer to the period containing the two sub-periods. A lower bar identifiesvalues of variables (signals, actions, prices, etc.) in the first sub-period and an upper baridentifies values in the second sub-period.112.2. The Expected Value ModelFigure 2.1: Trade Timing 1 2 3 4 Trader    1 3 2 4 Time 2a  2a  1a  3a  3a  4a  4a  Trade identifier Note: Arrows reflect the two periods in which each trader may trade. Trade identifiersubscripts denote period, not trader identity.prices, denoted Ht and Ht+1, are observed by all traders.11A risk-neutral market maker posts a single price equal to the publicbelief about the value of the asset, its expected value based upon all publicinformation, pt= E[V |Ht] = Pr[V = 1|Ht] or pt+1 = E[V |Ht+1] = Pr[V =1|Ht+1]. The expected payoffs to a trader who buys the asset at t or t + 1are then, E[V |Ht, st] − pt or E[V |Ht+1, st, st+1] − pt+1, respectively. Theexpected payoffs from selling are identical but with opposite sign. ptplaysa more prominent role in the analysis so I will generally refer to it as theprice at t and denote it pt, unless a distinction must be made.I emphasize that the market maker is not a strategic player: prices maybe best thought of as being determined by an entity that is more concernedabout information aggregation than profits (for example, the owner of aprediction market). Given the strategies of the traders, the market makercan compute the public belief on the equilibrium path, but at off-equilibriumhistories some assumption must be made. In particular, if the equilibrium11Formally, the histories can be defined recursively as At = At−1 ∪ at−1, At = At ∪ at,P t = P t−1 ∪ pt−1, and P t = P t ∪ pt, for t = 2, . . . where A1 = Ø, A1 = A2 = a1, andP 1 = P 1 = P 2 = p1. Ht = At ∪ P t and Ht+1 = At+1 ∪ P t+1 denote the joint action andprice histories.122.2. The Expected Value Modelstrategy is for traders with both st = 0 and st = 1 to rush, then the pricethat arises should a trader deviate to wait depends upon what informationabout st is assumed to be revealed by the deviation.12 Fortunately, a naturalassumption arises in the model. Lemma 2 shows that, independent of anyparticular assumptions about off-equilibrium prices, traders with st = 0 andst = 1 choose to wait with the same probability in any equilibrium. As aconsequence, after a delayed trade, no information about st is revealed andtherefore the public belief is unchanged. Given this, I assume that if a traderdeviates to wait, the public belief and price are also unchanged. I restrictthe analysis to equilibria that satisfy this restriction on prices, providingfurther justification after Lemma 2 in Section a dynamic game of incomplete information, the appropriate so-lution concept for the model is sequential equilibrium (Kreps and Wilson(1982)).14 In addition to the restriction on price formation, I require thatstrategies be a function of only the payoff-relevant state (i.e. Markov strate-gies). While the sequence of past trades and/or prices could be used ascoordination devices, I ignore such possibilities. The payoff-relevant statefor a trader trading at time t + 1 is simply the price she faces. But, for atrader trading at time t, the price she faces and whether or not the previoustrader, t−1, rushed are both relevant. The price reflects any information re-vealed by the decision of t−1 to wait, but when t−1 waits, t’s expected profitfrom waiting is impacted by t− 1’s delayed trade. Formally, an equilibriumof the expected value model is defined as follows.12Other off-equilibrium histories are possible but it turns out that the assumption madeabout prices in these cases is not critical, so I make no particular assumption.13The assumption can be thought of as a restriction on off-equilibrium beliefs as faras the other traders are concerned. Because the market maker is essentially a robot,however, formulating the restriction in terms of beliefs may lead to confusion. Underother assumptions about price formation, the general analysis is unaffected, with themain difference being that the cutoff prices in Theorem 2.1 are more difficult to compute.14Sequential equilibrium, as opposed to weak Perfect Bayesian equilibrium, places re-strictions on beliefs off-equilibrium which help to pin down the public belief, and thereforeprice. It is not, however, restrictive enough to completely pin down beliefs after a traderdeviates to wait and thus an additional assumption is still necessary.132.2. The Expected Value ModelEquilibrium Definition: An equilibrium of the expected value modelconsists of a set of behavioral strategies, σt : (Ht, st)→ {B,S,NT} andσt+1 : (Ht, st, st+1) → {B,S,NT}, a system of beliefs ν and a sequenceof prices, ptand pt+1, such that:1. σt and σt+1 are sequentially rational given beliefs ν;2. There exist a sequence of completely mixed strategies{σkt}∞k=1 and{σkt+1}∞k=1 with σt = limk→∞ σkt and σt+1 = limk→∞ σkt+1 suchthat ν = limk→∞ νk where νk denotes the beliefs derived from σktand σkt+1 using Bayes’ rule;3. Prices satisfy pt= E[V |Ht] and pt+1 = E[V |Ht+1] as derived fromσt and σt+1 using Bayes’ rule wherever possible;4. At any history: (i) σt must specify the same behavioral strategyfor any two traders who face the same price pt, the same timingdecision of their immediate predecessor I(at = NT ), and have thesame signal st; (ii) σt+1 must specify the same behavioral strategyfor any two traders who face the same price pt+1, and have thesame signals st and st+1;5. At an off-equilibrium history Ht reached by a trader waiting: (i) be-liefs ν are common among all traders and are such that the publicbelief does not change, E[V |Ht] = E[V |Ht]; (ii) prices are un-changed, pt = pt.Items 1-2 form the standard definition of sequential equilibrium. Item3 specifies that prices are set equal to the public belief as derived from thetraders’ strategies. Item 4 is the restriction of strategies to payoff-relevantstates. Finally, item 5 is the restriction on prices and beliefs after a traderdeviates to wait in the period she arrives. In the definition, I denotes theindicator function so that I(at−1 = NT ) is 1 if t’s predecessor waited and 0otherwise.Without loss of generality, strategies can be decomposed into a tradingstrategy (buy or sell) for each of the two periods in which a trader maytrade and a timing strategy (rush or wait). When strategies are restricted142.3. Analysis of the Expected Value Modelto payoff-relevant states, we can define the probability with which a traderobserving a particular first period signal (which I refer to as her type) waitsas βx(pt, I(at−1 = NT )) ≡ Pr(at = NT |st = x, pt, I(at−1 = NT )). When itdoesn’t lead to confusion, I drop the dependencies of βx.2.3 Analysis of the Expected Value ModelTo characterize the equilibria of the expected value model, I first proveseveral intermediate results that are of interest on their own. The firstlemma determines the optimal trading strategy taking the timing strategyas given. The second lemma proves that both types of traders must followthe same timing strategy in any equilibrium. Proposition 2.1 establishes themotivating intuition of the model: the presence of other traders reduces theexpected profit from waiting. Proposition 2.2 establishes the interesting factthat, as prices become certain, traders wait to obtain additional information.2.3.1 Optimal Trading StrategyLemma 2.1 provides the optimal trading strategy of a trader taking as giventhe timing strategies, β1(pt, I(at−1 = NT )) and β0(pt, I(at−1 = NT )). Allproofs are contained in Appendix A.3.152.3. Analysis of the Expected Value ModelLemma 2.1: In any equilibrium:1. Any trader who rushes buys if st = 1 and sells if st = 0.2. With timing strategies β1(pt, I(at−1 = NT )) and β0(pt, I(at−1 =NT )), at least one of which is strictly positive, for all pt andI(at−1 = NT ), any trader who waits:(a) buys if st = 1 and st+1 = 1; sells if st = 0 and st+1 = 0(b) buys (sells) if st = 0, st+1 = 1, and g0(q, q) ≡ (1− q)qNT0 −q(1− q)NT1 ≥ (<)0(c) buys (sells) if st = 1, st+1 = 0, and g1(q, q) ≡ q(1− q)NT0 −(1− q)qNT1 > (≤)0where NT0 = (1− q)β1(pt, I(at−1 = NT )) + qβ0(pt, I(at−1 = NT )), NT1 = qβ1(pt, I(at−1 = NT )) + (1 − q)β0(pt, I(at−1 = NT ))are shorthand for Pr(at = NT |V = 0) and Pr(at = NT |V = 1),respectively.For rushed trades, the result of Lemma 2.1 is intuitive: traders withgood signals buy and those with bad signals sell. For delayed trades, theoptimal trading strategy is also determined by comparing the trader’s pri-vate belief to the price she faces, but the price she faces depends upon thetiming strategies of the two types of traders, β0 and β1. If β0 6= β1, someinformation about st is revealed to the market when a trader waits and soher optimal trading strategy may depend upon these strategies. If β0 = β1,no information is revealed, so that the resulting optimal trading strategy isindependent of the timing strategies (as can be seen by setting β0 = β1 inLemma 2.1) . By assumption, a deviation to wait does not reveal informa-tion and thus the optimal trading strategy in the continuation game aftera deviation to wait corresponds to setting β0 = β1 in Lemma 2.1. Also,note that other traders’ strategies do not affect the delayed trading strategyof t because any information revealed by their trades is both known to thetrader and reflected in the price.An implication of Lemma 2.1 is that, except in the case of indifference162.3. Analysis of the Expected Value Modelwhen a trader’s signals oppose each other, a trader always trades.15 Thissimple fact is a consequence of the trader always having an informationaladvantage and hence a profitable trade.16 As a result, some private infor-mation is revealed by each trade. In Section 2.4, this fact will be used toestablish that prices converge asymptotically to the true asset value.2.3.2 The Benefit of Additional InformationTo determine the optimal timing strategy of a trader, one must calculatethe value of waiting to obtain additional information: the expected profitfrom trading at t + 1 less the profit from trading at t. When the benefit ispositive (negative), a trader’s best response is to wait (rush). The benefitfrom waiting depends upon the timing strategies of t because informationmay be revealed by her decision to wait. In equilibrium, the timing strategyfor each type of trader must be consistent with the sign of the benefit.The benefit from delaying also depends upon the timing and tradingstrategies of t − 1 and t + 1, because any trade that occurs while t waitsaffects the price she will trade at. Initially, I use generic notation to denotethe actions of the other traders because several properties of the benefit towaiting can be established for any possible strategies of others. I denote ageneric event that occurs during the time t waits as aˆ and the set of possiblesuch events as A, so that aˆ ∈ A. I also abbreviate Pr(aˆ|V = y) as aˆy,for y ∈ {0, 1}, so that, when summing over all possible events,∑aˆ∈A aˆ0 =∑aˆ∈A aˆ1 = 1.17 In Appendix A.1, I derive a general form of the benefit fromwaiting, which I denote as Bx(pt, β0, β1) for traders with st = x. It is givenby15For concreteness, I assume a trader that is indifferent trades according to st+1 but theanalysis is qualitatively robust to other assumptions, including that she does not trade.16Another implication is that herding and contrarianism are precluded: a trader neverignores her private information and either copies or trades contrary to her predecessors.17As an example, if t − 1 and t + 1 both rush, then A = {at+1 = B, at+1 = S} is theset of possible events. From Lemma 2.1, t + 1 buys if st+1 = 1 and sells if st+1 = 0, so∑aˆ∈A aˆ1 =∑aˆ∈A aˆ0 = q+1−q = 1. If instead, both t−1 and t+1 trade during the timet waits, then each possible (joint) event is more complex and may consist of, for example,t− 1 buying and t+ 1 waiting, etc.172.3. Analysis of the Expected Value ModelBx(pt, β0, β1) =pt(1− pt)Pr(st = x)[∑aˆ∈Aaˆ0aˆ1f(q, q, β0, β1)Pr(aˆ&at = NT )− (2q − 1)](2.1)wheref(q, q, β0, β1) =(2q − 1)(qNT0 + (1− q)NT1) if st = 1 & g1(q, q) ≤ 0qNT0 − (1− q)NT1 if st = 1 & g1(q, q) > 0qNT1 − (1− q)NT0 if st = 0 & g0(q, q) < 0(2q − 1)(qNT1 + (1− q)NT0) if st = 0 & g0(q, q) ≥ 0The formula for the benefit (2.1) is only defined when at least one type oftrader t waits with a positive probability. If both types rush with probability1, then each of their benefits from deviating to wait depends upon the priceset after such an off-equilibrium action. Without imposing any restrictionon these off-equilibrium prices, however, Lemma 2.2 establishes that bothtraders must follow the same timing strategy in equilibrium.Lemma 2.2: In any equilibrium, traders with st = 0 and st = 1 mustfollow the same timing strategy, β1(pt, I(at−1 = NT )) = β0(pt, I(at−1 =NT )) ∀pt, I(at−1 = NT ).The intuition behind Lemma 2.2 is instructive and the proof by contra-diction follows the intuitive argument closely. If, for example, a trader witha good signal were to wait more often than a trader with a bad signal, thenthe market maker would infer from the decision to wait that the first periodsignal is more likely to be good and thus raise the posted price. This increasein price benefits the trader with a bad signal (who is more likely to sell) moreso than the trader with a good signal (who is more likely to buy). Thus, ifthe trader with a good signal is waiting with some positive probability, thetrader with a bad signal must be waiting with certainty, contradicting theinitial assumption that the trader with a good signal waits more often.If both types of trader rush with probability 1, then their expected profit182.3. Analysis of the Expected Value Modelfrom deviating depends upon the information that is assumed to be revealedby the deviation. But, under the assumption made about off-equilibriumprice formation, no information is revealed, which corresponds to settingβ0 = β1 in (2.1). Therefore, Lemma 2.2 and this assumption together implythat we can set β0 = β1 = β in (2.1) for the remainder of the analysis.18 Asshown in Appendix A.1, with β0 = β1, the benefit function simplifies so thatit no longer depends upon the timing strategies of the two types of traders.Therefore, I denote the simplified function Bx(pt).Although Lemma 2.2 does not preclude mixed timing strategies, β1 =β0 ∈ (0, 1), it proves useful to first consider a trader’s benefit when all othertraders use pure timing strategies. Because both types of a trader mustfollow the same timing strategy, a pure timing strategy for a trader canbe described simply as rush or wait. There are six possible cases for the(joint) timing strategies of others from trader t’s perspective. First, t− 1 isobserved to either rush or wait. On the other hand, t+ 1’s timing decisionmust be anticipated by t and t may expect it to depend upon the price t+1 isexpected to face. If t−1 has rushed, t knows that if she waits, t+1 faces thesame price as t. But, if t−1 waits, t may rationally expect t+1’s decision todepend upon whether t− 1 buys or sells (which is observed by t+ 1 but nott). The formulas for the benefit in each case are provided in Appendix A.1.The benefit is denoted Bu,v1v2x (pt) where the superscript, u, v1v2, signifiesthe timing strategies, R or W , of the other traders. u corresponds to theobserved timing decision of t − 1. When u = R, v1 = v2 corresponds tot + 1’s expected timing decision. When u = W , v1 (v2) corresponds to theexpected timing decision of t+ 1 if t− 1 buys (sells).To determine the best response of a trader to the strategies of others, itis helpful to study the properties of the six benefit functions as a function of18Note that, were a different assumption to be made about the price and beliefs aftera deviation to wait, the expected benefit from deviating would depend upon the beliefsand price assumed and therefore the benefit function would be discontinuous in the timingstrategies of the two traders at β0 = β1 = 0. A different assumption would also intro-duce an asymmetry between the two types of trader that Lemma 2.2 establishes cannotbe present on the equilibrium path. These reasons provide further justification for theassumption made.192.3. Analysis of the Expected Value ModelFigure 2.2: Example Benefit Functions0 0.2 0.4 0.6 0.8 100.06PriceBenefit  BW,RWBW,RRpR,RR1-pR,RRBR,WWBR,RRBW,WWBW,WRNote: st = 1, q = 0.75, and q = 0.8. Only the zero-crossings for BR,RRx (pt), 1− pˆR,RRand pˆR,RR are labeled to reduce clutter.202.3. Analysis of the Expected Value Modelprice. To illustrate, Figure 2.2 plots all six benefit functions for a trader withst = 1, given parameters q = 0.75 and q = 0.8. In general, each functionis either always negative (not shown) or has one of the two shapes in theexample: always positive or positive only near pt = {0, 1}. Propositions 2.1and 2.2 prove two of the more interesting properties of the benefit functions,while the remainder are established in Lemma A.3.1 of Appendix A.3.Proposition 2.1 establishes that each additional trade that may occurduring the time a trader waits reduces her expected benefit from waiting.19The unconditional correlation of signals rationalizes the fear of expectedadverse price movements.20Proposition 2.1: Each additional, conditionally independent, infor-mative, potential trade between t and t + 1 strictly reduces the benefitto waiting, Bx(pt, β0, β1), for all pt ∈ (0, 1) and all timing strategies,β0(pt, I(at−1 = NT )) and β1(pt, I(at−1 = NT )).The implications of Proposition 2.1 can be observed in Figure 2.2. Specif-ically, the benefit, BR,WWx (pt), is largest because, during t’s waiting period,neither t − 1 nor t + 1 trade. BR,RRx (pt) and BW,WWx (pt) are next largestbecause only one or the other trades. The remaining benefits are smaller be-cause both trade, at least some of the time. From the formula for BR,WWx (pt)in Appendix A.1, we see that it is positive for all pt ∈ (0, 1) if and only ifq > q. When q ≤ q, it is zero for all pt because the additional signal neverchanges t’s trading decision and so is of no value. This fact, combined withProposition 2.1, immediately implies that all of the benefit functions are(weakly) negative for all pt when q ≤ q. For the more interesting case ofq > q, also note that BR,RRx (pt) is greater than BW,WWx (pt) at all prices due19The focus here is on the potential trades by t−1 and t+1, but the lemma is actuallybroader in application. Any realization of a random variable, xi, (for example, a publicearnings announcement) that is both informative (takes at least two possible values suchthat Pr(xi|V = 1) 6= Pr(xi|V = 0) for at least one value) and independent (conditionalon V ) similarly reduces the benefit to waiting.20Even when q = 12 , such that the best prediction of the price at t+ 1 is the price at t,informative trades by other traders reduce the benefit to waiting because others’ signalsare unconditionally correlated with st so that they are likely to trade in the same directionas you.212.3. Analysis of the Expected Value Modelto the weaker information revealed by rushed trades.Proposition 2.2 captures a property of the benefit to waiting as pricesbecome certain, pt → {0, 1}: the value of waiting approaches that of atrader that faces no possible trades between t and t + 1. This propertyarises because any intervening trade has a negligible impact on prices as theybecome certain. Importantly, however, it is not just the magnitude of thebenefit that approaches that of a single trader. Proposition 2.2 also showsthat the two benefits must have the same sign. Together, these propertiesensure that the best response of a trader facing other traders approachesthat of a trader who is alone in the market. Defining the benefit functionwhen no other informative intervening trades are possible as BSTx (pt, β0, β1),we have:Proposition 2.2: For all timing strategies, β0(pt, I(at−1 = NT )) andβ1(pt, I(at−1 = NT )), the benefit function, Bx(pt, β0, β1), for any possi-ble intervening informative trades satisfies:1. limp→{0,1}|Bx(pt, β0, β1)−BSTx (pt, β0, β1)| → 02. limp→{0,1}sgn (Bx(pt, β0, β1)) = sgn(BSTx (pt, β0, β1))When β1 = β0 and q > q, I previously established that the benefit for atrader alone in the market, BR,WWx (pt), is strictly positive for all pt ∈ (0, 1)and so an immediate consequence of Proposition 2.2 is that there exist pricessufficiently close to 0 and 1 such that all of the benefit functions are strictlypositive. Therefore, in any equilibrium of the model with q > q, as pricesbecome certain, panics cease to exist and all traders wait. This feature ofthe model may be surprising because, building on the intuition of Grossmanand Stiglitz (1980), one may think that there would be a tendency to freeride off of the strong public information contained in prices as they becomecertain. While it is true that the private value of information decreases tozero as prices become certain, the difference here is that the cost of additionalinformation also (endogenously) decreases to zero. As long as public beliefsare not perfectly certain, there remains a strictly positive benefit to obtainingthe stronger information of the second signal.222.3. Analysis of the Expected Value ModelNote that Propositions 2.1 and 2.2 did not impose the equilibrium restric-tion of Lemma 2.2 that the timing strategies of the two types of trader arethe same. This fact becomes important in Section 2.5 where the robustnessof results to the incorporation of bid and ask prices is analyzed.21In addition to the properties established by Propositions 2.1 and 2.2,Lemma A.3.1 establishes that each benefit function has, other than thoseat pt = {0, 1}, at most two zero crossing points, pˆ and 1 − pˆ, which aresymmetric with respect to pt = 12 .22 When the additional zero-crossingsexist, I denote the one for Bu,v1,v2x (pt) that occurs for pt ∈ [12 , 1) by pˆu,v1v2 .Under the restriction on prices and public beliefs on off-equilibrium histories,each pˆu,v1v2 is a function of q and q only.23 In the following section, I in turnconsider each of the price regions delineated by the zero-crossings, pˆu,v1v2 ,to establish the fixed point of best responses within the region. In thediscussion, I assume all of the price regions exist, as in Figure 2.2. However,Theorem 2.1 also applies when q and q are such that one or more of theregions does not exist (see footnote 24).2.3.3 Equilibrium Trading BehaviorIn this section, I use the properties of the benefit functions to determine thefixed point of best responses in timing strategies at any price. For some priceregions, the fixed point of best responses is particularly straightforward,while for others it is more complex because it involves considering the pricest + 1 is expected to face. Here, I work out two simple cases to provide theintuition, leaving the more complex cases to the proof in Appendix A.3.The discussion here is for the case of q > q. When q ≤ q, the unique21In fact, Lemmas 3 and 4 would continue to hold if the restriction that information isrevealed by a deviation to wait was replaced by some other reasonable assumption aboutwhat information is revealed by a deviation to wait (i.e. st = 0 deviates with a differentprobability than st = 1 ). It is in this sense that the fundamental nature of the equilibriaare not altered by different off-equilibrium assumptions.22The zero-crossings for BW,RWx (pt) and BW,WRx (pt) are not symmetric, but they dohave a symmetry property with respect to each other. If BW,RWx (pt) crosses zero at pt,then BW,WRx (pt) crosses zero at 1− pt, and vice versa.23Under different assumptions, pˆu,v1v2 would depend upon the assumed off-equilibriumbeliefs.232.3. Analysis of the Expected Value Modelequilibrium is for all traders to rush for the same reason that all tradersrush at prices, pt ∈ (1 − pˆR,RR, pˆR,RR), which is established below. First,however, I comment on the role of mixed timing strategies in equilibrium.In order for a trader to mix between rushing and waiting, she must beindifferent between the two strategies. Because the timing decision of t− 1is observed, whether or not she is using a mixed strategy is irrelevant for t.Therefore, if t + 1 is playing a pure strategy, t can only mix if pt happensto be at the zero-crossing of the appropriate benefit function. For genericparameters, however, a price equal to a zero-crossing is never reached. I ig-nore non-generic mixing possibilities in the analysis and assume that traderswait if indifferent. On the other hand, if t + 1 is mixing between rushingand waiting with the appropriate probabilities, t can be induced to mix.When t + 1 mixes, the benefit to waiting for t is a linear combination ofthe benefits that result when t + 1 uses each of her pure strategies, withthe weights being determined by the mixing probabilities. The possibility ofmixed timing strategies in equilibrium is accounted for in what follows. Im-portantly, I show that one can still proceed by individually considering theprice regions that are delineated by the zero-crossings of the pure strategybenefit functions.The equilibrium timing strategy for a trader that faces a price, pt ∈(1− pˆR,RR, pˆR,RR), nicely illustrates the idea of rational panic. If t−1 waits,t’s benefit is one of BW,WWx (pt), BW,WRx (pt), BW,RWx (pt), or BW,RRx (pt), allof which are negative in this price range, so she will rush regardless of whatt + 1 does. If, instead, t − 1 rushes and t were to wait, t + 1 would face aprice of pt and would rush because t waited. But, if t+ 1 rushes, t’s benefitis BR,RRx (pt) < 0, so t must rush. Thus, we see that all traders rush becauseof the off-equilibrium threat of the next trader rushing if a trader were towait. Mixing is not possible because t+1 can never mix when t waits. Panicripples throughout the sequence of traders as long as prices remain in thisrange.Next, consider pt ∈ (1− pˆW,WW , 1− pˆR,RR]∪ [pˆR,RR, pˆW,WW ). This pricerange differs from the previous one in that BR,RRx (pt) > 0 so that t bestresponds by waiting if t − 1 rushes, regardless of what t + 1 does. But, as242.3. Analysis of the Expected Value Modelin the previous case, if t − 1 waits, t′s benefit is negative, so she will rushregardless of what t + 1 does. Because t can condition her behavior on theobserved timing decision of t−1, we obtain what I call conditional rushing asthe equilibrium timing strategy in this price range: if t − 1 waits, t rushes;if t − 1 rushes, t waits. Again, mixing is not possible because a trader’sdecision does not depend upon what her successor does.For pt ∈ (0, 1 − pˆW,WR] ∪ [pˆW,WR, 1), the proof of Theorem 2.1 estab-lishes that all traders wait, reflecting Proposition 2.2. For the remainingprice range, pt ∈ (1 − pˆW,RR, 1 − pˆW,WW ] ∪ [pˆW,WW , pˆW,RR), the reasoningthat determines equilibrium timing strategies is more complex. We can ruleout rushing when t − 1 rushes because, in this case, the benefit is eitherBR,RRx (pt) or BR,WWx (pt), both of which are positive in this price range,so t will wait independent of what t + 1 does. Therefore, only conditionalrushing or waiting are possible equilibrium strategies. When t− 1 waits, t’sbest response depends upon what she expects t+1 to do, which may in turndepend upon the two possible prices reached after t− 1 trades. Although tdoes not know whether t− 1 will buy or sell, she knows the resulting pricesafter each possible trade because they are functions of pt and q only. I denotethe price reached after a buy decision as p+t = pt+1|at = B and the pricereached after a sell decision as p−t = pt+1|at = S. If p+t and p−t lie in priceranges for which it has previously been determined that t+ 1 must follow aspecific equilibrium timing strategy, then t can easily anticipate what t+ 1will do and thus best respond accordingly. However, for any given pt, p+tand p−t don’t necessarily lie in any particular price range so there are manycases to consider. Also, if either p+t or p−t lie again in a price range for whichthe best response of t+ 1 is not known, what t expects t+ 1 to do dependsupon what she expects t + 1 expects t + 2 to do. Of course, this reasoningmay extend indefinitely.Progress can be made by realizing that t+2 may in fact be in an identicalsituation to t. If all of t − 1, t, and t + 1 wait and the trading decisions oft−1 and t turn out to be opposite, t+2 faces an identical situation to t and,given the restriction that strategies must be the same in payoff-equivalentsituations, t+2 must follow the same timing strategy as t. Therefore, rather252.3. Analysis of the Expected Value Modelthan thinking of the game between a trader and her successor, one can re-frame the problem as a static game between the trader and the“neighboring”traders at prices which can be reached by delayed trades.To describe this translation of the dynamic game into a static gameformally, I introduce the concept of an unrestricted price chain, P(p˜): theset of all of possible prices that can be reached from delayed trades startingat some price, p˜ ∈ [12 , q).CU (p˜) ≡{p|p =p˜qkp˜qk + (1− p˜)(1− q)k, k = −∞ . . .∞}I also define a restricted price chain as the set of prices in CU (p˜) for whichthe equilibrium timing decision is not necessarily unique.CR(p˜) ≡ CU (p˜) ∩((1− pˆW,WR, 1− pˆW,WW ] ∪ [pˆW,WW , pˆW,WR))If pˆW,WR does not exist, I define the restricted price range to be the nullset. The union of restricted price chains for all p˜ ∈ [12 , q) consists of allprices, p ∈ (1− pˆW,WR, 1− pˆW,WW ]∪ [pˆW,WW , pˆW,WR) and no two restrictedprices chains have any price in common. These facts together imply that onecan partition this price range into price chains and describe the equilibriumstrategies for each price chain.Two examples of price chains are illustrated in Figure 2.3. The pricesin an unrestricted price chain for k greater than some finite kˆ (and smallerthan some finite −k˜) must lie in a price region in which traders always waitdue to Proposition 2.2. I loosely refer to this fact as the unrestricted pricechain “ending” in a region in which traders wait, although strictly speakingthere is no actual end to a price chain. The equilibrium timing decisions forthose regions in which the equilibrium timing decision is unique are labeledaccordingly.In Case 1, the unrestricted price chain does not pass through pt ∈ (1 −pˆW,WW , 1− pˆW,WW ) and therefore the only prices for which the equilibriumtiming strategy is known are at its ends. In this case, multiple possibleequilibrium timing strategies exist at prices in the associated restricted price262.3. Analysis of the Expected Value ModelFigure 2.3: Equilibrium Timing Strategies and Sample Price Chains tp  1 0 Wait Conditional Rush 21  Case 2 Rush Conditional Rush Wait price chain  tp  1 0 Wait Conditional Rush RRRp ,ˆ1−  RRRp ,ˆ  WWWp ,ˆ  WRWp ,ˆ  21  Case 1 WWWp ,ˆ1−  WRWp ,ˆ1− Rush Conditional Rush Wait price chain  p~  p~  RRRp ,ˆ1−  WWWp ,ˆ1−  WRWp ,ˆ1− RRRp ,ˆ  WWWp ,ˆ  WRWp ,ˆ  Note: The infinite set of prices at each “end” of the price chain are not shown. Thecutoff, pˆW,RW (between pˆW,WW and pˆW,WR) is not shown because it does not directlyplay a role in determining the equilibrium timing strategies. Similarly, 1− pˆW,RW(between 1− pˆW,WR and 1− pˆW,WW ) is also not shown.272.3. Analysis of the Expected Value Modelchain. Intuitively, if each of a trader’s neighbors in the chain are waiting,a trader faces BW,WWx (pt) and waits. If they are conditionally rushing, atrader faces a strictly negative benefit when t − 1 waits and conditionallyrushes. The static game is essentially a coordination game between a traderand her neighbors, so as one might expect, there also exists an equilibriumin mixed strategies, as the proof in Appendix A.3 demonstrates.In Case 2, the unrestricted price chain does pass through (1−pˆW,WW , 1−pˆW,WW ), and the fact that the traders in this interval rush or conditionallyrush forces all traders in the chain to conditionally rush. Theorem 2.1, themain result of the chapter, establishes these claims formally, characterizingthe equilibria of the trading model.2424When a price range in Theorem 2.1 does not exist, the next innermost price rangespecifies the timing strategies. For example, when pˆR,RR does not exist, part 2b appliesto all prices, pt ∈ (1 − pˆW,WW , pˆW,WW ) and when pˆW,WR does not exist, traders at allprices wait.282.3. Analysis of the Expected Value ModelTheorem 2.1: All equilibria are characterized by the trading strate-gies of Lemma 2.1 and the following timing strategies, β1(pt, I(at−1 =NT )) = β0(pt, I(at−1 = NT )) ≡ β(pt, I(at−1 = NT )).1. For q ≤ q: ∀pt ∈ (0, 1), a trader rushes2. For q > q:(a) For pt ∈ (1− pˆR,RR, pˆR,RR), a trader rushes(b) For pt ∈ (1 − pˆW,WW , 1 − pˆR,RR] ∪ [pˆR,RR, pˆW,WW ), a traderconditionally rushes(c) For pt ∈ (1− pˆW,WR, 1− pˆW,WW ]∪ [pˆW,WW , pˆW,WR), partitionthe interval into the restricted price chains CR(p˜) for eachp˜ ∈ [12 , q). Then,i. ∀CR(p˜) such that CU (p˜) ∩ (1 − pˆW,WW , pˆW,WW ) 6= Ø, atrader with pt ∈ CR(p˜) conditionally rushesii. ∀CR(p˜) containing only one price, ps, and such thatCU (p˜) ∩ (1− pˆW,WW , pˆW,WW ) = Ø, a trader at ps waitsiii. ∀CR(p˜) not satisfying (i) or (ii), a timing strategy is partof an equilibrium if and only if it is one of: (A) alltraders at prices pt ∈ CR(p˜) wait; (B) all traders at pricespt ∈ CR(p˜) conditionally rush; (C) two or more tradersat prices pt ∈ CR(p˜) mix between conditionally rush andwait and the remaining traders conditionally rush or wait(d) For pt ∈ (0, 1− pˆW,WR] ∪ [pˆW,WR, 1), each trader waitsHere, rush means β(pt, 0) = β(pt, 1) = 0, conditionally rush meansβ(pt, 1) = 0 and β(pt, 0) = 1, and wait means β(pt, 0) = β(pt, 1) = 1.Theorem 2.1 establishes that panics occur when uncertainty in the mar-ket is high, in accord with intuition. Panics are predicted to be more likelyto occur, for example, in hot new technology stocks than in stocks for whichthe fundamental value of the company is well known, such as well-researchedblue chip stocks. This result is driven by the feature of the model that theprice impacts of other traders are largest when uncertainty is high. With-out this effect, traders would be more likely to acquire more information292.3. Analysis of the Expected Value Modelwhen uncertainty is high because information is most valuable at this time.But, the fact that panics occur when uncertainty is high also means thattheir impacts on price convergence are significant because when informationis most valuable, it is not acquired. I study price convergence in detail inSection 2.4.The multiplicity of Theorem 2.1 2c, part (iii), allows for expectationsto affect optimal behavior and thus opens the door to the possibility thatpublic information that influences expectations (such as media coverage,etc.) may shift equilibria, inducing or calming panics. The multiplicityexists over certain price ranges only, but, for some parameterizations, theseprice ranges can span almost the entire range of possible prices.25 On theother hand, for some parameterizations, the equilibrium is unique.262c, part (iii), states that mixed timing strategies are possible but doesnot provide a complete characterization. I don’t pursue a complete char-acterization but, as Appendix A.2 shows, any equilibrium involving mixedstrategies is unstable in the pseudo-dynamic sense: any small change inthe strategy of one of the traders will tend to lead away from the mixedequilibrium towards one of the pure strategy equilibria. The pure strategyequilibria, on the other hand, are stable in that a small change in strategywill tend to reverse itself. Given their stability, the pure strategy equilibriaare the focus in the following analysis of price convergence.25When pˆW,WW doesn’t exist and q < pˆW,WR, then there are no restricted price chainsin which the equilibrium timing strategy is unique. Then, the equilibrium timing strategyis only unique for pt ∈ (0, 1− pˆW,WR] ∪ [pˆW,WR, 1), which may be very small.26When pˆW,WW exists and (pˆW,WW )− > 1 − pˆW,WW , it is possible to show that allunrestricted price chains contain a price p ∈ (1 − pˆW,WW , pˆW,WW ) and therefore 2c,part (ii) ensures conditionally rushing is the unique equilibrium timing strategy for pt ∈(1−pˆW,WR, 1−pˆW,WW ]∪[pˆW,WW , pˆW,WR) so that the equilibrium timing strategy is uniquefor all prices. If (pˆW,WR)− < 1 − pˆW,WR, the opposite is true: the unique equilibriumhas traders waiting at every pt ∈ (1− pˆW,WR, 1− pˆW,WW ] ∪ [pˆW,WW , pˆW,WR) because allrestricted price chains are singletons so 2c, part (i) applies to this entire range of prices.It is also possible to show that there is no parameterization in which both 2c, part(i) and2c, part(ii) apply.302.4. Effects of Panics2.4 Effects of PanicsIn this section, I discuss the impact of panics on the ability of prices toreflect fundamental values. In addition, I demonstrate the potential forpanic cycles.I begin with an illustrative example using q = 0.75 and q = 0.80. Underthis parameterization, the equilibrium is to rush for all pt ∈ (0.18, 0.82) andconditionally rush for all pt ∈ (0.10, 0.18] ∪ [0.82, 0.90). Figure 2.4 plots arandomly generated price path for the case of V = 0. For comparison, Iconsider a benchmark model in which all traders wait, ensuring all potentialinformation is incorporated into prices. Figure 2.4 illustrates two (related)detrimental effects of panics on asset prices. First, because panics implytrading on weaker information, prices are more likely to diverge away fromfundamental values, as seen at t = 1, 2. Second, although prices eventu-ally converge to V = 0, they converge more slowly than in the benchmarkmodel. While this example only illustrates the possibility of these negativeeffects, in the following subsections, I show that increasing deviations fromfundamentals and longer convergence times also occur in expectation.2.4.1 Price ConvergenceA standard result from the literature, beginning with Glosten and Milgrom(1985), is that prices converge to the true value of the asset. This result iseasily shown to extend to the model considered here. From Lemma 2.1, aslong as prices are not equal to 0 or 1, there always exists a trader who iswilling to buy and another who is willing to sell, and one of these tradersmust have a private belief about the value of the asset that is further fromthe true value than the public belief. These facts, as shown in Avery andZemsky (1998), Proposition 2.4, imply that public beliefs, and thereforeprices, must converge to the true value of the asset.Of more interest here is the rate of convergence. Panics cause tradersto trade on lower quality information than what they could have obtained,which can slow the rate of price convergence. I focus on the more interestingcase of q > q because when q ≤ q, although the unique equilibrium is to panic312.4. Effects of PanicsFigure 2.4: Sample Price Path0 2 4 6 8 10 12 1400.  Model with PanicsBenchmark ModelNote: n = 1, q = 0.75, q = 0.80, andV = 0. Solid dots correspond to the periods in whicha trader optimally panicked.322.4. Effects of Panicsat every price, panics have no impact on convergence because all trades arebased upon st in both models.27 When q > q, on the other hand, traders thatwait trade according to st, revealing information of quality q, but tradersthat panic trade according to st, revealing information of weaker quality, q.In trading models with binary signals, no single number defines the rateof convergence because it varies with the public belief. Thus, convergenceis typically measured by the expected number of periods it takes the price(or some convenient function of price such as the log-odds ratio) to reachsome specified value, conditional on knowing the true value of the asset.As in Glosten and Milgrom (1985), one can use Wald’s lemma to derive ananalytical expression for the expected number of periods, T˜ , for the log-oddsratio, log(pT˜1−pT˜), to reach a particular value in the benchmark model.28However, in the model with panics, there is no corresponding analyticalexpression. Knowing only the number of buys and sells is not sufficient todetermine the price because the price ranges that are passed through, andtherefore which traders rush or wait, depend upon the order of the buy andsell decisions. This complication makes it impossible to derive a closed formexpression for the expected price (or log-odds ratio), making it difficult toperform comparative statics on the effects of changes in parameters on therate of convergence.An indirect approach allows me to obtain an interesting result: an in-crease in the first period signal strength, q, can actually slow convergence.Basic intuition in a model with exogenous information would suggest thatproviding higher quality information would tend to increase the rate of con-vergence. But, this intuition does not apply here because increasing thequality of information in the first period induces traders to rush more often27Panics actually speed convergence through the mechanical effect of all trades beingone period earlier in time. I do not emphasize this effect because it relies on the specificassumption of a single period of delay to acquire information and there being an initialperiod.28Specifically, the expected number of periods for the log-odds ratio to exceed log(b1−b)for some public belief, b ∈ (0, 1), when the asset is worth V = 1 can be shown to beE[T˜ ] =log( b1−b )−log(p11−p1)(2q−1) log(q1−q) where p1 is the initial public belief.332.4. Effects of Panicsthrough two effects. First, it increases the profit from trading in the firstperiod, making waiting less attractive. In the absence of other traders, thiseffect on its own is insufficient to cause traders to rush. With the fear ofprice movements, however, there is a second effect: if one waits and the sub-sequent trader rushes, her trade has a larger impact on the price, reducingone’s profits in expectation. Together these effects may induce a trader totrade on lower quality information. Proposition 2.3 formalizes this intuitionby demonstrating that, for any second period signal strength, one can al-ways find first period signal strengths such that convergence is slower whenmore information is available.29Proposition 2.3: For all q ∈ (12 , 1) and all p1 ∈ (12 , 1), there existql, qh∈ (12 , q) with ql < qh, and a price, p˜ > p1, such that the expectedtime for prices to converge to any cutoff price p ≥ p˜, conditional onV = 1, is strictly larger under qhthan under ql.The reason Proposition 2.3 only guarantees convergence is slower toprices greater than or equal to some p˜ is because panics initially speed upconvergence mechanically by forcing trades to occur earlier in time. Thiscaveat is relatively innocuous because one is normally interested in the timeit takes prices to converge to a value close to the true asset value. The basicidea behind the proof is that, for q sufficiently close to 12 , all traders waitin the unique equilibrium because all benefit functions are strictly positive.Thus, at this low value of q convergence proceeds at the rate of the bench-mark model where all trades reveal signals of strength q. On the other hand,for q sufficiently large, any equilibrium involves price regions for which theunique trading strategy is to rush. Therefore, at least one trade must bebased on lower quality information causing prices to take longer to convergein expectation.To demonstrate that the quantitative impact of panics on convergencespeeds can be substantial, in the absence of a closed-form expression for29Proposition 2.3 is stated in terms of convergence when V = 1, but a symmetricproposition is easily proven for the case of V = 0.342.4. Effects of Panicsconvergence times, I rely on numerical simulation. Specifically, I simulatethe time necessary to reach an average price of 0.99 when the true value ofthe asset is 1 for both the expected value model and the benchmark model,then calculate the percentage difference relative to the benchmark model.30To understand how the quantitative effect of panics on price convergencevaries with the signal strengths, q and q, Figure 2.5 provides a heat mapof the simulated percentage slowdown for each combination of parameters,scaled such that black represents the largest slowdown (negative numbers)and white represents the largest speed up (positive numbers).From Figure 2.5, we first note that slowdowns in convergence can bedramatic, with the expected value model taking more than twice as long toconverge in the neighborhood of q ∈ [0.82, 0.84] and q ∈ [0.96, 0.98]. Also,there is clearly a complicated relationship between the parameters and theslowdown in price convergence, suggesting that no simple comparative staticresults exist. Convergence is noticeably discontinuous in the parameters atcertain boundaries due to the fact that a small change in parameters cancause the discrete prices reached by trades to jump across a cutoff price thatdelineates panicking and waiting. Generally, convergence slows down as qincreases for fixed q, as long as q < q, reflecting Proposition 2.3. Increasingq for a fixed q, on the other hand, has a non-monotonic effect. Althoughincreasing q intuitively increases the benefit to waiting, if one knows thatby waiting, the asset value may be revealed with near certainty because of adelayed trade by your predecessor, it can cause one to rush to avoid gettingalmost zero profit at t+ 1. Thus, initial increases in q raise the benefit butat higher values, the benefit begins to fall. This non-monotonicity carriesover to the size of the ranges of prices for which panics occur and, therefore,the rates of price convergence.While Figure 2.5 demonstrates substantial slowdowns in convergencespeeds due to panics, when only a single trader arrives each period (n = 1)30To be conservative, in all simulations, I assume all traders wait in the region forwhich a multiplicity of equilibria exists so that convergence results for the expected valuemodel are actually upper bounds on the speed of convergence. If traders instead rush, theaverage price would be lower because E[pt+1|pt, V = 1] is easily shown to be increasing inq and rushing implies trading on a signal of weaker strength, q < q.352.4. Effects of PanicsFigure 2.5: Heat Map of Simulated Percentage Slowdowns in Price Conver-genceqq  0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.950.550.60.650.70.750.80.850.90.95−100−80−60−40−20020Note: n = 1. The color reflects the percentage difference in average convergence times(to 99% of the asset’s true value) divided by the average convergence time of thebenchmark model. Negative numbers correspond to the expected value model convergingmore slowly. Each average is over 100,000 simulated price paths.362.4. Effects of PanicsFigure 2.6: Simulation of Average Price Paths0 1 2 3 4 5 6 7 Price  Model with PanicsBenchmark ModelNote: n = 4, q = 0.7, q = 0.80, and V = 1. Price paths are determined by averaging over100,000 simulations. Each price reflects all trades in the period (both rushed anddelayed).the most substantial slowdowns are only observed when the second signalis very strong. By increasing n so that multiple traders arrive each period,waiting costs are increased so that traders may be more willing to panic.With n > 1, a full equilibrium characterization is difficult to obtain, but inAppendix A.5 I establish an upper bound on the amount of waiting that canoccur in any equilibrium. Simulations then provide an upper bound on aver-age prices at each point in time. To illustrate an example with n > 1, Figure2.2 plots the average price over time when n = 4, q = 0.7, and q = 0.8. Un-der this parameterization, traders must rush at all pt ∈ (0.13, 0.87) and Iassume they wait outside this range.Figure 2.6 demonstrates that, although prices initially arise more rapidlyin the model with panics due to the mechanical effect of trading earlier,panics cause an overall slowdown in convergence. In the benchmark model,372.4. Effects of Panicsthe average price takes about 4.5 periods to reach 99% of the asset’s truevalue, whereas in the expected value model, the same price level is onlyreached after 7.5 periods, a slow down of about 66%. Thus, by increasingthe number of traders arriving in each period, one can demonstrate substan-tial slowdowns even with what might be considered more reasonable signalstrengths. Furthermore, n > 1 seems realistic in real markets with manytraders.2.4.2 Increases in MispricingIn addition to longer convergence times, Figure 2.4 illustrates that trading onweaker information during panics increases the probability that prices moveaway from fundamentals. Figure 2.7 plots the probability of observing a pricethat is farther away from the fundamental value (V = 1) than the initialprice (p1 = 0.5) for the same parameters for which average convergence wasstudied in Figure 2.6.31 Figure 2.7 clearly demonstrates that increases inmispricing are much more frequent when traders may panic and that thiseffect persists over time. In fact, at t = 9, the probability of observingpt < 0.5 is more than an order of magnitude higher in the expected valuemodel than in the benchmark model. Widening mispricing is important notonly theoretically, but also may impact any arbitrage activity that attemptsto correct for it. An arbitrageur that knows the true asset value and buysthe asset at t = 0 would face considerably larger arbitrage risk under theexpected value model where prices have a much higher chance of movingfurther away from fundamental value before they move towards it. Thus,arbitrageurs may be less active in attempting to correct for the mispricing(Shleifer and Vishny (1997)).31The simulated probability of observing a price, pt < 0.5, is a lower bound becausethe simulations assume the most amount of waiting possible. Because E[pt+1|pt, V = 1]is increasing in q, if traders actually rush more than assumed, mispricing would be morefrequent.382.4. Effects of PanicsFigure 2.7: Simulation of Probability of Increase in Mispricing0 1 2 3 4 5 6 7 800. of p t < 0.5  Model with PanicsBenchmark ModelNote: n = 4, q = 0.7, q = 0.80, and V = 1. Probability of pt < 0.5 is determined byaveraging over 100,000 simulations.392.4. Effects of Panics2.4.3 Panic CyclesThe illustrative example of Figure 2.4 also shows that panic cycles can occurin equilibrium. During the initial period of uncertainty about the value ofthe asset, all traders panic, acting on relatively weak information about thevalue of the asset. As the value of the asset becomes more certain, a traderoptimally waits to obtain additional information (t = 3). When a strongsignal is obtained, prices fall back into the region of uncertainty, generatingfurther panic and oscillations in prices. Although eventually panics cease toexist, such panic cycles arise naturally during the early life of an asset.Figure 2.4 also shows that there is a sense in which price crashes can ariseendogenously in the model when prices reach the point at which traders findit valuable to do additional research.32 The price path in the figure couldrepresent, for example, the stock issuance of a new technology firm, thevalue of which is initially uncertain. Imagine the technology is actuallynot viable such that the true value of the firm is low. Initial information,based on little research, happens to be favorable, so we observe an initialboom in the price of the stock as traders correctly infer from their positiveinformation that others also have positive information. No one is willing toperform further research due to the rational fear of continued upward pricemovements. However, as the market becomes more certain that the firmis of good quality, traders are induced to perform further research becauseprices are no longer increasing as rapidly. When they do further research,they discover the firm’s value is actually low, and a large drop in the stockprice results. Although it is unlikely that information acquisition is the solesource of price booms and crashes, the model demonstrates how endogenousinformation quality can play a role in their occurrence.32By crashes and booms, I simply mean large changes in the price as new (stronger)information is made available to the market. Crashes are often defined to be discontinuouschanges in prices (Barlevy and Veronesi, 2003). No such discontinuities arise here.402.5. The Zero-Profit Model2.5 The Zero-Profit ModelIn the preceding analysis, I assumed that the price of the asset was givenby a single value that does not incorporate the information contained inthe current trade order. While this assumption allowed me to focus on theinteraction between traders, it leads to a market maker who loses money inexpectation. In this section, I show that the main results of the expectedvalue model continue to hold when market makers instead earn zero profitsin expectation.33 Formally, I modify the model as follows. I consider onlythe case of n = 1: a single trader arrives each period.In each period, the trader that arrives at t is an informed trader withprobability µ ∈ (0, 1) and is an uninformed, noise trader with probability1−µ .34 If a trader is informed, she receives information as in the expectedvalue model, but if uninformed, she trades for reasons exogenous to themodel, such as to meet liquidity needs. Specifically, a noise trader buys orsells in the period she arrives or the next period, with each possible tradehaving probability 14 .35 Because the market maker takes into account theinformation contained in the current trade order, he posts separate prices forbuy and sell orders. For a rushed trade, the ask price at which the marketmaker is willing to sell the asset is given by pAt = Pr(V = 1|Ht, at = B)and the bid price at which he is willing to buy the asset is given by pBt =Pr(V = 1|Ht, at = S). The bid and ask prices are set similarly for delayedtrades. These bid and ask prices are easily shown to be optimal for a marketmaker that is constrained to earn zero profits.To analyze the model with noise traders, I proceed in the same manneras in the analysis of the expected value model. The equilibrium definition re-mains as in Section 2.2 except that item 5, the restriction on off-equilibrium33This standard zero-profit condition results from the assumption that the marketmaker faces unmodeled competition (Bertrand competition in prices, for example). For amore detailed discussion, see Glosten and Milgrom (1985).34Noise traders must be introduced once prices reflect the information contained in thecurrent trade because, without them, the no trade theorem of Milgrom and Stokey (1982)applies.35Allowing noise traders to not trade in each period with some probability does notaffect the qualitative results412.5. The Zero-Profit Modelprices and beliefs, is no longer required because noise traders ensure thatbeliefs and prices are pinned down after all histories. The main differencefrom the expected value model is that here, a trader’s timing strategy af-fects the prices she faces. Intuitively, if informed traders wait, then delayedtrades will have a lot of informational content and the bid/ask spread (thedifference between the bid and ask prices) will be large. Any rushed tradethen must be due to a noise trader so that the bid/ask spread at that timewill be zero. However, this difference in spreads reduces the value to waitingand encourages informed traders to rush. If informed traders always rush,the reasoning is reversed and waiting becomes profitable. Based upon thisintuition, one can anticipate that equilibria generally involve mixed timingstrategies on the part of the informed traders. The existence of bid/askspreads also creates several possibilities in terms of which types of traders,based on their signals, will buy or sell when waiting. To focus on only oneparticular case, I set q = q ≡ q in this section so that traders with contra-dictory signals do not trade because their expected value of the asset lieswithin the bid/ask spread.36The unique bid and ask prices are easily calculated using Bayes’ rule andare provided in Appendix A.4. For delayed trades, traders with st = st+1 = 1buy, those with st = st+1 = 0 sell. For rushed trades, traders with st = 1buy and those with st = 0 sell. These trading decisions can be shown to beoptimal by comparing a trader’s private belief to the appropriate price, asin the proof of Lemma 2.1.Given the optimal trading decisions, one can derive a general formula forthe benefit to waiting for each type of trader just as in the expected valuemodel. The formulas are provided in Appendix A.4. Unlike in the expectedvalue model, the benefit function for each type of trader, Bx(pt, βx), dependsonly on her own timing strategy because a delayed trade reveals both st andst+1. One can show that each of the benefit functions is strictly decreasingin the timing strategy of the trader so that the equilibrium timing strategy36Results for q 6= q are qualitatively similar as long as the probability of a noise traderis not too high because traders with contradictory signals will still have expectations thatlie within the bid/ask spread.422.5. The Zero-Profit Modelis uniquely determined for fixed strategies of the other traders.Comparing the formulas in Appendix A.4 for the benefit to waiting inthe zero-profit model with those of the expected value model in (2.1), onecan see that the general structures of the benefit functions are very similar.Because of this similarity, Propositions 2.1 and 2.2 are easily extended tothe zero-profit model, as stated in Corollary 2.1.Corollary 2.1: Replacing Bx(pt, β0, β1) with Bx(pt, βx) for x ∈ {0, 1},the statements of Propositions 2.1 and 2.2 hold in the zero-profit model.From Corollary 2.1, we know that the benefit to waiting decreases whenothers trade during the waiting period and that this effect diminishes asprices become certain, as in the expected value model. There is, however,no counterpart to Lemma 2.2 of the expected value model because equilibriagenerally involve the two types of traders following different timing strate-gies. Also, as discussed previously, equilibria generally involve mixing dueto the interaction between the market maker and the traders. These com-plications make determining the equilibria of the model more difficult thanin the expected value case and I do not have a complete characterization.However, by solving for the simplest case of a single trader (i.e. a finite timeversion of the model with T = 1) and using Propositions 2.1 and 2.2, I amable to derive several necessary properties of the equilibria for T →∞.Proposition 2.4 describes the equilibrium timing decisions of a singletrader facing the market maker. Together with the optimal trading deci-sions at each point in time discussed previously, it characterizes the uniqueequilibrium for the single trader case.Proposition 2.4: The unique equilibrium of the zero-profit model witha single trader is characterized by the following timing strategies for allpt ∈ (0, 1).βST0 (pt) =pt(1− q) + (1− pt)qpt(1− q) + (1− pt)q + pt(1− q)2 + (1− pt)q2βST1 (pt) =ptq + (1− pt)(1− q)ptq + (1− pt)(1− q) + ptq2 + (1− pt)(1− q)2432.5. The Zero-Profit ModelThe optimal timing strategies given by Proposition 2.4 are interior forall pt ∈ (0, 1) and all q so that each type of trader mixes in the uniqueequilibrium. Furthermore, the two types wait with the same probabilityonly at pt = 12 , which is intuitive given the symmetry of the problem at thisprice. One can also show that ∂β1∂pt < 0 and∂β0∂pt> 0 so that, when pt > 12 ,the trader with st = 0 is waiting with a higher probability. Intuitively, whenpt > 12 , the trader with st = 0 has the greater benefit from waiting dueto having more unexpected information. By waiting more often, however,negative information about the quality of the asset is revealed, which reducesthe price and thus decreases the benefit of type st = 0 relative to that ofst = 1. In equilibrium, this reduction in price is just sufficient to ensureboth types are indifferent between rushing and waiting. Also note that theequilibrium strategies of the traders in Proposition 2.4 are independent ofthe probability of an informed trader, µ. An increase in µ increases thebid/ask spreads at both t = 1 and t = 2 in such a way as to keep each typeof trader indifferent between rushing and waiting.Using Propositions 2.1 and 2.4, Proposition 2.5 shows that the presenceof other traders causes all traders to rush more often than they would if theywere alone in the market.Proposition 2.5: In any equilibrium of the infinite horizon zero-profitmodel, each type of each trader waits with probability β0(pt, I(at−1 =NT )) < βST0 (pt) and β1(pt, I(at−1 = NT )) < βST1 (pt) for all pt andI(at−1 = NT ), where βST0 (pt) and βST1 (pt) are given in Proposition 2.4.From Proposition 2.2, we also know that the equilibrium timing strate-gies approach those given in Proposition 2.4 as prices become certain. There-fore, the main insights of the expected value model hold in the zero-profitmodel: panics cause traders to rush more often when uncertainty is high,but as prices become certain they acquire information as if the other traderswere not present.442.6. Empirical Implications2.6 Empirical ImplicationsTheoretical analysis of the model has developed predictions with respect tothe quality of information traders will trade upon relative to time, uncer-tainty, and volume. Because the quality of information traders possess istypically unobservable, in this section I develop several testable predictionsin terms of observables. Some of these predictions have empirical backingand others suggest novel tests that could be used to validate (or falsify) themodel.The extension of the model to n > 1 traders arriving each period (seeAppendix A.5) demonstrates that an increase in n leads to more panics andthus trading on lower quality information. An increase in n corresponds toan increase in volume, so we have the following prediction.Prediction 1: In either a cross-sectional or time-series analysis, hold-ing the level of uncertainty constant, order flows are more balanced andpersistent price impacts are smaller when volume is higher.Trading on lower quality information causes more balanced order flowsbecause signal realizations are more likely to be incorrect when informationis of lower quality.37 Beginning with Easley et al. (1996), empirical workhas used the order flow imbalance to estimate the probability of informedtrading (PIN). In the model of Easley et al. (1996), traders are assumed to beeither uninformed or perfectly informed, so that more balanced order flowsoccur when the percentage of uninformed traders is higher (PIN is lower).In their model, the rates of arrival of informed and uninformed traders areexogenous, so any relationship between volume and the PIN can be captured.Here, because a lower estimated PIN results from more balanced order flows,the correlate of Prediction 1 is that the estimated PIN should be lower whenvolume is higher.38 In a cross-sectional study, Easley et al. (1996) in fact37Order flow imbalance is defined with respect to those initiating the trade as in Easleyet al. (1996).38Models that have been developed to explain intraday trading patterns in observablesalso make predictions regarding the relationship between volume and the PIN. In Admatiand Pfleiderer (1988) and Malinova and Park (2012), the PIN increases with volume,452.6. Empirical Implicationsfind that the PIN is lower in their sample of high volume stocks than intheir sample of low volume stocks, consistent with Prediction 1. They donot control for the level of uncertainty among the stocks, which is importantin this context because order flows are predicted to be more balanced whenuncertainty is higher, as discussed below. However, unless their sample ofhigh volume stocks happens to be those in which uncertainty is lower, theirresult can be considered a validation of Prediction 1.When market participants trade on lower quality information, we wouldalso expect the persistent price impact of trades to be lower. As first sug-gested by Hasbrouck (1991), informative trades should have longer lastingprice impacts than trades for other purposes, such as inventory re-balancing,etc. Using the vector auto-regressive approach developed by Hasbrouck(1991) to measure the persistent price impact of trades, trades of high vol-ume stocks have been found to have smaller price impacts in a sample ofNYSE stocks (Engle and Patton, 2004) and in foreign exchange markets(Payne (2003) and Lyons (1996)). These results validate Prediction 1, againassuming that high volume is not directly correlated with low uncertainty.Thus, we see that lower informational content of trades when volume ishigher is a robust empirical finding consistent with the model.Proposition 2.2 and the equilibrium characterization of Theorem 2.1 pre-dict that market participants trade on lower quality information when un-certainty is higher. As in Prediction 1, trades on lower quality informationare expected to result in more balanced order flows, so we have the followingprediction.39and in Foster and Viswanathan (1990) either relationship is possible. The informationalcontent of high volume stocks may also be reduced if high volume stocks attract noisetraders due to their salience (Barber and Odean, 2008).39The price impacts of trades are not necessarily predicted to be lower when uncer-tainty is higher due to two opposing effects. Holding the quality of information constant,price impacts are larger when uncertainty is higher. On the other hand, because marketparticipants trade on lower quality information when uncertainty is higher, price impactsare lower. Thus, the prediction of the model is only that price impacts due to not neces-sarily increase as uncertainty increases, which would be the case in models with constantinformation quality, such as Glosten and Milgrom (1985). Similarly, the model makes nostraightforward, monotonic prediction with respect to the relationships between volatilityor the bid/ask spread and uncertainty. An important consequence of the model is that462.6. Empirical ImplicationsPrediction 2: In either a cross-sectional or time-series analysis, hold-ing volume constant, order flows are more balanced when uncertainty ishigher.Prediction 2 is perhaps the most novel prediction of the model as it iscontrary to the predictions of models in which information is exogenous,such as Glosten and Milgrom (1985), and models with monetary costs suchas Nikandrova (2012) and Lew (2013). With a fixed quality of informa-tion, there should be no relationship between order flows and uncertainty.With fixed monetary costs, when uncertainty is low, trading should stopcompletely rather than become more informative. Lew (2013) constructs asequential trading model with increasing monetary costs of acquiring (per-fect) information. His model delivers a prediction opposite to Prediction2: the PIN is higher (and hence order flows are less balanced) when un-certainty is higher. Should Prediction 2 hold in the data, it would providestrong evidence for the mechanism suggested by the model, because it is theendogenous cost of acquiring information that delivers this prediction.40The relationship between the PIN and various measures of uncertaintyhas been explored (Kumar (2009) and Aslan et al. (2011)) with mixed re-sults. However, the main difficulty in performing a test of Prediction 2 isin finding a suitable proxy for uncertainty. Uncertainty here refers to aflatter distribution over the possible fundamental values of the asset beingtraded. The ideal proxy would be uncorrelated with volume and would alsonot be likely to affect the information traders possess through any otherchannel. In Kumar (2009), firm age, firm size, monthly volume turnover,the way in which the informational content of prices is measured (PIN, bid/ask spread,persistent price impact) is important: not all measures need necessarily deliver the sameresults.40The model also predicts more frequent deviations from fundamental values duringtimes of high uncertainty. Limits to arbitrage (see Shleifer and Vishny (1997) and Mitchellet al. (2002)) can also cause mispricing when volatility is high due to increased arbitragerisk. To the extent volatility is related to uncertainty, these two explanations providesimilar predictions. However, volatility and uncertainty are not necessarily related (seefootnote 39). Furthermore, the explanations can be distinguished by studying the infor-mational content of trades when uncertainty is high.472.6. Empirical Implicationsand idiosyncratic volatility are used as proxies in a cross-sectional study,but firm age and size are likely to be directly correlated with the quality ofinformation available, monthly volume turnover is closely related to volume,and volatility is an output of the model that is not necessarily related touncertainty (see footnote 39). In Aslan et al. (2011), PIN is regressed onmany accounting variables in a cross-sectional study. While some variablescan arguably be interpreted as proxies for uncertainty (industry, volatility,volatility in earnings), these proxies suffer from problems similar to those inKumar (2009).Because of the difficulty in finding a suitable proxy for a cross-sectionalstudy, a time-series study in which one can control for firm-specific effectsmay be better able to tease out the effect of uncertainty on the informationalcontent of trades. Potential proxies that could be used include the impliedvolatility index of the market (VIX) as a whole or dispersion in analyst fore-casts. Alternatively, one could look for particular news events that one canreasonably argue have increased (or decreased) uncertainty about firm val-uations (for example, results of drug trials in the pharmaceutical industry).Certainly there is scope for interesting empirical work in this direction.41Proposition 2.3 states that an increase in the initial quality of informationcan actually reduce the rate of price convergence: the increase, holding thesecond period information constant, can induce traders to panic. As panicsresult in more balanced order flows, we have:Prediction 3: Access to higher quality initial information, holding thelevel of uncertainty and volume constant, results in more balanced orderflows.41For other cross-sectional evidence that is suggestive of less-informed trades whenuncertainty is higher, it has been found that underreaction in stock prices is stronger instocks with higher uncertainty (Zhang (2006) and Jiang et al. (2005)). One interpretationof underreaction, as summarized by Zhang (2006), is that underreaction is “more likely toreflect slow absorption of ambiguous information into stock prices than to reflect missingrisk factors”. Under this interpretation, the fact the information appears to be moreslowly absorbed into stocks of higher uncertainty is consistent with the model. However,strictly speaking, the model does not capture underreaction (prices follow a martingale).Extending the model to capture underreaction is an interesting avenue for future research.482.6. Empirical ImplicationsFor evidence related to Prediction 3, in a study related to that of Easleyet al. (1996), Yan and Zhang (2012) provide a new algorithm to estimate thePIN. Interestingly, they find that estimates of the PIN on both the NYSEand the American Stock Exchange (AMEX) have significantly declined be-tween the years of 1993 and 2004. They do not put forward a theory as towhy this has occurred, but one possibility is that improvements in technol-ogy have made better quality information available sooner.42The novel empirical predictions of the model are driven both by theendogenous cost of acquiring information and the fact that weak informationcan result in misinformed traders that trade in a direction that moves pricesaway from fundamental values. Empirical work that estimates PIN or thepersistence of price impacts assumes (explicitly or implicitly) instead thatinformed trades always move prices towards fundamental values. Should thepredictions of the model be validated, it would suggest the importance ofrelaxing this assumption.42Recent increases in high-frequency trading as documented in Biais et al. (2013) andHoffmann (2013) may have led to what Hoffman calls an “arms race” in which investorsinvest in better and better technology to obtain faster access to information.49Chapter 3Rational and Heuristic-BasedTrading Panics in anExperimental Asset Market3.1 IntroductionIn real-world financial markets, traders decide not only what assets to trade,but also when to trade them. These timing choices are important and canfundamentally shape the nature of market outcomes. Perhaps most vividly,the ability to time trades can give rise to market panics - episodes in whichtraders rush to trade in order to avoid adverse price movements due topreemptive trades by others. Chapter 2 theoretically demonstrates thatrational panics - defined as rationally rushing to buy or sell prior to receivingfull information - have detrimental effects on the ability of prices to aggregateinformation, as traders rationally forgo acquiring additional information. Itis therefore instructive to understand the nature of observed panics. Arethey consistent with equilibrium behavior or do they result from other rules-of-thumb? Real-world evidence is difficult to interpret because we do nottypically observe the information traders acquire nor when they acquire it.43In this chapter, I conduct a laboratory experiment to understand the natureof panics and their consequences, benchmarking behavior to that predictedby the theory of Chapter 2.43Methods of inferring information from trades do exist. For example, Hasbrouck (1991)uses the persistence of price impacts and Easley et al. (1996) construct a structural modelbased on the arrival of news. Chapter 2 discusses the theoretical implications of rationalpanics for such indirect evidence.503.1. IntroductionI study the model of Chapter 2 in which traders make only a single timingdecision: they may trade immediately, based on their initial information, orwait to obtain more information before trading. Subjects trade sequentiallyin an overlapping sequence such that an additional trader arrives duringthe time it takes the first trader to acquire additional information. Thus,waiting is potentially costly because other market participants may tradein the interim, moving prices adversely in expectation. I design a pair oftreatments within this framework, one in which it is rational to panic, andthe other in which it is rational to wait for more information. Precise theo-retical predictions allow me to assess the rationality of panics in a controlledlaboratory setting, and allow for a significantly deeper understanding of sub-jects’ motivations. Comparing behavior across treatments, I provide the firstavailable laboratory evidence that subjects rationally panic to avoid adverseprice movements, suggesting that theory can be a useful guide to predictingpanics in the field. Behavior is not completely captured by equilibrium the-ory, however, as a certain fraction of subjects exhibit non-equilibrium panicbehavior.Careful study of the conditions under which subjects panic out of equi-librium reveals a novel trading heuristic used by almost half of subjects.Specifically, they tend to buy (sell) when their beliefs exceed (fall below) acritical threshold, and wait otherwise. This heuristic, although not optimalin any of the treatments, is quite intuitive. Subjects trade in the directionthat is more likely to provide a positive payoff. They fail to realize, however,that because prices already reflect all public information, trading in this waydoes not necessarily provide the highest expected payoff. At extreme prices,the heuristic prescribes trading against one’s private information, often re-ferred to as herding.44 Looking at individual behavior, I find that these44Various definitions of herding and contrarianism are present in the literature. I followthe definitions of Avery and Zemsky (1998) and the subsequent experimental literaturethat tested their model (Cipriani and Guarino (2005) and Drehmann et al. (2005)). Herd-ing refers to buying (selling) at a high (low) price when one’s information indicates tosell (buy). Contrarianism refers to selling (buying) at a high (low) price when one’s in-formation indicates to buy (sell). In these papers and the models considered here, suchbehavior is not rational. For a survey on herding in financial markets, see Devenow andWelch (1996).513.2. Model“herding” types are the most common in both treatments, so that they drivethe majority of non-equilibrium behavior in the aggregate.In addition to explaining behavior in this experiment, the heuristic canalso explain the herding trades observed in past sequential trading laboratoryexperiments (Cipriani and Guarino (2005) and Drehmann et al. (2005)).Because trade timing is exogenous in this literature, it is not possible torelate herding to trade timing in order to detect the underlying heuristictraders are using.45 Following the heuristic, subjects herd as a functionof their beliefs about the asset’s value, not due to conformity or beliefsabout the mistakes of others.46 Finally, the heuristic also provides a novelexplanation for behavior observed in experiments that study the informationherding environment of Banerjee (1992) and Bikhchandani et al. (1992).47Thus, it reconciles experimental findings across three different environments.3.2 ModelThe model is a finite-time version of the expected value model of Chapter2. In order to distinguish it from the richer model developed in Chapter 4,I refer to this model as the Basic model. In each period t = 1, 2, . . . , T , asingle new trader arrives to the market and may trade an asset of unknownvalue, V ∈ {0, 1}, at a single price, p. A market maker (the experimentalist)sets the price equal to the asset’s expected value based upon all publicinformation. When the asset value is realized at T , those who purchasedthe asset receive a payoff of V − p and those who sold (short) receive apayoff of p− V . There is no discounting. The overlapping timing structure45Park and Sgroi (2012) also relate herding to trade timing, but in an environmentwith three states and signals where rational herd behavior is predicted. While they donot have precise theoretical timing predictions, they argue (and observe) that those whohave signals that are more likely to herd move later than those with monotonic signals.In contrast, subjects that herd move earlier here.46Cipriani and Guarino (2005) consider and reject the idea that herding is due to beliefsabout mistakes by previous traders. Drehmann et al. (2005) consider and reject conformity.To my knowledge, no explanation of non-equilibrium herding that is supported by the datahas been put forth in the literature.47This literature originated with Anderson and Holt (1997). See also Ku¨bler andWeizsa¨cker (2004) and Goeree et al. (2007).523.3. Theoretical Predictionsis identical to that of Chapter 2 and all traders are informed with privatesignals as described there. Subjects are restricted to trade only once in theperiod they arrive, or the following period.Table 3.1: Basic Model TreatmentsTreatment Name q q Subjects PeriodsBasic Rush (BR) 17241624 n = 6 T = 6Basic Wait (BW) 13241724 n = 6 T = 6In designing the two treatments for the Basic model, Basic Rush (BR)and Basic Wait (BW), the goal is to provide a strong comparative statictest across treatments: subjects should rush in one, but wait in the other.To achieve this, I chose the parameters for the two treatments specified inTable 3.1. For both treatments, the initial prior, p1, is set to 12 . The twotreatments differ mainly in the quality of the initial information received,being higher in Basic Rush than Basic Wait. In both treatments, T = 6 sothat there are 6 subjects in each session of each treatment.3.3 Theoretical PredictionsA detailed derivation of the equilibria of the Basic model for general pa-rameters is given in Chapter 2. Here, I present only the results for the setsof parameter values used in the treatments. Equilibrium strategies can bedecomposed into an optimal trading strategy (buy or sell), taking the timeof trade as given, and an optimal trade timing strategy. The optimal tradingstrategies for the Basic model are provided in Proposition 3.1.Proposition 3.1: In the Basic model:a) traders who rush in either treatment and traders who wait in the BRtreatment buy if st = 1 and sell if st = 0b) trades who wait in the BW treatment buy if st = 1 and sell if st = 0Intuitively, the trading strategies of Proposition 3.1 are optimal because533.3. Theoretical Predictionsthey ensure one buys when one’s private belief is greater than the publicbelief (which is equal to the price), and sells otherwise. Given the parameterschosen, no trader is ever indifferent between buying and selling.Proposition 3.2 specifies the equilibria of the Basic model for the pa-rameterizations used in the experiment. Appendix B.1 provides the detailsneeded to apply the results of Chapter 2.Proposition 3.2:a) In any equilibrium of the Basic model with q = 1724 and q =1624 (BR)all traders rush at every history, except trader T , who is indifferent whentrader T − 1 rushes.b) In the unique equilibrium of the Basic model with q = 1324 and q =1724(BW) all traders wait at every history.Equilibrium trading strategies are given by Proposition 3.1.In the BR treatment, the optimal timing decision is to rush at everyprice, pt, with the exception of trader T who is indifferent when T−1 rushes.Intuitively, rushing is optimal because the second signal never changes one’strading decision: if the two signals contradict one another, trades are madeaccording to the first, stronger, signal. Thus, there is no benefit to waiting,but there is a cost due to trades by other traders. The final trader is anexception in that she is indifferent between rushing and waiting when T − 1rushed, due to the fact that no intervening trades are possible. In the BWtreatment, there is a strictly positive benefit to waiting at all pt, independentof whether or not t − 1 and t + 1 rush or wait, making the optimal timingstrategy to wait.Proposition 3.2 provides point predictions about subject behavior, butsubjects are unlikely to behave rationally 100% of the time. A weaker hy-pothesis is that subjects rush more often in BR than in BW. I formally statethis comparative static result as Corollary B1, as it follows immediately fromProposition 3.2.Corollary 3.1: Subjects rush more often in BR than in BW.543.4. Experimental Design3.4 Experimental DesignAll subjects were recruited from the University of British Columbia studentpopulation using the experimental recruitment package Orsee.48 Subjectscame from a variety of majors and no subject participated in more than onesession. Four sessions of each treatment, for a total sample of 48 subjects,were conducted. New randomizations were performed for each session’s assetvalues, signals, and subject ordering, in order to avoid the possibility of aparticular set of draws influencing the results. In each session, subjectsfirst signed consent forms and then the instructions (provided in AppendixB.5) were read aloud. Subjects were allowed to ask questions while theinstructions were read and then completed a short quiz. All quiz questionshad to be answered correctly by each subject before the experiment began,and this policy was common knowledge. Once the experiment began, nocommunication of any kind between subjects was permitted.49In each session, 42 trials, preceded by two practice trials, were run. Ineach trial, subjects made their trading decisions via computerized interfaces,an example of which is provided in Appendix B.5. Past prices and tradeswere available on an intuitive graphical display. Software was developedusing the Redwood package (Pettit and Oprea, 2013) which uses HTML5to allow very rapid updating of the computer interface. This feature allowsfor many more trials than would have been possible otherwise, which isimportant because learning in this relatively complex environment is seento play a role (see Appendix B.3). An additional benefit of a large numberof trials is that it provides a sufficient number of observations to studyindividual behavior, something which has not been fully explored in previoustrading experiments.The trading environments were framed as such: subjects were told thatthey would trade a stock with the computer. It was emphasized that theycould only trade once and that they had to trade (in each trial).50 Asset val-48http://www.orsee.org/49Subjects were separated by physical barriers so that they could not observe eachother’s information or decisions.50Both Cipriani and Guarino (2005) and Drehmann et al. (2005) allow subjects to553.4. Experimental Designues were represented visually as bins containing different numbers of coloredballs, with signals corresponding to draws from the appropriate bin.Subjects earned payoffs as described in Section 3.2 in each trial. Theasset value, V , and prices were scaled by a factor of 100 currency units.Subjects were endowed with 100 units with which to trade prior to each trial,for a maximum possible earning of 200 currency units per trial. In order toinduce risk-neutrality, each currency unit represented a lottery ticket witha 1/200 chance to earn $1.00 Canadian. After all trials were completed, acomputerized lottery was conducted for each paid trial and subjects werepaid according to the results of the lotteries. In addition, each subject waspaid $5.00 as a show-up fee. Average earnings were $28.67 (minimum $22.00,maximum $35.00) with a corresponding wage rate over an hour and a halfof $19.11/hour.Ex-ante assumptions about behavior must be made in order to set pricesin the experiment. I assume that no information is revealed by the decisionto wait, which must be the case in equilibrium as proven in Lemma 2.2 ofChapter 2. Therefore, if no trade occurs, pt = pt. After a trade, the priceis updated according to Bayes’ rule, assuming that traders follow equilib-rium buy and sell strategies. In the case of a trade that occurs after anoff-equilibrium timing decision, the price is set assuming traders make theoptimal buy or sell decision according to their private information after thedeviation.51Subjects were told that prices reflect the mathematical expected valueof the value of the asset, conditional on all public information. They werealso explicitly told that prices would increase (decrease) after buy (sell) deci-sions. The exact amount of each increase or decrease was not communicated.However, subjects participate over many trials so that they can learn thenot trade in some treatments, finding that a considerable fraction do so. Given that nottrading is never optimal and the additional complexity of the environment here, I choseto require subjects to trade in order to eliminate one potential source of noise.51As shown in Section 3.5, the second assumption is valid the vast majority of the time,but the first assumption is violated frequently in the data. In Section 4.5, I discuss theconsequences of traders believing that other traders may not be making optimal tradingor timing decisions when prices are set assuming that they do.563.5. Resultspossible price movements over time.3.5 ResultsTable 3.2: Basic Environment Trading ResultsTreatment Rational Herding Contrarian IrrationalBR 84.5% (284) 5.7% (19) 6.5% (22) 3.3% (11)BW 83.9% (282) 6.3% (21) 6.9% (23) 3.0% (10)Notes: Results reported for last 14 trials. Number of observations in parentheses: 336total observations per treatment.In all reported results, I focus on the last third of trials (14 trials), resultingin a total of 336 trading and 336 timing observations per treatment. Becausethe trading environment is relatively complex, focusing on the latter third oftrials eliminates some of the noise associated with early behavior as subjectshave had time to learn about their environment. Evidence of this learningis provided in Appendix B.3.Table 3.2 begins the analysis of the Basic model by reporting the tradingresults. Behavior is rational if it corresponds to that of Proposition 3.1.In this case, traders reveal their private information through their trades,contributing to market efficiency. Behavior is classified as herding if a subjecttrades against their private information but in the direction indicated by thecurrent price (i.e. buying at a price, pt > 0.5, with a signal or signals thatindicates one should sell, or the converse). Behavior is classified as contrarianif a subject trades against their private information and also in a directionopposite to the price. Finally, behavior is classified as “irrational” if onefaces a price of 0.5 and trades contrary to one’s signal or signals. In Table3.2, we see that a relatively high percentage of behavior is rational, 84.8%when pooled across treatments. This finding suggests that, at least alongthis dimension, traders have a good understanding of their environment.Finding 3.1 summarizes the trading behavior.573.5. ResultsFinding 3.1 (Proposition 3.1) In the Basic treatments, almost 85% ofsubjects rationally reveal their private information through their trades.Table 3.3: Basic Environment Timing ResultsTreatment Frequency of RushBasic Rush 47.6% (147)Basic Wait 31.3% (105)Notes: Results reported for last 14 trials. Number of observations in parentheses: 309(336) total observations in BR (BW). Rushing corresponds to rational behavior in BR,but not in BW.Turning to the timing decisions, Table 3.3 provides the percentage ofrush decisions observed in each treatment, aggregated across sessions. Inthe BR treatment, I omit the timing observations of the final trader whenthe previous trader rushed, leaving 309 observations.52 A t-test compar-ing the average frequency of rushing across treatment rejects the null ofequal means at the 5% level (p-value = 0.046), supporting Corollary 3.1.We conclude that subjects respond to the difference in equilibrium forcesacross treatments in a predictable way. This finding provides evidence ofrational panics in a theoretically-understood laboratory setting: traders un-derstand that waiting for others to trade is costly. Interestingly, behavioris not statistically different across treatments in the first third of trials (dif-ference in means is 9.0%, p-value = 0.24), indicating that subjects learn toavoid adverse prices movements as they acquire experience. I summarizethe comparative static result in Finding 3.2.Finding 3.2 (Corollary 3.1) In the Basic treatments, subjects rush moreoften in the treatment where the equilibrium prediction is to rush (BasicRush).Equilibrium panics in treatment BR do not result in informational losses,52When the final trader knows the previous trader rushed, she is indifferent. Of the 27times traders are indifferent, they wait 66.7% of the time.583.5. Resultsbecause even if subjects were to wait to acquire additional information, theirsecond signals would not be revealed by their trades: in equilibrium, theytrade according to their first signal. However, non-equilibrium panics intreatment BW result in informational losses over and above those predictedby theory, evidence of which is provided in Appendix B.4.Although subjects respond as predicted by Corollary 3.1, behavior is notperfectly rational. In particular, subjects both wait too much (Basic Rush)and rush too much (Basic Wait), depending on the treatment. It is temptingto attribute excess rushing to irrational fear and excess waiting to a cautiousover-gathering of information.53 However, it turns out that a more subtleheuristic lies behind both behaviors. I first describe the intuitive heuristicthat captures behavior and then provide strong evidence that subjects infact use it.A large proportion of subjects appear to make decisions based uponmaximizing their chances of earning a positive profit, rather than maxi-mizing their expected profit.54 In doing so, they place value upon publicinformation that provides additional certainty about the asset value, eventhough such information is fully reflected in prices. Thus, they are willing towait to observe others’ trades even when not optimal. When they becomesufficiently certain of the asset value (based upon all public and private in-formation), they then trade to maximize their chances of earning a positiveprofit. In the most extreme cases, subjects that follow this intuition herd,trading contrary to their private information. Because of this fact, I referto this heuristic as the τ -herding heuristic, where τ is the sufficiently highbelief threshold beyond which subjects immediately buy. More formally, Idefine the τ -herding heuristic as follows. In the definition and throughoutthe analysis of timing decisions, I assume subjects treat the asset values of 153Eliaz and Schotter (2010) provide experimental evidence of over-gathering of infor-mation. Subjects are willing to pay for “instrumental” information that is of no real valuein order to be more certain about their decisions.54When there are no prices, or prices are fixed, the two criteria are the same, but whenprices reflect all public information, they often differ. To take a simple example, considerp = 0.9 and one’s private belief is 0.8. Buying the asset is more likely to result in a positiveprofit than selling, but selling maximizes one’s expected profit.593.5. Resultsand 0 symmetrically.55 Under this assumption, the price range can be trans-formed to p′ ∈ [0.5, 1] where p′ ≡ max(p, 1 − p), and similarly for beliefs.For convenience, I refer to p′ as the price.Definition 1: A trader uses the τ -herding heuristic if she has a thresholdprivate belief, τ ∈ [0.5, 1], and uses the following strategy.1. With a private belief, b ∈ [1− τ, τ ], always wait for more informa-tion, if possible. If not possible, trade according to private infor-mation.2. With a private belief b > τ , buy immediately. With a private belief,b < 1− τ , sell immediately.Use of the heuristic, while non-optimal, implies that subjects are reactingto their environment in a sophisticated way. Moreover, their behavior iscompletely predictable, providing alternative predictions to those based onequilibrium theory. I test the aggregate predictions first, and then show thatalmost half of individual subjects follow the τ -herding heuristic.55Given the neutral framing of their environment, there is no reason to think subjectswould favor trading in one direction over the other. Formally, I assume symmetric behavioraround p = 0.5: a trader facing a price, p, with private information, I, makes the sametiming decision as a trader facing price, 1 − p, and complementary private information,IC . To test this assumption, I partition trials into those in which V = 1 and those inwhich V = 0. If there is an asymmetry between rising and falling prices, one would expectthe determinants of panics to be different in these two samples. However, the interactionterms are insignificant if one interacts a dummy for V = 1 with the other covariates inthe regression of Table 3.4 that follows.603.5. ResultsTable 3.4: Determinants of Rushed Trades in the Basic TreatmentsBasic Rush Basic Wait Basic Rush Basic Wait(τ -herding types)Private 0.30 1.69*** 0.76** 0.95***Belief [0.27] [0.25] [0.23] [0.22]Previous -0.12 -0.04 -0.07* -0.09**Rushed [0.07] [0.07] [0.04] [0.04]Notes: Dependent variable is a dummy variable: 1 indicates a rushed trade. 308 (294)observations in BW (BR). 168 (126) observations when restricted to τ -herding types. Logitmarginal effects reported. Subject and trial fixed effects are included. Robust standarderrors in brackets. Significance at the 10% level is represented by *, at the 5% level by **,and at the 1% level by ***.In the Basic environment, the aggregate prediction is that as prices (andbeliefs) become more extreme, so that more traders’ threshold beliefs aresurpassed, we should observe more rushed trades. Table 3.4 presents theresults of a logit regression of the probability of a rushed trade on a trader’sprivate belief for each treatment. In the regressions, subject and trial fixedeffects are included (not reported), as is an indicator dummy that indicateswhether or not the previous trader has rushed.56 Results are also shownfor the subsample of individuals classified as τ -herding types. I postponediscussion of these results until after the classification exercise later in thissection.In Table 3.4, we see that, consistent with the τ -herding heuristic, theprobability of a rushed trade increases with traders’ beliefs. This effect isstatistically significant and much stronger in the BW treatment: movingfrom complete uncertainty (belief is 0.5) to completely certainty (belief is1), the probability of panic increases by almost 0.85, a very large effect.5756The results are robust to clustering standard errors at the session level rather thanusing heteroskedastic robust standard errors. Because there are only a small number ofclusters (4) and because the clustered standard errors are typically smaller, I choose toreport robust standard errors instead.57The difference between the BW and BR treatments can be explained by consideringthe expected losses from deviating from the optimal strategies. These losses are largestat prices near one half, and they diminish towards zero as prices become certain where613.5. ResultsTable 3.4 also shows that the probability of panic, although not significant,is reduced when the previous trader rushed, as would be expected becausethere can be no price impact from the previous trader in this case.In addition to predicting when rushed trades occur, the τ -herding heuris-tic predicts their direction. Because I have sufficient data for each individual,I explore this dimension of behavior at the individual level, providing a moredetailed picture than an aggregate analysis. I attempt to classify each indi-vidual into one of four possible types: rational, τ -herding, τ -contrarian, andsimplistic. A trader is τ -contrarian if she follows a strategy identical to aτ -herding type except that she trades against her belief, rather than with it.Simplistic types rush when they should wait, or vice versa, and follow theirsignal when they trade. To classify subjects, I impose a rather stringentcriterion: their actions must be consistent with those of a particular typein 13/14 of the last trials.58 I allow subjects to be classified into more thantype in order to get a sense of the robustness of the classification, but I alsoassign a unique type based upon the following prioritization scheme (highto low): rational, simplistic, τ -herding, and τ -contrarian. Table 3.5 providesthe resulting classifications for each treatment, including the percentages ofexact matches and subjects that have ambiguous types. Overall, the clas-sification scheme is quite successful, with about 60% exactly matching asingle type and only 23% remaining unclassified. τ -herding types are thethere is no profit from trading. In the BW treatment, the differences in losses reinforcethe heuristic because the largest penalties from deviating from waiting are at prices nearone half where the heuristic also predicts waiting. In the BR treatment on the otherhand, the expected loss from deviating from rush is strongest where the heuristic specifieswaiting. Thus, one would expect behavior to be more noisy in the BR treatment where theheuristic and payoff incentives oppose each other. Furthermore, the fact that the heuristicis moderated or strengthened by the cost of implementing the heuristic suggests that itis not itself driven by changes in payoff incentives but is instead an innate tendency thatsubjects bring to the lab.58For the rational and simplistic classifications, I only consider timing decisions. Inorder to be identified as τ -herding (τ -contrarian), it must be that case that whenever onerushes, one trades with (against) one’s signal. Whenever one waits, I ignore the subsequenttrade direction because identifying a match would require an additional assumption aboutthe subject’s belief threshold. Furthermore, any combination of waiting and rushing isallowed because a high (low) frequency of waiting is consistent with a high (low) thresholdbelief, τ .623.5. Resultsmost common type, making up 45% of the population. In fact, about 40%of the population matches the τ -herding type exactly.59 The proportionof subjects classified as τ -herding types is very similar across treatments,providing some evidence of the robustness of the heuristic.Table 3.5: Frequency of Subjects of Each Type in the Basic TreatmentsType Basic Rush Basic Wait(multiple) (prioritized) (multiple) (prioritized)% Rational 8.3% 8.3% 8.3-33.3 % 33.3%% Simplistic 4.1-8.3% 8.3% 4.2% 4.2%% τ -Herding 45.8-54.2% 50% 37.5-66.7% 41.7%% τ -Contrarian 4.1-8.3% 4.1% 4.1-8.3% 4.1%% Unclassified 29.2% 29.2% 16.7% 16.7%% Exact 50.0% 70.8%% Ambiguous 8.3% 29.2%Notes: 32 subjects for each of BR and BW. Results of assigning a single type through theprioritization scheme are shown in blue.I perform two additional checks to ensure that τ -herding individuals ac-tually follow the prescribed heuristic, focusing on those that match the typeexactly.60 First, if a subject follows the heuristic, then the beliefs at whichthey trade should be higher on average than the beliefs at which they wait.This criterion is satisfied for 78% of the τ -herding types. Second, I return tothe regression results in Table 3.4 for the subsample of subjects classified asτ -herding. We observe that the previous relationship between the probabil-ity of rushing and beliefs is now stronger and significant in treatment BR, asexpected.61 When the previous trader rushes now significantly reduces theeffect of panic, consistent with τ -herding types being more sensitive to pricechanges than the average subject. I summarize the results of the subject5940% is almost certainly an underestimate. A large proportion of the rational typesin BW are likely herding types with high threshold beliefs and are classified as rationalonly because I prioritize rational over herding.60There are 14 subjects in BW and 9 in BR that match the τ -herding type exactly.61In the BW treatment, it is actually lower among τ -herding types, but running a singleregression with interaction terms establishes that the difference is insignificant.633.6. Discussionclassification exercise as Finding 3.3.Finding 3.3: Although subjects exhibit heterogeneity in their strategies,the proportion of subjects that can be described as τ -herding are mostcommon, making up about 45% of the population. On the other hand,τ -contrarian subjects are the least frequent, making up only 4% of thepopulation.3.6 DiscussionEndogenizing trade timing allowed the the τ -herding heuristic to reveal it-self. Here, I argue that it provides an explanation for behavior in pastexperiments in which use of the heuristic was not easily identifiable. Cipri-ani and Guarino (2005) and Drehmann et al. (2005) study trading behaviorin an exogenous timing version of the Basic model. Both papers documentherding and contrarian trades, neither put forth a satisfactory explanationfor herding. Cipriani and Guarino (2005) consider the possibility that sub-jects believe previous subjects made mistakes, but show that this explana-tion can only explain contrarian trades.62 Drehmann et al. (2005) insteadconsider, but reject, conformity: subjects trade in the direction of previ-ous subjects, regardless of signals or beliefs. My explanation for herding inthese previous papers is the τ -herding heuristic: subjects herd when theirbeliefs become extreme, as they do here. In Appendix B.2, I show how theexistence of τ -herding types can also explain the lack of rational herdingin yet another related environment, the informational herding environmentof Banerjee (1992) and Bikhchandani et al. (1992).63 Thus, the τ -herdingheuristic reconciles behavior across at least three related laboratory envi-ronments.Previous literature (Long et al. (1990) and Cipriani and Guarino (2005))62I similarly reject this explanation as a source of herding in the data here (see thediscussion in Chapter 4).63This finding has been demonstrated by numerous papers beginning with Andersonand Holt (1997). See also Ku¨bler and Weizsa¨cker (2004) and Goeree et al. (2007).643.6. Discussionpoints out that contrarian trades tend to stabilize markets by causing pricesto become less extreme, while herding trades do the opposite. Thus, thefinding that τ -herding types are much more frequent than τ -contrarian typessuggests that behavioral types may tend to destabilize markets overall. Thisconclusion stands in contrast to Cipriani and Guarino (2005) who, based ahigher frequency of contrarian trades in their data (Drehmann et al. (2005)find the same), conclude that contrarianism is more frequent. I suggesttwo reasons their data may have led to the wrong conclusion. First, in thedata here, the overall frequency of contrarian trades declines over time. Asthese two previous studies did not allow for as many learning opportunities,they picked up behavior only in non-experienced subjects. Second, partof the reason contrarian trades can be more frequent is that there are moreopportunities to act contrarian: signals are more likely to go in the directionof the price trend rather than against it. Conditioning on the opportunityfor herding or contrarian behavior, I find higher rates of herding behavior(21% vs. 14%).Results at both the aggregate and individual levels strongly suggest thata large proportion of subjects follow the τ -herding heuristic. If traders inactual financial markets follow such strategies, what would we expect to see?One prediction is clustering of trades as prices cross the common thresholdbelief of multiple traders.64 In addition, when trades cluster, they shouldall be in the same direction and in the direction of the initial price move.In this case, asset returns would exhibit short-term positive correlation. Asboth trade clustering and correlation in returns are robust features of actualfinancial markets, one wonders whether or not the τ -herding heuristic mightprovide an explanation for their existence.65 The Basic environment has64In real markets, asset values are not binary, so rather than having a threshold belief,one can think of a threshold expected value. Beliefs and expected values are equivalent inthe stylized models here.65Dufour and Engle (2000) provide empirical evidence of trade clustering. Short-termcorrelation in returns is often attributed to underreaction and has received considerableattention both theoretically and empirically. For theoretical explanations, see Barberiset al. (1998), Daniel et al. (1998), and Hong and Stein (1999). For empirical evidence, seeDaniel et al. (1998) for a review. For an experimental paper demonstrating underreactionin a double-auction environment in which asset values follow a stochastic process, see653.6. Discussionsufficed to demonstrate rational panics in the laboratory, but its simplicityalso prevents it from demonstrating clustering and correlation in returns dueto the fact it doesn’t allow subjects to trade simultaneously. This limitationmotivates an additional trading environment where these predictions can betested. Such an environment is developed in Chapter 4.Kirchler (2009).66Chapter 4Panics with SimultaneousTrading: Theoretical andExperimental Extensions4.1 IntroductionChapters 2 and 3 have developed and studied experimentally a simple modelin which trading panics lead to informational losses. However, when onethinks of panics, one typically thinks of many agents acting at (or nearlyat) the same time. The stylized model of Chapters 2 and 3 captures themain intuition behind the tradeoff agents face, but precludes simultaneoustrading in order to provide a tractable setting. In this chapter, a slightlyricher model which allows simultaneous trading, the Extended model, isdeveloped. Due to its additional complexity, only partial theoretical resultsare available, but they are sufficient to characterize the equilibria for theexperimental treatments considered.In the Extended model, all traders receive information simultaneouslyand are then given several opportunities to trade before receiving additionalinformation in the final trading period. As in Chapter 3, I construct twotreatments. Developing new theoretical results, I establish that subjectsshould theoretically rush to trade immediately in the first treatment, butshould wait to trade in the final period in the second treatment. Comparingbehavior in these treatments allows me to directly test and contrast theheuristic and theoretical (rational) predictions.When panics are rational, they lead to severe informational losses: sub-674.2. Extended Modeljects forgo acquiring perfect information about the asset’s value 99.7% ofthe time, instead rationally choosing to trade immediately with only weakinformation. 75% of trades occur in the period of arrival, leading to ex-treme clustering of trades as subjects rationally rush to be the first to trade.This treatment provides a robust demonstration of rational panics and theirpotential consequences.In the treatment in which subjects should wait for additional informa-tion, the heuristic-based predictions made at the end of Chapter 3 are val-idated: both trade clustering and short-term positive correlation in returnsare observed. Knowing the information traders possess, I can show that thepositive correlation in returns is not driven primarily by herding.66 Instead,the novel explanation for correlation is that it is driven by traders with dif-ferent private information choosing to trade at different times. Given thelarge proportion of subjects following the heuristic and the emergence of dis-tinctive naturally-occurring patterns in the laboratory data, it seems likelythat this heuristic is a potentially important driver of behavior in the field.4.2 Extended ModelThe asset structure, common prior, and payoffs are identical to the Basicmodel of Chapters 2 and 3. Time is discrete with trading periods t = 1 . . . T ,but rather than having a single trader arrive each period, n traders arepresent from the first trading period. Each trader, identified by i ∈ n,receives a private signal before the first trading period, si ∈ {0, 1}, whichhas a correct realization with probability q ∈ (12 , 1). Each trader may tradeonly once in any of the T trading periods. If a trades waits until time T totrade, she receives an additional private signal, si ∈ {0, 1}, immediately priorto T , which has a correct realization with probability q ∈ (12 , 1]. Note thatI allow for the second private signal to reveal the true asset value perfectly.Denote the action of trader i in period t as ai,t ∈ {B,S,NT}.66Herding, whether information-based or due to conformity, has has long been consid-ered an explanation for correlation in returns. See Hirshleifer and Hong Teoh (2003) foran extensive review.684.2. Extended ModelImmediately after each trading period, there is a public announcementperiod in which a binary public signal, sP,t ∈ {0, 1}, is revealed. It has acorrect realization with probability qP ∈ (12 , 1). A single price, pt, equal tothe expected value of the asset conditional on all publicly available infor-mation (including both trades and public signals), is set by a market makerprior to each trading period. Prior trades, timing decisions, and prices areobserved by all traders.This model can be thought of as a simplified version of the situationarising around a firm’s earnings announcement. Prior to the announcement,traders receive independent (weak) private information about the upcomingrevision in the firm’s valuation. As the announcement approaches, pub-lic information slowly becomes available through a series of public signalsand, finally, at the announcement date, the firm’s revised valuation becomespublicly known.67,68Table 4.1: Extended Model TreatmentsTreatment Name q q qP Subjects PeriodsExtended Rush (ER) 34 11724 n = 8 T = 8Extended Wait (EW) 1324 11724 n = 8 T = 8Specifications for the two treatments of the Extended model, ExtendedRush (ER) and Extended Wait (EW), are provided in Table 4.1. The exper-imental protocol for the Extended treatments is virtually identical to thatof the Basic treatments discussed in Chapter 3. Four sessions of each treat-ment, for a total sample of 64 subjects, were run. Each session consisted of30 trials plus two practice trials. Average earnings were $21.53 (minimum$12.00, maximum $30.00) with a corresponding wage rate over an hour and67The slow release of public information, while not unreasonable, has the additionalbenefit of providing a unique equilibrium prediction.68The theoretical model is not limited to public information at the announcement date.If one instead assumes the announcement generates further private information becausetraders have heterogeneous abilities to process information, or have different models of afirm’s dividend-generating process (Kandel and Pearson (1995)), theoretical predictionsare still available.694.3. Theoretical Predictionsa half of $14.35/hour. The instructions and an example of the trading in-terface are provided in Appendix C.4. Prices are set equal to the asset’sexpected value by the market maker (experimentalist). If no trade occurs,the change in price reflects the public signal only. In addition to being toldhow prices change after trades, subjects were told that when a public signalsuggests V = 1 (V = 0), the price would increase (decrease).4.3 Theoretical PredictionsThe optimal trading strategies for the Extended model are provided inProposition 4.1. They ensure one buys when one’s private belief is greaterthan the public belief (which is equal to the price), and sells otherwise. Alldetails and proofs are in Appendix C.1.Proposition 4.1: In the Extended model:a) traders who trade prior to period T buy if st = 1 and sell if st = 0b) traders who trade in period T buy if the true asset value is 1 and sellif the true asset value is 0Due to the richness of the strategy space of the Extended model, a com-plete characterization of all equilibria for general parameters is tedious toderive, although not conceptually difficult. I focus on the particular sets ofparameter values used in the two treatments. Proposition 4.2 provides anecessary condition for any equilibrium of treatment ER and a statement ofthe unique equilibrium of treatment EW.69Proposition 4.2:a) In any equilibrium of the Extended model with q = 34 , qP =1724 , andq = 1 (ER), all trades occur in the first trading period.b) In the unique equilibrium of the Extended model with q = 1324 , qP =1724 ,and q = 1 (EW), all trades occur in the final trading period.Equilibrium trading (buy or sell) strategies are given by Proposition 4.1.69Proposition 4.2 part a) does not specify the off-equilibrium timing strategies, butthe proof in Appendix C.1 ensures that all trades occur immediately regardless of thesestrategies.704.4. ResultsA weaker comparative static prediction follows directly from Proposition4.2: subjects trade earlier in treatment ER than in treatment EW.Corollary 4.1: Subjects trade earlier in ER than EW.The difference between ER and EW lies solely in the initial signal strength.In ER, the stronger initial signal increases the profit from trading immedi-ately. Furthermore, it increases the price impacts of others’ trades, shouldthey trade before you. Together, these effects are expected to produce ro-bust equilibrium behavior in which all trades occur immediately. On theother hand, in EW, one’s initial signal strength is so weak that it rationallypays to wait to learn the asset value in the final period before trading. How-ever, given the many opportunities to trade before additional information isreceived, one expects use of the heuristic to be tempting.4.4 ResultsAs in Chapter 3, I report data from only the last third of trials in orderto focus on behavior after subjects have gained experience (320 tradingobservations per treatment). Evidence of learning is provided in AppendixC.2.Equilibrium Behavior and Informational LossesTable 4.2: Extended Model Trading ResultsTreatment Rational Herding Contrarian IrrationalER 93.8% (300) 3.4% (11) 1.9% (6) 0.9% (3)EW 77.2% (247) 15.9% (51) 5.9% (19) 0.9% (3)Notes: Results reported for last 10 trials. Number of observations in parentheses: 320total observations per treatment.714.4. ResultsFigure 4.1: Cumulative Distribution of Trades Occurring at t ≤ t′1 2 3 4 5 6 7 800. PeriodCumulative Distribution of Trades  Extended RushExtended WaitTable 4.2 first demonstrates that trading behavior in the Extended modelis highly rational, as in the Basic model results of Chapter 3. In fact, theproportion of each type of trade is very similar to that in the Basic model(Table 3.2), with the exception of the increased frequency of herding in theEW treatment.70To compare timing behavior across the ER and EW treatments, Figure4.1 plots the empirical cumulative distribution functions (cdfs) of the fractionof trades that occur in periods t ≤ t′ as a function of t′ ∈ 1 . . . 8. The rationalprediction is that all trades occur at t = 1 in the ER treatment and at t = 8in the EW treatment, corresponding to step function cdfs that transitionfrom 0 to 1 at t′ = 1 and t′ = 8, respectively.Figure 4.1 provides clear evidence that trades occur earlier in treatmentER relative to treatment EW. The median trading periods in the four ses-70This increased frequency of herding, relative to the Basic environment, is a resultof τ -herding subjects having many more opportunities to trade: their beliefs have moreopportunities to cross their threshold belief, providing more opportunities for herding.724.4. Resultssions of ER and EW are {1, 1, 1, 1} and {2, 3, 3.5, 4}, respectively. Applyinga non-parametric Mann-Whitney U test, one rejects the null of equal me-dian trading periods (test statistic = 0, p-value = 0.05), supporting Corol-lary 4.1.71 I capture these trading and timing results in the following twofindings.Finding 4.1 (Proposition 4.1) In the Extended treatments, approxi-mately 85% of subjects rationally reveal their private information throughtheir trades.Finding 4.2 (Corollary 4.1) In the Extended treatments, subjects tradeearlier in the treatment for which the equilibrium prediction is to tradeat t = 1 (Extended Rush).Behavior in the ER treatment provides a vivid demonstration of pre-dictable, rational panics and their consequences. Traders rationally rush totrade in the first period almost 75% of the time, and over 90% of tradesoccur in the first two trading periods. Extreme clustering of trades is ob-served as subjects scramble to be the first to trade in all four sessions of thetreatment. However, this result does not contradict the fact that subjectsuse the τ -herding heuristic. In early trials of the ER treatment, there is ev-idence that subjects use the heuristic (results available upon request). But,the fact that it causes large decreases in expected payoffs means that it diesout in later trials.72Equilibrium behavior in ER produces large informational losses due totraders acting on weak information. Here, when all traders rush to be thefirst to trade, each trades with information that is correct only 75% of thetime, forgoing the opportunity to obtain perfect information. In fact, onlyone subject in one trial (out of 320 subject-trials) ever waits to obtain theasset value at t = 8, meaning prices reflect the true asset value perfectly71A Kolmogorov-Smirnov test also rejects the null of equal empirical cumulative distri-butions (p-value = 0.01).72When all others are trading at t = 1, if one delays one’s trade, the price at t = 2 islikely to be very close to the true asset value, so that trading produces almost zero profitin expectation. Trading immediately instead results in an expected profit of 0.75∗$1.50 +0.25 ∗ $.50 = $1.25 in each trial (relative to the endowment profit of $1.00).734.4. Resultsonly 0.3% of the time.Non-equilibrium behavior produces informational losses over and abovethose predicted, evidence of which I provide in Appendix C.3. Both toofrequent rushing and trading against one’s private information contributeto these additional losses. Finding 4.3 summarizes the results related toinformational losses.Finding 4.3: Rational panics result in considerable informationlosses, as predicted (Extended Rush). Non-equilibrium panics and non-equilibrium trading behavior can, in some environments, severely exac-erbate these losses, even after acquiring substantial experience.Heuristic-Driven Behavior (Extended Wait)Because behavior in the ER treatment is captured well by the rational pre-diction, I focus on the EW treatment to study the heuristic predictions. Ifsubjects follow the τ -herding heuristic, we’d expect to see trading at in-termediate periods when beliefs cross traders’ threshold beliefs, and this isexactly what is observed: 62% of trades occur in trading periods other thant = 1 and T = T .73 To investigate these trades in more detail, consider whatwe would expect to see with a population of τ -herding subjects with a dis-tribution of threshold beliefs. It is difficult to predict a simple relationshipbetween beliefs and the probability of trade at a particular price (or period)because of selection issues: traders with low threshold beliefs exit earlier intime. However, a simple prediction is that when beliefs exceed a particularlevel for the first time, we should observe all traders with threshold beliefsbelow that level trading in the following period. I use the exogenous publicsignals, which drive most of the changes in prices (and therefore beliefs),as a coarse predictor of when belief thresholds are crossed. Specifically, I73Trading at intermediate periods is difficult to reconcile with rationality. If one were tooverestimate the precision of one’s signal, one should trade immediately at t = 1. Waitingfor public information has an associated cost, but provides no benefit. Lemma C.1.1 inAppendix B.1 formalizes this intuition by showing that it is never optimal to wait until aperiod that produces no new private information.744.4. Resultsconstruct a set of indicator variables, each of which is set to 1 when theabsolute value of the difference in public signals reaches a new level for thefirst time in the period prior to the trading period of interest (and is zerootherwise). I denote these dummy variables PubDiff1 through PubDiff6 (6being the largest absolute difference in public signals arising in the data).One expects to see a higher probability of trade in the period immediatelyafter a new level is reached, as long as there are traders with threshold beliefsbetween adjacent levels.Table 4.3: Determinants of Rushed Trades in the Extended Wait TreatmentExtended Wait Extended Wait(τ -herding types)PubDiff1 0.14*** 0.38**[0.05] [0.17]PubDiff2 0.12* 0.50**[0.07] [0.24]PubDiff3 0.02 0.46[0.07] [0.29]PubDiff4 -0.08* -0.17[0.04] [0.23]PubDiff5 -0.01 -[0.12]PubDiff6 -0.06 -0.10[0.09] [0.34]Price (p′) 0.02 0.04[0.15] [0.98]Extreme Signal 0.03 0.25***[0.03] [0.08]Period 0.08*** 0.22***[0.01] [0.07]Notes: Dependent variable is a dummy variable: 1 indicates a rushed trade. 1077 obser-vations (229 when restricted to τ -herding types). Logit marginal effects reported. Subjectand trial fixed effects are included. Robust standard errors in brackets. Significance atthe 10% level is represented by *, at the 5% level by **, and at the 1% level by ***.Table 4.3 reports the results of a logit regression of the probability of754.4. Resultstrade on the public signal difference indicator variables described above. Iexclude t = 8 because I am interested in the determinants of rushed trades.I also control for the price, p′, whether a trader’s signal makes their beliefsmore or less extreme (i.e. a positive signal makes beliefs more extreme whenp ≥ 0.5, but makes beliefs less extreme when p < 0.5), and a linear timetrend (i.e. trading period).74 Results are also reported when the subsam-ple is restricted to only subjects identified as τ -herding types. I postponediscussion of these results until later in this section.Table 4.3 shows that when public signals first push prices to about 0.7(PubDiff1) and about 0.85 (PubDiff2), the probability of trade jumps, sug-gesting that subjects with τ values below ≈ 0.7 and between ≈ 0.7 and≈ 0.85 both exist in the data.75 The indicator variables corresponding tothree or more public signals in the same direction are not significantly pos-itive and one is in fact (marginally) significantly negative. Higher publicsignal differences are reached less often in the data so statistical power islow in these cases. For this reason, I cannot rule out other τ -herding traderswith threshold beliefs beyond 0.85.The fact that a signal which makes one’s belief more extreme does notincrease the probability of trade is surprising, although it is positive as onewould expect.76 The positive coefficient on the time trend shows that tradersare more likely to trade in later periods, all else equal.77The increased probability of trade when new levels of public signal dif-ferences are reached provide indirect evidence of trade clustering. As fur-ther evidence, consider the trading period (t = 2) after the first public isrevealed (PubDiff1), where the increase in trade probability is the largest.74Again, I choose to report heteroskedastic robust standard errors rather than standarderrors clustered at the session level. The significance of results remains unchanged withclustered standard errors.75These characterizations are coarse because any trades by other traders, as well asone’s own private information, also affect beliefs (to a smaller degree).76Here, the effect of one’s signal is averaged across cases in which a new public signaldifference level is reached and when it is not. In Table 4.5, we see that when we considerits effect conditional on reaching the first threshold, it is significant.77One possible explanation for this fact is that traders value public information moreso than is reflected in the price, leading to beliefs that are too extreme. I discuss thispossibility in more detail in Section 4.5.764.4. Results81/320 = 25.3% of all trades occur in this second trading period even thoughit makes up only 12.5% of all trading periods. Put another way, there arean average of 2.03 trades at t = 2. If the eight traders were randomlychoosing to trade in one of the eight periods, the expected number of tradesat t = 2 would be given by a binomial distribution, so that we’d expectonly 8 ∗ 0.125 = 1 trade, on average. Under the binomial distribution, theprobability of observing more than 2.03 trades is only 6.25%. Thus, we seestatistical evidence that trades cluster in time in a predictable manner.So far I have provided evidence that the timing of trades is predictable,but have said nothing about their direction. When τ -herding subjects rushto trade, they should trade in the direction of their private belief. Thisbehavior should produce positive correlation in returns which I provide evi-dence of now.Table 4.4: Correlation of Trading Returns in the Extended Wait TreatmentTrading Period 1 2 3 4 5 6 7Correlation 0.13 0.73*** 0.05 0.27 -0.13 -0.03 0.15Coefficient [0.42] [0.00] [0.74] [0.09] [0.44] [0.87] [0.35]Notes: Correlation is with respect to return due to first public signal. p-value of two-tailedt-test included in brackets. Each correlation is over 40 trials.I continue to focus on the second trading period after the first publicsignal has just been revealed. At this time, 85% of trades are in the directionof both the trader’s belief and the first public signal. Table 4.4 providesthe Spearman correlation coefficients between the return due to the firstpublic signal and that due to trades in each of the first 7 trading periods,excluding the last where the asset value is known.78 We observe a verystrong positive short-term correlation between the return due to the firstpublic signal and that due to the trades in the following trading period.Were traders simply rushing and trading according to their signals, priceswould form a martingale and such a correlation would be extremely unlikely.This finding provides solid evidence of trades being not only clustered, but78The results are very similar if the Pearson correlation coefficient is used, but there isno a priori reason to expect the correlations to be linear.774.4. Resultsalso occurring in the same direction.79 Finding 4.4 summarizes the clusteringand correlation results.Finding 4.4: Use of the τ -herding heuristic produces clustering of tradesand positive short-term correlation in returns, as predicted.I now dig deeper into the source of the correlation at t = 2 , providingtwo pieces of evidence to show that it is primarily driven by traders withdifferent private information choosing to trade at different times, and notby herding. Although herding has long been considered to be a sourceof correlation (Hirshleifer and Hong Teoh (2003)), this finding suggests adifferent explanation.First, of the 81 trades that occur, 65% are by those whose signal agreeswith the public signal even though only 51% have such signals. This factis consistent with τ -herding behavior because traders whose signals agreehave more extreme beliefs so that their critical threshold for trading is morelikely to be exceeded. Furthermore, all of those whose signals agree withthe public signal trade in the direction of their beliefs, while only 57% ofthe others do (i.e. 57% herd). Therefore, on average, the herding tradescontribute little to the price change.79One may expect negative correlation in later trading periods, should those with sig-nals that did not agree with the first public signal delay their trades and later trade inthe opposite direction. However, any such negative correlation would be spread over theremaining trading periods as there is no salient time at which to trade. In addition, someof these traders may actually delay until t = 8 when they learn the true asset value.784.4. ResultsTable 4.5: Signals and Rushed Trades in the Extended Wait TreatmentSimultaneous WaitPubDiff1 -0.00[0.05]PubDiff2 0.10[0.08]PubDiff3 -0.01[0.08]PubDiff4 -0.10**[0.04]PubDiff1 * Extreme Signal 0.26**[0.10]PubDiff2 * Extreme Signal -0.01[0.07]PubDiff3 * Extreme Signal 0.07[0.14]PubDiff4 * Extreme Signal 0.12[0.20]Extreme Signal -0.05[0.05]Notes: Dependent variable is a dummy variable: 1 indicates a rushed trade. 1077 observa-tions. Logit marginal effects reported. Subject and trial fixed effects are included. Robuststandard errors in brackets. Significance at the 10% level is represented by *, at the 5%level by **, and at the 1% level by ***.As a second piece of evidence, I now show that the increased probabilitiesof trade at new public signal levels are driven by those whose signals maketheir beliefs more extreme than the current price. To do so, I interact each ofthe public signal indicator variables in Table 4.3 with a dummy that indicatesa trader’s signal makes her belief more extreme. The results of Table 4.5,show that the increased probability of trade after the first public signal issolely due to traders whose signals make their beliefs more extreme.80 The80For brevity, I do not report the coefficients on all variables. The price coefficient re-mains insignificant and the period coefficient remains positive and significant. In this case,with standard errors clustered at the session level, the coefficient on the interaction termof the first public signal indicator becomes insignificant (p-value = 0.219), but significance794.4. Resultscoefficient on the first public signal indicator, which now represents thosewith signals whose beliefs are less extreme than the price, is a fairly precisezero, indicating that there is no increased probability of a potential herdingtrade.In summary, we have convincing evidence that the correlation in returnsarises because traders choose when to trade based upon their private infor-mation. This finding provides a new explanation for the empirical puzzle ofcorrelation in stock returns, specifically post-earnings-announcement drift,in which returns drift in the direction of the earnings surprise (here, the firstpublic signal).81Table 4.6: Frequency of Subjects of Each Type In Extended WaitType Percent of Subjects(multiple) (prioritized)% Rational 6.3% 6.3%% Simplistic 25.0-43.8% 43.8%% τ -Herding 31.3-50% 31.3%% τ -Contrarian 6.3% 6.3%% Unclassified 12.5% 12.5%% Exact 53.1%% Ambiguous 18.8%Notes: 32 subjects total. Results of assigning a single type through a prioritizationscheme are shown in blue.As a final piece of evidence that subjects use the τ -herding heuristic inthe Extended environment, I classify individuals as done in Chapter 3 forthe Basic environment. Table 4.6 provides the results of this classification.Comparing with Table 3.5 of Chapter 3, we see remarkably comparableof the other coefficients remains unchanged.81See Daniel et al. (1998) for a list of papers providing evidence of post-earnings-announcement drift. Although the Extended model was framed as one in which theearnings announcement occurs at the final time period, one can think of the earningsannouncement generating private information in the first time period. The final timeperiod may then refer to some future date when additional public information becomesavailable, such as the filing of the firm’s 10-K.804.5. Discussionproportions of each type of subject, with τ -herding types again making up alarge proportion of subjects.82 As an additional check, among these types,the belief at which they trade should be higher than any previous belief atwhich they could have traded. In the 90 subject-trials in which τ -herdingtypes participate, there are only 5 violations of this prediction. Returningto the determinants of panic in Table 4.3, for the 9 subjects that match τ -herding types exactly we see that the increased probabilities of trade at thenew public signal difference levels are around three times as large as in theregression with all subjects. Although not significant, even the third publicsignal difference level has a large, positive coefficient, suggesting types withbelief thresholds between ≈ 0.85 and ≈ 0.92 also exist. These results confirmthat a fairly heterogeneous mix of belief thresholds are used among thoseidentified as τ -herding types. Also, among these types, a subject’s signalnow has a significant positive effect, indicating that traders with signalsthat make their beliefs more extreme are significantly more likely to trade,consistent with the findings on return correlation. Given the evidence thatsubjects use the τ -herding heuristic, we have Finding 4.5.Finding 4.5: Use of the τ -herding heuristic explains the majority ofnon-equilibrium behavior when it is observed.4.5 DiscussionA critical question facing any laboratory study of markets is the degree towhich behavior exhibited by university students can be extrapolated to real-world financial settings with experienced professionals. In terms of rationalpanics, Chapter 2 cites a variety of empirical evidence that is consistent withthe fact that traders rationally rush to trade. Data from the laboratorybacks up this evidence by providing a clear demonstration of rational panicsin the Extended Rush treatment, where subjects clearly understand that82A subjective evaluation of the large number of subjects classified as simplistic suggeststhat most are likely τ -herding types who made a single mistake too many to be classifiedas such.814.5. Discussionwaiting to trade can be extremely costly. Although individually rational,the resulting rational panic is severely costly for the market as a whole dueto large informational losses. It also produces acute trade clustering, which,interpreted in terms of an earnings announcement, is consistent with thefact that volumes spike around these announcements (Lamont and Frazzini(2007)).The laboratory data can also provide insight as to when rational panicsare more likely to occur in actual financial markets. Comparing observedbehavior across treatments, we should expect to see rational panics whenthere are smaller differences in quality between initial private informationand the information that can be acquired through research. Also, panics aremore likely when many other traders are present in the market (as in theExtended environment) than when only one or two other traders may trade(as in the Basic environment of Chapter 3). Finally, I note that experience inthe environments where rushing is optimal drives behavior towards rationalpanics. Given that equilibrium behavior involves preempting other traders,it seems likely that experienced professionals understand the benefit of actingquickly and do so.Although heuristics found in the laboratory should generally be cau-tiously exported to the field, there are good reasons to suspect the τ -herdingheuristic operates in real-world markets. First, as already noted, the heuris-tic, although not optimal, is far from irrational behavior: it demonstratesa reasonable level of understanding of one’s environment. Second, the per-centage of τ -herding types in the data actually increases over time.83 As inthe case of rational panics, this finding suggests that traders may actuallylearn over time that prices move against them in expectation, causing themto (rationally or heuristically) panic. Finally, the fact that such behaviorgenerates the well-known phenomena of short-term positive correlation inreturns and trade clustering in the laboratory data suggests that it is agood candidate explanation for these findings.8483The percentage of τ -herding types increases from 6.3% in the first third of trials to31.3% in the last third in the EW treatment. Similar increases also occur in the Basicenvironment of Chapter 3.84Although not detectable in the laboratory data, given that traders with signals that824.5. DiscussionAs further suggestive evidence that traders may exhibit τ -herding behav-ior in actual markets, consider the fact that one of the pillars of technicalanalysis is momentum trading: wait until prices appear to be moving in aparticular direction and then trade in that direction.85 In the absence ofprivate information, momentum trading and τ -herding are identical, so, atleast among those that subscribe to technical analysis, something akin toτ -herding strategies is natural.86 Furthermore, τ -herding strategies can po-tentially generate positive-feedback trading (Long et al. (1990)). As initialtraders cause a trend in prices, those with more extreme threshold beliefs areencouraged to join in, reinforcing the trend and setting off further trading.Given the explanatory power of the heuristic, two additional questionsnaturally arise about this behavior. Do traders understand that others usethe heuristic? And, what is the underlying behavioral force that generatesthe use of the heuristic? Prices are set in the experiment assuming that ob-serving a trader wait reveals no information and that traders always tradeaccording to their private information. τ -herding behavior causes violationsof both of these assumptions so that prices are seen to be too extreme expost.87 If traders understand this fact and the mispricing is severe enough,oppose the price trend delay their trades, it is not inconceivable that τ -herding behaviorcould also be responsible for the longer-term negative correlation in returns (overreaction)observed in real markets (see Appendix A of Daniel et al. (1998) for a review of papersfinding such evidence).85For a description of momentum trading, seehttp://www.investopedia.com/articles/trading/02/090302.asp.86Note, however, that there is a subtle difference between τ -herding strategies andmomentum strategies. In a momentum trading strategy, traders that observe a positiveprice change (or series of positive price changes) purchase the stock regardless of theirprivate information, so there is no predicted relationship between traders’ signals and whenthey choose to trade. Instead, through the use of a threshold belief, τ -herding behaviorallows for differences in trade timing precisely because traders with different signals havedifferent beliefs. As shown in Section 4.4, the correlation in returns in the data is primarilydriven by differences in timing strategies and not by herding. If momentum strategies wereinstead the correct explanation, herding would be a more significant contributor to returncorrelation in the laboratory data. Several behavioral finance papers posit that tradersfollow momentum strategies in order to explain post-earnings-announcement drift (seeLong et al. (1990) and Hong and Stein (1999)), but the data here suggests that τ -herdingbehavior may be a better explanation.87For example, consider the case in which the price trend is upwards. A decision towait reveals that a trader is more likely to have had a negative signal. Furthermore, a834.5. Discussionthen they may be induced to act contrarian and sell regardless of their signal(perhaps rushing to do so). Because such contrarian behavior is very infre-quent, it would seem either that the majority of traders do not understandothers are τ -herding types, or that the drive to follow the τ -herding strategyis so strong that traders follow it even after accounting for mispricing.Can τ -herding behavior be generated by alternative preferences or com-mon behavioral game theoretical models? Without presenting a formal anal-ysis, I argue informally that the answer is no. One possibility is tradershave risk preferences different from than those induced (risk neutrality wasinduced by paying subjects in lottery tickets). However, if subjects are risk-averse, it can be shown that they would tend to trade in a contrarian manner.Risk-seeking behavior could potentially explain τ -herding strategies, but itdoes not seem plausible that over 40% of subjects are risk-seeking. Similarly,loss aversion also leads to contrarian behavior. Intuitively, at a high price,there is little to gain (relative to one’s endowment) from buying the asset,but a lot to lose. Finally, common behavioral game theoretic models alsoall lead toward contrarian behavior, rather than the more common herdingbehavior observed.88Two possible candidate theories can, at least intuitively, generate τ -buy decision may be taken by a trader with a negative signal. In both cases, the resultingprice is too high.88Behavioral game theory has developed several theories that have proven successful atdescribing experimental behavior: quantal response equilibrium (McKelvey and Palfrey(1995, 1998)), level k reasoning (Stahl and Wilson (1994); Nagel (1995)), cognitive hier-archy (Camerer et al. (2004)), and cursed equilibrium (Eyster and Rabin (2005)). Thesetheories are based upon models of subjects’ beliefs about other subjects behavior. For-mally analyzing any of these theories in the environment here is a non-trivial task, butintuitively they all lead to contrarian behavior. Quantal response equilibrium assumesthat subjects probabilistically take actions in proportion to the payoff of each action, andthat this behavior is common knowledge. While there is some evidence of mistakes pro-portional to payoff differences (see footnote 57 in Chapter 3), if subjects believe othersmake such mistakes, then they would believe prices are too extreme. In this case, con-trarian behavior would be prescribed. In level k and cognitive hierarchy, subjects believeother subjects are boundedly rational and best respond to this behavior. The lowest levelof rationality is typically assumed to be random behavior, but this would again lead toprices being too extreme and contrarian behavior. Finally, in cursed equilibrium, subjectscorrectly predict the frequencies of actions, but believe that actions are not connectedto underlying signals. Again, this belief leads to believing prices are too extreme andcontrarian behavior.844.5. Discussionherding behavior. The first is base rate neglect, a version of the representa-tiveness heuristic.89 In base rate neglect, one underweights one’s prior. Assuch, one’s beliefs can become extreme relative to the price and herding maybecome optimal. Also, if one’s belief is too extreme, one would expect pricemovements to be greater in expectation, potentially inducing rushing.90,91The second candidate explanation is a preference for certainty, as formu-lated by Eliaz and Schotter (2010). Intuitively, if one places explicit valueupon being certain about the state before trading, one could be induced towait when uncertain. If, in addition, there are diminishing returns to thispreference for certainty, then subjects may choose to rush as beliefs becomemore certain. Detailed development of a psychologically-based model forτ -herding behavior is an interesting direction for future research.89See Barberis and Thaler (2003) for a discussion of the representativeness heuristic ina financial context and for references to the psychological literature.90To explain the fact that many traders herd after only a single public signal in theEW treatment, one’s belief must become extreme relative to the price after only a singlesignal. Models in which beliefs become extreme only after a sequence of signals (such asBarberis et al. (1998) and Rabin (2002)) cannot generate such behavior.91A related concept is overconfidence in which one places too much weight upon one’sown signal. While similar, overconfidence can not generate τ -herding behavior becausea trader with a signal opposite to the price trend would always trade according to hersignal. For a model of underreaction based on overconfidence, see Daniel et al. (1998).Overconfidence has many different interpretations. More precisely, overweighting one’sprivate information may be referred to as overprecision (Moore and Healy (2008)).85Chapter 5ConclusionThis dissertation has studied informational losses due to trading panics, boththeoretically and experimentally. The fear of adverse price movements dueto others’ trades can induce traders to rush to trade as soon as possible.Because traders panic, they forgo doing research about the asset they aretrading, resulting in trades based upon relatively weak information. As aconsequence, prices take longer in expectation to converge to fundamentalvalues and more frequent increases in mispricing are predicted.The main theoretical result is that markets are susceptible to panicswhen uncertainty is high, so that information is forgone when it is mostvaluable. This finding stands in stark contrast to models with monetaryinformation costs and suggests that future work which models informationacquisition in financial markets should consider the time costs of acquiringinformation. The theoretical work also highlights the coordination aspectof trading panics. A multiplicity of equilibria exists due to strategic com-plementarity between traders in the market arises, allowing expectations ofpanics to become self-fulfilling. As such, there may be a role for marketintervention to calm fears during times of panic.The theoretical ideas may be used in future work on market design.Rushing to trade is only optimal when there is a rational fear of otherstrading before one has a chance to acquire information. In markets thatare continuously open, this fear can be justified. On the other hand, inmarkets that are cleared through call auctions, as long as the auctions arenot too frequent, such fears are alleviated. Thus, the model suggests thatmarkets that are cleared through call auctions may be less prone to traderstrading on weak information and the associated consequences.92 Similarly,92For a recent paper on the debate about market design that summarizes previous86Chapter 5. Conclusionthe model justifies the practice of regulatory trading halts on stock exchangeswhen news is to be released. The temporary trading halt ensures tradershave time to process information, which prevents rushing to trade on weakinformation and, assuming the accumulated information is sufficient, ensuresno panics occur when trading resumes because prices will adjust to valuesnear the fundamental value (where not panicking is optimal).The experimental work investigates the sources and consequences of trad-ing panics in a laboratory setting, using precise theoretical predictions as aguide. The theoretical results successfully predict the relative frequencyof rational panics and allow non-equilibrium panics to be identified. Non-equilibrium behavior creates informational losses over and above those dueto rational panics. I develop a simple heuristic to explain behavior observedin the first environment and use it to generate predictions that are confirmedin a second, richer environment. The heuristic also reconciles findings in pre-vious experimental studies with related environments and puts forth a newexplanation for the trade clustering and short-term correlation in returnsobserved in real financial markets.A clear direction for future experimental research is to understand inmore detail the trading heuristic used by almost half of traders. Of particularinterest may be to determine whether or not subjects understand that othersfollow the heuristic and try to exploit such behavior to their advantage. Forexample, in settings in which multiple trades are possible, rational tradersmay buy in an attempt to push up prices such that they can then profitablysell the asset at an even higher price. If so, the heuristic may help to explainprice bubbles, both in the field and in laboratory experiments (e.g. Smithet al. (1988)).The trading heuristic has additional implications outside of the labora-tory. In particular, it suggests that short-term correlation in returns shouldbe more dramatic after earnings announcements that generate disagreementamong traders, as opposed to those that are unambiguous in their impli-cations. In the former case, there is more scope for traders with differentbeliefs to choose to trade at different times. Similarly, return correlationtheoretical and empirical work, see Kuo and Li (2011).87Chapter 5. Conclusionshould be more pronounced in stocks with higher volumes, where increasedfears of adverse price movements cause traders to rush to trade on weakerinformation. 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(2012), “An improved estimation method and empir-ical properties of the probability of informed trading,” Journal of Banking& Finance, 36(2):454–467.Zhang, X. (2006), “Information uncertainty and stock returns,” The Journalof Finance, 61(1):105–137.Ziegelmeyer, A., My, K.B., Vergnaud, J.C., and Willinger, M. (2005),“Strategic delay and rational imitation in the laboratory,” mimeo, MaxPlanck Institute of Economics.96Appendix AAppendix to Chapter 2A.1 Benefit FunctionsHere, I show that the general form of the benefit to waiting can be writtenas (2.1). From Lemma 2.1, a trader trading at t with st = 1 always buysand so her profit is given byPr(V = 1|st = 1, Ht)−pt ⇐⇒ptqptq + (1− pt)(1− q)−pt ⇐⇒pt(1− pt)(2q − 1)ptq + (1− pt)(1− q)The profit for a trader with st = 0 is calculated similarly, using the factthat she always sells. The profit for both types of traders can be writtenpt(1−pt)(2q−1)Pr(st).The expected profit from waiting depends upon the timing strategies ofthe two types of trader and the strategies of the other traders. I calculate theprofit explicitly for a trader with st = 1 who buys when receiving st+1 = 1and sells when st+1 = 0. The other cases are calculated similarly and areomitted for brevity. The expected profit is∑aˆ∈A{Pr(st+1 = 1&aˆ|st = 1) (Pr(V = 1|st = 1, st+1 = 1, aˆ)− Pr(V = 1|at = NT, aˆ))+ Pr(st+1 = 0&aˆ|st = 1) (Pr(V = 1|at = NT, aˆ)− Pr(V = 1|st = 1, st+1 = 0, aˆ))}where all probabilities are also conditional on Ht. Here, I sum over eachpossible generic combination of events that result from the timing and trad-ing strategies of t−1 and t+1. The first term corresponds to the profit frombuying the asset after receiving st+1 = 1 and the second term from sellingafter receiving st+1 = 0. Using Bayes’ rule and the independence of signals,the above becomes∑aˆ∈A{Pr(st+1 = 1&st = 1&aˆ)Pr(st = 1)(ptqqPr(aˆ|V = 1)Pr(st+1 = 1&st = 1&aˆ)−ptNT1Pr(aˆ|V = 1)Pr(at = NT&aˆ))+Pr(st+1 = 0&st = 1&aˆ)Pr(st = 1)(ptNT1Pr(aˆ|V = 1)Pr(at = NT&aˆ)−ptq(1− q)Pr(aˆ|V = 1)Pr(st+1 = 0&st = 1&aˆ))}97A.1. Benefit Functionswith NT0 and NT1 as in Lemma 2.1. Combining each of the terms in the firstand second expressions, canceling and factoring out common terms givespt(1− pt)Pr(st = 1)∑aˆ∈Aaˆ0aˆ1Pr(at = NT&aˆ)(qqNT0 −NT1(1− q)(1− q) +NT1q(1− q)− q(1− q)NT0)⇐⇒pt(1− pt)Pr(st = 1)∑aˆ∈Aaˆ0aˆ1Pr(at = NT&aˆ)((2q − 1)(qNT0 + (1− q)NT1))Finally, subtracting the profit at t from the expected profit and factoringout common terms gives the benefit formula stated in (2.1).As discussed in the main text, we can set, β0 = β1, simplifying the benefitfunction. f(q, q, β0, β1) simplifies to β(2q−1) when q > q and β(2q−1) whenq < q, reflecting the fact that a trader follows her stronger signal when theyare contradictory. The denominator simplifies to Pr(aˆ&at = NT ) = βPr(aˆ)so that β cancels in the numerator and denominator. When q = q, a traderwith contradictory signals is indifferent between trading or not, but becauseI assume that such a trader follows her second period signal, f(q, q, β0, β1) =β(2q − 1) and β cancels in this case as well. Denoting q = max(q, q), onecan then write the general benefit function asBx(pt) =pt(1− pt)Pr(st = x)[∑aˆ∈Aaˆ0aˆ1(2q − 1)Pr(aˆ)− (2q − 1)]where the function no longer depends on β0 or β1.For specific (pure) strategies of the other traders, one can substitutethe information revealed by each possible event into the general formula toobtain a specific formula for those strategies. As shown in Lemma 2.1, anytrader that rushes trades according to her first period signal and so, for arushed trade by t + 1, Pr(at+1 = B|V = 1) = Pr(st+1 = 1|V = 1) = qand Pr(at+1 = S|V = 1) = 1 − q. For a delayed trade by t − 1, Pr(at =B|V = 1) = q and Pr(at = S|V = 1) = 1 − q where q denotes max(q, q).The unconditional probabilities of buy and sell decisions by t − 1 are thenPr(at = B) = pq + (1 − p)(1 − q) and Pr(at = S) = p(1 − q) + (1 − p)q.The reason only the stronger signal is revealed comes from Lemma 2.1 whenβ0 = β1: if q > q, a trader with st = 1 (st = 0) buys (sells) regardless of st,so that no information about st is revealed by her trade. Similarly, if q > q,trades don’t depend on st. If the two signal strengths are the same, since Ihave assumed a trader follows her second period signal, only it is revealed.Substituting the information revealed, we get the following formulas, wherePr(at = x&st+1 = y) is abbreviated Pr(x, y), Sx ≡pt(1−pt)Pr(st=x)is a scale factorcommon to all of the benefit functions, Q ≡ q(1 − q), Q ≡ q(1 − q), and98A.2. Stability of EquilibriaQ ≡ q(1− q).BR,WWx (pt) = Sx[(2q − 1)− (2q − 1)](A.1)BR,RRx (pt) = Sx[(2q − 1)Q(1Pr(st+1 = 1)+1Pr(st+1 = 0))− (2q − 1)]BW,WWx (pt) = Sx[(2q − 1)Q(1Pr(at = B)+1Pr(at = S))− (2q − 1)]BW,RWx (pt) = Sx[(2q − 1)Q(1Pr(at = S)+ q(1− q)(1Pr(B, 1)+1Pr(B, 0)))− (2q − 1)]BW,WRx (pt) = Sx[(2q − 1)Q(1Pr(at = B)+ q(1− q)(1Pr(S, 1)+1Pr(S, 0)))− (2q − 1)]BW,RRx (pt) = Sx[(2q − 1)QQ(1Pr(B, 1)+1Pr(B, 0)+1Pr(S, 1)+1Pr(S, 0))− (2q − 1)]A.2 Stability of EquilibriaIn this section, I argue that any possible mixed strategy equilibrium of theexpected value trading model is unstable in the sense that pseudo-dynamicswould tend to destabilize it, and that the pure strategy equilibria are insteadstable. The discussion is relatively informal, as the purpose is to simplyjustify a focus on the pure strategy equilibria.Consider first an equilibrium that involves mixed strategies. That suchan equilibrium can exist was demonstrated in the proof of Theorem 2.1.In the equilibrium, two neighbors in the restricted price chain are mixingbetween conditionally rushing and waiting with probabilities such that theother is indifferent. Without loss of generality, label the two traders as theprices they face, pt and p+t . No assumptions are made about the strategiesof the traders at p−t and (p+t )+: those traders may be mixing or playingpure strategies of either conditional rushing or waiting. p+t is mixing suchthat pt has a benefit of zero given the strategy of p−t . If p+t changes herstrategy to wait slightly more often, the benefit of pt then becomes strictlypositive because her benefit is linear in the probability with which p+t mixesand increases when p+t waits more often. For example, if p−t is waiting,then pt faces a benefit of BW,WWx (pt)>0 when p+t waits and BW,RWx (pt) < 0when p+t conditionally rushes. Therefore, her benefit is βBW,WWx (pt) + (1−β)BW,RWx (pt) = 0 where β is the equilibrium probability with which p+twaits. When p+t waits more often, pt’s benefit is strictly positive becausemore weight is placed upon BW,WWx (pt) > 0 which will cause her to change99A.3. Omitted Proofsher strategy to wait with probability 1. But, when this happens, by the sameargument, p+t will be induced to change her strategy to wait with probability1. The small change to waiting more often by one trader leads both tradersto wait with probability 1, so the equilibrium is unstable. A small changeto rushing more often leads to both traders (conditionally) rushing withprobability 1 by very similar arguments.Consider instead what happens when the equilibrium is such that alltraders in the restricted price chain wait. In this case, each trader facesa strictly positive benefit to waiting (ignoring the non-generic case of theprice being exactly equal to one of the cutoff prices). Now, if a traderbegins to conditionally rush with some small probability, since the othertraders have strictly positive benefits, they will continue to wait. In addition,the trader who conditionally rushes is strictly worse off because any timeshe conditionally rushes, her profit is reduced given her positive benefitfrom waiting. Therefore, it is in her best interest to return to waiting withprobability 1: the equilibrium is stable. A similar argument establishes thatan equilibrium with all traders conditionally rushing is also stable, so thatall pure strategy equilibria are stable.A.3 Omitted ProofsProof of Lemma 2.1:The proof when rushing is similar to the proof when waiting, only sim-pler, so is omitted for brevity. Let aˆ denote any information that becomespublic due to the decisions of traders other than t between t and t + 1and abbreviate Pr(aˆ|V = y) as aˆy, for y ∈ {0, 1}. A trader, t, who waitsbuys if her expected value of the asset is greater than the price she faces,pt+1 = Pr(V = 1|aˆ, at = NT,Ht):Pr(V = 1|st, st+1, aˆ, Ht) > Pr(V = 1|aˆ, at = NT,Ht)⇐⇒ptPr(st|V = 1)Pr(st+1|V = 1)aˆ1ptPr(st|V = 1)Pr(st+1|V = 1)aˆ1 + (1− pt)Pr(st|V = 0)Pr(st+1|V = 0)aˆ0>ptaˆ1Pr(at = NT |V = 1)ptaˆ1Pr(at = NT |V = 1) + (1− pt)aˆ0Pr(at = NT |V = 0)⇐⇒ Pr(st|V = 1)Pr(st+1|V = 1)NT0 > Pr(st|V = 0)Pr(st+1|V = 0)NT1 (A.2)where the first equivalence follows from applying Bayes’ rule to each side ofthe inequality and using the fact that the public belief, Pr(V = 1|Ht) = pt.100A.3. Omitted ProofsUsing (A.2), a trader with st = 1, st+1 = 1 buys ifqqNT0 > (1− q)(1− q)NT1⇐⇒ q(1− q)(2q − 1)β1 +(q2q − (1− q)2(1− q))β1 > 0which is true ∀q, q ∈ (12 , 1). Similarly, a trader with st = 0, st = 0 alwayssells. Finally, the conditions in Lemma 2.1 under which traders with st =1, st+1 = 0 and st = 0, st+1 = 1 buy or sell are easily obtained by substitutingfor the appropriate probabilities in (A.2).Proof of Lemma 2.2:The proof is by contradiction. Assume there is an equilibrium in whichβ1 > β0 (the case of β0 > β1 similarly leads to a contradiction) at somept and for some arbitrary strategies of the other traders. I first show thatβ1 > β0 and B1(pt, β0, β1) ≥ 0 together imply B0(pt, β0, β1) > 0 for allpossible strategies of the other traders and for all pt.One can see from the general form of the benefit in (2.1), that the sign ofBx(pt, β0, β1) is determined by the term in square brackets and that the onlydifference between the bracketed terms in B0(pt, β0, β1) and B1(pt, β0, β1) isdue to differences in the function, f(q, q, β0, β1). I show that this function isstrictly greater in B0(pt, β0, β1) than in B1(pt, β0, β1) whenever β1 > β0so that if B1(pt, β0, β1) ≥ 0, we must have B0(pt, β0, β1) > 0 (becausef(q, q, β0, β1) is always positive). There are four possible cases dependingupon the optimal strategies of buying and selling from Lemma 2.1.Consider first the case of g1(q, q) ≤ 0 and g0(q, q) ≥ 0. The comparisonof f(q, q, β0, β1) in B0(pt, β0, β1) relative to B1(pt, β0, β1) is qNT1 + (1 −q)NT0 > qNT0 + (1 − q)NT1 ⇐⇒ (NT1 − NT0)(2q − 1) > 0 ⇐⇒ (β1 −β0)(2q − 1)2 > 0 so f(q, q, β0, β1) is strictly greater in B0(pt, β0, β1). Wheng1(q, q) > 0 and g0(q, q) < 0, we have qNT1 − (1 − q)NT0 > qNT0 − (1 −q)NT1 ⇐⇒ NT1−NT0 > 0⇐⇒ (β1−β0)(2q−1) > 0 so again f(q, q, β0, β1)is strictly greater in B0(pt, β0, β1).When g1(q, q) ≤ 0 and g0(q, q) < 0, I begin by noting thatg0(q, q) < 0 =⇒ qNT1 − (1 − q)NT0 > (2q − 1)(qNT1 + (1 − q)NT0).This can be seen algebraically or simply by noting that B0(pt, β0, β1) mustbe larger when t follows her optimal trading strategy at t + 1 instead ofthe optimal strategy for the other case, g0(q, q) ≥ 0. But, then we have,qNT1− (1− q)NT0 > (2q− 1)(qNT1 + (1− q)NT0) > (2q− 1)(qNT0 + (1−q)NT1) where the second inequality was shown in the first case above, sof(q, q, β0, β1) is strictly greater in B0(pt, β0, β1). Similarly, when g1(q, q) >0 and g0(q, q) ≥ 0, g0(q, q) ≥ 0 implies (2q − 1)(qNT1 + (1 − q)NT0) >101A.3. Omitted ProofsqNT1 − (1 − q)NT0 by optimality of the second period trading decisionand qNT1 − (1− q)NT0 > qNT0 − (1− q)NT1 as shown in the second caseabove. Thus, f(q, q, β0, β1) is strictly greater in B0(pt, β0, β1) in all cases andtherefore B1(pt, β0, β1) ≥ 0 implies B0(pt, β0, β1) > 0. Furthermore, withβ1 > β0, we must have β1 ∈ (0, 1] and therefore B1(pt, β0, β1) ≥ 0. Then, wemust have B0(pt, β0, β1) > 0 by the established implication. However, wemust also have β0,t ∈ [0, 1) and therefore B0(pt, β0, β1) ≤ 0, a contradiction.Proof of Proposition 2.1:To establish Proposition 2.1, I first prove the following mathematicalclaim:Claim: The following inequality holds for any x, y ∈ R+, n ≥ 1, andany ai, bi ∈ [0, 1] ∀i = 1 . . . n satisfying∑ni=1 ai =∑ni=1 bi = 1 and at leastone of ai or bi greater than zero ∀i = 1 . . . n. Furthermore, it holds withequality if and only if ai = bi 6= 0 ∀i = 1 . . . n.n∑i=1aibiaix+ biy≤1x+ yWhen n = 1, we must have a1 = b1 = 1 due to the constraints that eachset of ai and bi must sum to one. In this case, the inequality is easily seento be satisfied with equality. So, consider n > 1. I show that the lhs of theinequality reaches its global maximum of 1x+y when ai = bi 6= 0 ∀i = 1 . . . nto show both that the inequality is always satisfied and that it only holdswith equality when ai = bi 6= 0 ∀i = 1 . . . n.I first show that f(ai, bi) =aibiaix+biyis concave and use the fact that thesum of any number of concave functions is also concave so that∑ni=1aibiaix+biyis concave. With simple algebra, we have∂2f(ai,bi)∂a2i= −2b2i xy(aix+biy)3≤ 0, ∂2f(ai,bi)∂b2i= −2a2i xy(aix+biy)3≤ 0, and ∂2f(ai,bi)∂aibi=−2aibixy(aix+biy)3. Therefore, ∂2f(ai,bi)∂a2i∂2f(ai,bi)∂b2i−(∂2f(ai,bi)∂aibi)2= 0. Thus, the Hes-sian of f(ai, bi) is negative semi-definite and therefore f(ai, bi) is concave.If any ai or bi is equal to zero, the corresponding term in the summationof the lhs is zero and thus contributes nothing. So, consider the n∗ ≤ n non-zero terms of the summation. We must have∑n∗i=1 ai = v and∑n∗i=1 bi = wfor some v, w ≤ 1, due to the constraints. Consider the unconstrainedmaximization of the non-zero terms of the lhs of the inequality after usingthese constraints to substitute out an∗ and bn∗ . Because we are maximizinga concave function, the first-order conditions are necessary and sufficient102A.3. Omitted Proofsfor determining the global maximizer(s) of the function. We have, ∀i =1 . . . n∗ − 1, the first-order conditions with respect to ai:b2i y(aix+ biy)2=b2n∗y(an∗x+ bn∗y)2⇐⇒bi(aix+ biy)=bn∗(an∗x+ bn∗y)⇐⇒aibi=an∗bn∗The first-order conditions with respect to bi result in the same set of equa-tions. Substituting for each ai in the constraint∑n∗i=1 ai = v, we havean∗bn∗∑n∗i=1 bi = v ⇐⇒ an∗ = bn∗vw which then implies ai = bivw ∀i = 1 . . . n∗.Using this relationship, we have∑n∗i=1aibiaix+biy= vwvx+wy ≤1x+y , where theinequality holds with equality only when v = w = 1 which requires nonon-zero terms in the summation and implies ai = bi 6= 0. Therefore,∑ni=1aibiaix+biy≤ 1x+y is always satisfied and is satisfied with equality if andonly if ai = bi 6= 0 ∀i = 1 . . . n, as claimed.Now, to see that any additional informative potential trade reduces thebenefit from waiting, one can apply the mathematical claim. In particular,looking at the general structure of the benefit functions, (2.1), one can seethat an additional, independent, trade, c, which results in one of n possibleactions, modifies each contribution to the benefit from an event, dj , of traded by replacingdj,1dj,0ptNT1dj,1 + (1− pt)NT0dj,0withn∑i=1dj,1dj,0Pr(c = ci|V = 1)Pr(c = ci|V = 0)ptNT1dj,1Pr(c = ci|V = 1) + (1− pt)NT0dj,0Pr(c = ci|V = 0)≤dj,1dj,0ptNT1dj,1 + (1− pt)NT0dj,0where dj,y is shorthand for Pr(d = dj |V = y), y ∈ {0, 1}. The inequalityfollows from applying the claim with ai = Pr(c = ci|V = 1), bi = Pr(c =ci|V = 0), x = ptNT1dj,1, and y = (1 − pt)NT0dj,0. Thus, the additionaltrade reduces the benefit to waiting, where the inequality is strict if the tradeis informative (Pr(c = ci|V = 1) 6= Pr(c = ci|V = 0) for some i ∈ 1 . . . n aslong as pt 6= {0, 1}.93 Proof of Proposition 2.2:By continuity of the benefit functions, the limit as pt → 1 must be the93If event d were perfectly informative, the inequality would not be strict, but I haveassumed signals (and therefore trades) are not perfectly informative.103A.3. Omitted Proofsvalue of the function evaluated at 1. Therefore, consider the bracketed termin the general formula for the benefit function given in (2.1), evaluated atpt = 1. We have∑aˆ∈Aaˆ0aˆ1f(q, q, β0, β1)Pr(aˆ&at = NT )− (2q − 1)=∑aˆ∈Aaˆ0aˆ1f(q, q, β0, β1)ptaˆ1NT1 + (1− pt)aˆ0NT0− (2q − 1)=∑aˆ∈Aaˆ0aˆ1f(q, q, β0, β1)aˆ1NT1− (2q − 1)=f(q, q, β0, β1)NT1− (2q − 1) (A.3)where q ≡ max(q, q) and using∑aˆ∈A aˆ0 = 1. (A.3) is the benefit of atrader that faces no informative intervening trades, BSTx (pt, β0, β1), evalu-ated at pt = 1 (set aˆ0 = aˆ1 = 1 in the general formula for the benefit). Atpt = 1, and therefore, by continuity in the limit as pt → 1, we have then es-tablished that the sign of the benefit, which is determined by the bracketedterm, is the same as when no informative intervening trades occur, establish-ing part 2 of the proposition. For part 1, it is easy to see that the magnitudeof the benefit function approaches zero whether there are informative inter-vening trades or not because the part of the benefit outside of the bracketedterm approaches zero as pt → 1 and the bracketed term remains finite, asshown above. The proof of both properties for pt approaching 0 is identical.Proof of Theorem 2.1:Before proving Theorem 2.1, Lemma A.3.1 establishes the remainingproperties of the benefit functions exhibited in the example of Figure 2.2,specifically the number and locations of the zero-crossings. Theorem 2.1assumes the restrictions on off-equilibrium public beliefs and prices and thusthe simplified benefit functions given in (A.1) are the relevant ones for thetheorem and lemma.104A.3. Omitted ProofsLemma A.3.1: For q ≤ q, each of the benefit functions, Bu,v1v2x (pt), is≤ 0 for all pt. For q > q, the benefit functions satisfy:1. Bu,v1v2x (pt) = 0 for pt ∈ {0, 1}2. For pt ∈ (0, 1):(a) BR,WWx (pt) > 0(b) BR,RRx (pt),BW,WWx (pt), and BW,RRx (pt) are either > 0 overthe full range or< 0 pt ∈ (1− pˆu,v1v2 , pˆu,v1v2)= 0 pt ∈ {1− pˆu,v1v2 , pˆu,v1v2}> 0 pt ∈ (0, 1− pˆu,v1v2) ∪ (pˆu,v1v2 , 1)(c) BW,RWx (pt) is either > 0 over the full range or< 0 pt ∈ (1− pˆW,WR, pˆW,RW )= 0 pt ∈ {1− pˆW,WR, pˆW,RW }> 0 pt ∈ (0, 1− pˆW,WR) ∪ (pˆW,RW , 1)(d) BW,WRx (pt) is either > 0 over the full range or< 0 pt ∈ (1− pˆW,RW , pˆW,WR)= 0 pt ∈ {1− pˆW,RW , pˆW,WR}> 0 pt ∈ (0, 1− pˆW,RW ) ∪ (pˆW,WR, 1)3. 1− pˆW,RR < 1− pˆW,WR < 1− pˆW,RW < 1− pˆW,WW < 1− pˆR,RR <pˆR,RR < pˆW,WW < pˆW,RW < pˆW,WR < pˆW,RR where the zero cross-ings nearer to pt = {0, 1} exist if the next innermost zero crossingexists (but not necessarily the converse).Proof of Lemma A.3.1:With q ≤ q, the fact that all benefit functions are weakly less thanzero follows from the fact that BR,WWx (pt) = 0 for all pt and Proposition2.1. So, consider q > q. In this case, Pr(at = B) = Pr(st = 1) andPr(at = S) = Pr(st = 0), so these substitutions can be made in equations105A.3. Omitted Proofs(A.1).Part 1 follows from evaluation of the benefit functions at pt = {0, 1).Part 2a follows from the fact that BR,WWx (pt) > 0⇐⇒ q > q.For part 2b, Proposition 2.2 ensuresBR,RRx (pt), BW,WWx (pt), BW,RRx (pt) >0 as prices become close to 0 and 1 because each of these functions musthave the same sign as BR,WWx (pt), which is positive from 2a. It remains tobe shown that these functions cross zero at at most two additional prices,pˆu,v1v2 and 1 − pˆu,v1v2 . Considerable algebraic manipulation allows each ofthe three functions to be writtenBR,RRx (pt) = Sx[2(q − q)Q− (2q − 1)3PtPr(st+1 = 1)Pr(st+1 = 0)]BW,WWx (pt) = Sx[2(q − q)Q− (2q − 1)(2q − 1)2PtPr(st = 1)Pr(st = 0)]BW,RRx (pt) = Sx[(2q − 1)QQA− (2q − 1)DD]whereA = QQ+ Pt(Q+Q− 8QQ), D = P 2t(Q−Q)2+ PtQQ(1− 2Q− 2Q)+(QQ)2,and defining Pt ≡ pt(1− pt).Given that the zeros of each of the three benefit functions can be writtenas a function of Pt, it follows immediately that any zero, pˆu,v1v2 must bepaired with another zero, 1− pˆu,v1v2 .94 For BR,RRx (pt) and BW,WWx (pt), wehave a linear function of Pt and thus there is at most one additional pairof zeros. Furthermore, we can see that there are cases in which such a pairof zeros exist by, for example, taking q → 1. For BW,RRx (pt), the additionalzeros are determined by a quadratic function of Pt, so that there may existtwo additional pairs of zeros. But I now show that one of the pairs alwayslies outside pt ∈ (0, 1) by showing that one of the solutions to the quadraticis always negative, P ∗t < 0, implying p∗t < 0 or p∗t > 1. The quadraticwhich determines the additional zeros of BW,RRx (pt) can be algebraically94For specific parameters, the two zeros can be equal, pˆu,v1v2 = 1− pˆu,v1v2 = 12 .106A.3. Omitted Proofsmanipulated to obtain(2q − 1)QQA− (2q − 1)D = 0⇐⇒ −(2q − 1)P 2t(Q2−Q2)2+ Ptf(q, q) + 2(q − q)(QQ)2= 0where f(q, q) is a function which proves to be inconsequential. The solutionsto the quadratic are, Pt = −b±√b2−4ac2a where a = −(2q−1)(Q2−Q2)2, b =f(q, q), and c = 2(q−q)(QQ)2. Now, a < 0 and c > 0 for q > q and therefore−4ac > 0. But, this means that√b2 − 4ac > |b| and therefore the root −b−√b2 − 4ac is always strictly negative. Thus, there are at most two additionalzeros of BW,RRx (pt) and furthermore we know that there are exactly two rootsin some cases because, by Proposition 2.1, BW,RRx (pt)<BW,WWx (pt) and Ihave already established that BW,WWx (pt) < 0 for pt ∈ (1− pˆW,WW , pˆW,WW ),for some parameterizations.For parts 2c and 2d, I first note that Proposition 2.1 implies thatBW,RRx (pt) < BW,RWx (pt), BW,WRx (pt) < BW,WWx (pt)for all pt ∈ (0, 1). Because of this fact and the fact that the equation thatdetermines the additional zeros of BW,RWx (pt) and BW,WRx (pt) (beyond thosethat occur at pt = {0, 1}) is cubic, one can easily see graphically that at mosttwo of the additional zeros lie within pt ∈ (0, 1). That is, it is not possible tohave either of these functions cross zero three times within pt ∈ (0, 1) and stillhave it remain within the envelope of BW,RRx (pt) and BW,WWx (pt), given theirestablished properties. It also implies that, for some parameter ranges thesefunctions are always positive and for others they must cross zero exactlytwice.95 Therefore, what remains to be shown is that if pˆW,WR and pˆW,RWare zeros for BW,WRx (pt) and BW,RWx (pt), respectively, then 1− pˆW,WR and1− pˆW,RW are zeros for BW,RWx (pt) and BW,WRx (pt), respectively, as claimedin the lemma.Combining terms in BW,RWx (pt), we see that for it to be zero at pˆW,RW ,we must have(2q−1)Q(Pr(1, 1)Pr(1, 0) +Q (Pr(1, 0) + Pr(1, 1))(pˆW,RW (1− q) + (1− pˆW,RW )q))− (2q − 1)Pr(1, 1)Pr(1, 0)(pˆW,RW (1− q) + (1− pˆW,RW )q)= 0where Pr(1, 1) = Pr(st = 1&st+1 = 1) and Pr(1, 0) = Pr(st = 1&st+1 =95For specific parameters, the two zeros may be degenerate with both equal to 12 .107A.3. Omitted Proofs0) are understood to be evaluated at the price, pˆW,RW . We want to showthat 1− pˆW,RW is a zero for BW,WRx (pt). Combining terms in BW,WRx (pt), wesee that for it to be zero at 1− pˆW,RW requires(2q−1)Q(Pr(0, 0)Pr(0, 1) + q(1− q) (Pr(0, 0) + Pr(0, 1))((1− pˆW,RW )q + pˆW,RW (1− q)))− (2q − 1)Pr(0, 1)Pr(0, 0)((1− pˆW,RW )q + pˆW,RW (1− q))= 0where, here, Pr(0, 0) == Pr(st = 0&st+1 = 0) and Pr(0, 1) = Pr(st =0&st+1 = 1) are understood to be evaluated at the price, 1− pˆW,RW . Now,note that Pr(0, 0) evaluated at 1− pˆW,RW is equal to Pr(1, 1) evaluated atpˆW,RW and Pr(0, 1) evaluated at 1 − pˆW,RW is equal to Pr(1, 0) evaluatedat pˆW,RW so the two conditions are identical. Therefore, if pˆW,RW is a zeroof BW,RWx (pt), 1 − pˆW,RW is a zero of BW,WRx (pt). The reverse is shown inthe same manner.For part 3, the properties established in parts 2b-d and Proposition 2.1ensure thatpˆR,RR, pˆW,WW < pˆW,RW , pˆW,WR < pˆW,RRso it remains to be shown that pˆR,RR < pˆW,WW and pˆW,RW < pˆW,WR. Thefact that1− pˆW,RR < 1− pˆW,WR < 1− pˆW,RW < 1− pˆW,WW < 1− pˆR,RRthen follows by the established symmetry properties of the zero-crossings.To show pˆR,RR < pˆW,WW , I show BR,RRx (pt)>BW,WWx (pt) for all pt ∈(0, 1). For q > q, this inequality is equivalent toq(1− q)Pr(st+1 = 1)+q(1− q)Pr(st+1 = 0)>q(1− q)Pr(st = 1)+q(1− q)Pr(st = 0)Substituting for the probabilities of each signal and simple algebra showsthat this inequality is equivalent to (q − q)(1 − q − q) < 0 which holds forq > q and q, q > 12 .To show pˆW,RW < pˆW,WR, I show that BW,RWx (pt)>BW,WRx (pt) for pt >108A.3. Omitted Proofs12 . This comparison simplifies toQPr(st = 0)+QQ(1Pr(1, 1)+1Pr(1, 0))>QPr(st = 1)+QQ(1Pr(0, 1)+1Pr(0, 0))⇐⇒Pr(st = 1)− Pr(st = 0)Pr(st = 0)Pr(st = 1)+Q(Pr(st = 1)Pr(1, 1)Pr(1, 0)−Pr(st = 0)Pr(0, 1)Pr(0, 0))> 0⇐⇒ (Pr(st = 1)− Pr(st = 0))Pr(1, 1)Pr(1, 0)Pr(0, 1)Pr(0, 0)+q(1− q)Pr(st = 0)Pr(st = 1) (Pr(st = 1)Pr(0, 1)Pr(0, 0)− Pr(st = 0)Pr(1, 1)Pr(1, 0)) > 0Considerable algebraic manipulation of the second term allows one tofactor out Pr(st = 1)−Pr(st = 0), resulting in this difference multiplied bya term which can be shown to be strictly positive. Thus,BW,RWx (pt)>BW,WRx (pt)⇐⇒ Pr(st = 1)−Pr(st = 0) = (2q−1)(2pt−1),which is strictly positive for all pt > 12 . Given Lemma A.3.1, I now prove Theorem 2.1 by considering each ofthe price ranges in turn. The proofs for the cases of q ≤ q and for pt ∈(1 − pˆW,WW , pˆW,WW ) have already been established in the main text, sohere the focus is on the case of q > q and the remaining price ranges (parts2c and 2d of the theorem). Also, as discussed in the main text, for theremaining price ranges, any equilibrium timing strategy involves waitingwhen t− 1 rushes, so I consider only the best response to t− 1 waiting here.2c, part (i). When the unrestricted price chain passes through (1 −pˆW,WW , pˆW,WW ), if there exist any traders in the associated restricted pricechain and also in the price range, [pˆW,WW , pˆW,WR), then one of them, p, mustbe such that the trader at p− faces a price in (1 − pˆW,WW , pˆW,WW ). Then,because the trader at p− rushes or conditionally rushes, p must conditionallyrush because her benefit is either BW,WRx (pt), or BW,RRx (pt), both of whichare negative (ruling out p mixing as well). But then any trader in therestricted price chain at prices greater than p must also conditionally rushfor the same reason: her neighbor at the next lowest price in the price chainconditionally rushes. Similarly, any trader in the restricted price chain andalso in the price range, (1 − pˆW,WR, 1 − pˆW,WW ], must conditionally rushbecause her neighbor at p+ rushes or conditionally rushes, so her benefit iseither BW,RWx (pt) or BW,RRx (pt), which are both negative in this price range.2c, part (ii). When the unrestricted price chain does not pass through(1− pˆW,WW , pˆW,WW ) and the restricted price chain contains only one price,ps, then, by the definition of the restricted price chain, p+s and p−s must bothbe such that the trader at those prices waits. Thus, the trader at ps has a109A.3. Omitted Proofsbenefit of BW,WWx (pt) > 0 and must wait.962c, part (iii). When a restricted price chain contains more than oneprice and its associated unrestricted price chain does not pass through(1− pˆW,WW , pˆW,WW ), multiple possible timing strategies can be sustained inequilibrium. Note that only the two traders nearest the ends of the restrictedprice chain have neighbors that lie outside it and these neighbors wait. Con-sider first case A in which all traders facing prices in the restricted price chainwait. Then, each trader in the price chain has a benefit of BW,WWx (pt) > 0so that waiting is a best response and thus an equilibrium.I next show that if any trader in the restricted price chain conditionallyrushes and all use pure strategies, then they must all do so. This fact simul-taneously proves that all of the traders conditionally rushing (case B) canbe sustained in equilibrium and that no other combination of pure strategiesis possible.First, note that any trader in (1− pˆW,RW , 1− pˆW,WW ]∪ [pˆW,WW , pˆW,RW )must conditionally rush if either of her neighbors in the restricted price chainconditionally rushes because she then faces either BW,RWx (pt) or BW,WRx (pt),both of which are negative in this price range. Second, any trader in[pˆW,RW , pˆW,WR) must follow the same timing strategy as her neighbor atthe next lowest price because if this neighbor conditionally rushes, she faceseither BW,WRx (pt) or BW,RRx (pt) which are both negative and, if this neighborwaits, she faces either BW,RWx (pt) or BW,WWx (pt) which are both positive.Similarly, any trader in (1− pˆW,WR, 1− pˆW,RW ] must follow the same timingstrategy as her neighbor at the next highest price. Therefore, it is immediatethat if any trader in (1− pˆW,RW , 1− pˆW,WW ]∪ [pˆW,WW , pˆW,RW ) conditionallyrushes, then all traders in the price chain must conditionally rush and thatthis set of timing strategies can be sustained in equilibrium. The remainingpossibility is that a trader in either (1−pˆW,WR, 1−pˆW,RW ] or [pˆW,RW , pˆW,WR)conditionally rushes but some other trader in the restricted price chain waits.I now show this is impossible. Assume that the trader that conditionallyrushes is in [pˆW,RW , pˆW,WR). Then, by the second property above, all tradersin the restricted price chain at higher prices must conditionally rush too, sothe only remaining possibility is that a trader at a lower price in the re-stricted price chain waits. But, by the second property, her neighbor at the96The conditions for this part of the theorem are only satisfied if (pˆW,WW )− < 1− pˆW,WR.In this case, it can be shown that no non-empty restricted price chain has an associatedunrestricted price chain with a price contained in (1− pˆW,WW , pˆW,WW ). Therefore, thereis no parameterization for which 2c, parts (i) and (ii) both apply.110A.3. Omitted Proofsnext lowest price must be conditionally rushing, otherwise she herself wouldwait. This argument can be repeated until the neighbor at the next lowestprice is at a price lower than pˆW,RW . This neighbor must be conditionallyrushing. If this neighbor lies within (1−pˆW,RW , 1−pˆW,WW ]∪[pˆW,WW , pˆW,RW )then again, all traders must be conditionally rushing. If this neighbor is in-stead at a price less than 1 − pˆW,RW , then all traders less than 1 − pˆW,RWmust conditionally rush because they must all follow the same timing strat-egy as their neighbor at the next highest price. Finally, if the neighborat the next lowest price is outside the restricted price chain, she can’t beconditionally rushing which creates a contradiction. In all cases then, alltraders in the restricted price chain must be conditionally rushing if anytrader in [pˆW,RW , pˆW,WR) conditionally rushes. A symmetric argument es-tablishes that if any trader in (1− pˆW,WR, 1− pˆW,RW ] conditionally rushes,then all must do so.To demonstrate the possibility of mixed timing strategies in equilibrium(case C), consider the simplest case of only two prices in the restricted pricechain, p1 < 12 and p2 >12 . The trader facing p1 mixes between conditionallyrushing and waiting such that the trader at p2 has a benefit which is a linearcombination of BW,WRx (p2) < 0 and BW,WWx (p2) > 0. The trader facing p2mixes such that the trader at p1 has a benefit which is a linear combinationof BW,RWx (p1) < 0 and BW,WWx (p1) > 0. It is easy to see that such mixingprobabilities can always be found and therefore, there exists an equilibriumin which traders at both prices mix.2d. For pt ∈ (0, 1− pˆW,RR] ∪ [pˆW,RR, 1), no matter what t− 1 and t+ 1do, t has a dominant strategy to wait because all of the benefit functions arestrictly positive. Thus, the only equilibrium timing strategy in this range isfor all traders to wait (mixing is again precluded). Also note that if pˆW,RRdoes not exist, then all benefit functions are strictly positive so that theunique equilibrium consists of all traders waiting at every price.Next, consider pt ∈ [pˆW,WR, pˆW,RR). For prices in this range, when t− 1waits, t’s best response is to rush if t + 1 rushes after both a buy and sellby t− 1 and to wait otherwise. Thus, knowing that t+ 1 waits if t− 1 buysis sufficient to ensure t’s best response is to wait. Mixing is also precludedin this case because even if t+ 1 mixes after a sell decision, t faces a linearcombination of BW,WRx (pt) and BW,WWx (pt) which is always strictly positivein this price range, so t won’t mix. Now, note that when t − 1 buys, theprice increases so that, for some pt ∈ [pˆW,WR, pˆW,RR), pt+1 exceeds pˆW,RR sothat t + 1 waits, as already established. Thus, the trader at this pt waits.The following algorithm extends this reasoning to prove that the uniqueequilibrium timing strategy for pt ≥ pˆW,WR is to always wait.111A.3. Omitted Proofs1. Define pˆi such that pˆ+i = pˆi−1 for i = 1 . . . k where k ≥ 1 is the smallestinteger for which pˆk < pˆW,WR. Also, define pˆ0 = pˆW,RR.2. If k = 1, jump to step 4. Otherwise, when i = 1, we have pt ≥ pˆ1 issuch that p+t ≥ pˆ0 = pˆW,RR so that t + 1 waits after a buy by t − 1.Therefore, t must wait in equilibrium ∀pt ∈ [pˆ1, pˆ0].3. Repeating step 2 for i = 2 . . . k − 1, at each step we establish thatwaiting is the unique equilibrium ∀pt ∈ [pˆi, pˆi−1] because for pt ≥ pˆiwe know from the previous step that t + 1 waits when t − 1 buys(p+t ≥ pˆi−1).4. At step k, for pt ≥ pˆW,WR, t must wait because p+t ≥ pˆk−1 and stepk − 1 established t + 1 waits for pt+1 ≥ pˆk−1. Thus, after k steps,we have established that each trader facing pt ∈ [pˆW,WR, pˆW,RR) mustwait.The fact that k is finite follows from the fact a finite number of price increasesare sufficient to move the price from pˆW,WR to pˆW,RR. For pt < pˆW,WR,the argument no longer goes through because knowing that t + 1 waitsafter a buy by t − 1 is not sufficient for t to wait. A symmetric argumentfor pt ∈ (1 − pˆW,RR, 1 − pˆW,WR] proves that the unique equilibrium timingstrategy in this price range is also for t to wait. Here, a sufficient conditionfor t to wait is to know that t+1 will wait after a sell decision by t−1 becauseBW,RWx (pt) > 0 for pt < 1 − pˆW,WR. Finally, note that if pˆW,WR does notexist, the argument above can be applied to all pt ∈ (1 − pˆW,RR, pˆW,RR) sothat the timing strategy in any equilibrium must be to wait over this range.Proof of Proposition 2.3:The proof consists of three parts. In the first part, I establish that, for allq ∈ (12 , 1), there exists a ql ∈ (12 , q) such that the unique equilibrium involvestraders waiting at all prices. In the second part, I show that there exists aqh∈ (12 , q) such that at least some traders rush for any p1 ∈ (12 , 1). Finally,using these facts, I show that there must exist a p˜ > p1 such that prices takestrictly longer to converge in expectation to p˜ under qhthan under ql.Fix q to be any value in (12 , 1). By Theorem 2.1, part 2d, when theparameters are such that pˆW,WR does not exist, then all traders must waitin any equilibrium. Given the shape of BW,WRx (pt) as determined by LemmaA.3.1, pˆW,WR does not exist if BW,WRx (12) > 0. So, it suffices to show that forany q we can find a value of q such that BW,WRx (12) > 0. From the formulafor BW,WRx in (A.1), one can easily show that BW,WRx (12) > 0 at q =12 forany q ∈ (12 , 1). Also, Lemma A.3.1 established that BW,WRx (pt) < 0 at q = q112A.3. Omitted Proofsfor any pt ∈ (0, 1) which includes pt = 12 . Therefore, because BW,WRx (12) iscontinuous in q, there must exist a ql∈ (12 , q) such that BW,WRx (ql) > 0.A sufficient condition for at least one trader to rush is BR,RRx (p1) < 0so that the first trader rushes in any equilibrium. From Lemma A.3.1,BR,RRx (pt) < 0 at q = q for any pt ∈ (0, 1), including pt = p1. Therefore,because BR,RRx (p1) is continuous in q, there must exist a qh ∈ (12 , q) such thatBR,RRx (p1) < 0. Furthermore, qh must be strictly greater than ql identifiedin the previous step. To see this, we have BR,RRx (12) > BW,WRx (12) > 0when the benefits are evaluated at qland the first inequality follows fromProposition 2.1. We also have BR,RRx (12) < 0 when the benefit is evaluatedat qh. But, BR,RRx (12) is monotonically decreasing in q so qh must be strictlygreater than ql:BR,RRx (12) =12[4(2q − 1)q(1− q)− (2q − 1)]=⇒∂BR,RRx∂q(12) =12[4(2q − 1)(1− 2q)− 2]< 0where I have used q < q in evaluating the benefit function and its derivative.At the value of ql, we have established that all traders wait and thereforeconvergence is as in the benchmark model. At the value of qh, (at least) thefirst trader rushes, implying that the expected value of the price after thefirst trade is strictly less than the in the benchmark case because it is easilyshown that E[pt+1|pt, V = 1] is increasing in q. Therefore, if the number oftrades were equal after some amount of time, the expected price would belower with qhthan with ql. However, due to the first trader rushing, thereis initially one more trade with qhthan with qland so the expected price isinitially higher with qh. But, if we set p˜ > p1 sufficiently close to 1, tradersmust begin to wait before reaching p˜ because Lemma 4 established that, forq < q , at sufficiently high prices, all traders wait. Once traders wait, thenumber of trades becomes equal under qland qhfor any realized signals.Thus, there is no price path reaching p˜ for which the number of trades ishigher under qhand so the effect of any trades being earlier disappears. Butthen convergence to p˜ must be slower in expectation under qhthan ql. Proof of Proposition 2.4:I prove the proposition for the case of st = 1. The other case is provensimilarly. With only a single trader in the market, her benefit can be written(set aˆ0 = aˆ1 = 1 in the general form of the benefit function):113A.3. Omitted ProofsB1(pt) =pt(1− pt)(2q − 1)(1−µ4 )Pr(st = 1)[1Pr(at+1 = B)−1Pr(at = B)]The term outside the brackets is always strictly positive, so the bracketedterm determines the sign of the benefit. Cross-multiplying the terms insidethe brackets, we see that that sign is determined byPr(at = B)− Pr(at+1 = B)⇐⇒ pt(1− µ4+ µq(1− β1))+ (1− pt)(1− µ4+ µ(1− q)(1− β1))−pt(1− µ4+ µq2β1)+ (1− pt)(1− µ4+ µ(1− q)2β1)⇐⇒ µ((ptq + (1− pt)(1− q))− β1(ptq2 + (1− pt)(1− q)2 + ptq + (1− pt)(1− q))From the above expression, we see that the benefit is strictly positive whenβ1 = 0 and strictly negative when β1 = 1 so that neither corner may bean equilibrium. The unique mixed strategy equilibrium is then obtained bysetting the expression to zero, which corresponds to the strategy given inProposition 2.4. Proof of Corollary 2.1:The proof of Proposition 2.1 in the case of the zero-profit model isidentical to that in the expected value model except that, in applying themathematical claim, NTx = Pr(at = NT |V = x) is replaced with eitherPr(at+1 = B|V = x) or Pr(at+1 = S|V = x), depending on which type oftrader’s benefit is being considered.The proof of Proposition 2.2 in the case of the zero-profit model is iden-tical to that in the expected value model. Proof of Proposition 2.5:By the application of Proposition 2.1 to the zero-profit model, if thereis any possibility of either type of trader t − 1 or t + 1 trading betweent and t + 1, then each type of t’s benefit must be strictly less than whenalone in the market, Bx(pt, βx) < BSTx (pt, βx) ∀βx ∈ [0, 1] and ∀pt. But,given that Bx(pt) is strictly decreasing in βx, this implies that t must bewaiting with βx(pt, I(at−1 = NT ) < βSTx (pt) in equilibrium. The inequalityis strict because βSTx (pt) is interior for all pt. It then follows immediatelythat βx(pt, 1) < βSTx (pt) for both types of t because, in this case t − 1 haswaited and will potentially trade between t and t+ 1. Therefore, it remainsonly to show that βx(pt, 0) = βSTx (pt) is impossible for either type of t and,114A.4. Formulas for the Zero-Profit Modelby the above argument, it is only possible if both types of t+ 1 always wait.But, if we assume βx(pt, 0) = βSTx (pt) for either type of t and both types oft + 1 always wait, then we arrive at a contradiction: the type of t that isusing βx(pt, 0) = βSTx (pt) is waiting with strictly positive probability and,when she does, each type of t+ 1 must rush with positive probability by thesame argument as above, so both cannot be always waiting. A.4 Formulas for the Zero-Profit ModelThe unique bid and ask prices for delayed trades are:pAt+1 = Pr(V = 1|Ht, at+1 = B) =pt( 1−µ4 + µq2β1)pt( 1−µ4 + µq2β1)+ (1− pt)( 1−µ4 + µ(1− q)2β1)pBt+1 = Pr(V = 1|Ht, at+1 = S) =pt( 1−µ4 + µ(1− q)2β0)pt( 1−µ4 + µ(1− q)2β0)+ (1− pt)( 1−µ4 + µq2β0)For rushed trades, the prices are:pAt = Pr(V = 1|Ht, at = B) =pt( 1−µ4 + µq(1− β1))pt( 1−µ4 + µq(1− β1))+ (1− pt)( 1−µ4 + µ(1− q)(1− β1))pBt = Pr(V = 1|Ht, at = S) =pt( 1−µ4 + µ(1− q)(1− β0))pt( 1−µ4 + µ(1− q)(1− β0))+ (1− pt)( 1−µ4 + µq(1− β0))The benefit functions for generic actions of the other traders are given by:B1(pt, β1) =pt(1− pt)(2q − 1)(1−µ4 )Pr(st = 1)[∑aˆ∈Aaˆ0aˆ1Pr(aˆ&at+1 = B)−1Pr(at = B)]B0(pt, β0) =pt(1− pt)(2q − 1)(1−µ4 )Pr(st = 0)[∑aˆ∈Aaˆ0aˆ1Pr(aˆ&at+1 = S)−1Pr(at = S)]The timing strategies of the informed traders, β0 and β1, enter the benefitfunctions through the probabilities of observing a buy and sell decision at115A.5. Multiple Arrival (n > 1)each point in time:Pr(at = B) = pt(1− µ4+ µq(1− β1))+ (1− pt)(1− µ4+ µ(1− q)(1− β1))Pr(at = S) = pt(1− µ4+ µ(1− q)(1− β0))+ (1− pt)(1− µ4+ µq(1− β0))Pr(aˆ&at+1 = B) = ptaˆ1(1− µ4+ µq2β1)+ (1− pt)aˆ0(1− µ4+ µ(1− q)2β1)Pr(aˆ&at+1 = S) = ptaˆ1(1− µ4+ µ(1− q)2β0)+ (1− pt)aˆ0(1− µ4+ µq2β0)A.5 Multiple Arrival (n > 1)In this section, I extend the expected value model to one in which ntraders arrive in each period. Each trader receives an independent signal inthe period they arrive and in the next period if they choose to wait. Withmultiple traders arriving each period, waiting costs are increased such thattraders may rush to trade with even greater differences in signal quality. Thisextension is used in the numerical simulations of Section 2.4 and to derivethe comparative static prediction with respect to volume of Section 2.6. Afull equilibrium characterization is difficult to obtain and I do not attemptto derive one here. Instead, I rely on a combination of theoretical resultsand numerical analysis to determine an upper bound on the average pricereached and a lower bound on the probability of increasing mispricing asfunctions of time. Because numerical analysis is used, I focus on the specificcase of n = 4, q = 0.7 and q = 0.8 used in Section 2.4. However, most of theanalysis is general in nature and applies to a wide range of parameters.I look for a symmetric equilibrium in which traders with the same signalsfollow the same strategies. Because traders who trade at a time t do notaffect the price each other faces, the optimal trading strategies are identicalto the n = 1 case. Furthermore, the general form of the benefit function,(2.1) continues to apply because it was derived for any possible combinationof trades while one waits. Therefore, analogous statements to Lemma 2.2and Propositions 2.1 and 2.2 can be easily derived. In particular, Lemma2.2 implies that both types of traders must follow the same timing strategyso that the simplified benefit function of Appendix A.1 applies. With theaddition of the symmetry assumption, we then have that all traders arrivingin the same period must follow the same timing strategy.The main difficulty in obtaining a complete characterization of the equi-librium for n > 1 is that when the t−1 traders wait, there are many possible116A.5. Multiple Arrival (n > 1)prices that the traders at t+1 may face and thus many possible benefit func-tions a trader at t may face. Thus, the approach used for the n = 1 case,looking at each of the price ranges determined by the zero-crossings of eachof the benefit functions, becomes extremely tedious. My approach is to in-stead characterize the equilibrium within a specific price range and assumeall traders wait outside this range. I can then simulate an upper bound onthe average price reached and a lower bound on the probability of wideningmispricing as argued in Section 2.4.More specifically, for the parameters given, I claim that there existsa price, pˆR,Rn , such that any equilibrium involves all traders rushing whenpt ∈ (1− pˆR,Rn , pˆR,Rn ). In particular, 1− pˆR,Rn and pˆR,Rn are the prices at whichthe benefit to waiting when all t−1 and t+1 traders rush, BR,Rn (pt), is zero.The analytic formula for this benefit is easily obtained using the simplifiedgeneral form of the benefit in Appendix A.1. However, establishing theproperties corresponding to those of Lemma A.3.1 is more difficult so I relyon numerical simulation to show that it in fact crosses zero at exactly twoprices which are symmetric around p = 12 , 1− pˆR,Rn and pˆR,Rn , where pˆR,Rn ≈0.866. The benefit function is positive near p = 0, 1 and negative otherwise.The benefit function when all t − 1 and t + 1 traders wait, BW,Wn (pt) alsoplays a role. Because q > q, the price impact of a t−1 trader is greater thanthat of a t trader, so BW,Wn (pt) < BR,Rn (pt) < 0 over pt ∈ (1− pˆR,Rn , pˆR,Rn ).97To establish the claim, I show that there exists an equilibrium in which alltraders rush, independent of the number of t − 1 traders that wait (withn traders, the number of t − 1 traders observed to have waited is payoff-relevant) and that there does not exist an equilibrium in which all traderswait on the equilibrium path. The proof is as follows.Consider first the claim that there does not exist an equilibrium in whichall traders wait on the equilibrium path. If all t − 1 traders rush and thetraders at t were to all wait at some pt ∈ (1− pˆR,Rn , pˆR,Rn ), then the traders att+ 1 face the same price with a benefit of BW,Wn (pt) < 0 or less (dependingon the strategies of the t+2 traders: by Proposition 2.1, if any t+2 trader isrushing, t+1’s benefit would be reduced from BW,Wn (pt)) and so each wouldhave an incentive to deviate to rush, a contradiction. Alternatively, if all t−1traders wait, then all t traders have a benefit of BW,Wn (pt) < 0 or less, andso would rush, a contradiction. Thus, the only possible symmetric equilibriaare those in which all traders rush on the equilibrium path. So, consider the97It can be generally shown that one’s benefit is strictly decreasing in the signal qualityof any trade that occurs while one waits (proof available upon request). This fact ensuresthat a trade by t− 1 impacts t more than a trade by t+ 1 when q > q.117A.5. Multiple Arrival (n > 1)strategy profile in which all traders rush regardless of the number of t − 1traders that wait. One must show that no single trader has an incentive todeviate to wait under this strategy profile. There are two possible cases.In the first case, all t− 1 traders were observed to rush. In this case, if asingle trader at t deviates to wait, the t+1 traders face pt ∈ (1− pˆR,Rn , pˆR,Rn )and therefore rush under the specified strategy profile. Thus, the trader at tthat is considering deviating faces the price impacts of 2n−1 traders. Giventhat the price impact of n traders gives her a benefit of BR,Rn (pt) < 0 andthat Proposition 2.1 ensures additional traders reduce her benefit further,she will not deviate.In the second case, one or more t− 1 traders were observed to wait. Inthis case, if t deviates, nothing can be said about the timing decisions ofthe t+ 1 traders because the trades by the t− 1 traders may drive the priceoutside of pt ∈ (1 − pˆR,Rn , pˆR,Rn ). However, t still faces the price impacts ofthe n − 1 other t traders as well as those of the one or more t − 1 traders.Because q > q, the price impact of a t− 1 trader is greater than that of a ttrader, so t’s benefit from waiting is strictly less than BR,Rn (pt) < 0, so shewill not deviate.In summary, there exists an equilibrium in which all traders rush andthere does not exist an equilibrium in which all traders wait for all pt ∈(1 − pˆR,Rn , pˆR,Rn ). Having established these facts, I can then simulate themodel under the additional assumption that all traders wait outside pt ∈ (1−pˆR,Rn , pˆR,Rn ). This simulation provides an upper bound for the average pricereached and a lower bound for the probability of increases in mispricing. Ifin fact traders rush more often than assumed, the average price would in factbe lower and the probability of increases in mispricing higher. Simulationresults are presented in Section 2.4.Finally, the analysis for n > 1 justifies the claim in Section 2.6 thathigher volume leads to more panics and therefore less informational contentin trades. Proposition 2.1 guarantees that an increase in n decreases thebenefit to waiting and thus makes the interval over which traders mustpanic, (1− pˆR,Rn , pˆR,Rn ), larger. It is in this sense that an increase in volumeleads to more panics. As an example, when q = 0.7 and q = 0.8, going fromn = 1 to n = 4 changes the interval from not existing to about 73% of theentire price range.118Appendix BAppendix to Chapter 3B.1 Analysis DetailsThe equilibrium characterization for the Basic model is derived in Chapter 2.Here, I comment on the specific results for each of the two parameterizations.As in Chapter 2, define the probability of waiting for a trader with signalst = x at price pt who has observed whether or not t − 1 traded or notas βx(pt, I(at−1 = NT )), where I is the indicator function. To determineoptimal timing decisions, I look at the net benefit to waiting (expected profit- current profit) for a trader with st = x, Bx(pt). One optimally rushes ifthis benefit is negative, and waits otherwise.Proposition 3.1 is a simple restatement of Lemma 2.1 after applyingβ1(pt, I(at−1 = NT )) = β0(pt, I(at−1 = NT )) from Lemma 2.2.Proposition 3.2 part a) is almost a direct application of Theorem 2.1 forthe case of q < q except that here T is finite. Finite T affects only the finaltrader who faces no potential trade from a subsequent trader if she waits.Because of this fact, if trader T − 1 rushes, rather than T rushing becausethe next trader will rush if she waits, trader T is indifferent between waitingand rushing. Thus, a multiplicity of equilibria exist, differing only in thebehavior of T . In the experimental results, I ignore the timing behavior ofT when T − 1 rushes for this reason.For treatment BW, Theorem 2.1 again applies, but it remains to bedetermined which case is relevant. To establish Proposition 3.2 part b), Iproceed to establish that t’s net benefit to waiting is positive at all pt, nomatter what the strategies of the other traders are. From Proposition 2.1, ifwe know that the benefit is positive when t− 1 waits and t+ 1 rushes, thenwe know that it must be positive for all combinations of strategies becausethey result in less potential trades while t waits. Thus, consider the benefitfor this set of strategies of the other traders, BW,RRx (pt). Its formula fromAppendix A.1 is119B.2. Rational Herding and the τ -Herding HeuristicBW,RRx (pt) = Sx[(2q − 1)QQ(1Pr(1, 1)+1Pr(1, 0)+1Pr(0, 1)+1Pr(0, 0))− (2q − 1)](B.1)where Sx ≡pt(1−pt)Pr(st=x), Q ≡ q(1 − q),Pr(x, y) ≡ Pr(st = x&st+1 = y),and I have used the fact that q > q for the BW treatment. Lemma A.3.1establishes that BW,RRx (pt) crosses zero at at most two places which mustbe symmetric about p = 12 . And, Proposition 2.2 establishes that, as pricesapproach 0 and 1 BW,RRx (pt) must approach that of a trade alone in themarket, who in this case has a positive benefit from waiting given q > q.Together these facts imply that if BW,RRx (12) > 0, then BW,RRx (pt) > 0 forall pt ∈ (0, 1) which means all traders must wait regardless of the strategiesof other traders or price they face. Numerically evaluating (B.1) at pt = 12 ,q = 1324 , and q =1724 gives BW,RRx (12) ≈ 0.130 > 0. Thus, Proposition 3.2 partb) is established.B.2 Rational Herding and the τ-HerdingHeuristicThe informational herding environment of Banerjee (1992) and Bikhchan-dani et al. (1992) has been well-studied in experimental economics.98 In thisenvironment, subjects sequentially take one of two actions after observinga private signal and the past actions of previous subjects. One can con-sider this environment a version of a sequential trading environment withexogenous timing in which prices are fixed: the cost of taking an action isindependent of the previous actions taken (and is usually set to zero). Ra-tional herding, in which one follows one’s predecessors instead of one’s signalis predicted to occur. Intuitively, after two previous subjects take the sameaction, their information outweighs one’s own, so it becomes rational to fol-low their actions. At this point, an informational cascade develops in whichall subsequent subjects take the same action as the first two. In experimen-tal evidence, rational herding occurs, but not as strongly as predicted. Thetwo robust findings are that rational herds are frequently broken by subjectsthat instead follow their own signal, and that this “breaking” of a cascade98Anderson and Holt (1997) provide the first study. See also Ku¨bler and Weizsa¨cker(2004) and Goeree et al. (2007).120B.2. Rational Herding and the τ -Herding Heuristicdecreases with the number of previous subjects that have taken the sameaction.Goeree et al. (2007) show that quantal response equilibrium, in whichsubjects understand that previous subjects may have made mistakes whenthey chose their action, can explain both the fact that cascades break andthat they break less frequently as the length of the cascade increases. Ku¨blerand Weizsa¨cker (2004) have a similar explanation based on a limited depthof reasoning model. However, while these explanations explain behavior inthe fixed price environment, the discussion in Section 4.5 shows that be-lieving past trader’s made mistakes cannot explain behavior observed in thelaboratory data obtained here. Cipriani and Guarino (2005) and Drehmannet al. (2005) also show that these explanations are inconsistent with herdingin their sequential trading environments.Having shown that the τ -herding heuristic can explain both the behaviorin the environment considered here and in the sequential trading environ-ment with endogenous prices, one wonders whether it can also explain be-havior in the fixed-price environments. Under the assumption that subjectsunderstand that other subjects may be τ -herding types, I show here that theanswer is yes. The discussion is informal, but suffices to show the reasoningby which the heuristic can explain the observed behavior.Consider the standard informational herding environment in which thereare only two actions, A and B, each of which is equally likely ex ante.Subjects receive private binary signals which indicate the true state (A orB) with some probability, typically 0.6−0.75. If a subject chooses an actioncorresponding to the true state, she receives a payoff of 1, otherwise shereceives 0. Subjects observe all past actions. Now consider the behavior thatwould be observed if the majority of subjects follow the τ -herding heuristicand understand that others do so. That is, subjects have threshold beliefsthat the true value is A above (below) which they will choose action A (B)regardless of their private signal. For less extreme beliefs, they follow theirsignal.The first subject simply follows her signal, regardless of her thresholdbelief because this is the only information she has. Without loss of generality,say the first subject took action A. The second subject will take action Aregardless of her threshold belief if her private signal indicates A. If, instead,her signal indicates B, she is indifferent. The typical assumption in this caseis that she follows her own signal. If the third subject observes the pasttwo subjects took action A, rationally she should choose A regardless of her121B.3. Learningprivate signal, since her belief will indicate A is more likely.99 However, if shefollows the τ -herding heuristic (which is a misnomer in this environment!),she will choose action B if her private signal indicates B and her thresholdbelief, τ , is greater than b∗, where b∗ is the belief corresponding to twosignals indicating A and one indicating B. Therefore, if some subjects areτ -herding types, we obtain the first experimental finding that informationalcascades may break.Now consider the belief of a fourth subject that observes three A choicesin a row. If she believes that the third subject uses the heuristic, then sheplaces a higher belief on state A than after observing only two A actions,because she realizes that the third subject would have chosen B if her thresh-old belief is above b∗ and she observed a signal indicating B. If the fourthsubject also uses the heuristic, she will take action A if her signal indicatesA, but she may take action B if her signal indicates B and her belief does notexceed her own threshold belief. Importantly, she is less likely to take actionB than the third subject because her belief that the state is A is higher thanthat of the third subject. Therefore, we obtain the prediction that, as thecascade lengthens, a subject is less likely to break the cascade, which is thesecond robust experimental finding.While the above analysis is admittedly informal, it demonstrates how amore formal analysis that makes concrete assumptions about the thresholdbeliefs of subjects, and their beliefs about the threshold beliefs of othersubjects, can generate behavior consistent with both experimental findingsin the fixed price environment. In particular, as long as some subjects use theheuristic and also believe that some other subjects use the heuristic and havethreshold beliefs above b∗, the argument holds. In the experimental datahere, evidence of considerable heterogeneity in threshold beliefs exists (seeSection 4.4) and, in particular, some subjects use sufficiently high thresholdbeliefs for the above argument to hold. Therefore, the τ -herding heuristicis capable of simultaneously explaining behavior across three related sociallearning environments, something which none of the other existing theoriesis capable of.122B.3. LearningFigure B.1: Proportion of Rational Behavior as Trials Progress (Basic)1−7 8−14 15−21 22−28 29−35 36−420.40.450.50.550.60.650.70.750.8TrialsProportion of Rational Timing Decisions  Basic Rush (BR)Basic Wait (BW)B.3 LearningB.1 plots the evolution of perfectly rational behavior in each treatment overthe full course of each session. There is considerable evidence of learning inthe BW treatment, where subjects learn that an additional signal is valuable.In the BR treatment, however, learning is difficult because waiting is onlycostly if either the previous trader waited or the subsequent trader rushesand the trade(s) move prices adversely. If one’s predecessors mostly rush,it may take time to learn that prices on average move adversely. Learningmay have perhaps been easier had subjects maintained the same positionin the sequence in treatments BR and BW, but the concern in this case isthat subjects would be learning about particular subjects rather than theirenvironments.99For concreteness, I consider only the case in which all of one’s predecessors took thesame action. However, the implications can be generalized to other cases.123B.4. Non-equilibrium Informational LossesB.4 Non-equilibrium Informational LossesNon-equilibrium behavior exacerbates the informational losses predicted byrational theory in both treatments. I measure informational losses by com-paring final prices in each trial to actual asset values. B.2 plots the empiricalcumulative distribution function (cdf) of the absolute value of the differencein these two values for each of the treatments. In each case, the differencebetween the asset value and the theoretical final price, had all traders actedrationally, is compared to the difference between the asset value and theactual final price in the data. In the treatment in which rushing is optimal,BR, we see that the theoretical and actual losses are very similar: any dif-ference is due solely to trades not revealing private information. Therefore,when panicking is rational, almost all information losses are due to rationalbehavior. Importantly, non-equilibrium waiting does not cause additionalinformation to be aggregated because waiting to obtain an additional signaldoes not reveal additional information.In treatment BW, in which waiting is optimal, we observe large infor-mational losses due to non-equilibrium behavior. Here, traders determineall information aggregation, and we observe the large losses due to non-equilibrium behavior. On average, the observed final price differs from thetheoretical by 9.6%. The average difference in the BR treatment is 3.5%.B.5 InstructionsThe instructions for the BW treatment, are provided. The instructions forthe BR treatment are identical except for the differences in parameters.124B.5. InstructionsFigure B.2: CDFs of Difference Between Final Prices and Asset Values byTreatment (Basic)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9|Final Price − Asset Value|CDFBasic Rush (BR)  ActualTheoretical0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9|Final Price − Asset Value|CDFBasic Wait (BW)  ActualTheoretical125Contents ofbin if value =100Contents ofbin if value =0First Clue13 blue11 green11 blue13 greenSecond Clue17 blue7 green7 blue17 greenInstructions7 his is a research e[ S eriment designed to understand hoZ  S eoS le make stock trading decisions in a simS le trading environment 7 o assist Z ith ourresearch  Z e Z ould greatly aS S reciate your full attention during the e[ S eriment Please do not communicate Z ith other S articiS ants in any Z ay andS lease raise your hand if you have a T uestion<ou Z ill S articiS ate in a series of  trials In each trial  you and  other S eoS le Z ill each trade a stock Z ith the comS uter 7 he trades areseT uenced so that each S erson arrives to the market one at a time <ou Z ill be S aid 00 for comS leting all trials In addition  in each trial you Z illearn lottery tickets as described ne[ t 7 he more lottery tickets you have  the more you may earn%efore each trial  the comS uter Z ill randomly select Z hether the stocks value  V  is 100 tickets or 0 tickets  Z ith eT ual S robability :hen it is yourturn  you Z ill be given 100 tickets to begin and then you may choose either to buy or sell the stock <ou Z ill only trade once If you buy the stockyou Z ill gain its actual value minus the S rice  P  you S ay for it If you Z ant to sell the stock  you must first borroZ  it from the comS uter and laterS ay back its actual value So  if you sell  you Z ill gain the S rice you receive minus its value In summary  your total S rofit is100+V-P lottery tickets if you buy100+P-V lottery tickets if you sellSo  for e[ amS le  if you sell the stock at a S rice of 0 and it turns out to be Z orth 100  you Z ould earn 100+0-100=0 tickets for that trial %ut ifit turns out to be Z orth 0  you Z ould earn 100+0-0=10 tickets7 o helS  you guess the value of the stock the comS uter chose  you Z ill get a First Clue Z hen it is your turn SS ecifically  there Z ill be tZ o S ossiblebins one that the comS uter Z ill use if the value is 100 and another that the comS uter Z ill use if the value is 0 %ins contain some number of blueand green marbles as shoZ n beloZ  7 he comS uter Z ill draZ  a marble randomly from the bin and shoZ  it to you ( ach marble in the bin is eT uallylikely to be draZ n <ou can use this clue to get a better idea of Z hat the stocks value is$fter observing your First Clue  you can choose to buy or sell the stock immediately $lternatively  you can choose to Z ait and trade in the folloZ ingS eriod If you choose to Z ait  you Z ill get a Second Clue the comS uter Z ill draZ  a marble from another bin 7 he bin used Z ill again deS end on thestocks true value but the S ossible bins used for the Second Clue are different from the bins used for the First Clue as shoZ n beloZ  $fter observingthe Second Clue  you may choose to buy or sell the stock 1ote that you must trade in either the S eriod you arrive or the ne[ t S eriod and that youcan only trade onceImS ortantly  if you decide to Z ait  other S articiS ants Z ill have a chance to trade before you have your second chance to trade SS ecifically  if theS articiS ant before you Z aited  they Z ill trade before you $nd  the ne[ t S articiS ant Z ill arrive to the market and have a chance to tradeimmediately or to Z ait  M ust like you If they trade immediately  they Z ill also trade before you 7 he timing of trades is shoZ n in the figuresbeloZ  ( ach oS en circle indicates a S ossible trade time and the arcs indicate the tZ o S ossible times at Z hich a S articular S articiS ant can trade 7 hered  highlighted arc corresS onds to the S articiS ant Z hose turn it is the second S articiS ant in this e[ amS le:hen you arrive in the market  you Z ill observe Z hether or not the S erson before you hasalready traded In the first e[ amS le to the right  the second S articiS ant has M ust arrived to themarket and the first S articiS ant has already traded they didnt Z ait 7 his is indicated by thearroZ  S ointing to the time at Z hich they traded $lso  the oS en circle contains a + sign for abuy or a - sign for a sell after a trade has comS leted In this e[ amS le  if the secondS articiS ant Z aits  there is only one S ossible trade by the third S articiS ant if they tradeimmediately before the second S articiS ant gets a chance to trade againIn the second e[ amS le to the right  Z hen the second S articiS ant arrived in the market  she saZthat the first S articiS ant has Z aited arroZ  to the right In this case  the oS en circle Z here thefirst S articiS ant arrived disaS S ears because no trade occurred $lso  a shaded circle aS S ears Z herethe first S articiS ant must trade So  if the second S articiS ant Z ere to Z ait  the first S articiS antZ ould trade for certain before the second S articiS ant has a chance to trade again $nd  the thirdS articiS ant Z ould also trade before the second  if they trade immediately$ll S ast trading decisions and S rices Z ill be disS layed in a figure as shoZ n beloZ  In this e[ amS le  the first S articiS ant Z aited and then bought thestock 7 he second S articiS ant sold the stock immediately 7 he third S articiS ant has Z aited and it is currently the fourth S articiS ants turn 7 heS rice history is disS layed in the graS h along Z ith the current S rice at Z hich you can trade 77 in ths e[ amS le 7 he current time S eriod isindicated by the dashed red line $fter choosing buy or sell or Z ait in the S eriod you arrive  you Z ill S ress the confirm button not shoZ n126 7 he S rice of the stock Z ill change over time based uS on the trades that occur 7 herefore  if you Z ait  and another S articiS ant trades  the S riceyou Z ill face Z hen you have a second chance to trade Z ill be different than the S rice you could have traded at if you traded immediately7 he initial S rice of the stock is 0 tickets  reflecting the fact that it is eT ually likely to be Z orth 100 tickets or 0 tickets Similarly  after a tradethe S rice of the stock Z ill be uS dated so that it continues to be eT ual to the stocks e[ S ected value given the information that can be inferred fromthe trades  but no other information 7 he stocks S rice Z ill therefore increase after a buy and decrease after a sell  and the change Z ill be smallerfor immediate trades than for trades that take S lace after Z aiting as you can see in the e[ amS le aboveSummary1 $t the beginning of each trial  the comS uter randomly selects the stocks value 0 or 100 7 he first S articiS ant is shoZ n their First Clue and then chooses to buy  sell  or Z ait3 If the first S articiS ant chooses to buy or sell  they are done for this trial If they choose to Z ait  the second S articiS ant arrives to the marketobserves their oZ n First Clue  and chooses to buy  sell  or Z ait If the first S articiS ant chose to Z ait  they Z ill then buy or sell after observing a Second Clue SteS s - are reS eated for all si[  S articiS ants ( ach S articiS ant observes their oZ n First Clue and Second Clue if they choose to Z ait from thesame bins as S revious S articiS ants 0arbles are alZ ays reS laced so later S articiS ants may see the same or different marbles $t each S oint in time  all S ast S rices and trading decisions are available to use to helS  guess the value of the stock in addition to the Clues$fter all S articiS ants have traded  the trial is comS lete 7 he true value of the stock Z ill then be revealed and you Z ill be told hoZ  many tickets youearned for the trial <ou Z ill S ress ne[ t trial to S articiS ate in the ne[ t trial It is imS ortant to remember that  in each trial  the value of the stockis indeS endently randomly selected by the comS uter -- there is no relationshiS  betZ een the value selected in one trial and another 7 he order ofS articiS ants in the seT uence may be different from trial to trial you may be S articiS ant 1 in one trial  S articiS ant  in another  and S articiS ant  inyet another :hich S articiS ant you are in each trial Z ill be told to you before you make your trade$fter all  trials are comS lete  one lottery Z ill be conducted for each trial For each lottery  a random number less than 00 Z ill be chosen by thecomS uter If the number is smaller than the number of lottery tickets you earned for the trial  you Z ill get 100 7 herefore  the more lotterytickets you earn in each trial  the more you can e[ S ect to make S artial tickets are S ossible and count too For e[ amS le  if you earn 100 tickets ineach trial  you can e[ S ect to make 01=100 over the  trials %ut  if you earn 10 tickets in each trial  you can e[ S ect to make01=0Please try to make each decision Z ithin 1 seconds so that the e[ S eriment can finish on time $ timer counts doZ n from 1 to helS  you keeS  trackof time Z hen it is your turn 1ote  hoZ ever  that if the timer hits zero  you can still enter your trading decision and Z ill still have the same chanceto earn money %efore beginning the S aid trials  Z e Z ill have tZ o S ractice trials for Z hich you Z ill not be S aid 7 hese trials are otherZ ise identicalto the S aid trials e[ ceS t that they are not timedQuizPlease ansZ er the folloZ ing T uestions 7 o ensure you understand the instructions  you must ansZ er all of the T uestions correctly before Z e beginthe e[ S eriment1 <ou are the first S articiS ant and have Z aited <our First Clue is agreen marble and your Second Clue is a blue marble %ased only onthis information  :hat is the most likely value of the stock" 100  0  <ou are the second S articiS ant and observe the first S articiS ant green 127sold the stock immediately :hat color marble are they most likely tohave seen" blue 3 <ou are the first S articiS ant and your First Clue is a blue marble:hat color marble is the second S articiS ant most likely to see" green  blue  If you choose to sell the stock at a S rice of 0 and its value turnsout to be 100  hoZ  many total tickets Z ould you get for that trial" 0  0  10  If you choose to buy the stock at a S rice of  and its value turnsout to be 100  hoZ  many total tickets Z ould you get for that trial"   7  17  7 he stocks current S rice is 0 :hich value of the stock is morelikely" 100  0 7 <ou are the second S articiS ant and you observe that the firstS articiS ant did not trade immediately + oZ  many trades can occurbefore you trade again if you Z ait" 0  1 or   0 or 1  1 2nce you have comS leted the T uiz  S lease S ress Check ansZ ersCheck answers128Appendix CAppendix to Chapter 4C.1 Analysis Details and Omitted ProofsIn this section, I describe the theoretical analysis of the Extended model,proving Proposition 4.2. The details of the analysis borrow heavily fromthe results in Chapter 2, but do not immediately follow from the analysisthere. Here, I focus on q > q. For q ≤ q, it is easily shown that the uniqueequilibrium is for traders to trade immediately for the same reason as in theBasic model: the additional information, si, is of no value because it neverchanges one’s trading decision.The solution concept is sequential equilibrium and I focus on Markovstrategies that depend only upon the payoff-relevant state. Because the priceis a sufficient statistic for all prior public information, a Markov strategydepends upon the price and not upon the specific realizations of past tradesor public signals. The number of traders that have traded in prior periods,n˜ ∈ 0, . . . , n − 1, is also payoff relevant because, should one wait to trade,it determines the number of potential trades that could occur before onetrades.I restrict attention to off-equilibrium beliefs such that no informationis revealed by the decision to wait to trade. This restriction is satisfiedwhen one’s timing decision does not depend upon one’s private information.As shown below, any equilibrium of the Basic model must involve timingstrategies which do not depend upon private information, so that it seemsnatural to assume that any deviation from the optimal timing strategy alsodoes not.A behavioral strategy for a trader in the Extended model is a mappingfrom her initial private signal, the number of other traders who have pre-viously traded, and the current price, to an action: buy, sell, or not trade.Without loss of generality, this strategy can be decomposed into a tradingstrategy (buy or sell) and the probability of postponing one’s trade (waiting).Denote the probability of waiting, βx(pt, n˜), where x denotes the realizationof the trader’s initial signal, si.129C.1. Analysis Details and Omitted ProofsConsider first the optimal trading strategy taking the timing strategiesas given. The proof of Lemma 2.1 can be reproduced exactly to show thatthe optimal trading strategy for a trade in any period, t < T , is to buy ifsi = 1 and sell if si = 0. In period T , when V is known, it is simple toshow that one must buy if V = 1 and sell if V = 0.100 These facts establishProposition 4.1.The next step in characterizing equilibrium strategies is derive the benefitto waiting to some future period and trading then. To do so, one must derivethe optimal trading strategy after waiting. One can follow the derivation inLemma 2.1 to determine whether buying or selling is optimal. As is the casethere, events that occur during the waiting period are irrelevant because theyaffect both the trader’s private belief and the price in the same way. But,the optimal trading strategy does depend upon trader i’s timing strategiesbecause, if β0(pt, n˜) 6= β1(pt, n˜), information is revealed by waiting and thusaffects the price. The end result is that, if waiting until a period in whichno further information is obtained, a trader with si = 1 buys and a traderwith si = 0 sells, for all possible timing strategies. If waiting until T wherean additional signal, si, is obtained, the optimal trading strategy is given byLemma 2.1, simply relabeling the timing strategies.Given the optimal trading strategies, one can derive a general form ofthe benefit to waiting identical to that of Chapter 2. This general form doesnot depend on the particular types of events that occur while one waits, andso is equally applicable for trades by other traders or public signal events.The general form of the benefit is:Bx(pt, β0, β1) =pt(1− pt)Pr(si = x)[∑aˆ∈Aaˆ0aˆ1f(q, q, β0, β1)Pr(aˆ&ai,t = NT )− (2q − 1)](C.1)wheref(q, q, β0, β1) =(2q − 1)(qNT0 + (1− q)NT1) if st = 1 &g1(q, q) ≤ 0qNT0 − (1− q)NT1 if st = 1 &g1(q, q) > 0qNT1 − (1− q)NT0 if st = 0 &g0(q, q) < 0(2q − 1)(qNT1 + (1− q)NT0) if st = 0 &g0(q, q) ≥ 0g0(q, q) ≡ (1− q)qNT0 − q(1− q)NT1 ≥ (<)0, g1(q, q) ≡ q(1− q)NT0 −100Because in a finite game prices are always strictly different from 0 and 1, one is neverindifferent.130C.1. Analysis Details and Omitted Proofs(1−q)qNT1 > (≤)0, NT0 = (1−q)β1(pt, n˜)+qβ0(pt, n˜), NT1 = qβ1(pt, n˜)+(1 − q)β0(pt, n˜), aˆy ≡ Pr(aˆ|V = y), and aˆ is a particular event in the setof possible events, A, that can occur while one waits. This benefit functionapplies only when waiting to the final period where si is obtained. Whenwaiting until any other period, g0(q, q) < 0 and g1(q, q) > 0, which pinsdown the formulas for f(q, q, β0, β1) that apply. Note that (C.1) implicitlydepends on n˜ by constraining the set of events, A, that one must sum over.Because the form of the benefit function is identical to that in the Ba-sic model, Lemma 2.2 also immediately applies, ensuring that traders witheither realization of si must follow the same timing strategy in any equilib-rium, β0(pt, n˜) = β1(pt, n˜). To determine the benefit when one is consideringa deviation to waiting (as opposed to waiting being an equilibrium strategy),the off-equilibrium beliefs of other traders and the expected price matter,so some assumption must be made as to what these are. I assume thatbeliefs about si after such a deviation are such that either signal is equallylikely. This assumption seems natural since it implies traders with eithersignal follow the same timing strategy both in and out of equilibrium. Con-sistent with these beliefs, I also assume prices are unchanged after observinga timing deviation.Under these assumptions, the benefit function can be simplified by set-ting β0(pt, n˜) = β1(pt, n˜) in (C.1):Bx(pt) =pt(1− pt)Pr(si = x)[∑aˆ∈Aaˆ0aˆ1(2q − 1)Pr(aˆ)− (2q − 1)](C.2)where q = q if waiting to a period with no new private informationand q = q if waiting until T when si arrives. Note that the benefit nolonger depends on the timing strategies of trader i. Propositions 2.1 and2.2 also immediately apply given that the benefit function is identical. Animmediate consequence of Proposition 2.1 is that waiting to trade in a futureperiod in which no new information arrives can never be optimal. In sucha case, Proposition 2.1 states that the (one or more) public signal(s) thatarrive during the waiting time strictly decrease the benefit to waiting, butit is easily seen that (C.2) is identically zero when there are no interveningpublic signals or trades by others, and therefore negative when there are.Because there is at least one public signal between trading periods, if atrader waits, it must be until period T . Intuitively, if one receives no newprivate information from waiting, one never waits because it is costly due tothe public signals and potential trades by other traders which move pricesagainst the trader in expectation (due to the unconditional correlation in131C.1. Analysis Details and Omitted Proofssignals). This intermediate result is formalized in Lemma C.1.1:Lemma C.1.1: At any history, an equilibrium behavioral strategy forthe simultaneous model specifies either trading immediately or waitinguntil t = T .To determine whether trading at t or waiting until t = T is optimal, onemust evaluate (C.2) at pt and n˜, given the strategies of the other traders.If positive, the trader waits, and, if negative, she trades immediately. Anequilibrium is characterized by finding the fixed point in strategies. Ratherthan pursuing a full characterization here, I focus on two sets of sufficientconditions that allow the equilibria to be easily characterized.101I first derive a sufficient condition to ensure all trading occurs at t = 1.Define the benefit of waiting from t = 1 until t = T when all other tradersalso wait until t = T as BWx (p1). Then, the cost of waiting is determinedby the T − 1 public announcements only. I claim that if BWx (p1) < 0 forx ∈ {0, 1}, then any equilibrium involves all trades at t = 1. The proof isstraightforward. When BWx (p1) < 0 for x ∈ {0, 1}, then even if all othertraders were to trade at t = T so that their price impacts have no effect ontrader i, she faces a negative benefit of waiting until T and thus trades att = 1. Then, by Proposition 2.1 of Chapter 2, if any other trader were toinstead trade before t = T , i’s benefit would be strictly less than BWx (p1) andtherefore negative. Thus, no matter what the strategies of the other tradersare, i must trade at t = 1. Because this argument applies to all traders,i = 1, . . . , n, they must all trade immediately in any equilibrium. As notedin the main text, the claim does not completely specify the equilibriumstrategies because it leaves open the strategy after off-equilibrium historiesin which a trader waits until t > 1. In particular, it does not say that atrader must rush at every (off-equilibrium) history. In fact, by Proposition2.2, as prices approach either 0 or 1, the benefit of waiting until T mustbecome positive and so there are histories at which waiting becomes optimal,conditional on reaching such a history.In general, the sets of parameters (q, q, qP , p1, T ) that satisfy BWx (p1) < 0for x ∈ {0, 1} can be quite wide and, given the complexity of this benefitfunction, it is difficult to provide a simple characterization. However, itis a simple calculation to evaluate BWx (p1) for any particular parameters,including the ones used in treatment ER, q = 34 , q = 1, q∗ = 1724 , p1 =12 , and101For parameter ranges that do not satisfy either of the sufficient conditions, my conjec-ture is that the game becomes a version of a coordination game with multiple symmetricequilibria.132C.1. Analysis Details and Omitted ProofsT = 8. Because only public announcements affect BWx (p1), we can rewriteit asBWx (p1) =p1(1− p1)Pr(si = x)[T−1∑k=0C0(k)C1(k)(2q − 1)p1C1(k) + (1− p1)C0(k)− (2q − 1)](C.3)where C0(k) =(T−1)!k!(T−1−k)!(1−qP )kqT−1−kP and C1(k) =(T−1)!k!(T−1−k)!qPk(1−qP )T−1−k are the probabilities of observing k public signal realizations equalto 1, conditional on V = 0 and V = 1, respectively. Under the parametersof the experiment, we find BW0 (p1) ≈ BW1 (p1) ≈ −0.075 < 0. Therefore, theunique equilibrium is for all traders to trade immediately, proving Proposi-tion 4.2 part a).I now turn to a sufficient condition to ensure waiting until T and tradingthen is the unique equilibrium. Define the benefit of waiting from t = 0until t= T when all others trade prior to T as BRx (pt). I claim that ifBRx (pt) > 0 for x ∈ {0, 1} and for all pt ∈ (0, 1), then all traders mustwait to trade until T .102 The reasoning is as follows. If BRx (pt) > 0 atsome pt, then even if all other traders will trade prior to T , trader i wouldwait until T . Then, by Proposition 2.1, if some of the other traders havealready traded (n˜ > 1) or if we have reached a trading period where less thanT − 1 public announcements remain, i’s benefit to waiting would be strictlygreater than BRx (pt) and therefore greater than zero. Thus, no matter whatthe strategies of the other traders are or what trading period i is in, i waitsuntil T . Because this holds for all i at every history, the unique equilibriumis for all traders to wait to trade until T .As with BWx (p1) < 0, BRx (pt) > 0 can be satisfied for many combinationsof parameters, but deriving the set of such parameters is non-trivial. Thus,I proceed to evaluate BRx (pt) for the parameters of treatment EW, q =1324 ,q = 1, q∗ = 1724 , p1 =12 , and T = 8. To do so, I first rewrite the benefit whenn− 1 trades and T − 1 public announcements occur while waiting as102I believe the weaker condition, BRx (p1) > 0, is actually sufficient because it appearsto be a general property of the benefit function that it crosses zero at most twice atsymmetric prices in the interval (0, 1). For up to two events, this fact was established inChapter 2, but I have not yet been able to extend the proof to a general number of events.If true, then BRx (p1) > 0 is sufficient because Proposition 2.2 of Chapter 2 ensures thebenefit function must be positive for prices sufficiently close to 0 and 1.133C.2. LearningFigure C.1: Proportion of Rational Behavior as Trials Progress (Extended)1−5 6−10 11−15 16−20 21−25 26−3000. of Rational Timing Decisions  Extended Rush (ER)Extended Wait (EW)BWx (p1) =p1(1− p1)Pr(si = x)[T−1∑k=0n−1∑j=0C0(k)C1(k)D0(j)D1(j)(2q − 1)p1C1(k)D1(j) + (1− p1)C0(k)D0(j)− (2q − 1)](C.4)where D0(k) =(n−1)!j!(n−1−j)!(1 − q)jqn−1−j and D1(j) =(n−1)!j!(n−1−j)!qj(1 −q)n−1−j are the probabilities of observing j buys by the n− 1 other traders,conditional on V = 0 and V = 1 respectively, and C0(k), C1(k) are asabove. Numerically evaluating (C.4) as a function of pt shows that it is infact positive everywhere except at pt = 0 and pt = 1, where it is identicallyzero. Therefore, for this parameterization, all traders must wait to tradeuntil t = T , establishing Proposition 4.2 part b).C.2 LearningC.1 plots the evolution of perfectly rational behavior in each treatment overthe full course of each session. Clear evidence of learning is observed in theER treatment, where robust convergence to rational rushing is observed.On the other hand, subjects do not appear to learn that the additional134C.3. Non-equilibrium Informational Lossessignal is valuable in the EW treatment. It may be that the large numberof opportunities to trade makes learning difficult: if one never tries waitinguntil the final period, one doesn’t observe its benefits.C.3 Non-equilibrium Informational LossesAs in Appendix B.4, I measure informational losses due to non-equilibriumbehavior by comparing final prices in each trial to actual asset values. C.2plots the empirical cumulative distribution function (cdf) of the absolutevalue of the difference in these two values for each of the treatments. Whenrushing is optimal (ER), the theoretical and actual losses are very similar:almost all information losses are due to the robust rational panics. Non-equilibrium waiting to obtain an additional signal does not contribute infor-mational gains because only one subject in one trial waited long enough toobtain additional information.In the treatment, EW, in which waiting is optimal, we observe large infor-mational losses due to non-equilibrium behavior. Because all traders shouldobtain perfect information, informational losses are particularly surprisingbecause, should any one of the 8 traders wait to obtain perfect information,the final price would reflect the true asset value. Thus, in the trials in whichinformation is lost, no trader waited. On average, the observed final pricediffers from the theoretical by 6.8% in EW versus only 2.6% in ER.C.4 InstructionsThe instructions for the EW treatment, are provided. The instructions forthe ER treatment are identical except for the differences in parameters.135C.4. InstructionsFigure C.2: CDFs of Difference Between Final Prices and Asset Values byTreatment (Extended)0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9|Final Price − Asset Value|CDFExtended Rush (ER)  ActualTheoretical0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9|Final Price − Asset Value|CDFExtended Wait (EW)  ActualTheoretical136Contents of binif value = 100Contents of binif value = 0Private Clue13 blue11 green11 blue13 greenPublic Clue17 blue7 green7 blue17 greenInstructions7 his is a research e[ S eriment designed to understand hoZ  S eoS le make stock trading decisions in a simS le trading environment 7 o assist Z ith ourresearch  Z e Z ould greatly aS S reciate your full attention during the e[ S eriment Please do not communicate Z ith other S articiS ants in any Z ay andS lease raise your hand if you have a T uestion<ou Z ill S articiS ate in a series of 30 trials In each trial  you and the 7 other S articiS ants Z ill each trade a stock Z ith the comS uter <ou Z ill beS aid 00 for comS leting all trials In addition  in each trial you Z ill earn lottery tickets as described ne[ t 7 he more lottery tickets you have  themore you Z ill earn on average7 rading and Profits%efore each trial  the comS uter Z ill randomly select Z hether the stocks value  V  is 100 tickets or 0 tickets ( ach is eT ually likely to be selected $tthe start of each trial  you Z ill be given 100 tickets <ou may choose either to buy or sell the stock in any one of  trading S eriods <ou can onlytrade 21C(  and 08 S7  trade in one of the S eriods If you buy the stock  you Z ill gain its actual value minus the S rice  P  you S ay for it If you Z antto sell the stock  you must first borroZ  it from the comS uter and later S ay back its actual value So  if you sell  you Z ill gain the S rice you receiveminus its value In summary  your total S rofit is100+V-P lottery tickets if you buy100+P-V lottery tickets if you sellSo  for e[ amS le  if you sell the stock at a S rice of 0 and it turns out to be Z orth 100  you Z ould earn 100+0-100=0 tickets for that trial %ut ifit turns out to be Z orth 0  you Z ould earn 100+0-0=10 ticketsClues about Value7 o helS  you guess the value of the stock the comS uter chose  you Z ill get a single Private Clue at the start of the trial 7 his Private Clue Z ill beknoZ n only by you SS ecifically  there Z ill be tZ o S ossible bins one that the comS uter Z ill use if the value is 100 and another that the comS uterZ ill use if the value is 0 %ins contain some number of blue and green marbles as shoZ n beloZ  7 he comS uter Z ill draZ  a marble randomly from thebin and shoZ  it to you as your Private Clue ( ach marble in the bin is eT ually likely to be draZ n 7 he color of the ball you see can give you a hint asto the stocks value$fter observing your Private Clue  you can choose to trade immediately in the first trading S eriod $lternatively  you can choose to Z ait and tradein one of the folloZ ing 7 trading S eriods %etZ een trading S eriods  there are S ublic announcement S eriods In each annoucement S eriod  a PublicClue Z ill become available 7 he Public Clue  unlike your Private Clue  is seen by everyone For the Public Clue the comS uter Z ill draZ  a marblefrom another bin 7 he bin used Z ill again deS end on the stocks true value but the S ossible bins used for the Public Clue are different from the binsused for the Private Clue  as shoZ n beloZ  1ote that the bins used for both clues are fi[ ed throughout the trial - they deS end only on the initiallychosen random value of the stock $lso  marbles for both tyS es of clues are alZ ays reS laced before another is draZ nIf you decide to Z ait to trade until the last trading S eriod  the true value of the asset Z ill be revealed to you before you trade 2therZ ise  you Z illonly learn the true asset value after the trial is comS lete ImS ortantly  hoZ ever  each time you choose to Z ait  the S rice is likely to change beforeyour ne[ t chance to trade  as described ne[ tPrices7 he S rice of the stock is set by a Price Setter S layed by the comS uter 7 he Price Setters M ob is to set the S rice eT ual to the stocks mathematicale[ S ected value given all of the S ublic information available 7 herefore  the S rice Z ill alZ ays be betZ een 0 and 100 7 he Price Setter can observethe trades made by you and the other S articiS ants and the Public Clues + oZ ever  she can not observe any of the Private Clues nor the true assetvalue even Z hen it is revealed in the final S eriod%ecause the S rice changes Z ith the available information  if you decide to Z ait  the S rice at Z hich you can trade is likely to change 7 he initialS rice of the stock is 0 tickets  reflecting the fact that it is eT ually likely to be Z orth 100 tickets or 0 tickets $fter a S articiS ant buys  the PriceSetter Z ill increase the S rice and after a S articiS ant sells  shell decrease the S rice $fter a Public Clue is revealed  the S rice Z ill increase if itsuggests on its oZ n that the stock is more likely to be Z orth 100 and decrease if it suggests it is more likely to be Z orth 07 rading Screen7 he trading screen you Z ill use to trade is as shoZ n beloZ  7 he eight trading S eriods are indicated by the numbers 1- Public Clues are revealedbetZ een trading S eriods at the times indicated by the megaS hone symbol the clues are not shoZ n in the figure - they aS S ear elseZ here on yourtrading screen $ll S ast trading decisions and S rices are disS layed 7 rading S eriods in Z hich one or more trades occur are indicated by a solid dot7 he number of buys is indicated by a + and then a number and the number of sells by a - and then a number 7 he current S rice at Z hich you cantrade 71 in ths e[ amS le is disS layed at the current time Z hich is indicated by the dashed red line 7 he dashed red line Z ill S rogress to the right137as Z e move through the S eriods In this e[ amS le  one S articiS ant bought in the first trading S eriod  so the S rice increased In the second trading S eriod  one S articiS ant bought andone sold  so the S rice did not change In the third trading S eriod  no one traded 7 he first and third Public Clues suggested the stocks value is 100and the second that the stocks value is 0 It is currently the fourth trading S eriod 1ote that this is an e[ amS le only and is not meant to suggestZ hen you should tradeIn each trading S eriod in Z hich you havent already traded  you must choose buy or sell or Z ait and then S ress the confirm button If you choose toZ ait  the red arc S oints to the ne[ t trading S eriod in Z hich you can trade In trading S eriods after you have traded  you do not have to do anything- you Z ill simS ly be notified that you have already traded In the S eriods Z ith Public Clues  you must S ress 2. to acknoZ ledge having seen theclueSummary1 $t the beginning of each trial  the comS uter randomly selects the stocks value 0 or 100 ( ach S articiS ant is shoZ n their Private Clue from the same bin 0arbles are reS laced so each S articiS ant may see the same or differentmarbles3 ( ach S articiS ant chooses to buy  sell  or Z ait in the first trading S eriod $fter all S articiS ants have made their decisions  a Public Clue isrevealed and Z e move to the second trading S eriod SteS  3 is reS eated until all  trading S eriods are comS lete <ou must trade in one of the eight trading S eriods and may only trade once -ust before the th trading S eriod  the true value of the stock Z ill be revealed to you if you have not already traded $t each S oint in time  all S ast S rices and trading decisions are available to use to helS  guess the value of the stock in addition to the Clues$fter all S articiS ants have traded  the trial is comS lete 7 he true value of the asset Z ill be revealed to all S articiS ants and you Z ill be told hoZ  manytickets you earned for the trial <ou Z ill S ress ne[ t trial to S articiS ate in the ne[ t trial It is imS ortant to remember that  in each trial  the valueof the stock is indeS endently randomly selected by the comS uter -- there is no relationshiS  betZ een the value selected in one trial and another$fter all trials are comS lete  one lottery Z ill be conducted for each trial For each lottery  a random number less than 00 Z ill be chosen by thecomS uter If the number is smaller than the number of lottery tickets you earned for the trial  you Z ill get 100 7 herefore  the more lotterytickets you earn in each trial  the more you can e[ S ect to make S artial tickets are S ossible and count as Z ell For e[ amS le  if you earn 100tickets in each trial  you can e[ S ect to make 0301=100 over the 30 trials %ut  if you earn 130 tickets in each trial  you can e[ S ect to make0301=10Please try to make each trading S eriod decision Z ithin 1 seconds so that the e[ S eriment can finish on time $ timer counts doZ n from 1 to helSyou keeS  track of time 1ote  hoZ ever  that if the timer hits zero  you can still enter your trading or Z ait decision and Z ill still have the samechance to earn money %efore beginning the S aid trials  Z e Z ill have tZ o S ractice trials for Z hich you Z ill not be S aid 7 hese trials are otherZ iseidentical to the S aid trialsQuizPlease ansZ er the folloZ ing T uestions and S ress the Check ansZ ers button to see Z hether or not you ansZ ered all T uestions correctly 7 o ensure allS articiS ants understand the instructions  everyone must ansZ er all of the T uestions correctly before Z e begin the e[ S eriment1 <our Private Clue is a blue marble %ased only on this information:hat is the most likely value of the stock" 100 138 0  It is the second trading S eriod <ou observe another S articiS antsold the stock in the first trading S eriod :hat color marble is theirPrivate Clue most likely to be" green  blue 3 <our Private Clue is a blue marble :hat color marble is anotherS articiS ants Private Clue likely to be" green  blue  If you choose to sell the stock at a S rice of 0 and its value turnsout to be 100  hoZ  many total tickets Z ould you get for that trial" 0  0  10  If you choose to buy the stock at a S rice of  and its value turnsout to be 100  hoZ  many total tickets Z ould you get for that trial"   7  17  7 he stocks current S rice is 0 :hich value of the stock is morelikely" 100  0 2nce you have comS leted the T uiz  S lease S ress Check ansZ ersCheck answers139


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