quality signals in information cascades and the dynamics of ...directory.umm.ac.id/data...

*Corresponding author.E-mail address: [email protected] (A. De Vany).

Journal of Economic Dynamics & Control25 (2001) 593}614

Quality signals in information cascades andthe dynamics of the distribution of motion

picture box o$ce revenues

Arthur De Vany*, Cassey Lee

Economics and Mathematical Behavioral Sciences, University of California, Irvine CA, 92697, USA

Accepted 22 February 2000

Abstract

The movies are an ideal testing ground for information cascade models. One canobserve the dynamics of box o$ce revenues from the &opening' to the end of the run. Tostudy the possibility that information cascades occur in the movies one has to incorpor-ate the fact that agents can observe box o$ce revenues (as reported on the evening news)and communicate &word of mouth' information about the quality of the movies they haveseen. One must consider many movies rather than the mere two choices assumed ininformation cascade models. We develop an agent-based model of information cascadeswith these extensions. Our objective is to see if the conclusions of information cascademodels hold with these extensions (they do not) and to see if the agent-based model cancapture the features of the motion picture box o$ce revenues (it does). ( 2001 ElsevierScience B.V. All rights reserved.

JEL classixcation: C63; D83; L82

Keywords: Quality information; Information cascade; Complex dynamics; Motionpictures; Box o$ce revenues

0165-1889/01/$ - see front matter ( 2001 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 5 - 1 8 8 9 ( 0 0 ) 0 0 0 3 7 - 3

1. Introduction

Information cascades occur when an agent ignores personal preferences orinformation to conform to the choices of others. As Bikhchandani et al. (1992)de"ne it, an information cascade is a sequence of decisions where it is optimal foragents to ignore their own preferences and imitate the choice of the agent oragents ahead of them. Models of cascades answer two key questions: Can theyoccur and are they fragile? When agents choose among two alternatives, theanswer to both questions is yes. In this paper, we want to examine thesequestions when there are many choice alternatives available to agents. Ourpurpose in extending the binary choice model of information cascades tomultiple choices is to determine if these conclusions are robust in the world ofextended choices that we actually live in. Do we still obtain the sort of &lock in'and &follow the leader' behavior that information cascade models predict whenagents are confronted with a wide range of choices? In addition, we extend themodel to include quality and quantity information. Do we still get cascadeswhen agents are able to communicate quality information with one another?

A second purpose in our research is to determine if some speci"cation of aninformation cascade model is capable of capturing the dynamics of motionpicture box o$ce revenues. When agents choose among many alternatives andobserve not just the choices of other agents their assessment of quality, whatsorts of distributions of choices do information cascades produce? This isa natural question to ask because the dynamics of cascades must have certainstatistical properties. If the cascade &locks onto' a single choice among several, asthe theory posits, then the resulting distribution of choices must be skewed andhighly uneven. If information cascades are fragile, then they should exhibit&large' #uctuations and they may have complex basins of attraction. Both ofthese questions come down to the same one: What are the properties of thestatistical distributions to which information cascades converge when there aremany choice alternatives and agents process a mixture of local and globalsignals?

The movies business is a natural laboratory in which to evaluate informationcascade models. The Bose}Einstein model used by De Vany and Walls (1996) tocapture the distributional dynamics of box o$ce revenues is a (stochastic)information cascade. Movie fans do observe the actions of other agents in theway that is typically assumed in information cascade models; that is theyobserve box o$ce reports that tell them what other agents are choosing. Onecould even argue that the movie studio casting, marketing, and release decisionsare an attempt to initiate information cascades. If a studio succeeds in drawinga large crowd to its "lm's &opening' and, if movie fans choose movies accordingto the number who chose each movie before them, a movie with a large openingaudience might initiate an information cascade. Once a cascade begins, agentsmight then ignore their personal information so that a movie with a large

594 A. De Vany, C. Lee / Journal of Economic Dynamics & Control 25 (2001) 593}614

opening might go on to capture most of the revenue, even if it is a bad movie.How likely is such a cascade in the context of the motion picture business? Caninformation cascade models give a reasonable account of the box o$ce revenuedynamics and the probability distribution of revenue outcomes?

Section 2 surveys the related literature brie#y. In Section 3, a binary choiceand a multiple choice model of the Bikhchandani et al. (1992) informationcascade are developed and simulated. This model is extended further to a qual-ity- and quantity-signal model in Section 4. The dynamics of competition amonggood and bad movies and the in#uence of the opening week are examined inSection 5. Section 6 concludes.

2. Literature

A seminal paper by Bikhchandani et al. (1992) explains the conformity andfragility of mass behavior in terms of informational cascades. In a closely relatedpaper Banerjee (1992) models optimizing agents who engage in herd behaviorwhich results in an ine$cient equilibrium. Anderson and Holt (1997) are able toinduce information cascades in a laboratory setting by implementing a versionof the Bikhchandani et al. (1992) model. To our knowledge, no models haveextended the choice set beyond two choices. Lee (1998) shows how failures ininformation aggregation in a security market under sequential trading result inmarket volatility. Lee argues that &informational avalanches' occur when hiddeninformation (e.g. quality) is revealed and reverses the direction of the informa-tion cascade.

De Vany and Walls (1996) show that a Bose}Einstein statistical informationcascade generates a Pareto distribution of box o$ce revenue. Walls (1997) andLee (1999) have veri"ed this "nding for other markets, periods, and movies. DeVany and Walls (1999a) and Lee (1999) show that the tail weight parameter ofthe Pareto}LeH vy distribution implies that the second moment (variance) of boxo$ce revenues may not be "nite. Lastly, De Vany and Walls (1999b) have shownthat motion picture information cascades undergo a transition from an imitativeto an informed cascade after enough people have seen the movie to exchange&word of mouth' information.

3. The BHW model of action-based information cascades

Bikhchandani et al. (1992) model information cascades as imitative decisionprocesses. In their model, Bayesian agents choose to adopt or reject an actionbased on their own private information and information about the actions of allthe other agents who chose before them. The agents are essentially in a line, theorder of which is exogenously "xed and known to all, and they are able to

A. De Vany, C. Lee / Journal of Economic Dynamics & Control 25 (2001) 593}614 595

observe the binary actions (adopt or do not adopt) of all the agents ahead ofthem. The agents infer the signal of the other agents in the sequence from theiractions: the agent reasons that if an agent adopts, then they must have got a highsignal. If enough agents adopt, then the agent may ignore her own signalbecause the weight of the evidence of previous adoptions overcomes the weightthe agent places on her own signal. Thus, according to the model, a rationalagent may ignore her own information and imitate the choices of agents whochose before her.

3.1. An agent-based model of information cascades

In the BHW model agents choose sequentially between adopting or rejectinga single product (binary choice). Each individual privately observes a condi-tionally independent signal X about the value of the product. Every individual'ssignal X can be High or Low. A signal High is observed with probability p'0.5when the true value is High and a signal High is observed with probability 1!pwhen the true value is Low. Each agent in the choice sequence observes thechoices of all the agents ahead of them and uses this information to infer thesignal of the preceding agent.

In our "rst extension of the BHK model, agents choose between m products,which we will hereafter call movies (though they could be software applications,clothing, stocks, or what have you). As in the BHK model, each agent getsa private signal that is either high or low. This is not a signal speci"c to anymovie, rather it is a positive or negative assessment of the prospects among themovies that are playing. Assume that agent i sees movie j (i.e. i's private signalabout movies is high). Agent i#1 will infer agent i's private signal X

jfrom i's

decision. If the second agent gets a high signal (Xi`1

is high) this con"rms theinference drawn from the "rst agent's choice and the agent will see the moviechosen by the "rst agent. If the second agent gets a low signal then she will #ipa coin to choose movie j or the movie with the highest market share (whichmight be movie j).

If the previous agent chose no movie, the next agent will choose not to seea movie if her private signal is low, or, if her signal is high, she will #ip a coin tochoose between no movie or the movie with highest market share. The choicelogic is diagrammed in Fig. 1.

3.2. Signal accuracy and information cascades with action signals

This model is simulated with 2000 agents who are sequentially allocatedamong 20 movies. Before the simulation begins, each movie is given one agent.Then the "rst agent is allocated randomly to one of the 20 movies and the choicelogic sets in. The second agent observes the prior agent's choice of moviej"1,2, 20 or no movie. We carry out 20 simulation runs for values of the


Table 1Mean values of the proportion of correct choices (C) as a function of signal accuracy (p)

Accuracy of private signal (p) C

0.95 0.640.90 0.490.85 0.450.80 0.610.75 0.470.70 0.440.65 0.490.60 0.500.55 0.530.50 0.49

probability p that an agent gets a high signal from 0.5 to 0.9 with an increment of0.05. A correct choice is de"ned as one in which a good movie is chosen. In oursimulation, half of the movies are good movies and half are bad movies, thoughthe agents are unable to communicate this information and only observe theaction of their immediate predecessor. The mean numbers of correct choicesover each sequence are tabulated for p in Table 1.

In the BHW model the probability of a correct cascade is increasing in theaccuracy of the private signals. The statistics of our simulation results in Table 1cannot con"rm this result. A higher value of p is not necessarily associated witha higher mean value of correct choices. The reason for this result is that highsignal accuracy gives too much credibility to the choice of the agent ahead and,thus, may steer the agents following onto an inferior movie.

The BHW model also implies that the probability of a cascade is almostsurely one (the probability of no cascade is vanishing for N'5 for any value ofp, see Fig. 2). The simulations indicate that, at low values of p, cascades seldomoccur and when they do they are extremely fragile. As an example, considerFig. 3(a) and (b) which depict movie choices over time for two extreme cases:p"0.55 (low accuracy) and p"0.95 (high accuracy).

In the "gures, time evolves from left to right, each time step corresponding toa decision. The integers from 1 to 20 on the vertical axis correspond to the eachof the 20 movies. When an agent chooses a movie that event is indicated by a dotin the "gure corresponding to time of the decision on the horizontal axis and themovie chosen on the vertical axis. When signal accuracy is high, the choicesindicated in the "gure form a path at movie 10. But, these choice sequences areinterrupted by leaps to other movies. The intermittent jumps do not persist andthe agents eventually return to a cascade on movie 10 because it has the highestmarket share. It is the global information about market share that pulls the


Fig. 1. A simple BHW model.

audience back to movie 10. The random element in choice, and local informa-tion about the agent just ahead in the sequence, are capable of moving a cascadeaway from the leading movie temporarily, but the global information (marketshares) eventually pulls it back if the shares are su$ciently uneven. Thishighlights how important the global view posited in standard models is to theirresults (recall that, in these models, each agent sees all the choices before shemakes hers).

When the accuracy of the signal is low, the choice sequences jump randomlyamong movies and the "gure is typical of nearly all the simulations. There is norapid convergence to a cascade. In fact, there are no cascades when signals are


Fig. 2. The ex ante probability that a cascade will not occur as a function of signal accuracy and thenumber of agents choosing.

below a threshold of accuracy. When signals have high accuracy, informationcascades do appear but they are fragile* they break and then reappear* andthe lengths of cascades correspond to &bursts' that appear to have a power lawdistribution.

3.3. Action-based information cascades and the distribution of choices

Does this simple action-based information cascade converge to a distributionof box o$ce revenues? How closely does the distribution to which the informa-tion cascade converges reproduce the empirical distribution of box o$ce rev-enues? A number of studies show that the distribution of motion picture boxo$ce revenues is well "tted by a Pareto}LeH vy (stable) distribution with in"nitevariance. A standard approach when dealing with such distributions is toexamine their upper tail, typically the top 10%, where they are power laws andto estimate a measure of weight in the upper tail. The value of the weight in thetail is called the tail index a.

We estimate the tail index by applying least-squares regression to the upper10% of the observations generated by the simulations. In this upper tail, a LeH vystable distribution is asymptotically a Pareto distribution of the following form:

P[X'x]&x~a for x'k, (1)

where x is the number of agents and k is &large'. A low value of the tail indexa corresponds to a slow decay of the tail and, hence, to what is called a heavytail. For a LeH vy distribution with a "nite mean, a must be in the interval (1, 2).


Fig. 3. Informational cascades among 20 movies.

We want to see if the action-based information cascade model is capable ofgenerating outcomes similar to the empirical "ndings. We allocate 2000 agentsto 200 movies via our agent-based version of the BHW model. This largenumber of movies is required to give the degrees of freedom to estimate the valueof a from only the top 10 percent of movies. For each value of p, we run 20simulations. The statistics of our simulations are in Table 2.

Our simulation results indicate that higher values of signal accuracy areassociated with heavier-tailed distributions of agents over movies (i.e. lower a)which place more weight on extreme values of box o$ce revenue outcomes. This


Table 2Summary statistics of tail index a from simulations

p Min a Max a Mean a S.D. a Median a

0.5 1.02 3.60 2.23 0.65 2.110.6 0.86 3.56 2.21 0.73 2.300.7 0.27 2.60 1.66 0.63 1.670.8 0.31 2.13 1.34 0.43 1.360.9 0.14 1.20 0.56 0.29 0.51

result appears to be stronger for higher values of signal accuracy, which isconsistent with our earlier simulation results that show that informationalcascades are less fragile for higher values of p. The a estimates have a lowerstandard deviation at high values of p, which further supports this view. Theseresults suggest that action-based informational cascades produce highly skewedand heavy-tailed distributions, provided that signal accuracy is su$ciently high.

A value of p of 0.7 gives a value of a of 1.5 that is virtually the same as thevalue found in De Vany and Walls (1996, 1999a), Lee (1999), Walls (1997) andSornette and Zajdenweber (1999) for the movies. This consensus regarding thevalue of a can be considered to be fairly reliable as the aforementioned studiescover di!erent samples, countries, and time frames, yet they all "nd a value ofa close to 1.5.

It is known that the variance of the Pareto}LeH vy distribution is in"nite whenthe value of a is less than 2 and that the mean is in"nite when a is less than 1. Asthe results reported in the table indicate, at a signal accuracy of 0.7 and higher,the second moment of the distribution is in"nite and, at accuracy 0.9, even themean is in"nite. An information cascade model is capable of generating both theform and the values of the parameters of the distribution of box o$ce revenues.We obtain the LeH vy-stable form of the distribution and the right value of a.

4. Models of action and quality-based information cascades

In the preceding model, the agents know nothing about the quality of eachmovie. They receive a private signal about the quality of the movies available,but they only observe the actions of the other agents. They learn nothing fromthe other agents about the quality of the movies they saw. Given this limitedinformation, a string of bad choices can become such a powerful signal thata cascade might lock the audience on an inferior choice. Is it the lack of qualityinformation from other agents that determines this outcome? Or, do the agents&see too much' when they observe the actions of all the other agents so thataggregate information becomes a too-compelling statistic? How robust are


information cascades when agents can pass quality assessments to one another?What happens when the con"dence that agents place in the assessments of otheragents is not exogenously set but evolves endogenously with the accumulationof evidence?

We extend our agent-based model in several ways to investigate these ques-tions. We assume, as above, that an agent can observe the choice of a local agent(the agent just ahead of her in the line). In addition, agents can communicatetheir assessment of the quality of their selection to a neighbors concerningproducts they have chosen. The con"dence an agent attaches to another agent'sassessment is uniform among agents and exogenously "xed at n (this is madeendogenous in the extended version of the model below). This model capturesthe interaction of &word of mouth' and other sources of quality information withbox o$ce reports.

The structure of the model is as follows. Let there be a sequence of agentsi"1, 2, 3,2 each deciding whether to see one of the m movies or no movie atall (see Fig. 4). Agent i#1 observes the action of agent i before her. If agenti chose movie j, agent i#1 will ask her about its quality. If agent i tells agenti#1 that the movie is good, then there is a probability n that agent i#1 will seethe movie. We can interpret n as the con"dence that agent i#1 attaches toagent i's evaluation of movie j. A low n implies that agent i#1 might not havecon"dence in agent i's evaluation. If this happens, there is a greater chance thatagent i#1 will #ip a coin to decide either to see one of the m movies or not tosee a movie at all. If agent i#1 decides on the former, she will choose a moviefrom a list movies with the top n market shares (n(m and n takes values 5, 10and 20).

The decision process for a bad movie is modeled similarly (see Fig. 4). If agenti did not see a movie, agent i#1 #ips a coin to decide either to see a movie ornot to see any movie. If she decides on the former, she will choose randomly onemovie from the top n movies.

If agent i#1 lacks her neighbor's evaluation or has no con"dence in herneighbor's evaluation, she resorts to the market shares of the leading n movies tohelp her decide which movie to see. If agents are evenly distributed amongmovies, aggregate information is di!use and uninformative. In contrast, if agentsare unevenly distributed over movies, the top n movies readily can be identi"ed.

4.1. Accuracy of signals and quality and action-based information cascades

As in the simulations for the previous model, 2000 agents are sequentiallyallocated to 20 movies. Each movie is allocated one agent at the beginning. Tobegin the simulation, the "rst of the 2000 agents is allocated randomly to one ofthe 20 movies. When agents do not rely on a neighbor's evaluation, they choosea movie from the top-"ve (n"5). We carry out 20 simulation runs for each valueof n from 0.5 to 0.9 with an increment of 0.05. Half the movies are good and half


Fig. 4. The quantity- and quality-signals model.

are bad. The mean number of correct choices made by the agents for each valueof n is given in Table 3.

The mean proportion of correct choices declines as the con"dence leveln declines; in fact, the proportion of correct choices is nearly identical to theaccuracy of the signal. The proportion of correct choices is higher when qualityinformation is available than when it is not* the proportion of correct choiceswhen there is no quality information was only about 70% of signal accuracy,whereas it is about 100% of signal accuracy when there is quality information.The transmission of information about quality increases the proportion of correctchoices and this &word of mouth' information appears to have a strong e!ect.

Do informational cascades occur more often when there is quality andquantity information than there is only quantity information? Fig. 5(a) and (b)are two examples from many simulations that illustrate a general conclusion.


Table 3The proportion of correct choices (C) under di!erent con"dence levels!

Con"dence level (n) Mean C

0.95 0.950.90 0.860.85 0.850.80 0.820.75 0.770.70 0.740.65 0.660.60 0.640.55 0.630.50 0.58

!Note: Global information involves top-5 market shares.

Information cascades occur only when n is high. Cascades are brief and morethan one movie has a cascade. Low accuracy in quality evaluation does not leadto herding on the box o$ce numbers. Low accuracy seems to keep people fromgoing to the movies in the "rst place. The ability to choose &none of the above' isa powerful constraint that prevents audiences from #ocking to the box o$celeaders. If quality information is accurate and a movie is good, then thedynamics can look like imitative #ocking because as audience size grows, morequality information is transmitted and market share grows. The growth inmarket share reinforces the #ow of quality information but does not drive it.

4.2. Quality and quantity-based information cascades and heavy tails

Now we investigate the e!ect of the con"dence that agents place in word ofmouth information on the shape of the distribution by varying the value ofn and estimating a. Two thousand agents are allocated to 200 movies. For eachvalue of n, we run twenty simulations. The summary statistics of our simulationsare in Table 4.

The values of a generated by the quantity and quality signals model are lowerthan those obtained with the quantity signal model. This means the distributionis more unequal and skewed and that more mass lies on extreme values of boxo$ce revenues. Because all the a values estimated are less than 1, both the "rstand second moments of the box o$ce distribution do not exist. An interestingresult is that the value of a is a U-shaped function of n (it initially declines andthen increases as n increases). The minimum a occurs around n"0.7. Thissuggests that there may be a critical value of p at which the distribution of agentsis maximally heavy-tailed. Clearly, the cascades simulated with this model resultin distributions that are too heavy-tailed to "t the movies. In part, this is due to


Fig. 5. Information cascades with quality and quantity signals.

the assumption that half the movies are good and half are bad. When most themovies are good or bad, the values of a are closer to the empirical value of 1.5.

But, there is another factor that seems to be driving the cascade * thecon"dence that agents place in quality information is exogenously set andindependent of the weight of the evidence contained in the aggregate informa-tion. In our next model, we let that con"dence evolve with the accumulation ofevidence.


Table 4Summary statistics of tail index a from simulations

n Min a Max a Mean a S.D. a Median a

0.5 0.343 2.195 0.691 0.449 0.4930.6 0.014 1.297 0.577 0.272 0.5560.7 0.002 1.114 0.562 0.278 0.4910.8 0.019 1.655 0.647 0.365 0.5270.9 0.321 2.139 0.735 0.427 0.578

4.3. A model of endogenous transition between local and global information

In the quality and quantity-based model agent i#1 switches from quality tomarket share information when she has little con"dence in agent i's evaluationor when she cannot get information from agent i about any of the moviesavailable. In these cases, she #ips a coin to choose between a movie with a largemarket share and no movie.

The agent's choice between local and aggregate information is in#uenced byher con"dence level in the local information about quality that she gets fromother agents. We extend that model now to let the con"dence that an agentattaches to another's assessment be endogenous. We assume that when a moviehas a large market other agents will place con"dence in an agent evaluation of it(because it is con"rmed by many other choices). Agent i#1 places con"dencenij" movie j's market share in agent i's quality assessment of j. The same

con"dence is placed in all agents. Thus, if movie j's market share is very small,agent i#1 might not have con"dence in agent i's evaluation of it. We alsoassume that agents have bounded capabilities and can keep track only of the top5 movie shares (this is about what is reported on the evening news).

We do twenty simulations in which 2000 agents are allocated to 200 moviesand obtain the mean value of the proportion of correct choices made using thismodel. Table 5 summarizes our results for both the mean value of the propor-tion of correct choices (C) and the tail index (a).

The proportion of correct choices in this model with endogenous nij

seems tobe low in comparison to the earlier models with n (con"dence level) "xed orp (signal accuracy) "xed. In the twenty simulation runs, none of the correctchoices exceeded 58%. Similarly, the distribution is more heavy-tailed than inthe previous models.

We plot the proportion of correct choices, C, against a in Fig. 6. The tail indexa does not appear to be correlated to the proportion of correct choices made.This is a model of extreme cascades because, even though quality informationcan be communicated, it carries weight only when market share is high. Theprocess spends a good deal of time making random choices until market shares


Table 5Summary statistics of the proportion of correct choices (C) and the tail index a

Min Max Mean S.D. Median

C 0.05 0.58 0.32 0.16 0.27a 0.11 1.16 0.45 0.22 0.37

Fig. 6. Tail index a relative to the proportion of correct choices C.

begin to di!erentiate, if they do. If the shares do become di!erentiated enoughfor a top 5 to emerge, a good movie in that group quickly captures a command-ing lead because the quality evaluations reinforce the market share signal. Byadding a weight that is non-linear in a movie's market share to the con"dencethat an agent attaches to another's quality assessment, we obtain a dynamic thatstrongly selects good movies. But, it drifts so long at low resolution that it makesa low percentage of correct choices.

Does this outcome depend on the number of movies whose market sharesenter into the agent's decision? To examine this question, we ran simulationswhen a di!erent number of movie market shares are reported (n"5, 10, 15, and20). Tables 6 and 7 summarize the results. We observe that there is no discerniblerelationship between the number of movies whose shares are reported and theproportion of correct choices (Table 6). There is a slight increase in a (re#ectingless tail weight) as n increases (Table 7), so that a broader reporting of the &topmovies' on the evening news might well reduce the concentration of box o$cerevenues.


Table 6Summary statistics for the proportion of correct choices (C) under di!erent top-n market shares

n Min C Max C Mean C S.D. Median C

5 0.05 0.58 0.32 0.16 0.2710 0.07 0.49 0.29 0.11 0.3015 0.08 0.43 0.29 0.10 0.2820 0.16 0.38 0.26 0.06 0.27

Table 7Summary statistics for the tail index a under di!erent top-n market shares

n Min a Max a Mean a S.D. Median a

5 0.11 1.16 0.45 0.22 0.3710 0.04 0.89 0.40 0.18 0.3715 0.06 0.90 0.48 0.18 0.4920 0.14 1.47 0.78 0.36 0.86

5. A closer look at dynamics and initial conditions

We assumed that agents were uniformly distributed among movies to beginthe simulations. This is in sharp contrast with the actual situation in the movies,where the distribution of theater screens and opening week revenue is highlyuneven. We also assumed that quality (good and bad) was evenly distributedamong movies. These assumptions may be too restrictive if we are interested inmodeling the observed dynamics and distributions of box o$ce revenues in themotion picture industry. In this section, we brie#y explore the importance ofthese assumptions.

In analyzing these issues, we use the most complete model that contains thequantity and quality signals with an endogenous transition between local andglobal information. For simplicity, we simulate this model with 1000 agents and5 movies. With only 5 movies we cannot estimate the stability index, but we canobserve the dynamics in more detail. The simulations then are tournamentsbetween good and bad movies in which we vary the number of good versus bad.

We assume all movies are bad. Then we seed the movies with opening marketshares that are di!erent from one another as shown in the "gures. Figs. 7 and8 show two di!erent dynamics from the same initial conditions. None of themovies achieve a very high market share and there are evident crossing pointswhere one movie overtakes another. The dynamics do settle down and the "naldistribution of audiences is quite uneven, and unrelated to the initial conditions.


Fig. 7. Distribution of agents when all "ve movies are bad.

Fig. 8. Distribution of agents when all "ve movies are bad.

Of course, there is no room for quality information to distinguish among moviessince they are all bad. Most people end up not going to a movie in this case.

In contrast, if we assume that all movies are good then we begin to see caseswhere a movie captures a substantial market share. Figs. 9 and 10 summarizetwo extreme examples where we assume that all movies are good. In Fig. 9, the"rst movie begins with an initial market share of 50% but ends with a "nalmarket share of only 0.1%! However, a movie that began with a large initialmarket share can increase its share further as shown in Fig. 10. What thesesimulations show is that extreme inequality in the distribution of agents canoccur when only there are good movies. In this case, quality information doesnot distinguish among movies, but market shares are reinforced by positiveevaluations and become highly credible signals of quality. Now informationcascades do occur and are readily detected. This suggests that cascades are morelikely when they are based on correct information. This point is driven home inthe following simulations of tournaments between bad and good movies.


Fig. 9. Distribution of agents when all "ve movies are good.

Fig. 10. Distribution of agents when all "ve movies are good.

Fig. 11 shows that a bad movie with large initial market share may gain thehighest market share, but generally will not. Its market share will never be verylarge (typically well less than 50% and never higher than 60%). Similarly, a goodmovie with small initial market share may (Fig. 13) or may not (Fig. 14) gain themarket. A good movie's market share can be very large (close to 99% as in Fig. 13).

Fig. 11 illustrates the &Godzilla' e!ect where a bad movie with a large openingshare of theater screens gains a substantial share, but one that is well below itsshare at the opening. Fig. 12 illustrates the &Full Monty' e!ect where a goodmovie with a tiny opening gains a large following. Fig. 13 illustrates the &Titanic'e!ect where a good movie with a large opening share gains a large market shareand kills the other movies playing against it. Fig. 14 tests your faith in people,just as the movies always seem to do, for it depicts a good movie opening big andplaying against only bad movies that, nonetheless, goes on to mediocre results.Nearly every producer can name a movie like this in her/his portfolio with highpromise that died for inexplicable reasons. Our answer is, &It's the dynamics,


Fig. 11. Distribution of agents when one movie is bad and four are good.

Fig. 12. Distribution of agents when one movie is bad and four are good.

Fig. 13. Distribution of agents when one movie is good and four are bad.

stupid.' The individual characteristics of a movie are not su$cient to determinehow it will fare in the complex dynamics of the motion picture market. Nota very satisfying answer for a movie executive to hear, but one that is closer tothe truth than any other answer that can be given.


Fig. 14. Distribution of agents when one movie is good and four are bad.

How did a bad movie with only 2% of the opening go on to gain 58% of themarket against a good movie that opened with 50%? It happens; that is themovies. In this simulation there is an early period of extreme variation of marketshares that weakens the credibility of quality signals. That was enough to startan up cascade in the bad movie and a down cascade in the good one. Once a badmovie gets a large share, the shares of good movies are too small for word-of-mouth evaluations to be credible. So, even poor quality evaluations are notsu$cient to overcome the power of the signal contained in market share. A badmovie can gain a large market share from purely random circumstances. But, itcannot become a monster hit like a good movie can. Moreover, when a goodmovie gains a commanding market share total box o$ce revenues rise becausemore people choose to go to the movies. &Titanic' increased the total box o$cerevenues of all movies during its run even while it drained revenue away fromthe "lms it played against.

How true to life are these sorts of complex dynamics? Very true, it turns out.To give a sense of how variable this market is we plot the data in Fig. 15. The"gure shows the evolution of market shares for 115 "lms on Variety's top-50 listover a period of 55 weeks (from 24 May 1996 to 12 June 1997). The "gure lookslike a mountain range; there is a lot of low-level variation in the foothills toppedwith a few intermittent peaks. In fact, like a mountain range, it is a fractal andthe height variations are a power law, often taken to be a signature self-organized behavior.

6. Conclusion

Information cascades are more complex objects than standard models suggestthem to be. Their dynamics are richly varied and intermittent and they can goalmost anywhere. Cascades do not occur at high frequency and they usually do


Fig. 15. Market shares of 115 "lms on the Variety's top-50 list over 55 weeks (24 May 1996 to 12June 1997).

not converge rapidly or narrowly on one or two movies. The complex #ows ofchoices revealed in our models suggest that the agents are processing a high levelof information. The high information content of the choice sequences is shownby the fact that each sequence appears to be a random series. In other words,there is no model or algorithm that will generate the series that is shorter thanthe series itself. A series whose shortest description is its self is algorithmicallyincompressible so that each element in the series reveals new information.An information cascade is algorithmically compressible; once one is started,subsequent choices are easily predicted (the algorithm is that each choice is thesame as the previous choice). Thus, information cascades do not reveal newinformation.

Our main "nding is that cascades may coexist in an &punctuated equilibrium'in which a cascade bursts to the lead and then gives way to a rival and thenreturns to the lead again. When there are many choice alternatives, cascades areintertwined and do not persist; this intermittence may make it impossible toidentify information cascades empirically. And, it is impossible to predict thedynamic path or "nal outcome of a competition among cascades because theirbasins of attraction are the stable distributions that have an in"nite varianceand may have an in"nite mean. The mixing of information cascades of many


scales among competing choices results in a turbulent #ow to outcomes thatcannot be predicted. Our results suggest that the pessimistic conclusions ofmodels of information cascades may come from the restrictions they impose onthe number of choices available and on the ability of the agents to communicateto others the quality of the products they chose. When these restrictions areremoved, information cascades are less likely to occur and when they do theyselect the best products.

7. For further reading

The following references are also of interest to the reader: Bak, 1996; Chen,1978; Cont and Bouchaud, 1998; Hill, 1974; Mandelbrot, 1997.

References

Anderson, L.R., Holt, C.A. Holt, 1997. Informational Cascades in the Laboratory, AmericanEconomic Review 87, 847}862.

Bak, P., 1996. How Nature Works. Copernicus, New York.Banerjee, A.V., 1992. A simple model of herd behavior. Quarterly Journal of Economics 107,

797}817.Bikhchandani, S., Hirshleifer, D., Welch, I., 1992. A theory of fads, fashion, custom, and cultural

change as informational cascades. Journal of Political Economy 100, 992}1026.Chen, W.-C., 1978. On Zipf 's law. Ph.D. Dissertation, University of Michigan.Cont, R., Bouchaud, J.-P., 1998. Herd behavior and aggregate #uctuations in "nancial markets.

Unpublished.De Vany, A., Walls, D., 1996. Bose}Einstein dynamics and adaptive contracting in the motion

picture industry. Economic Journal 106, 1493}1514.De Vany, A., Walls, D., 1999a. Uncertainty in the movie industry: does star power reduce the terror

of the box o$ce?. Journal of Cultural Economics 23, 285}318.De Vany, A., Walls, D., 1999b. Screen wars, star wars, and turbulent information cascades at the box

o$ce. Unpublished.Hill, B.M., 1974. The rank-frequency form of Zipf 's law. Journal of American Statistical Association

69, 1017}1026.Lee, C., 1999. Competitive dynamics and the distribution of box o$ce revenues in motion pictures:

the Stable}Paretian hypothesis. Ph.D. Dissertation, University of California, Irvine.Lee, I.H., 1998. Market crashes and informational avalanches. Review of Economic Studies 65,

741}759.Mandelbrot, B., 1997. Fractals and Scaling in Finance. Springer, New York.Sornette, D., Zajdenweber, D., 1999. Economic returns of research: the Pareto law and its implica-

tions. Unpublished.Walls, D., 1997. Increasing returns to information: evidence from the Hong Kong movie market.

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