Sunspots in the Laboratory

Sunspots in the Laboratory John Duffy; Eric O'N Fisher The American Economic Review; Jun 2005; 95, 3; ABI/INFORM Global pg. 510

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95 NO.3 DUFFY AND FISHER: SUNSPOTS IN THE LABORATORY 511

appears necessary for sunspot equilibria toobtain.

Sunspot equilibria occur reliably only in aclosed-book call market, with its centralizeddetermination of price. They happen less oftenin the double auction, where many differentbids, offers, and transaction prices may be observed. This difference has to do with the tiowof information under the two mechanisms andwith how agents learn to believe in the efficacyof using the sunspot as a coordinating device.In the closed-book call market. the marketclearing price is the only feedback that subjectsreceive. Since market participants cannot communicate, the sunspot variable plays a criticalrole in coordinating actions. In the double auction, the best bids and offers are always visible.and they are being updated almost continuously.Further, the different prices at which transactions occur within a period give information thatmay reinforce or obfuscate the meaning of thesunspot variable. Thus, in the double auction.individuals communicate indirectly and maynot need to condition their actions on a publicsunspot variable.

An important implication of our findings isthat theorists ought to pay more attention toinformation flow in future extensions of sunspottheory. Furthermore. theories relying on extrinsic uncertainty ought to make explicit how people interpret sunspot variables. Learning tobelieve in sunspots is a lot like learning a language of the market; it is essentially a socialphenomenon, and the salience of a potentialsunspot variable depends upon a shared common culture of the market.

I. Related Literature

We are not the first to use the laboratory to tryto obtain direct evidence of sunspot equilibria.Ramon Marimon et al. (1993) designed an experiment based on an economy with overlapping generations where sunspots might play arole. Their environment had two steady-stateperfect foresight equilibria and one where pricesfollowed a two-period cycle. This multiplicityallowed for the possibility that prices mightdepend upon a sunspot variable.

Those authors placed a blinking square thatcycled between red and yellow on subjects'

computer screens in an effort to coordinateexpectations on the cyclical equilibrium. Without any correlation between sunspot realizationsand actual price movements, subjects ignoredthe different colors of the blinking squarethe sunspot variable-and coordinated on oneof the steady states. So the authors tried toinduce a correlation between price movementsand sunspot realizations by alternating the number of subjects playing the role of "young"agents in each period of the training phase.This design amounted to an endowment shockthat was perfectly correlated with realizationsof the sunspot variable; it induced a cycle inprices in the experiment's training phase. Oncethis phase was over, the shock to economicfundamentals was eliminated. Afterward, thesubjects usually coordinated near one of thesteady-state equilibria. In the sessions whereprices remained volatile after the initial trainingphase. the actual price path deviated substantially from the predicted sunspot equilibrium.Thus, while Marimon et a1. (1993) made a significant effort to get subjects to condition theirexpectations on a sunspot variable, they did notobserve a sunspot equilibrium in any of their/lve sessions.

Our experimental design differs considerablyfrom theirs. Sunspot equilibria in macroeconomics are often modeled in dynamic generalequilibrium environments that are difficult toimplement in the laboratory. This considerationled us to use treatments based on two certaintyequilibria in an economy without asset markets," Indeed, one should think of our design ascorresponding to the simplest case that Cass andShell ( 1983) studied: a randomization over certainty equilibria.

In most of our treatments. the sunspot variable is a random, public announcement as towhether a high or a low price is likely to occur.The announcement serves as a coordination device that subjects are free to use or to ignore.Our sunspot variable provides the social andcultural context that was missing from Marimonet al.·s (1993) blinking square. Indeed. an important implication of our work is that the

, As Azariadis and Roger Guesnerie (19H6. p. 726) observe. "Sunspot phenomena. of course. are not necessarilydynamical: the related concept of 'correlated equilibrium'does not require the passage of time."


THE AMERiCAN ECONOMiC REVIEW JUNE 2005

semantics of the language (~f sunspots matters;if it is not immediately clear to everyone how asunspot variable is to be interpreted, then it isunlikely to play any role in coordinating expectations. Common knowledge about the sunspotvariable also appears to be important. So apublic announcement that everyone understandsis a good candidate for a sunspot vatiable.While our announcement may seem contextladen, it remains a genuine sunspot variable,since it has nothing to do with economicfundamentals.4

We think that our framework is the simplestone in which to observe a sunspot equilibrium;the equilibrium involves randomization overtwo certainty equilibria in an economy in whichthere are no assets at all. Thus asset markets areentirely incomplete, and the only important dynamic in our framework is whether the subjectslearn to believe in the sunspot as the experimental session evolves. We use a very stringentcriterion for judging whether sunspots matter:we say that a sunspot equilibrium obtains only ifsubjects coordinate on the random announcement in every period, beginning from the veryfirst period. Subjects must start off believing inthe sunspot, and their beliefs must always beself-fulfilling. We say that a sunspot equilibrium fails to obtain when subjects do not followthe sunspot announcement in one or more of theten periods of a session.

There is some related experimental work ontwo-person games with multiple equilibriawhere subjects respond to recommendationsmade by the experimenter.5 While the aim ofthis literature is different from ours, one interesting finding is that pairs of subjects will folIowa recommendation to playa strategy that is

4 Indeed, Farmer (1999, p. 225) suggests a very similarexample of a context-laden sunspot. He states, "I like tothink of the sunspot as the predictions for the economy ofthe Wall Street Journal."

5 See, for example, Jordi Brandts and Charles A. Holt(1992); Brandts and W. Bentley MacLeod (1995); John B.Van Huyck et al. (1992). Andrew Schotter (2003) andassociates examine how subjects play games when theyreceive advice from other subjects. Hakan Holm (2000)examines how announcements of an opponent's genderaffect play in coordination games. Colin F. Camerer andMarc Knez (1997) examine the effect of announcing bonuspayments to groups of subjects who all choose the sameaction in a "weak-link" coordination game.

not dominated, as long as it does not result inasymmetric payoffs. By contrast, we observethat large groups of subjects are willing to coordinate on our announcements unerringly andimmediately, even though some subjects strictlyprefer one equilibrium over the other. This finding may occur because each subject has lessinfluence in a market than in a two-persongame.

There is an experimental literature that considers whether asset markets can be manipulated (e.g., Colin F. Camerer, 1998) or whethersuch markets are susceptible to price bubblesand crashes (e.g., Vernon L. Smith et aI., 1988,and Vivian Lei et aI., 200 I). While this literature is similar in spirit to our experiment, thenotion of a sunspot equilibrium is quite distinctfrom that of a price bubble. Among other differences, stationary sunspot equilibria are ofindefinite duration, whereas most price bubbleseventually burst. Further, in those experiments,price bubbles are not equilibrium phenomena.Still, we view this literature as being complementary to our paper.

There is also an experimental literature insocial psychology on self-fulfilling propheciesbeginning with Robert Rosenthal and LenoreJacobson's seminal (1968) study demonstrating how teachers' false expectations abouttheir students' abilities shaped the students'subsequent performance. This strand of research differs in many respects from the sunspot literature in economics. Psychologistsdefine self-fulfilling prophecies as false beliefs that are nevertheless fulfilled, whereaseconomists are agnostic about the verity ofnonfundamental beliefs. Psychologists focuson the "expectancy effects" of individuals,whereas economists are concerned with the"madness" of crowds. Experimental psychologists typically do not offer their subjectssalient monetary incentives and sometimesuse deception to induce false beliefs; teachersin Rosenthal and Jacobson's study were toldthat some of their students had "unusual potential for intellectual growth," even thoughthis claim was spurious. In contrast, we didnot practice any kind of deception and werequite explicit about how subjects' choicestranslated into monetary rewards.

Finally, we note that there is a relationshipbetween our sunspot equilibria and the notion of


95 NO.3 DUFFY AND FISHER' SUNSPOtS IN THE LABORATORY 513

a correlated equilibrium. (, Our sunspot variableprovides a common signal about which certainty equilibrium will actually occur, and eachsubject takes an action conditional upon it. Unilateral deviations are unprofitable, and the sunspot equilibrium is a correlated equilibrium of asimple game. Since there are no previous experimental tests of this concept. our results areof independent interest to game theorists.

II. Hypotheses

We explore three fundamental hypotheses.The first is:

HYPOTHESIS I: SUlIspot equilibria exist.Further, they call be easily replicated.

While the logical foundations of equilibriumtheory based upon endogenous expectations ofextrinsic uncertainty are quite well founded, theeconometric evidence from field data is mixedat best. Indeed, Robert P. Flood and Peter M.Garber's seminal work (1980) showed how difficult it is to find price bubbles using econometric tests based upon a well-specified model.Hence, there is compelling need for evidencefrom the laboratory. As we show below, whenthere is a clear semantic mapping from sunspotvariable realizations to actions, a sunspot equilibrium always occurs in the call market andoften happens in the double auction. Our findings suggest that sunspot equilibria in call markets are easily replicated: others will now beable to build upon our design.

Our second hypothesis is subtler, and it isperhaps of greatest interest to both economictheorists and policymakers.

HYPOTHESIS 2: SUlIspot equilibria are .11'11

sitil'e to the floll' or informatioll.

The usual Walrasian framework that servesas the foundation for any theory of extrinsicuncertainty is based upon a static notion of theflow of information. It actually obviates an important element of many field markets, where

(, Sec Robert J. Aumann (1974) for thc definition ofcorrelated equilibrium, and Jamcs Pcck and Karl Shell(1991) on the relationship between correlated and sunspotequilibria.

there is nearly continuous trading betweenevents that signal the advent of important newinformation. The simplest way to allow for adifferential flow of information in asset marketsin the laboratory is to highlight the differencebetween a double auction, in which severaltransactions can occur in a period, and a callmarket. in which only one price clears the market in each period. Of course, in a double auction, all the inframarginal bids and offersbecome a part of the information set of everytrader as the period unfolds, while only oneprice becomes public knowledge in a call market, once the price fixing has occurred.

Our third hypothesis may be of interest toboth philosophers and social scientists.

HYPOTHESIS 3: The sell/antics o(the SUlIsPOtl'ariahle II/atter.

When we speak of semantics, we have threethings in mind. First. a sunspot variable can bea coordinating device only if its meaning istransparent. Second, a sunspot variable musthave realizations that are public: thus it is common knowledge that everyone in the marketsees the same random variable, and you and Ibelieve that everyone else ascribes the samemeaning to it that we each do. Third, there mustbe some "training period" during which we allcome to believe that the sunspot variable isactually correlated with market outcomes. Inbrief, learning to believe in sunspots has manyof the same elements as learning a language.Seeing a group of children playing soccer, Ludwig Wittgenstein (1953) is alleged to have hadthe remarkable insight that language is actuallya game that serves a useful social function onlyif everyone has a shared sense of its conventions. Hypothesis 3 states that a sunspot variablewill serve as a coordinating device if it is easyto understand and a useful guide for one'sdecisions. 7 ~

7 David K. Lewis (1969) might think of our sunspotvariable as a convcntion of truthfulncss. Suppose that thesunspot variable is "thc forccast is high'" Then each agcntmay well takc an action (like dcciding to bid or offer at ahigh pricc) that in the aggregatc will lead to a high-priccdequilibrium. Each buyer or seller docs so because he or shethinks that others will do the same. and thc community hasa CO III III on interest in achieving maximal ex post economic


514 THE AMERICAN ECONOMIC REVIEW JUNE 2005

TABLE I-ExpERIMENTAL DEStGN

Announcements

Predetennjnedby experimenter

Publiccoin flips

Cell 25 Sessions

Cell 43 Sessions

Cell 14 Sessions

Cell 33 Sessions

Cell 53 Sessions

Double auction

Call market with "sunshine"and "rain" announcelnents

Call market

Market mechanism

Sunspot semantics

III. Experimental Design

For most of our research, we used a 2 X 2design where the treatment variables were themarket mechanism and the forecast announcement. 8 The two different market mechanismswere a double auction and a call market.9 Therewere two kinds of random forecast announcements: an opaque one where realizations of thesunspot variable had been determined in advance and a transparent one where a coin wasflipped publicly each period. The four cells ofthe main design are presented in Table 1.

We used an extremely simple market designwith two different demand and supply curvesthat intersected near a price of 100 or a price of200. In either case the equilibrium quantity wassix. The instructions told the subjects that theirmarginal costs and valuations depended uponthe end-of-period median price, and this statisticwas determined solely by their actions duringthe trading period. Suppliers made money byselling above their marginal costs and buyersmade money by purchasing below their marginal valuations. At first blush, it may seemunusual that costs and valuations "dependupon" a statistic based upon transactions prices,but this formulation is just a reduced form for an

surplus. Lewis argues that this "convention of truthfulness"is the essence of language as a social norm.

8 We urge the interested reader to retrieve the instructions for any treatment at http://economics.sbs.ohio-state.edulefisher/duffyfisher/docs. The files are named celll.PDF,ceIl2.PDF, ceIl3.PDF, ceIl4.PDF, and ceIl5.PDF.

9 The double auctions were conducted using the MUDAsoftware described in Charles R. Plott and Peter Gray(1990). The software for the call market was developedspecifically for this project and is available upon request.

economy with multiple equilibria. 1o Of course,there is no fundamental uncertainty because noone's payoffs actually depend on any randomvariable. In this framework, an equilibriumprice is just one that clears the market, giventhat it is anticipated correctly.

Figure 1 shows the actual demand and supply curves that were used in all sessions. Thesteps indicate precisely which valuations andcosts accrue to which subjects. The five buyers in Figure I are labeled B1 through B5 andthe five sellers are labeled S I through S5.Notice that each subject has two valuations orcosts along a given (price-contingent) demand or supply curve; they correspond to thetwo units he or she could trade. Buyers weretold that their profits were the difference between their valuation and the price they paidfor a unit, while sellers were instructed thattheir profits were the difference between theirsales price and the cost of that unit. FigureI reveals that there is a continuum of equilib-

10 Consider an Edgeworth hypercube for an economywith ten agents and two goods. Assume that this economyhas three equilibria, two of which are stable. Each agentprefers the equilibrium where the tenns of trade favor his orher exports, and thus neither is Pareto ranked. Of course, theexcess demand function has three roots, and it would bealmost impossible to describe it simply. Our treatments arejust the reduced fonn for this economy. The "suppliers" arethe agents who export the first good, and the "demanders"are those who import it. Since the first market clears, thesecond (notional) one must as well. The marginal valuationsof the demanders and the "marginal costs" of the suppliershave nothing to do with the sunspot variable. Indeed, theshape of the aggregate excess demand functions is independent of any announcement. Hence the instructions area convenient reduced fonn summarizing each person's excess demand (or supply) in a neighborhood of each stableequilibrium.


VOL. 95 NO.3 DUFFY AND FISHER: SUNSPOTS IN THE LABORATORY 515

s

0B5

I B40

I B3 I 85High 8upply

0I B2 I 84

0

0- I Bl Bl I 83

820

0- B2

I 81 81t B3

0 I 82 I B40 I 83 I B50

85 84High Demand

0

0Bl B2

Low 8upplyI B3 I 81

0I B4 I 82

0

0- J B5 B5 I 83

840

B4

I 85 85 I B3I 84 I B2

I 83 I B1Low DemandI 82

81

25

24

23

22

21

20

19

18

17

16

15

14

13

12

11

10

90

80

70

60

50

PriceIn

Franc

1 2 3 4 5 6 7 8 9 10 Quantity

FIGURE 1. INDUCED HIGH AND Low DEMAND AND SUPPLY, BUYERS: B I-B5, SELLERS: 51-55

rium prices in the interval [90, 1001 and another in the range 1190, 210], depending onwhether the high or low demand and supplycurves were in effect. In all the treatments, wetold the subjects that low valuations and costswould occur if the end-of-period mediantransaction price was less than 150 and highcosts and valuations would occur otherwise.Every subject had to make decisions based onuncertainty, not knowing which equilibriumwould occur.

The two equilibria are not Pareto comparableby design; some agents' profits are higher in one

equilibrium than in the other. If one were Paretodominant. subjects might coordinate on it as afocal point for their expectations. On the otherhand, if each had the same inframarginal rents,then sunspots would matter only in a trivialsense, since every subject's payoff would beindependent of the sunspot realization.

In this framework, a sunspot equilibrium isjust a selection among the certainty equilibria inwhich the agents' actions are conditioned uponthe realization of a publicly observed randomvariable. The canonical example is to transact atprices in a neighborhood of 100 if one sunspot


5i6 THE AMERiCAN ECONOMiC REViEW

TABLE 2-PREDETERMINED SEQUENCE OF ANNOUNCEMENTS

JUNE 2005

PeriodAnnouncement

7Low

8High

9Low

10Low

11High

12High

13Low

14High

ISHigh

16Low

realization occurs and at those near 200otherwise.

We would like to emphasize that this environment allows for the simplest type of sunspotequilibrium that Cass and Shell (1983) study:randomization over certainty equilibria. It isobviously good science to use the cleanest possible treatment to test the empirical existence ofan equilibrium concept whose theoretical implications are profound. Using a time-dependentstate variable, as is the norm in some macroeconomic models of sunspots, would complicate the treatments unnecessarily. Finally, thefundamentals for this economy are obviouslyuncorrelated with the sunspot variable sincethey are the underlying (constant!) preferencesand endowments that give rise to an economywith two stable equilibria.

During the first three periods of every session, we trained subjects by eliminating the lowequilibrium; buyers had only high valuations,and sellers had only high costs. During the nextthree periods, we eliminated the high equilibrium analogously. The primary purpose of thetraining periods was to allow the subjects tolearn how to use the computerized softwarewhile replicating two different static environments. A secondary purpose of the training was tomake the two equilibria focal points for the subsequent periods in which extrinsic uncertainty wasallowed full and free rein. In essence, subjectswere being exposed to the language of the sunspotequilibrium at the start of every session.

Of course, in periods 7 through 16, eitherequilibrium was possible. At the beginning ofeach period, there was a public announcementbased upon a random variable. The usual announcement was: "The forecast is high," or"The forecast is low."

It is important to give the exact text of therelevant instructions. In the treatment where theexperimenter made the announcement, the instructions read:

"Beginning with period 7, the experimenter will make an announcement at the

beginning of each period. The announcement will be either that 'the forecast ishigh' or that 'the forecast is low.' It isimportant that you understand that theseannouncements are only forecasts; theymay be wrong, and they do not determinein any way your actual costs or values inthat period. Indeed, the experimenter doesnot have any more information than youdo. Remember that your actual costs andvalues depend only upon the official median price for that period."

In keeping with the spirit of the literature onsunspots, we used a random number generatorto determine the sequence of announcements.Since the sequence of announcements is an obvious control variable, the sequence in Table 2was used in every session in cells I and 3.

In the treatment where a public coin flip wasused, the instructions read:

"Beginning with period 7, an announcement will be made at the beginning ofeach period. The announcement will beeither that 'the forecast is high' or that'the forecast is low.' This forecast will bedetermined by flipping a coin. Anyonewho wants to can come up and look atthe coin and how it landed. If the coinlands heads up, the person who flippedit will announce that the 'forecast ishigh.' If it lands tails up, that personwill announce that the 'forecast is low.'The experimenter will ask each of you totake a tum flipping the coin. When it isyour tum, flip the coin in the air and let itland on the floor. Anyone can come up atany time and make sure that the personmaking the announcement is telling thetruth. I will now let everyone see that thisis a fair coin, and I will keep the coin inplain view at every moment during theexperiment. Come up and look at the coinnow.

It is important that you understand thatthe forecasts based on the coin flips maybe wrong and do not determine in anyway your actual costs or values in that


95 NO.3 DUFf-T AN/) FISHER: Sl'.\Sl'mS IN TIIF IAHOR:\TORY 517

trading period. Remember that your actual costs and values depend only on theofficial median price for that period."

In this treatment, the random sequence of announcements will almost surely differ acrosssessions, so that replication of any particularsequence is highly unlikely.

We chose this coin-flip treatment for tworeasons. First, a public randomization devicemight matter. Second, our findings for the treatment where announcements were made by theexperimenter might be subject to a Clever-Hanseffect. 1

I We were concerned that subjects mightplace undue reliance on the experimenter's announcement because they were afraid thatsomething nasty might be in store if they tried todeviate from it. Perhaps subjects blindly followed the experimenter's announcement because they wanted to please the professor or hadtrust in him or her. It is of interest to seewhether the outcomes differ when the stochasticprocess determining announcements is transparent and more obviously beyond the control ofthe experimenter.

The instructions for all treatments made itclear that announcements were not binding inany way. Indeed, subjects were reminded thattheir actual costs and values depended (mlrupon the official end-of-period median price forthat period. Each trader was faced with a decision fraught with uncertainty about which equilibrium would actually occur, and some of themlearned by hard knocks that the official pricecould be quite different from the announcementor from what they had hoped would occur.

In the double auction, subjects were allowedto submit bids or offers as long as they had unitsleft to buy or sell. A trading period lasted forfour minutes. Subjects observed the current bestbid and offer on their screens and could instantly transact at these prices. The best bid andoffer were updated in real time according to the

II This effect is discussed widdy in experimental I"Ychology. It captures the notion lhal suhjecls may respondunconsciously to unwilling cues hy the experimenter. Indeed. in the nineteenth century. Clever Hans was a famousGerman horse who could do arilhmetic hy tapping outanswers \I.,'ith his hoof. Rigorous investigation revealedeventually that he was responding to suhtle (often unconscious) cues that his (human) audience gave him.

standard improvement rules: a buyer had toincrease the standing bid and a seller had toundercut the standing offer. Subjects saw eachtransaction as it oceurred, and the bid and offeron the computer screen were cleared when adeal was struck. (Thus no order book was maintained.) Also. the experimenter reported the current median price based on all transactions thathad occurred in that period up to that point.

[n the call market. subjects typed positiveintegers for their two bids or offers. The computer program then sorted all bids and all offers,creating demand and supply schedules. [1' therewas an interval of prices that cleared the market.the software chose the greatest integer not higherthan the midpoint. 1:' Each subject was informed ofthe market-clearing price as well as the number ofunits he or she had bought or sold. The callmarket mechanism was carefully explained tosubjects, using several illustrative examples.

We also had a fifth treatment in cell 5 thatfocused on whether the semantics of the sunspotvariable mattered. I' This treatment was identical to that in cell 4, but we changed only two\\'(II'{I.\ in the entire instructions. Now the relevant paragraph reads:

"Beginning with period 7, an announcement will be made at the beginning ofeach period. The announcement will beeither that 'the forecast is slinshin£" orthat 'the forecast is min.' This forecastwill be determined by flipping a coin.Anyone who wants to can come up andlook at the coin and how it landed. If thecoin lands heads up, the person whoflipped it will announce that the 'forecastis slIll.\hin£'.' If it lands tails up, that person wi II announce that the 'forecast is!'a Ill.

(We have added emphasis to help the reader.)The instructions changed "high" to "sunshine"and "low" to "rain." We chose these words fortwo reasons. First. we wanted the sunspot announcement to make grammatical sense in common English. Second, we wanted to use a

1- If therc was no such intcnal. then a IOllery wasconducted among those on the long side of the market todetermine who eot to trade.

I.' We arc ve;y grateful to an editor and the referees whosuggested that we design and conduct this extra treatment.


THE AMERICAN ECONOMIC REVIEW JUNE 2005

mapping from the flip of a coin to the twoannouncements that did not correspond tooclosely to high or low. 14 We also made animportant change during the first six trainingperiods. Instead of saying that the announcement is "sunshine" during the first three periodsand the announcement is "rain" during the nextthree, we just left these announcements blank.The subjects still converged quickly to the competitive equilibria in the training periods, butthey had received no lessons in the simple twoword language of the sunspot variable.

In every treatment, after the sunspot announcement, we asked each subject to recordhis or her forecast of the median price for thecoming period before we opened the market.Although we did not pay them for their forecasts, we explained that recording these datawould help them focus on what might happen inthe market in the coming period. Also, we kepta running tally of all past announcements andofficial end-of-period median prices on theblackboard. Thus the subjects' expectationswere reinforced by the common public historyof the market that was in plain view as thesession progressed.

In all treatments, the buyers' payoffs were thedifference between their unit values and purchase price, and sellers' payoffs were the difference between their sales price and unit cost.Subjects could-and occasionally did-losemoney if they bought too dear or sold too cheap.They were paid in cash, and earnings for atwo-hour session averaged around $29 per subject, including a $5 show-up fee.

IV. Experimental Results

We conducted 18 sessions using the fivetreatments described in Table 1. 15 Sixteen ofthese sessions involved ten inexperienced subjects recruited from the undergraduate populations of The Ohio State University or the

14 We did not say "the forecast is heads" and "the forecast is tails" precisely because writing an "H" on the blackboard is just too close to a natural symbol for 'high."

15 To keep this section short. we will describe datafrom five representative sessions. The interested reader canfind all the data from all I8 sessions in a compressed fileat http://economics.sbs.ohio-state.edu/efisher/duffyfisher/data. zip

University of Pittsburgh. 16 Three of the sessionsin cell I and four of those in cell 2 were conducted in Columbus, and the rest were held inPittsburgh. We obtained very similar findingsusing both subject pools, so we concluded thatthe subject pool was not important. Therefore,the nine call market sessions in cells 3, 4, and 5were done at the University of Pittsburgh.

The experiment has produced three importantresults. First, sunspot equilibria really exist.These equilibria can be implemented and replicated in the laboratory; thus we find strongsupport for Hypothesis I. Second, sunspot equilibria appear to be sensitive to the flow of information. We observed sunspot equilibria inevery session with a call market where themeaning of the sunspot variable was clear; inthe double auction, they occurred slightly lessthan half the time. Since a call market has amuch more restricted flow of information than adouble auction, we conclude that there is support for Hypothesis 2. Third, consistent withHypothesis 3, the semantics of the languageof sunspots matters. Our sunspot variable-theannouncement of the likely equilibriumprovides the context that enables all agents tocondition their expectations appropriately.

To be precise, we shall claim that a sunspotequilibrium occurs only if every time the announcement is high, the median transactionprice is 150 or greater, and if every time it islow, the median transaction price is less than150. 17 We would like to emphasize just howstringent this definition is. We report a successonly if ten subjects coordinate on the sunspotvariable in every single period. Thus the subjects have to believe in the sunspot variablefrom the very beginning, and their beliefs mustbe confirmed in every period. We do, however,allow for some noise in the data. The designpredicts that equilibrium quantities are alwayssix, but we occasionally observed market vol-

16 A referee suggested that we examine the effect ofexperience on these markets. and we ran two sessions inwhich the subjects had participated in the same treatmentbefore.

17 We recognize that there is another "contrarian" sunspot equilibrium. where an announcement of "high" leads tocoordination on the low equilibrium and an announcementof "low" leads to coordination on the high equilibrium. Wefocus on the more natural sunspot equilibrium because itwas the only type we ever observed.


95 NO.3 DUFFY AND FISHER: SUNSP07S IN THE LABORATORY 519

umes that were slightly different. So we willstill say that a sunspot occurred even if quantitydiffers slightly from this benchmark.

A. Double Auction, PredeterminedExperimenter Announcements

We ran four sessions for the treatment in cellI of Table I. Among these four sessions therewas one success (a sunspot equilibrium) andthree failures. Failures of the theory typicallyoccur in one of two ways: either the subjects donot coordinate on the sunspot in just a fewperiods; or they coordinate on a certainty equilibrium in every period. These two possibilitiesare illustrated in Figures 2 and 3, which showdata from the sessions I and 2 of cell I.

The top panel of these and all subsequentfigures shows transaction prices and the sunspotpredictions. By sunspot prediction, we meanthat a forecast of high is associated with thecompetitive equilibrium where prices lie in theinterval [190, 210], and a forecast of low isassociated analogously with prices in [90, 110].For simplicity, the sunspot prediction is represented in our figures as either 200 or 100. Onecan see in Figure 2 that the sunspot predictionsfailed in periods 7, 12, 14, 15, and 16 of thissession. Figure 3 presents an even starker example, where following the training period,subjects ignored the sunspot announcementsand coordinated on the low equilibrium in everyperiod.

The bottom panel of these figures describesaspects of subjects' beginning-of-period forecasts of the median transaction price that wouldobtain at the end of the period. For each period,the line segment shows the range of subjects'beginning-of-period forecasts. The square represents the median of the ten subjects' beginningof-period forecasts, and the cross is the actualmedian transaction price at the end of the period;this end-of-period median transaction price determined whether high or low values and costswould be used in calculating eamings. One cansee that the markets converged to the uniqueequilibrium during the first six training periodsand that agents' expectations about the actual median transaction price were being formed appropriately. In the bottom panel of Figure 2, we seethat, with few exceptions, subjects' forecastsnever converged enough to conform to the sun-

spot announcements during periods 7 through 16when both equilibria were possible. In the bottompanel of Figure 3, we see that in period 8 themedian subject beginning-of-period forecastturned out to be quite wrong; this was the firstperiod in which the sunspot announcement washigh. After that, subjects chose to ignore the sunspot, and their forecasts became largely consistentwith the low equilibrium that occurred in periods7 through 16. Still, at least one subject in everyperiod appears to have maintained a belief that thesunspot variable had predictive power, as evidenced by the range of subjects' beginning-ofperiod forecasts.

In analyzing the first bids and offers from allthe sessions in this treatment, we came to believe that demanders who benefited the most inthe high equilibrium often tried to induce it bymaking a high opening bid; likewise supplierswho benefited the most in the low equilibriumoften made a low first offer. With these openingbids or offers, the standard improvement rulefor a double auction (which is public information) constrains subsequent bids to be higher orsubsequent offers to be lower. Thus the flow ofinformation in a double auction does seem toallow inframarginal bids and offers to serve assignals independent of the sunspot realization.Initial transactions are very important, a factthat has been found in field data as well. In theworking paper version of this manuscript(Duffy and Fisher, 2003), we provide a modelshowing how early transactions can lead to acascade that mitigates the need for a sunspotvariable as a coordinating device.

B. Call Market, Predetermined ExperimenterAnnouncements

There were three sessions in cell 3, and asunspot equilibrium occurred in everyone. Figure 4 shows the data from the first session ofthis treatment. All of the data from the sessionsin cells 3 and 4 look like those in Figure 4, inthe sense that there is perfect coordination onthe sunspot announcement in every period ofevery session. Figure 4 plots the single marketclearing price, the volume of units traded ineach period, and the theoretical sunspot predictions. This figure contrasts sharply with Figures2 and 3; the evidence for sunspot equilibria isquite clear. A close analysis of the data suggests



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VOL. 95 NO.3 DUFFY AND FISHER: SUNSP07S IN THE LABORATORY 521

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that a few subjects submitted bids and offers inearly periods in a strategic attempt to influencethe current price, but this is a much more formidable task in a call market than in a doubleauction. Indeed, it takes about five or six independent bids or offers-each betting in essenceagainst the sunspot announcement-to movethe price across the threshold that defines theequilibrium. A subject knows only her ownactions and the price that is revealed at the endof the period. This paucity of informationmakes it extremely difficult to influence theequilibrium strategically, and the risk of makingthe wrong bid or offer is just too great to try tobuck the sunspot announcement. Also, it is obvious that the agents learn to believe in theseannouncements because the distribution of forecasts becomes very accurate after the eighthperiod.

In Duffy and Fisher (2003), we consider asimple game and show that the strategy wheresubjects follow the announcement and truthfully reveal their values and costs is a correlatedequilibrium. We argue that the data from thecall market sessions are consistent with the notion that subjects have played this correlatedequilibrium.

C. Double Auction, Announcements by PublicCoin Flips

There were five sessions in cell 2, andwe had three sunspot successes and two failures. Figure 5 shows the data from the firstsession. This treatment is a double auctionwhere each of the subjects takes a turnflipping a coin and making the sunspotannouncement. It is quite interesting thatin this treatment we observe sunspot equilibrium in three of five sessions. In the session shown in Figure 5, it is obvious thatthere was a great deal of uncertainty about theefficacy of the sunspot announcement as acoordinating device during the ninth period.After that, the subjects came to believe in thesunspot. A similar result obtains in the othertwo sessions where sunspot equilibria occurred. In the two sessions of cell 2 wheresunspot equilibria did not happen, the subjects quickly coordinated on either the lowequilibrium or the high one, so illustrations ofthose data would be analogous to Figure 3.

We conclude that sunspot equilibria in doubleauctions are possible, but this outcome may bedelicate.

D. Call Market, Announcements by PublicCoin Flips

There were three sessions in this cell, andwe observed a sunspot equilibrium in everyone. In this treatment, the subjects took turnsflipping a coin and making an announcementbased on the outcome. The data for thesethree sessions look exactly like those in Figure 4, except that the announcement sequenceis random. We conclude that sunspot equilibria in the call market are robust to the mannerin which random announcements are made.Also, the volume of transactions and theagents trading are efficient, given that thesunspot announcements act as a coordinationdevice.

E. Call Market, Announcements of"Sunshine" and "Rain" by Public Coin Flips

The sessions in this cell changed two wordsfrom the treatment in cell 4. The words "Theforecast is high" became "The forecast is sunshine," and "The forecast is low" became "Theforecast is rain." Also, the subjects were nottrained on any announcement in the initial sixperiods. There were three sessions in this cell,and not a single one had a sunspot. Figure 6 shows data from the first session in cell 2.It is clear that the equilibrium and the forecastsin the training periods were not affected by thelack of an announcement. But once the subjectshad to rely on the sunspot announcements during the last ten periods, they were at a completeloss as to how to interpret them. 18 Indeed, exchange became fully efficient only in the lastthree periods in this session, when all six unitswere traded and the agents had come to believewith high precision that the low equilibrium wasgoing to occur in every period. The bottompanel of Figure 5 shows a beautiful example ofmonotone convergence of expectations, as the

'K Indeed. the third session in this cell was conducted onI April 2004. and one of the subjects asked if the "sunshine"and "rain" announcements were an April Fools' Day joke'



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VOL. 95 NO.3 DUFFY AND FISHER: SUNSPOTS IN THE LABORATORY 525

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agents learned to believe that only the low equilibrium would matter.

The data from the other two sessions in thiscell were very similar. The subjects in the second session ignored the sunspot announcementsand coordinated on the high equilibrium; theirforecasts also ignored the announcements andshowed a nice monotone convergence to thehigh equilibrium. The third session had a mix ofexperienced subjects from the earlier two sessions, with a majority having experienced coordination on the high equilibrium. Again, thesubjects ignored the sunspot announcement andcoordinated on the high equilibrium.

F. The Effect of Subject Experience

What was the effect of subject experience inthese markets? The fifth session of cell 2 (thedouble auction with coin flips) and the thirdsession of cell 5 (the call market with sunshineand rain forecasts) had subjects who had been inalmost identical prior treatments. It seemed asthough the experience of the majority wasbrought to bear in the new sessions. In particular, there was a second sunspot equilibrium inthe fifth session of cell 2, perhaps because themajority of subjects in this session (seven often) had experienced a sunspot equilibrium theweek before. Thus the actions of the "gangof seven" may have been able "to persuade"the minority of three-who had experienceda session with coordination on the highcertainty equilibrium-to follow the sunspotannouncement.

The third session of the call market withsunshine and rain announcements had theagents coordinating on the high equilibrium.In this case, a majority of six subjects,who had seen the very same outcome in aprior session, were able to "persuade" theminority of four, who had seen coordinationon the low-price certainty equilibrium, to follow along with them to the high-price equilibrium. They did this even though there wasno means of communication at all! If mostpeople are submitting high bids and offers,you buck this trend at your peril. Indeed,many subjects "sat out" periods 7 through 9by offering very low bids or high offers, andthe market became fully efficient only in thelast few periods, when it became evident

that everyone was coordinating on the highequilibrium.

G. Discussion of the Results

The data provide clear support for our threehypotheses. First, sunspot equilibria do exist ina controlled environment, and we are the first tohave produced them. Indeed, sunspots occurredin ten of the 15 sessions in the main treatmentsin cells I through 4. Second, the market mechanism is also important. In the call market, wefound that sunspot equilibria always occurred.This finding is readily replicated, no matter howthe announcement is made. By contrast, sunspots do not occur every time in the doubleauction. Third, the semantics of the sunspotlanguage clearly makes a difference. In cell 5,we disrupted the use of the sunspot variablesimply by changing two words and not trainingsubjects in the meaning of the new sunspotannouncements. Also, in the main treatments,the "contrarian" sunspot equilibrium, where thesubjects coordinate on the equilibrium that isthe exact opposite of the common languageannouncement, was never observed.

The outcomes from the 18 sessions are summarized in Table 3. Let us focus on the datafrom cells 1 through 4, the treatments where thesemantics of the sunspot variable are clear. Wesee that there were four sunspot successes andfive failures in the double auction treatments.There were six successes and no failures in thecall market treatments. Imposing the null hypothesis of a random assignment of successesacross the type of market, Fisher's exact test l9

has a p-value of 0.044. Thus the hypothesis canbe rejected for a test of size 5 percent. Whataccounts for this difference across kinds of markets? It is likely that success breeds success inlearning the language of sunspots. The call market is so effective in creating and reinforcing thebelief in a semantically salient sunspot variablebecause the common language interpretation ofthe sunspot variable realization, and the publicnature of the announcement, effectively resolveany doubt about how others will solve the coordination problem.

19 See Sidney Siegel and N. John Castellano Jr. (1988)for the precise details of this nonparametric test.


VOL. 95 NO.3 DUFFY AND FISHER: SUNSPOTS IN THE LABORATORY

TABLE 3-ExPERIMENTAL OUTCOMES

Announcements

527

Market mechanism Double auction

Call market

Predetenninedby experimenter

Cell II Success3 Failures

Cell 33 SuccessesoFailures

Publiccoin flips

Cell 23 Successes2 Failures

Cell 43 SuccessesoFailures

Sunspot semantics Call market with "sunshine"and "rain" announcements

CellSoSuccesses3 Failures

On the other hand, the use of a public coinflip or private randomization device (predetermined by the experimenter) does not seem tomatter. As cells 1 through 4 in Table 3 reveal,there were four successes and three failures ofsunspot theory when the sequence of announcements was predetermined, and six successes andtwo failures when a public coin flip was used.Imposing the null hypothesis of no differenceacross the type of randomization device, Fisher's exact test has a p-value of 0.61.

V. Conclusions

Experimental economics has been perhapsmost successful in illustrating how marketswork. Experimentalists have repeatedly foundthat institutions matter: different kinds of markets give rise to different outcomes. Until now,models whose equilibria relied on the existenceof "animal spirits" have been useful and eleganttheoretical curiosities. But we have given thesemodels real empirical bite. We are the first toprovide direct evidence that extrinsic uncertainty can be an important source of volatility inreal markets. Furthermore, we have shown thatthe efficacy of sunspot variables in coordinatingexpectations depends on the flow of information, and this finding has important theoreticalimplications. Our sunspot announcements serveas a reliable coordinating device only wheninformation flows slowly, as in a closed-bookcall market; in the double auction, sunspotsoccur often enough, but much depends on thefaster and sometimes confounding flow of information in this environment. The theory ofsunspot equilibrium has been developed typi-

cally in a Walrasian framework, where the flowof information is no slower or faster than thespeed at which a market clears. Our findingsindicate that it is important to model information flows while the market is clearing.

Much of the interest in correlated equilibriaand sunspots stems from the possibility thatpublic signals allow one to achieve more thanjust the convex hull of the certainty equilibria.Thus we have tested for the least interesting andperhaps least important sunspot possibilities.This is a weakness of our experiments, and wethink that analyzing treatments with more nuanced sunspots is an important area for futurework. These treatments represent only the firststep in what we hope will grow into a broaderbranch of experimental research in macroeconomics and finance.

Our experiments also suggest that it is important to consider the semantics of the language ofsunspots. When we had subjects flip a coin, weasked them to say that "the forecast is high" ifheads came up and "the forecast is low" if tailscame up. This is quite different from flipping acoin, observing that it lands heads, announcing"the forecast is sunshine," and then expectingsubjects to coordinate on some equilibrium bythemselves, as though every person can simultaneously and independently come up with asemantic interpretation of what the event headsqua "sunshine" might mean. Thus it may not beenough to train subjects with a blinking squarecycling between two colors on the computerscreen, but it certainly is adequate to state publicly that "the forecast is high" or "the forecastis low." Likewise, there may be a sunspot in theperformance of the U.S. stock market based



upon whether the National Football Conferenceor the American Football Conference wins theSuper Bowl, but there will be no empiricallytestable hypothesis until it has become commonknowledge that the language of that sunspotdepends upon everyone knowing which teamsbelong to each conference.

The fact that sunspot announcements serve ascoordinating devices has important implicationsfor financial markets in the field. Indeed, onemight argue that an important aspect of monetary policy in the United States in the last fewyears has been trying to anticipate and perhapsmitigate the effects of sunspot equilibria in major financial markets. Thus, showing that sunspot equilibria may well depend upon the flowof information has very real implications for thearchitecture of these markets. For example, ourfindings would seem to indicate that stock tradesuspension rules that are commonly used in thefield may actually increase the possibility ofsunspot equilibria.

In these experiments, the sunspot realizationshelp, paradoxically, to ensure that the marketis conditionally fully efficient. Indeed, whenagents try unsuccessfully to manipulate information strategically, they pay a price becausethe wrong inframarginal bids or offers are submitted. This implies that the resultant equilibrium does not maximize ex post social surplus.In field markets, the connection between welfare and sunspots is not so apparent, but it maybe a very real part of any financial system.

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Sunspots in the Laboratory