Observational Learning in Congested Environments with MultipleChoice Options: The Wisdom of Majorities and Minorities

Chen Jin

Department of Industrial Engineering and Management Sciences Northwestern University

Laurens Debo

Booth School of Business, University of Chicago

Seyed Iravani

Department of Industrial Engineering and Management Sciences Northwestern University and

Mirko Kremer

Frankfurt School of Finance and Management

April 15, 2015

Abstract: We study the e↵ect of observational learning in a congested environment in which customers choose

among several alternatives (“locations”). While the locations’ quality is known to informed customers, unin-

formed customers infer quality from the locations’ congestion level (i.e., queue length at each location). We

introduce two new rational joining strategies, that stand in contrast to the “join the shortest queue” strategies

typically studied in the queuing literature. We characterize the conditions under which uninformed customers

should “join the longest queue” (i.e., the majority) and when they should “join the shortest non-empty queue”

(i.e., the minority), such as the single customer that seemingly chose against the wisdom of the majority (the

“contrarian”). Congestion costs “mutes” the observational learning process, as uninformed customers may

choose a short queue even when they expect higher quality from a location with a long queue, obfuscating what

subsequent customers can learn from the queue length at di↵erent locations. While the presence of informed

customers increases the overall welfare of the system, the average utility of uninformed customers may actually

decrease. We test our rational model in the laboratory, and find strong evidence for observational learning from

congestion levels. However, relative to theoretical predictions, uninformed subjects in a low congestion cost

environment tend to ignore the wisdom of minorities, and tend to follow more often the wisdom of majorities.

This observation is consistent with the notion of “random choice”: the possibility that minority queues may

have been formed only by mistake, significantly diminishes their signalling value in favor of longer queues.

Key Words: Queueing Game, Observational Learning, Wisdom of the Minorities, Contrarianism, Experiments

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1 Introduction

The rise of large online market places presents customers with unprecedented choice opportunities. At the same

time, product (or service) proliferation leads to information proliferation, which renders choice di�cult. On

eBay.com, for example, customers can choose from literally hundreds of options in every imaginable product

category. Even after narrowing down the market’s o↵erings based on criteria such as price, functionality and

reviews, customers are still facing a large choice set. For instance, parents who search children’s books can find

many books that are suitable for their children’s age, are within the price range, and have comparable reviews

(e.g., “top rated plus”), as shown in Figure 1 (left panel). Another example is ZocDoc.com, an online medical

care scheduling service, that provides free medical search for patients who look for physicians. There are several

physicians to choose from, even after patients limit their search based on speciality, geographical location and

insurance provider. For example, after inserting her zip code and insurance plan on the ZocDoc website, a

patient will see a list of several dentists in her neighbourhood along with their available appointment times as

shown in Figure 1 (right panel). Importantly, patients can make appointments directly on the website.

Figure 1: (Left) Information for Children’s Books on eBay.com; (Right) Dentist Search Results on ZocDoc.com

Although online platforms such as eBay.com and ZocDoc.com allow customers to access and reduce vast

amounts of information quickly, the resulting choice sets typically include multiple options (e.g., books within

same price range, functionality and comparable reviews; dentists under the same insurance plan, geographical

location, and comparable reviews), with residual quality uncertainty among them. In such circumstances,

observational learning from other customers’ choices can become a powerful driver of market shares, as customers

may infer quality from high sales (eBay) or crowded schedules (ZocDoc). A key antecedent for such learning

processes is the type of information asymmetry that characterizes many markets—some customers have more

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accurate information about the quality of options than other customers. For instance, preschool teachers might

be more informed about the quality of children’s books, relative to other parents. Similarly, after many years

of living in the same neighbourhood, a patient is more informed about the quality of local dentists, relative to

her neighbours who just moved in. Under such conditions of quality uncertainty and the presence of informed

customers, other relatively uninformed customers may rely on the popularity of each option (e.g., books sold

or doctors booked) to learn (infer) its quality. Naturally, such learning depends on what can be observed about

other customers’ choices, and many firms use available technology to make such information readily available

to their customers, see e.g. Figure 1. eBay.com displays the number of books already sold, which indicates

the popularity of each book and may a↵ect the uninformed customers’ perception of quality. Similarly, on

ZocDoc.com, the patient can visit Dr. G1 immediately since no patients have booked this doctor but she cannot

see Dr. J for at least one week, since Dr. J’s schedule is full in the next week. Dr. J’s busy schedule may also

a↵ect the uninformed patients’ perception of his quality.

While informed customers exert positive information externalities, as they render aggregate choice measures

(sales or schedules) into informative signals, the ZocDoc example highlights an important situation in which

previous customers’ choices may impose negative congestion externalities on subsequent customers as well—an

uninformed patient may infer from Dr. J’s full schedule that she is likely to be a good dentist, but the cost for

waiting for a week is high (e.g., if su↵ering from toothache). In this case, an uninformed patient might want

to schedule with another less popular dentist so as to get immediate treatment. In general, customers’ choices

may exert only a positive information externality in some systems (e.g., books on eBay), while other systems

may be characterized by both positive and negative externalities (e.g., physicians on ZocDoc).

How do customers choose among several options after observing the popularity of each option? And how does

the presence of congestion cost modulate the observational learning of uninformed customers? We study these

questions both theoretically and experimentally. We consider a simple setting where a population of customers

visits a market place sequentially, and choose among several options. The quality of each option is uncertain,

and a priori (i.e., before choosing) known only to informed customers, but not to uninformed customers. Upon

arrival, customers observe aggregate sales (queue length) for each of the options. Furthermore, our model

incorporates congestions cost— when it is non-negligible, and all else (e.g., quality) being equal, customers have

a preference for options with shorter queue (i.e., low aggregate sales).

We characterize the equilibrium choice strategy for both customer types, under di↵erent congestion cost

levels. While the informed customers’ decision is generally trivial, we show that the equilibrium strategy for

uninformed customers is complex without a general structure. We characterize the conditions under which

uninformed customers should: (1) avoid empty queues; (2) follow the crowd, i.e., join the longest queue, and (3)

1We covered the name, address and shaded the face of the dentists in Figure 1 to keep anonymity.

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avoid both the longest queue and the empty queues and choose the shortest non-empty queue. On the aggregate

level, we find that high quality options attract more customers than low quality options. Furthermore, while

the presence of informed customers makes the uninformed customers better o↵ when the congestion cost is

negligible, surprisingly, the presence of informed customers does not o↵er any benefit to uninformed customers

when the congestion cost is high.

We test the key predictions from our equilibrium analysis with human subjects under controlled laboratory

conditions. Our experimental data shows that uninformed customers overwhelmingly deviate from the shortest

non-empty queue strategy and flock to the longest queue. To adjust for this deviation, we extend our rational

model to Quantal Response Equilibrium (QRE) model that incorporates the possibility that customers make

errors when choosing an option. The QRE model provides a plausible explanation for why uninformed subjects

choose the longest queue, in situations for which our rational model predicts that they should choose the shortest

non-empty queue. We also find that the introduction of congestion costs “tames” the observational learning of

uninformed customers, resulting in a more balanced demand allocation, which is consistent with our theoretical

prediction. Finally, as predicted by our model analysis, uninformed customers do not benefit from the presence

of informed customers when congestion cost is significant—an e↵ect that disappears when congestion costs are

negligible.

2 Related Literature

We focus on situations where each customer’s choice a↵ects beliefs and choices of subsequent customers in two

ways. On the one hand, a particular choice may impose a negative “payo↵” externality, e.g., congestion cost

when customers derive disutility from the fact that other customers already chose the same location. On the

other hand, informed customers exert a positive “informational” externality as their choice may help subsequent

uninformed customers refine their beliefs about each choice option’s value. Our study sits in between two

extensive streams of literature. While the Operations Management literature on queuing systems is primarily

concerned with negative externalities, the economics literature on herding and information cascades has been

primarily concerned with positive externalities.

Since the seminal work of Naor (1969), the queuing literature predominantly model customers arriving at

a service system as decision makers that maximize their expected utility by selecting whether to join a queue

or not, after observing its length. As the expected net utility naturally decreases in queue length, due to

congestion costs, a rational customer joins only when the queue is short enough, i.e., below some threshold.

Subsequent literature (Winston 1977; Weber 1978; Whitt 1986) has extended this notion to environments that

require customers to select one out of many queues, and shows that a variation of join the shortest queue

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strategy is rational, depending on the service time distribution. A central assumption in the literature, and

underlying the preference for shorter queues, is that the quality of service is the same for all servers and is

common knowledge. Our study relaxes the assumption that quality is commonly known, yielding a new class

of (queue) joining strategies that is driven not by assumptions related to service times, but by the positive

informational externalities due to other customers’ choices upon arrival.

To sharpen our focus on the resulting observational learning process, we do not consider a stochastic service

departure process, such that each arriving customer can observe the total number of customers that have chosen

each location (e.g., eBay.com and ZocDoc.com). Together with information asymmetry that transforms queues

into quality signals, the resulting transient regime connects our research with the settings typically studied

in the literature on observational learning. In two seminal contributions, Banerjee (1992) and Bikhchandani

et al. (1992) model a sequence of rational Bayesian decision makers with private information about two choice

alternatives, and characterize conditions under which decision makers rationally ignore their own private infor-

mation and instead follow the crowd, i.e., the choices of their predecessors. A central assumption in these early

papers is that decision makers possess perfect information about the entire history of their predecessors’ deci-

sions. Celen and Kariv (2004b) considers the case when customers only observe the decision of their immediate

predecessor. Smith and Sørensen (2008) assumes that customer only sees unordered random samples from the

action history. Similarly, Acemoglu et al. (2011) assume that decision makers have access to a random sample

of previous choices, but know their own position in the sequence, as well as the position of their randomly

sampled predecessors. Monzon and Rapp (2014), Guarino et al. (2011), and Herrera and Horner (2013) relax

the assumption that customers know their position in the sequence. In terms of what can be observed about

predecessors’ choices, our work falls in between the extreme settings previously studied: like Callander and

Horner (2009) and Hendricks et al. (2012), we assume that decision makers observe the aggregate choices (e.g.,

total number of customers who chose each option), which is less informative than the entire choice history (e.g.,

Banerjee 1992; Bikhchandani et al. 1992), but more informative than a random sample of previous choices (e.g.,

Smith and Sørensen 2008).

Our study departs from the traditional observational learning literature reviewed above in two major ways:

congestion cost and multiple options to choose from. Studies on observational learning under congestion are

scarce. Modeling both congestion costs and a stochastic service process, Veeraraghavan and Debo (2008),

Veeraraghavan and Debo (2011) (with two queues), and Debo et al. (2012) (with a single queue) demonstrate

that non-threshold strategies emerge in equilibrium, which is in stark contrast to the above mentioned queue-

joining literature without information externalities. Veeraraghavan and Debo (2011) and Callander and Horner

(2009) identify “wisdom of minorities”in environments with congestions cost, customer turnover and capacity

constraints. Eyster et al. (2013) provide conditions on congestion costs that result in bounded learning in the

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original setup of Bikhchandani et al. (1992).

Almost exclusively, the literature has considered social learning environments where exactly one out of two

options has high value, and the other option has low value. The analysis of observational learning with arbitrary

many options in the choice set such as books sold on eBay or doctors on the ZocDoc website requires a more

general value model. We consider a system where the value of each of several options is independent of the

value of the other options and either high or low (with a commonly know prior probability), such that the

observational learning process is not predicated on the identification of the single high-value option, but on

finding one from a random set of high value options.

A small number of experimental studies have tested whether and how theoretical predictions translate into

observational learning among human subjects. Anderson and Holt (1997) find that subjects’ actions generally

align with the Bikhchandani et al. (1992) model, and Goeree et al. (2007) find evidence that information cascades

occur much less frequently than predicted by Bikhchandani et al. (1992). Similarly, Allsopp and Hey (2000) find

that herding occurs less frequently than predicted in the setting of Banerjee (1992). Celen and Kariv (2004a)

and Celen and Kariv (2005) studies observational learning in the Celen and Kariv (2004b) setting where decision

makers can only observe the action of their immediate predecessors. Celen and Kariv (2005) find less imitative

behavior under such “imperfect information” conditions, compared to settings where agents observe the entire

choice history. Experimental work on settings with positive informational and negative payo↵ externalities

is relatively scarce and is limited to situation with two options including exactly good option. Kremer and

Debo (2014) show that, in a single queue setting with informed customers and congestion costs, uninformed

customers infer quality from queues, may indeed join longer queues, but also tend to balk from empty systems.

A recurring theme in these experimental studies is the notion of “random noise” (also considered by Smith and

Sørensen 2000 in their theoretical developments). We explicitly incorporate random choice in our data analyses,

to account for the fact that observational learning may be imperfect not only because of structural properties

of the learning environment (such as the extent to which previous choices can be observed), but also because of

bounded rationality on the part of the decision makers involved in the learning process.

3 The Model

We consider a system with K 2 N locations, where N is the set of natural numbers (we define N0

, {0} [ N).

For instance, in the eBay and ZocDoc examples, a location refers to a book and a physician, respectively. The

quality of location k for 1 k K, denoted by ✓k, can be either H igh or Low, i.e., ✓k 2 {H,L}. Nature

determines the quality of each location before the game starts. Each location has probability p0

2 (0, 1) of

being high quality and, hence, with probability 1 � p0

of being low quality. The quality of each location is

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independent of the quality of any other locations. The high (low) quality location has gross value vH (vL) with

0 vL < vH . For ease of reference, we introduce v0

, vL + p0

(vH � vL) as the prior expected value of any

location. A total number of N 2 N customers arrive in a random sequence that assigns a unique rank r to

each customer, i.e., r 2 {1, 2, ..., N}. Each arriving customer can be either informed (with probability q) or

uninformed (with probability 1� q). We use � 2 {i,u} to denote the type of a customer. An informed customer

(“i”) knows the quality of each location, while an uninformed customer (“u”) only knows the prior probability

p0

. Upon arrival, a customer observes the total number of customers at each location, which we refer to as

the location’s “queue length” or “congestion level”. To accommodate the idea that customer’s utility depends

both on the gross value of the location (v✓k) and the congestion level, we model the utility function as follows.

Suppose a customer selects the kth location (where 1 k K) whose quality is ✓k and has been chosen by nk

customers already, then her utility is:Uk , v✓k(1� �)nk ,

where � 2 (0, 1) captures the cost of congestion. For instance, � close to 1 means extremely high congestion cost,

while � = 0 means no congestion cost. The multiplicative utility function implies a positive expected utility for

0 vL < vH and p0

> 0. Thus, customers always prefer joining one of the K locations over the outside option

(of not choosing any location), which we assume to have zero value.2 Furthermore, we assume that customers

cannot renege (i.e., leave a location after having joined it) nor jockey (i.e., change from one location to another

location). We assume both informed and uninformed customers are rational and risk-neutral. In summary, this

game is fully characterized by the set of parameters (K,N, q, p0

, vL, vH , �).

State space: Uninformed customers only observe the queue length at each location upon arrival, but not

their quality. Let n = (n1

, n2

, ..., nK) 2 NK0

denote the state observed by uninformed customers, where nk is

the queue length at location k and 1 k K. Informed customers, on the other hand, observe both queue

length and quality of each location. Therefore, let n , ((n1

, ✓1

), (n2

, ✓2

), ..., (nK , ✓K)) 2 (N0

⇥ {L,H})K be the

state observed by informed customers, where nk = (nk, ✓k) and ✓k 2 {H,L} is the quality of location k.

Decision Problem of Informed Customers. We indicate the queue-joining strategy of an informed

customer by ai(n) : (N0

⇥ {H,L})K ! [0, 1]K . Hence, strategy ai(n) is a probability vector whose kth entry,

aik(n), specifies the probability of choosing location k. Then the queue selection problem for informed customers

is straightforward:M i(n) , argmax

1kK

⇢(1� �)nk

vL + (vH � vL) {✓k=H}

��, (1)

where {z} , 1 if event z occurs, and zero otherwise. Notice that M i(n) is the set of locations with maximum

utility. Therefore, the probability of of informed customers choosing location k in state n is:

2A log transformation, which preserves preference, makes the congestion costs linear in the queue length (and the value of theoutside option is �1).

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aik(n) ={k2M i

(n)}

|M i(n)| , (2)

where | · | is the cardinality of set ·. Notice that 1 |M i(n)| K for all n. Essentially, an informed customer

simply chooses the location that yields the highest utility. If there are multiple locations generating maximum

utility (i.e., |M i(n)| > 1), then an informed customer randomly choose one of these locations.

Decision Problem of Uninformed Customers. As opposed to informed customers who choose among

locations based on the exact utility of each location, uninformed customers select a location based on the expected

utility of each location, after observing the current queue length at each location (n) and updating their quality

beliefs. We indicate the queue-joining strategy of an uninformed customer by au(n) : NK0

! [0, 1]K , where

au(n) is a probability vector whose kth entry auk(n), specifies the probability of choosing location k. Uninformed

customers, therefore, solve the following optimization problem:

Mu(n) , argmax1kK

⇢(1� �)nk [vL + (vH � vL)Pk(n)]

�,

where Pk(n) , Pr{✓k = vH |n} is the posterior probability that location k is of high quality given that the state

observed upon arrival is n. The probability of uninformed customers choosing location k in state n is:

auk(n) ={k2Mu

(n)}

|Mu(n)| . (3)

The posterior probability Pk(n) is, via Bayes’ rule,

Pk(n) =

P✓2{L,H}K

:✓k=H

Pr{n|✓}Pr{✓}P

✓2{L,H}K

Pr{n|✓}Pr{✓} , (4)

where ✓ , (✓1

, ✓2

, ..., ✓K) is the realization of the quality for each location. For location k, uninformed customers

consider all possible realizations of the quality among all locations such that the location k is of high quality,

i.e., the set {✓ 2 {L,H}K : ✓k = H}. For each given possibility in this set, uninformed customers evaluate the

probability of observing the current state n, which is Pr{n|✓}. The posterior is then obtained via (i) weighing

Pr{n|✓} by the prior probability Pr{✓} of having ✓ as the quality vector of all locations3 and (ii) via scaling byP

✓2{H,L}K

Pr{n|✓}Pr{✓}. The probability Pr{n|✓} can be computed recursively by:

Pr{n|✓} =X

1kK:nk>0

q aik(n

pk) + (1� q) auk(npk)

�Pr{npk |✓}, (5)

where npk = (npk ,✓), and npk is the predecessor of the current observed state n with one customer less at

location k with at least one customer (i.e. nk > 0), i.e., npk

k = nk � 1 � 0 and npk

k0 = nk0 for all k0 6= k. If the

previous customer is informed, which occurs with probability q, then with probability aik(npk) she chose location

3which is binomially distributed with parameter K (the number of trials) and success probability p0—binomial(K, p0) in short)

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k. If the previous customer is uninformed, which occurs with probability 1� q, then with probability auk(npk),

she chose location k. As the first customer observes for sure an empty system, denoted by 0, Pr{0|✓} = 1

initializes equation (5) for all possible realizations of ✓ in {L,H}K . We also define 0 , (0,✓) as the initial

state observed by an informed customers who happens to be the first arrival of the system. This completes the

description of the decision problem faced by informed and uninformed customers.4

A numerical example. The nature of the choice problem discussed above results in complex and coun-

terintuitive queue-joining strategy for uninformed customers. For illustrative purpose, consider a system with

K = 4 locations. Assume that each location has either value vH = 1000 (with probability p0

= 0.25) or vL = 200

(with probability 1 � p0

= 0.75); the proportion of informed customers in the population is q = 0.3 and the

congestion costs is negligible, � = 0.01. Suppose an uninformed customer arrives and observes the state in

Figure 2, i.e., n = (8, 1, 0, 0). Which queue should she join? Intuition may suggest to join the longest queue

(i.e., the most popular one), given that the negligible congestion cost (� = 0.01) would not reduce the value

of a location (vH or vL) by much. After all, 30% of customers are informed and there is a higher chance of

having informed customers at location 1, implying it is high quality. However, this is not the case. In the next

section, we provide the underlying rationale for why an uninformed customer should choose the location with

the shortest non-empty queue—location 2—instead of the longest queue. For expositional purpose and without

loss of generality, in this paper, we label the locations according to their queue length in descending order, i.e.,

nk � nk0 , 8 k0 > k.

Figure 2: A Possible State Faced by Customer with Rank r = 10

1 2 3 4 5 6 7 8 9 10 n1(0,0)

1

2

3

4

5

6

7

8

9

10

n2

=10, q=0.1

While it is rational for the tenth arriving customer to choose the shortest non-empty queue in this specific

4The complexity of the problem grows with the number of locations and customers in the system. In order to precisely computethe posterior probability, an uninformed customer with rank r has to consider 2K possible realizations of the location quality and2r�1 possible realizations of the customers’ type. Even after removing the redundant cases, the total number of possibilities is stillcomputational challenging. For example, with K = 10, N = 26. One needs to consider in total 7, 533 possible states. For each ofthese states, one has to enumerate all possible underlying compositions of high and low quality locations. This leads to 980, 419compositions in total. In addition, for each composition, there are a certain number of possible predecessors ranging from 1 to K.Hence it finally results in 4, 469, 860 states that needs to be created, examined and modified.

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state n = (8, 1, 0, 0), the queue-joining strategy is generally state-dependent. For our example, Table 1 lists

all eighteen possible states that can potentially be observed in equilibrium by our tenth customer, and maps

these equilibrium states (including the one from Figure 2, in bold) into four generic equilibrium strategies of

uninformed customers (underlined): (1) join empty locations; (2) join the minority (locations with shortest

non-empty queue); (3) join the majority (locations with the longest queue); (4) join locations that are not

described via (1) to (3). As this simple example shows, even when the congestion � = 0.01 is negligible, the

queue-joining strategy is not straightforward.

Table 1: State-dependent Strategy of an Uninformed Customer with Rank r = 10

Equilibrium Strategy: Joining States on the Equilibrium Path

(1) Empty Queues Null

(2) Minority Queues

(3,2,2,2),(5,2,1,1),(6,1,1,1),(3,3,2,1),(5,2,2,0)

(6,2,1,0),(4,2,2,1),(5,3,1,0),(7,1,1,0)

(4,3,1,1),(5,4,0,0),(8,1,0,0),(7,2,0,0)

(3) Majority Queues (3,3,3,0), (4,4,1,0), (6,3,0,0), (9,0,0,0)

(4) Other Queues (4,3,2,0)

4 Equilibrium Analysis

The central goal of our study is to characterize how observational learning shapes the equilibrium queue-joining

strategy of uninformed customers, and how the presence of congestion costs modulates the observational learning

dynamics.

We first briefly characterize the strategy of informed consumers, as this is the natural prerequisite for

uninformed consumers’ observational learning—after all, it is the presence and choices of informed consumers

that renders queues into informative quality signals for uninformed consumers. The proofs of all of our analytical

results are presented in Appendix A.

Proposition 1 The queue-joining strategy of informed customers is as follows:

(i) if there is no high quality locations in the system, informed customers join the shortest queue;

(ii) if there is at least one high quality location in the system, informed customers join a high quality location

with the shortest queue if that queue has no more than D customers than the shortest queue among low

quality locations, where D , inf{n 2 N : vH(1� �)n < vL}; otherwise, they join a low quality location with

the shortest queue.

Intuitively, informed customers favor high quality locations over low quality locations when they have the same

queue length. However, for any � > 0, it is not rational for informed customers to join a high quality location

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when its queue length is too long relative to the queue length at a low quality location. Let the shortest queue

among low (high) quality locations be of length m (length n+m). It is obvious that informed customers will not

join the shortest queue among high quality locations if vH(1� �)m+n < vL(1� �)m. Hence, D is the maximum

di↵erence between the queue length at any high quality locations and any low quality locations.

With the understanding of the queue-joining strategy of informed customers, together with the fact that each

location has non-zero probability of being high quality and that each customer already in the system is informed

with probability q, an uninformed customer is able to conclude that every customer in a queue increases the

location’s posterior probability of being of high quality, relative to a location without any customers in line.

The important implication is that non-empty queues are signalling high quality.

Proposition 2 With informed customers (0 < q < 1), the posterior probability (of being high quality) of any

location with a non-empty queue is strictly greater than the posterior probability of empty locations.

To gain qualitative insights into observational learning, in Section 4.1 we first study the case with vanishing

congestion costs, � ! 0+. We exclude the case � = 0 to ensure that the informed customers’ strategy is

uniquely defined in the sense that they prefer the one with the shortest queue when facing multiple locations

(with di↵erent queue lengths) of the same quality. In Section 4.2, we next study the e↵ect of congestion cost

(� 2 (0, 1)) on observational learning.

4.1 Vanishing Congestion Cost (� ! 0+)

Negligible, but non-zero, congestion cost (� ! 0+) implies D ! +1, i.e., informed customers never choose a

low quality location provided that there are at least one high quality location in the system. In the absence

of informed customers (i.e., q = 0), uninformed customers cannot infer quality from queue lengths, and hence,

uninformed customers assign the prior probability p0

to each location regardless of its queue length. Hence,

they simply choose the shortest queue. However, the presence of informed customers (q > 0) changes the queue-

joining strategy of uninformed customers systematically. In general, when the congestion cost is negligible,

choices of uninformed customers are driven entirely by their quality inferences. As informed customers never

choose a low quality location (if there is at least one high quality location), locations with non-empty queues

have higher chance of being high quality relative to empty locations. Except for the uninformed customer of

rank r = 1, who chooses randomly, other uninformed customers never choose an empty location.

Proposition 3 With vanishing congestion cost (� ! 0+), uninformed customers with rank r > 1 never join an

empty queue.

As the posterior probability of any location with non-empty queue is always no less than that of any location

with empty queue (Proposition 2), it is clear that Proposition 3 follows. While it is clear that uninformed

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customers never join an empty queue, the important question is: which locations do they choose? When the

congestion cost is negligible, the longest queue may look most attractive since the longer the queue is, the more

likely that there is at least one informed customer standing in the longest queue. Thus, the uninformed customer

should join the longest queue. However, this is not the case, as illustrated in the earlier example (Figure 2). To

gain more insights into the underlying drivers of joining the majority or minority, we focus on two particular

sets of states: the “single-queue state” and the “contrarian state”.

Wisdom of the Majority: Consider the single-queue state with only one non-empty queue in the system,

which implies that all previous customers have chosen the same location. Let nsn , (n, 0, ..., 0) 2 NK0

, i.e.,

the only non-empty queue in the system has length n. Recall that we label locations according to their queue

length in an descending order, i.e., nk � nk0 , 8 k0 > k. Also recall that Pk(n) is the probability of location k

being of high quality given the current observed state is n. As there is a non-zero probability that at least one

informed customer is in the single non-empty queue, we have the following proposition.

Proposition 4 With vanishing congestion cost (� ! 0+), in the single-queue state (nsn),

(i) uninformed customers always join the longest queue in state nsn for all n � 1;

(ii) the ratio P1(nsn

)

Pk(nsn)

is increasing in n, where k > 1.

Part (i) of Proposition 4 follows immediately from Proposition 3 since in the single-queue state, the longest

queue is the only non-empty queue. Part (ii) of Proposition 4 is also intuitive. Any customer who joins the single

queue in the system will not decrease the posterior probability of that single queue being of high quality relative

to the empty locations. Namely, if all the customers in that single queue are uninformed, then the posterior of

that queue equals that of empty locations. As each customer is informed with probability q, the likelihood of

having at least one informed customers in that single queue increases with the queue length. Hence, the signal

sent by that single queue intensifies with the queue length. This results in the “wisdom of the majority”, i.e.,

every uninformed customer joins the crowd—the queue that everyone else joined.

While not surprising in light of Proposition 2 and 3, Proposition 4 serves an important purpose. First, it

captures well the intuition that uninformed customers should “follow the crowd”, if there is a unique one. More

importantly, it provides a stark contrasts with the following analysis of states where uninformed customers

should not follow the crowd.

Wisdom of the Minority: Consider the “contrarian state”, denoted by ncn , (n, 1, 0, ..., 0), where the

first location has n > 1 customers, the second location has only one “contrarian” customer and the remaining

locations (from location 3 to location K) are all empty. The contrarian customer indicates that she made a

choice that, through the lens of an arriving uninformed customer, stands in contrast to the choices of all other

customers. The following proposition characterizes the conditions under which uninformed customers join the

12

contrarian queue, and describes the opposite result of Proposition 4 (ii): when a contrarian is present, the

intensity of the longest queue’s quality signal relative to the contrarian queue weakens in the queue length of

the longest queue.

Proposition 5 With vanishing congestion cost (� ! 0+), in the contrarian state (ncn), if q < 1

3

,

(i) uninformed customers always join the contrarian queue for all n > 1;

(ii) the ratio P1(ncn

)

P2(ncn)

is decreasing in n.

For the intuition behind part (i), remember that uninformed customers with rank r > 1 never join an empty

queue. Since customers do not observe the queue-joining history, both the contrarian queue and the longest

queue signal quality relative to an empty queue (Proposition 3). To discuss the relative intensity of these signals,

it is helpful to consider the di↵erent sample paths that lead to the contrarian state. Consider the contrarian

state with the smallest number of customers, nc2 = (2, 1, 0, 0), i.e., location 1 with two customers, and location

2 with one customer. In this state, we discuss the cases summarized in Table 2.

Table 2: Cases Leading to State nc2 = (2, 1, 0, 0)

Case 1 (0, 0, 0, 0)�! (1, 0, 0, 0)

u! (2, 0, 0, 0)i! (2, 1, 0, 0)

Case 2.1 (0, 0, 0, 0)�,i�! (1, 1, 0, 0)

u�! (2, 1, 0, 0)

Case 2.2 (0, 0, 0, 0)�,i�! (1, 1, 0, 0)

i�! (2, 1, 0, 0)

Above the arrow is an indication of the type and sequence of the arriving customers: informed (i) or uninformed (u) and �

represents that it can be customers of either type.

If the contrarian queue is created by the third customer, then the third customer must be informed since

uninformed customers never choose empty locations (Case 1). In this case, the contrarian queue is either of high

quality or none of the locations in the system is of high quality. Therefore, in Case 1, uninformed customers

should join the contrarian. If the contrarian queue is not created by the third customer, the state observed by

the third customer (regardless of her type) must be two locations with one customer in line, i.e., (1, 1, 0, 0) in

Cases 2.1 and 2.2. The second arrival must then be informed, since uninformed customers never choose empty

locations.5 The third arrival, however, can be uninformed or informed. If she is uninformed (Case 2.1), then

her decision does not a↵ect the posterior probabilities, such that joining the contrarian is as good as joining

the longest queue (as both are equally likely to be a high quality location). If the third arrival is informed

and chose location 1 (Case 2.2), then the longest queue at location 1 must be of high quality, and joining the

contrarian queue is not a good decision. Pulling these arguments together, joining the longest queue is better

than joining the contrarian queue in only one (Case 2.2) out of the three cases. Furthermore, Case 2.2 requires

one more informed customer than the other two cases, which is less likely to occur when the fraction of informed

5The type of the first arrival could be either informed or uninformed. Hence we see �, i above the arrow from state (0, 0, 0, 0) to(1, 1, 0, 0) in Case 2.1 and 2.2 in Table 2.

13

customers is low, i.e., q is small. Hence, joining the contrarian queue is rational when q is below a threshold

(i.e., q < 1/3). A similar argument can be made to explain the intuition behind joining the contrarian queue

when the longer queue has more than two customers.

Part (ii) of Proposition 5 seems counterintuitive, compared with part (ii) of Proposition 4 and hence, requires

more explanation: in the presence of a contrarian, the quality signal of the longest queue becomes weaker as the

length of the longest queue increases (i.e., as n becomes larger). In other words, the contrarian becomes more

“salient” in presence of a long queue. For the intuition, consider n = 3 with contrarian state nc3 = (3, 1, 0, 0).

Notice that the predecessor of state (3, 1, 0, 0) can be either (3, 0, 0, 0) or (2, 1, 0, 0), as illustrated in Table 3.

Table 3: Cases Leading to State nc3 = (3, 1, 0, 0)

Case 1 (0, 0, 0, 0)�! (1, 0, 0, 0)

u! (2, 0, 0, 0)u! (3, 0, 0, 0)

i! (3, 1, 0, 0)

Case 2.1 (0, 0, 0, 0)u! (1, 0, 0, 0)

i! (1, 1, 0, 0)u! (2, 1, 0, 0)

i! (3, 1, 0, 0)

Case 2.2 (0, 0, 0, 0)u! (1, 0, 0, 0)

i! (1, 1, 0, 0)i! (2, 1, 0, 0)

i! (3, 1, 0, 0)

Above the arrow is an indication of the type and sequence of the arriving customers: informed (i) or uninformed (u) and �

represents that it can be customers of either type.

If the predecessor is (3, 0, 0, 0), then the first arrival can be of any type, the last (the fourth) arrival has to be

informed, and all intermediate customers are uninformed (Case 1). If the predecessor is (2, 1, 0, 0), then the last

(the fourth) arrival must be informed—if the last arrival is uninformed, then by part (i) of Proposition 5, she

should choose the contrarian queue. The fact that the last arrival is informed further implies that the longest

queue corresponds to the only high quality location in the system, otherwise, the last customer should choose a

di↵erent location. This in turn indicates that the contrarian queue can only be formed by the first arrival who is

uninformed and happened to choose a low quality location randomly, and the second customer is informed who

chose the only high quality location in the system. Comparing Table 3 and Table 2, we see that as the queue

length of the longest queue in contrarian state increases by one customer, the number of informed customers in

those sample paths that support choosing the longest queue also increases by one. Notice that when the fraction

of informed customers is small, sample paths supporting the longest queue are less likely, hence, the contrarian

becomes more attractive. In a general contrarian state with n � 4 customers in the longest queue, the same

argument applies. Either the contrarian queue is created by the last arrival, in which case all intermediate

arrivals are uninformed (Case 1), except for the last arrival and possibly the first one; or the contrarian queue is

not created by the last arrival (Case 2), in which case, all customers in the longest queue starting from the third

place must all be informed (i.e., the last n� 2 customers in the longest queue have to be informed). Note that

when the fraction of informed customers (q) is small, as n increases, the probability of having n � 2 informed

customers in the longest queue decreases. Therefore, the signal strength of the contrarian queue grows with the

length of the crowd in the contrarian state. This explains the intuition behind the seemingly counterintuitive

14

result of Proposition 5 (ii).

We next leverage the intuition obtained from Proposition 5 (ii) and extend the rationality of joining a

“contrarian queue” to the more general case of joining a “minority queue”. By joining the minority queue,

we mean joining the shortest non-empty queue. While the equilibrium strategy for uninformed customers

is complex for general cases,6 we fully characterize the equilibrium behavior for the case with finitely many

customers selecting from infinitely many locations, i.e., N < K = +1. Interestingly, joining the shortest

non-empty queue is rational in that case:

Proposition 6 With vanishing congestion cost (� ! 0+), if N < K = +1,

(i) the posterior probability (of being high quality) of a location is non-increasing with the queue length. In

particular, uninformed customers with rank r > 1 always join the shortest non-empty queue;

(ii) any non-empty queue with at least two customers less than the longest queue corresponds to a location of

high quality with certainty.

Note that informed customers always choose the empty high quality location. Because N < K = +1, it is

guaranteed that a fraction p0

of locations are of high quality by the law of large numbers. Hence, for a finite

population of customers, there always exists an empty high quality location. Consequently, all customers who

are not in the first position of a queue must be uninformed.

To understand the intuition behind part (i) of Proposition 6, we use a simple example of a state with one

queue of length two, another queue of length one and all other locations are empty (i.e., (2, 1, 0, ...)). The queue-

joining history can be either: (1) the first two customers formed a queue at location 1 and the third customer

created a new queue at location 2. Then the shortest non-empty queue corresponds to a high quality location

since only informed customers join empty locations and they always choose high quality locations. Hence,

joining the shortest non-empty queue is rational; (2) the first two customers created two queues at location

1 and 2, then the third customer chose location 1, in which case, the third customer is uninformed and her

decision does not a↵ect the posterior of these two non-empty queues. Hence, the shortest non-empty queue is as

good as the longest queue. Combining the two cases, (1) and (2), the shorter queue has a higher chance of being

of high quality than the longer queue. This intuitive argument can be extended to more general case in which

joining the shortest non-empty queue of length n is rational: either the shortest non-empty queue was created

by the last n arrivals and, hence, is of high quality; or it is not formed by the last n arrivals, implying that all

non-empty queues (including the shortest non-empty queue) have equal probability being of high quality.

For part (ii) of Proposition 6, it is now clear that for any given state, e.g., (5, 3, 0, ...), in which any shorter

queues that have at least two customers less than the longest queue, the queue with three customers in line in

6It depends on the specific values of the set of parameters (K,N, q, p0, vL, vH , �) and the specific state observed by the uninformedcustomers

15

this example, must be created after the completion of the longest queue. Because if this were not the case, i.e., if

we had (4, 3, 0, ...)u! (5, 3, 0, ...), then the customer in the last position of the longest queue (who is uninformed)

should join the shortest non-empty queue. This would lead to state (4, 4, 0, ...) and not (5, 3, 0, ...). Therefore,

the shorter queue must be formed after the completion of the longer queue. This implies that the shorter queue

must be initiated by an informed customer, since uninformed customers never choose empty locations. Hence, as

informed customers only select high-quality locations, the uninformed customer infers that shorter, non-empty

queue must be of high quality for sure.

Summary. With negligible congestion cost, we find that uninformed customers avoid empty locations. They

always join the single queue, whose quality signal strength increases in its length. However, when the fraction of

informed customers is small and a “contrarian” (i.e. a single customer at some location) is present, uninformed

customers join the contrarian instead of joining the longest queue or the empty locations. The quality signal

strength of the longest queue now decreases in its length. This phenomenon of joining the contrarian queue

generalizes to always joining the shortest non-empty queue, i.e. following the wisdom of minority, in the system

with finitely many customers selecting from infinitely many locations.

4.2 Non-vanishing Congestion Cost (� > 0)

The previous analysis described how uninformed customers learn from their predecessors’ choices when con-

gestion cost is negligible (� ! 0+). In particular, we characterized the conditions under which they follow

the majority or the minority. We next study how congestion cost modulates this observational learning pro-

cess. When congestion costs are non-negligible (� > 0), and there are no informed customers (i.e., q = 0), the

equilibrium strategy of uninformed customers obviously is to simply join the shortest queue (as studied in the

Operations Research literature, Winston 1977). With q > 0, however, all customer types face a tradeo↵ between

quality and congestion. Informed customers join a low quality location only if the queue di↵erence between a

high quality location and low quality location is D (defined in Proposition 1), provided that there is at least

one high quality location in the system (Proposition 1). Similarly, uninformed customers have to trade o↵ the

potentially positive quality signal of a long queue against the location’s high congestion cost.

Wisdom of the Majority: We first study the single-queue state, nsn . Recall from Proposition 4 that

uninformed customers should always join the longest queue when congestion cost is negligible (� ! 0+). The

following proposition shows that congestion costs dampen this phenomenon.

Proposition 7 With non-vanishing congestion cost (� > 0), in the single-queue state (nsn), if q < �, there

exists queue length n (where n D) such that uninformed customers join the longest queue when n < n or

n < n < D, and join an empty location when n = n or n = D.

16

Proposition 7 indicates that there exists a “hole” in the uninformed customers’ joining strategy when facing

state nsn ; customers join the longest queue if its length is strictly below the hole (i.e., below a certain queue

length) and also above the hole, but they do not join at the hole. Intuitively, each additional customer (informed

with probability q) joining the only non-empty queue intensifies its quality signal, but increases congestion as

well. When the marginal intensity of the signal cannot o↵set the congestion cost increase of an additional

customer (q < �), there will be a threshold n such that the accumulated congestion cost overwhelms the signal

sent by that single queue. Hence, in nsn , an uninformed customer joins an empty location instead. Observing

state nsn+1 , a customer perfectly infers that her predecessor was informed since, otherwise, the predecessor

would have joined an empty location resulting in “contrarian” state ncn 6= nsn+1 . The uninformed customer

can also infer that the only non-empty queue corresponds to the only high quality location in the system since,

otherwise, the informed predecessor would have chosen an empty high quality location. Beyond n, all customers

will continue to join the longest queue until its length reaches D.

Wisdom of the Minority: We next study the contrarian state ncn . For vanishing congestion costs, recall

from Proposition 5 that uninformed customers should always join the contrarian in state ncn (regardless of the

queue length of the longer queue), if the fraction of informed customers is su�ciently small (q < 1

3

). However,

this conclusion no longer holds with non-vanishing congestion costs, which runs counter to the intuition that

the contrarian queue should become even more attractive as congestion costs increase, since it has smaller queue

length and greater posterior probability of being high quality than the longest queue.

Proposition 8 With non-vanishing congestion cost (� > 0), if n < D� 1, there exists a contrarian state (ncn)

with the longest queue of length n such that uninformed customers do not join the contrarian queue.

For the intuition behind this proposition, we first assume that it is rational for uninformed customers to join

the contrarian queue if the longest queue in the contrarian state is no greater than n, i.e., the “hole” defined

in the single-queue state in Proposition 7. If this assumption does not hold, then Proposition 8 is valid, i.e.,

uninformed customers do not always join the contrarian queue. If this assumption holds, we will construct a

contrarian state with n > n in which joining the contrarian queue is not rational for uninformed customers.

As a simple example, consider state nc3 = (3, 1, 0, 0) with n = 2 and D = 4. Now the uninformed customer

with rank r = 5 facing state (3, 1, 0, 0) will reason the formation of this observed state. If this state is formed via

(2, 1, 0, 0) �! (3, 1, 0, 0), then the last arrival must be informed since we assume that uninformed customers join

the contrarian when facing state (2, 1, 0, 0). This implies that the longest queue is the only high quality location

in the system and its length is still strictly less than D(= 4); hence, worth joining. Notice that the current

observed state cannot be formed via (3, 0, 0, 0) �! (3, 1, 0, 0), since state (3, 0, 0, 0) unambiguously signals that

the longest queue is the only high quality location in the system by Proposition 7. Since the current queue

17

length at location 1 is 3 (< 4 = D, defined in Proposition 1), the next arrival joins the longest queue regardless

of her type, which would have prevented us from seeing state (3, 1, 0, 0). Consequently, in state (3, 1, 0, 0) is

formed after (2, 1, 0, 0), and thus, it is not rational to join the contrarian queue. The same argument applies to

the general case of n for which n < D � 1.

Similar to the case of vanishing congestion cost, we next consider the case with finitely many customers

and infinitely many locations (N < K = +1), in order to shed some light on the more general case of a

“minority” queue with more than a single customer. For vanishing congestion cost, recall from Proposition 6

that uninformed customers always join the shortest non-empty queue. To characterize the conditions under

which this result continues to hold under non-vanishing congestion costs, we first prove the following Lemma:

Lemma 1 With non-vanishing congestion cost (� > 0) and N < K = +1,

(i) the maximum queue length for the single-queue state is Ds = inf{n : v1

(1 � �)n < v0

}, where v1

,

vL + (q + (1� q)p0

)(vH � vL);

(ii) the upper bound of the maximum queue length for all states is Dmax

= inf{n : vH(1� �)n < v0

}.

For part (i) of Lemma 1, as in Proposition 6, the expected utility of any empty location is always v0

. In the

single-queue state, we know that all the customers in that queue starting from the second position must be

uninformed since informed customers only join empty high quality location. Thus, the posterior probability of

that single queue being high quality (which is q + (1 � q)p0

) only depends on the first customer’s type. With

probability q, she is informed, in which case that queue is of high quality with probability one; with probability

1� q, she is uninformed, in which case the queue is of high quality with probability p0

. This posterior will not

change as the queue length increases, but the utility generated from that queue will decrease with the queue

length. Eventually, the utility from that queue will be lower than the utility of an empty location, which leads

to Ds.

For part (ii) of Lemma 1, note that the value of any non-empty queue is at most vH . Hence, the queue

length cannot grow beyond Dmax

, otherwise the empty location will yield a higher expected utility.

Proposition 9 With non-vanishing congestion cost (� > 0), if N < K = +1 and Ds = Dmax

, uninformed

customers with rank r > 1 join the shortest non-empty queue if there exists a queue with length strictly less than

Ds, otherwise they join the empty queue.

As discussed in Lemma 1, when there is only one non-empty queue in the system, its length cannot grow beyond

Ds. However, when there are multiple queues of length Ds, and no other non-empty queues, it is possible that

uninformed customers would join a queue of length Ds—as an increasing number of non-empty queues makes it

more likely that they were created by informed customers, their posterior probability of being high quality could

be greater than q+(1� q)p0

and hence the expected value of those queues could be greater than v1

. This opens

18

the possibility that uninformed customers join a queue of length Ds. The condition Ds = Dmax

eliminates this

possibility, since Dmax

is the upper bound for the maximum queue length in the system. Therefore, under this

condition, uninformed customers keep on joining the shortest non-empty queue until all non-empty queues are

of length Ds, then they have to join empty locations. Hence, under the conditions of Proposition 9, we fully

characterize the equilibrium queue joining behavior under congestion. When the conditions do not hold, the

equilibrium queue-joining strategy is more complex.

The results for Proposition 9 can be generalized to the asymptotic case with both infinitely many locations

and customers using fluid limit approximation, where we show that under mild congestion cost, always joining

the shortest non-empty queue is an equilibrium strategy for uninformed customers. We omit the construction

of the fluid limit approximation here to maintain the focus of this paper, and refer to Appendix B for details.

Summary. Even with non-negligible congestion cost, uninformed customers continue to join the single

queue in the system, except at the “hole”, or if it is too long. While the contrarian is not as attractive anymore

as with vanishing congestion cost, with finitely many customers choosing from infinitely many locations, and

under additional assumptions on the parameters, the join the shortest non-empty queue strategy carries over

to the case with congestion cost, provided that the queue is not too long.

5 System Performance

Besides equilibrium behavior at the individual strategy level, we are also interested in how the key parameters

of our environment a↵ect equilibrium performance at the aggregate level. In this section, we study how the

presence of informed customers and the congestion cost a↵ect the average customer value and welfare (in Section

5.1), as well as average queue lengths (in Section 5.2).

5.1 Average Value and Welfare

In order to compute the average value and welfare of customers, we present a small example for illustration.

Figure 3 displays a possible realization of locations’ and customers’ types. In this example, the total number

of customers at the high quality locations (H ) is seven, and two customers are at the low quality locations

(L). The total value received by customers is computed as 3vH for the informed customers and 2vL + 4vH

for the uninformed customers. Customers’ welfare is more complicated to compute, because customers’ queue

positions matter. For the informed customers who selected the high-quality location, their total welfare is

2vH +vH(1� �)2. The uninformed customers’ total welfare is vL[1+(1� �)]+vH [2(1� �)+(1� �)2+(1� �)3].

Formally, we define EV �,H(n) as the expected total value received by the ��type customers at high quality

locations by the end of the game when the current state is n = (n,✓). Then, EV �,H , P✓2{H,L}K

EV �,H(0)Pr{✓}

19

Figure 3: An Example: Queue Lengths and Customer “Types” (informed vs. uninformed customers)1 2 3 4 5 6 7 8 9 10 n1(0,0)

1

2

3

4

5

6

7

8

9

10

n2

=10, q=0.1

is the expected total value received by the ��type customer at high quality location in any given game. The

quantities EV �,H(n) can be computed recursively as follows: by the end of the game, there are N customers in

the system. Now consider a fictitious arrival with rank r = N + 1 who randomly chooses a location. We let

EV �,H(n) = 0 for any high quality allocations after the fictitious arrival made her choice. Then, for any state

with strictly less than N + 1 customers, we define nHk

as the successor of current state n, i.e., the next state

after n when one customer is added to location k, which is a high quality location. Then:

EV �,H(n) =KX

k=1

(1� q)auk(n)

⇥vH {�=u,✓k=H} + EV �,H(nHk

)⇤+ qaik(n)

⇥vH {�=i,✓k=H} + EV �,H(nHk

)⇤�. (6)

Given the current state n, the next customer can choose any queue depending on her equilibrium strategy.

With probability 1 � q, the next customer is uninformed. Then with probability auk(n), she choose location k.

If location k is indeed a high quality location (i.e., ✓k = H), then she contributes vH to the total value received

from of high quality locations. Then the state changes to nHk

, at which the total expected total value received

from high quality locations is EV �,H(nHk

). The quantity EV �,L(nLk

) is defined the same way as EV �,H(nHk

)

by simply changing H to L. Finally, we define the average value of a ��type customer at the end of the game

and the average value of all customers are defined as:

V �A , EV �,H(n) + EV �,L(n)

[(1� q) {�=u} + q {�=i}]Nand VA ,

P�2{u,i}

EV �,H(n) + EV �,L(n)

�

N.

The average welfare of a ��type customer as well as all customers can be defined in a similar fashion by

replacing V in equation (6) by W and replacing vH (or vL) by vH(1 � �)j (or vL(1 � �)j). Then we have the

following proposition:

Proposition 10 With vanishing congestion cost (� ! 0+) and with informed customers (q > 0),

(i) the average value and welfare of uninformed customers (V uA and W u

A)

(ii) the average value and welfare of all customers (VA and WA)

are no less than those in system without informed customers (q = 0).

In the absence of informed customers (q = 0), uninformed customers join the shortest queue in equilibrium,

which results in a balanced queue allocation across all locations. Therefore, every location regardless of its type,

20

is identical to all customers. Ultimately, every location receives same number of customers if N is a multiple

of K, otherwise the queues di↵er by one customer. In contrast, when q > 0, uninformed customers never join

empty locations according to Proposition 3, implying that there can be at most one low quality location that

attracts positive queue length provided that there is at least one high quality location in the system (the case

of one low quality location with positive queue length only happens when the first arrival is an uninformed

customer who chooses a low quality location). Consequently, customers cluster around the set of high quality

locations. Hence, uninformed customers’ ability to identify high quality locations is improved with the presence

of informed customers, implying that the average value (welfare) of uninformed customers as well as of all

customers increases due to the presence of informed customers.

We observe some fundamental changes in the presence of congestion cost. The following proposition implies

that congestion cost might mute the observational learning process, and decrease both average value and welfare

of uninformed customers.

Proposition 11 With � > 1� vL/vH and with informed customers (q > 0),

(i) the average value and welfare of uninformed customers (V uA and W u

A) are no greater than those in systems

without informed customers (q = 0);

(ii) the average value and welfare of all customers (VA and WA) are no less than those in systems without

informed customers (q = 0);

When the congestion cost is high enough, � > 1 � vL/vH , even informed customers cannot tolerate having

one customer ahead of them. Then, all customers in the system should choose the shortest queue, resulting

in a balanced system. Importantly, the queue dynamics appear as if there are no informed customers in the

system at all, with the following subtle di↵erence. When informed customers are factually absent (q = 0),

uninformed customers always choose the shortest queue. Since queues do not signal value, all customers choose

a high quality location with the prior probability p0

. In contrast, in systems with informed customers (q > 0),

although both informed and uninformed customers always choose the shortest queue, informed customers will

first choose the high quality locations, while uninformed customers randomly choose a location. This behavior

of informed customers largely reduces the probability of uninformed customers choosing a high quality location.

As a result, even though the presence of informed customers increases the overall average value (welfare),

uninformed customers su↵er from the presence of informed customers in terms of both value and welfare.

5.2 Average Queue Length

Similar to the definition of average value, EQ�,H(n) or EQ�,L(n) are the expected total number of ��type

customers at a high or low quality location, respectively, by the end of the game when the current state is

21

n. The EQ�,H(n) can be computed recursively via expression (6) by replacing vH (or vL) by one. Then, the

average queue length at a high quality location by the end of the game is defined as

QHA ,

P�2{u,i}

EQ�,H

[(1� p0

) {✓=L} + p0 {✓=H}]K

and QLA is defined in a similar way, i.e., replacing H by L. When the congestion cost is negligible, we have the

following proposition:

Proposition 12 With vanishing congestion cost (� ! 0+) and with informed customers (q > 0), the queue at

a high (low) quality location, QHA (QL

A), is no less (no greater) than in a system without informed customers.

Proposition 12 is intuitive. As mentioned in Section 5.1, when there is no informed customers, all customers

should join the shortest queue and their posterior belief about the quality of locations is the prior p0

. Hence,

a high and low quality location will, on average, attract same number of customers. With informed customers

in the system, observational learning will drive uninformed customers towards high quality locations since no

uninformed customers chooses an empty location (Proposition 3). This leads to at most one low quality location

with positive queue length, provided that there is at least one high quality location in the system. Therefore,

the presence of informed customers increases (decreases) average queue length at high (low) quality locations.

As discussed in the previous section, the presence of congestion cost may completely mute the observational

learning process of uninformed customers; in fact,it reduces average value and welfare. From the perspective

of system the manager, the question is whether the congestion cost will “hurt” the high quality location (e.g.,

decreases its sales)? The following proposition answers this question.

Proposition 13 With � > 1� vL/vH , and with informed customers (q > 0), the average queue at a high (low)

quality location, QHA (QL

A), is greater (less) than in a system without informed customers.

Under such high congestion cost, all customers will choose to join the shortest queue, regardless of their type.

Without informed customers (q = 0), all customers randomly choose between the shortest queues. This results

in a balanced system, where any location has the same average queue length. Howewer, with informed customers

(q > 0), when facing the state with equal queue length across all locations, high quality locations have higher

chance of being chosen over low quality locations. Therefore, with informed customers under heavy congestion

cost, the average queue length at high quality locations is still greater than that in the low quality locations.

6 Laboratory Experiment

We designed a laboratory experiment to test the key predictions of our observational learning theory. At the

individual strategy level, we investigate the two distinguishing features from the above model analysis. First, for

22

vanishing congestion costs, our model predicts that uninformed customers avoid empty locations (Proposition

3), but join “contrarian” queues when the fraction of informed customers is low (Proposition 5). Second, our

analyses suggests that congestion cost “dampen” observational learning dynamics. In particular, our model

predicts that uninformed customers avoid the contrarian when congestion cost is non-negligible (Proposition 8).

At the aggregate level, we investigate (1) whether the high (low) quality location will attract longer (shorter)

queues in the presence of informed customers (Propositions 12 and 13); (2) whether the presence of informed

customers will increase the average value and welfare of all customers in the system (Propositions 10 (ii) and

11 (ii)), and (3) whether the presence of informed customers will decrease the average welfare of uninformed

customers when the congestion cost is high (Proposition 11 (i)).

6.1 Design and Implementation

6.1.1 Task

In each of a total of 33 rounds, a cohort of N = 10 subjects arrive sequentially at a system with K = 4 locations.

At the beginning of each round, before subjects arrive, nature determines the quality of each location. A location

is either of high quality (vH = 1000, with probability p0

= 0.25), or of low quality (vL = 200, with probability

1 � p0

= 0.75). Similarly, at the beginning of each round, nature determines the arrival order of these 10

subjects7 as well as the type of each subject (informed with probability q). If informed, the subject knows

the quality of each location before she makes a decision. Upon arrival, a subject observes the queues at each

location, i.e., the aggregate outcome of all her predecessors’ choices. She then chooses a queue to join. After all

10 subjects have made their choices, the system reveals the quality of all locations, and calculates each subject’s

payo↵ from the round as V✓k(1 � �)nk , where V✓k is the quality of location k chosen by the subject, and nk is

her position in the queue. The system then moves on to the next round.

6.1.2 Design

To address our main research questions, we implement a 2⇥2 between-subject design that varies the fraction of

informed customers, q 2 {0, 0.3}, as well as congestion cost, � 2 {0.01, 0.3}. Representing the case of “vanishing

congestion cost” (� ! 0+), � = 0.01 ensures that, in equilibrium: (1) informed rational customers choose the

shortest queue among high quality locations if there is at least one; otherwise, they choose the shortest queue

among all locations, and (2) uninformed rational customers with rank r > 1 never choose an empty location.

As a notation mnemonic, we use Q· (corresponds to � = 0.01) versus Qcc· (corresponds to � = 0.3) to

distinguish experimental conditions with di↵erent congestion costs, respectively. Similarly, we useQ·u

to indicate

7Within each round, the 10 subjects’ arriving order is uniformly drawn from the set {1, 2, ..., 10}, e.g., a subject could be thefirst arrival in round 1 and the last arrival in round 2 with same probability, which is 1

10 .

23

that all customers are uninformed (corresponds to q = 0), and Q·ui

to indicate a mix of uninformed and

informed customers (corresponds to q = 0.3). We include the two experimental conditions without informed

subjects (Qu

,Qccu

) as experimental baselines. Both cases predict a balanced queue allocation (because customers

always simply pick the shortest queue as predicted by our rational model) and provide a rather simple test of

observational learning: if uninformed subjects do not infer any value from queue length at all, then their

behavior in treatments with informed customers (Qui

,Qccui

) and without informed customers (Qu

,Qccu

) would

be indistinguishable. In turn, any statistical di↵erences between experimental conditions with and without

informed customers would provide direct evidence that uninformed subjects infer quality from queue length,

and base their choices on those of their predecessors. Table 4 summarizes the experimental design.

Table 4: Experimental Design (Sample Sizes in Parentheses)

�

q 0.01 0.3

0 Qu

(30) Qccu

(30)

0.3 Qui

(30) Qccui

(30)

We include Qui

(q = 0.3, � = 0.01) because the rational model predicts uninformed customers with rank

k > 1 never choose empty locations. Under this condition, uninformed customers join the longest queue about

65.7% of the time, join the shortest non-empty queue 24.6% of the time, and join other non-empty queues 9.7%

of the time. Furthermore, we include Qccui

(q = 0.3, � = 0.3) since it allows us to test the idea that congestion

cost “dampens” such observational learning—indeed, our rational model predicts that uninformed customers

join the longest queue only 27.4% of the time, and join the empty location 41.1% of the time.

6.1.3 Prior Information and Sample Information

We provide subjects with full knowledge about the relevant parameters of the environment (K, N , q, p0

, vL, vH ,

�). The stochastic elements in the game consist of three random variables. First, the distribution of the quality

among all K locations, which follows a binomial(K, p0

) with K = 4 and p0

= 0.25. Second, the distribution

of the type among all N subjects, which follows a binomial(N, q) with N = 10 and q = 0.3 in conditions Qui

and Qccui

(q = 0 in conditions Qu

and Qccu

). Third, the distribution of the arrival rank of all 10 subjects in

each round, i.e., each subject in each round has equal chance of being rank r = 1, 2, 3, ..., 10. We use the term

“scenario” to represent a sample drawn from the joint distribution of the three random variables—location

qualities, subject types, and subject ranks. Because each cohort of 10 subjects would play for 33 rounds, we

independently generated 99 scenarios in each experimental condition.

We pre-generated the stochastic elements of our model in a systematic fashion to satisfy two requirements.

24

First, the random outcomes (location types, subject types and arrival rank) need to be representative of the

environment they are sampled from, i.e., the empirical distribution constructed from these 99 scenarios will

resemble binomial(K, p0

) for the quality of locations, binomial(N, q) for the type of subjects and each subject

should have equal chance to be in one of the ten ranks in each round. Second, as our key research questions

concern comparisons between experimental conditions, we use the same pre-generated random seeds of uniform

random [0, 1] variables in all four experimental conditions. These uniform variates were then converted into

events using the appropriate distributions, such as binomial for location value (high value with probability p0

),

subject type (informed with probability q) and arrival rank. As a consequence, subjects in two experimental

conditions would face identical sequences of the realization of the stochastic element in each round. For example,

cohort c in experimental condition Qui

would observe the same set of locations (in terms of their quality) in

round t as the corresponding cohort c in condition Qccui

. And for the cohort c in round t in both conditions

Qui

and Qccui

, the arrival rank as well as the subject type of each subject will be the same. In Appendix C, we

provide the table of uniform [0, 1] variables used to determine the quality of each location, type of each subject

and arrival order of each subject in each round.

6.1.4 Rational benchmark

Besides a comparison of observed (i.e., empirical) behavior and system performance across conditions, we

are interested in comparisons with the predictions from our rational equilibrium model. As our experiments

implement only 99 scenarios drawn from the joint distribution, we calculate the relevant “rational” benchmarks

as those predicted by the application of the equilibrium strategy (i.e., our analytical results) on the same set of

scenarios as we implemented in the experimental conditions.

6.1.5 Software, Recruitment and Payment

The experiment was implemented in the experimental software zTree (Fischbacher 2007). When a subject

arrives at the system to make a choice, the screen would indicate the subject’s type, i.e., either informed or

uninformed in the current round. The left (right) panel of Figure 4 shows the screen observed by an uninformed

(informed) subject with rank r = 4.

For each location,8 the screen would display the queue length, n = 0, 1, 2, ..., N . Furthermore, for each loca-

tion k, informed subjects (right panel of Figure 4) would see the gross value and payo↵ (discounted by congestion

costs) defined in the rational model, respectively, inside the oval representing the location. Uninformed subjects

(left panel of Figure 4) would see the expected value and payo↵ of each location, using the prior probability p0

and the actual queue length n.

8In experiment, i.e., in the screen observed by subjects, we use the word “server” instead of “location”.

25

Figure 4: A Typical Screenshot Observed by Uninformed (Left) and Informed (Right) Subjects

A total of 120 students participated (30 for each experimental condition). Subjects were undergraduate

students at a large public university in the US. After arriving at the laboratory facilities, participants read

written instructions (see Appendix C). Cash was the only incentive o↵ered, and subjects were paid based on

their mean payo↵ across all 33 rounds. To calculate the earnings, we applied a conversion factor that was

adjusted for each experimental condition. In particular, a subject would earn $0.044, $0.048, $0.024, $0.038

US dollars per laboratory token in condition Qccui

, Qccu

, Qui

, Qu

, respectively. The average payment across all

experimental conditions, including a $5 showup fee, was $18.95, with a standard deviation of $3.44.

6.2 Results: Individual Strategy Level

Table 5 summarizes our data. In total, we have 990 observations in each experimental condition. In order to

compare actual choices with what a rational customer would have chosen had she arrived in the same state,

we exclude from the subsequent analysis all “o↵-equilibrium” states (i.e., where the rational model makes no

predictions). From the remaining set (“on equilibrium path” in row 1), we exclude all states in our data where

all queues have the same length, i.e., states where any choice can be rationalized. For the remaining set (“in

our analysis” in row 2), for informed customers (rows 3 and 4) and uninformed customers (rows 5 and 6),

we count the number of rational choices, where “rational” is defined by applying the equilibrium strategy to a

given state. Furthermore, for uninformed customers, we define the following mutually exclusive (and collectively

exhaustive) generic strategies for all states in our analysis: (a) join the empty queue; (b1) join the contrarian

queue; (b2) join the minority queue that is not a contrarian; (c) join the longest queue, and (d) join other

queues, i.e., all cases not included in (a)-(c). Using this classification, the bottom part of the table contrasts

what uninformed subjects actually chose (“Obs.”) with what a rational decision maker would have chosen in

equilibrium (“Pred.”).

26

Table 5: Deviations from Equilibrium Predictions

Number of States: Qu Qccu Qui Qcc

ui

On equilibrium path 841 862 990 983

In our analysis 768 756 919 891

Faced by informed subjects —– —– 292 292

rational choice —– —– 274 282

Faced by uninformed subjects 768 756 627 599

rational choice 656 670 231 358

Uninformed subjects: breakdown by strategy Obs. Pred. Obs. Pred. Obs. Pred. Obs. Pred.

(a) Join empty queue 297 345 297 336 128 0 207 186

(b1) Join contrarian queue 2 0 2 0 14 105 2 0

(b2) Join minority - noncontrarian queue 366 423 380 420 62 275 152 126

(c) Join longest queue 91 0 71 0 408 242 223 255

(d) Join other queue 12 0 6 0 15 5 15 32

6.2.1 Experimental Conditions with No Informed Customers (q = 0)

Consider first our baseline conditions Qu

and Qccu

. In the absence of informed customers, the equilibrium

prediction is to join the shortest queue, in particular, to join an empty queue (if there is one). While uninformed

customers make the correct choice in the majority of states, we observe 768�656

768

=15% (in Qu

) and 756�670

756

=11%

of choices (in Qccu

) that deviate from the predictions of rational model. In particular, subjects occasionally

choose the longest queue even though, in the absence of informed customers, queues cannot signal value.

Observation 1. Without informed subjects in the system, most of the uninformed subjects understand that

queues are uninformative, and they join the shortest queue.

6.2.2 Experimental Conditions with Informed Customers (q = 0.3)

In the presence of informed customers (Qui

and Qccui

), the rational model predicts observational learning of

uninformed customers. The results in Table 5 show that informed subjects make the correct choice in the

vast majority of states ( 274292

= 93.8% in Qui

and 282

292

= 96.6% in Qccui

). While expected given the trivial

task of informed subjects, this result is important as correct choices of informed subjects render queues into

informative signals. Not surprisingly, we observe more deviations from equilibrium predictions for uninformed

subjects, specifically 627�231

627

=63.2% of all choices (in Qui

) and 599�358

599

=40.2% of all choices (in Qccui

) contradict

the equilibrium strategy.

27

Observation 2: Almost all informed subjects play rational strategy, which turns queues into informative signals.

A significant proportion of uninformed subjects deviates from the rational strategy.

Negligible Congestion Cost (� = 0.01). For condition Qui

, we observe 128

627

= 20.4% of uninformed subjects

choosing an empty queue, contrary to the prediction that an empty queue is never worth joining in systems

with vanishing congestion cost (Proposition 3). This tendency to join empty queues can be rationalized only if

an uninformed subject does not infer any value from non-empty queues. On the other hand, we observe that

uninformed subjects rarely choose a contrarian (14 cases) or a minority (62 cases) queue, even though the rational

model predicts many such choices for this experimental condition (105 contrarians and 275 minorities). Among

these 105 contrarian states, 77

105

= 73.3% of the uninformed subjects choose the longest queue. Overall, the data

provides strong evidence against the hypothesis that uninformed subjects follow the wisdom of contrarians or,

more generally, minorities. When facing a minority queue that is worth joining (through the lens of the rational

model), subjects rather choose an empty queue or the longer queue (i.e., the majority).

Next we analyze uninformed subjects’ choices when they are facing the single-queue state (Proposition 4),

where theory predicts that uninformed customers always choose a single non-empty queue regardless of its

length. Contrary to this prediction, the data in Figure 5 (solid line) shows that uninformed subjects join the

single queue of length n = 1 only in 42% of the cases, suggesting that they consider the quality signal of such a

queue as too weak to overcome even the negligible congestion cost in condition Qui

. Still, Figure 5 shows that

the probability of joining the single non-empty queue increases in its length, which is consistent with the result

that the signal strength of a single non-empty queue increases in its queue length (Proposition 4 (ii)).

Observation 3: With the presence of informed subjects and negligible congestion cost, uninformed subjects

rarely join a contrarian queue. Instead, they flock to the longest queue.

Figure 5: Joining Frequencies in Single-Queue State (Solid Line: Qui

, Dashed Line: Qccui

)

Non-negligible Congestion Cost (� = 0.3). When the congestion cost is non-negligible (Qccui

), Proposition 7

28

predicts a non-monotone queue-joining pattern for the single-queue state. In particular, uninformed customers

join the single non-empty queue when its length n is strictly below a certain threshold n, and choose an empty

queue if n = n. When n > n, they can perfectly infer that the queue is the only high quality location in the

system, and join this queue until congestion costs render it less attractive than empty low quality locations. In

condition Qccui

, we have n=1, implying that uninformed subject should not join the single queue at n = 1, but

join at n = 2. Further, no customers should join the longest queue in the single-queue state if its length is five

or larger (since vH(1� �)5 = $1000⇥ (1� 0.3)5 = $168 < vL = $200). The joining pattern in Figure 5 (dashed

line) is qualitatively consistent with these predictions from Proposition 7. Finally, Proposition 8 predicts no

states where the equilibrium strategy is to choose a contrarian queue. Indeed, we observe only two cases (out

of 46 cases) where subjects choose to join a contrarian.

Observation 4: When facing single-queue state in a system with congestion cost, the majority of uninformed

subjects do not join the single queue at the hole (Proposition7). Almost no uninformed subjects join a contrarian

queue, which is consistent with the rational model prediction (Proposition 8).

6.2.3 Quantal Response Equilibrium (QRE)

Observations 1, 2 and 4 show that the experimental data are qualitatively consistent with the rational model

prediction. However, Observation 3 is strongly against the rational model prediction, i.e., uninformed subjects

flock to the longest queue rather than choosing the contrarian queue when they should. Why do subjects in

condition Qui

choose contrarian queues significantly less frequently than predicted by the rational model? This

result is surprising given that minority queues signal high value through the lens of rational model and result

in low congestion costs (compared to longer queues).

To illustrate the theoretical value of choosing contrarian queues, we apply the equilibrium strategy to the 99

scenarios used in experimental condition Qui

. We then calculate, for all contrarian states that emerge (under

equilibrium play), the average value resulting from each of the three generic choice options in such states (i.e.,

the longest, contrarian, and the empty queue). The rational model predicts that the average value of the

contrarian queue is 573, which is higher than the average value of the longest queues (405), and much higher

than the average value of the empty queues in those states (245). This provides overwhelming evidence that a

contrarian queue is significantly more attractive than the others, in a population of fully rational customers.

In the experiment, however, we have no reason to believe that subjects would play rational strategies, and

we indeed observe a number of obviously erroneous choices in our data. Essentially, the contrarian queue may

loose some appeal if one accounts for the possibility that it was erroneously formed, either by an informed

or an uninformed subject. To render this idea more precisely, we next incorporate decision error into our

model in Section 3, based on the idea of Quantal Response Equilibrium (QRE) (McKelvey and Palfrey 1995).

29

Formally, when an informed customer observes state n upon arrival, the utility of location k will be U i

k(n) ,

v✓k(1� �)nk + ✏k, where ✏k is a noise term with mean zero. We indicate the strategy of an informed customer

under QRE by ai(n) : (N0

⇥ {H,L})K ! [0, 1], i.e., ai(n) specifies the joining probability of location k, where:

aik(n) = Pr

⇢U i

k(n) = max1k0K

U i

k0(n)

�(7)

With the standard assumption that the noise term ✏k follows an extreme-value distribution with parameter �

(McFadden 1980), equation (7) has an explicit expression:

aik(n) =exp{v✓k(1� �)nk/�}

KPk0

=1

exp{v✓k0 (1� �)nk0 /�}. (8)

Note that � scales the variance of the noise term, and can be interpreted as the extent of bounded rationality

(Su 2008; Chen et al. 2012), i.e., when � ! 0, customers are completely rational and hence apply equilibrium

strategy; when � ! 1, they randomly choose any location with equal probability.9

Analogously, when an uninformed customer observes state n upon arrival, the utility of location k will

be Uu

k (n) , (1 � �)nkVk(n) + ✏k, and we indicate the strategy of an uninformed customer under QRE by

au(n) : NK0

! [0, 1]. This yields:

auk(n) =exp{(1� �)nkVk(n)/�}

KPk0

=1

exp{(1� �)nk0Vk0(n)/�}(9)

where Vk(n) = vL + (vH � vL)Pk(n). Pk(n) is defined by Equation (4) and can be computed recursively by

Equation (5), except that in Equation (5) auk(n) and aik(n) has to be replaced by auk(n) and aik(n).

Let nj (nj0) be the state observed by the jth (j0th

) informed (uninformed) customer in condition Qui

and

Qccui

. Let N be the set of all states faced by customers (of either type) upon arrival in an experiment condition.

Then the likelihood function is given by:

L(�|N) =

"niY

j=1

KY

k=1

aik(nj)

#"nuY

j0=1

KY

k=1

auk(nj0)

#(10)

where ni = 292, nu = 698 in conditionsQui

andQccui

and ni = 0, nu = 990 in conditionsQu

andQccu

, respectively.

We fit the model to our experimental data, using standard maximum likelihood methods, and bootstrapped

confidence intervals for the estimated parameters �.10

Results. Table 6 shows the estimated �s across all experiment conditions. As an interpretation of the

9Our data shows that informed customers also make occasional mistakes. For sake of parsimony, we assume for both type ofcustomers, they made mistakes at same magnitude, i.e., they have the same value of �.

10In conditions Qui and Qccui , we draw 1000 stratified bootstrap samples with replacement from the pool of states observed by

informed and uninformed subject separately, i.e., each bootstrap sample contains 292 states observed by informed subject and 698states observed by uninformed subject. In conditions Qu and Qcc

u , there are no informed subjects, hence we draw 1000 bootstrapsamples with replacement from all states observed by uninformed subjects, i.e., each bootstrap sample contains 990 states byuninformed subjects.

30

Table 6: QRE Estimation Results

Condition � 95% lower 95% higher

Qu

2.23 1.96 2.50

Qccu

46.04 40.48 51.60

Qui

174.62 154.40 194.84

Qccui

30.54 10.85 50.23

estimates in absolute terms is meaningless (as are cross-condition comparisons), we directly move on to use

the predictions of the estimated QRE models. To this end, we develop a simulation model that applies the

estimated QRE model, as well as the rational model, to the 99 scenarios (of location qualities and customer

types) implemented in the experiment. The simulation allows us to compute quantities such as the fraction of

contrarian queues formed by informed customers, and the average value of the longest, contrarian and empty

queues in contrarian states. Technical details can be found in Appendix C. Our goal is to shed light on our

original conjecture that decision noise on the part of the subjects renders minority queues much less informative

in our data, than in the rational model (i.e., Observation 3). In particular, we focus on uninformed subjects

facing states with a contrarian queue, in condition Qui

(where such states are plentiful, i.e., 105 instances).

While the rational model predicts that uninformed customers always join the contrarian queue, simulation

results for the QRE model show that only 19.0% of the customers join contrarian queues, which is close to

the experimental data ( 14

105

= 13.3%). That the QRE model matches the pattern of “rejecting the wisdom

of minority” in our experiment data much better than the rational model makes eminent sense, of course.

Remember that rational customers (in a rational world) would choose a contrarian queue because it is most

likely created by an informed customer and, hence, has higher expected value than the longest queue and empty

queues. Following this reasoning, we compute the fraction of contrarian queues formed by an informed customer

in the experimental data as well as from the simulation of the QRE model. The experimental data shows that

only 30.5% (= 32

105

) of contrarian queues are created by informed customers, which deviates sharply from the

predictions of the rational model (84.3%), but aligns closely with the QRE model (28.8%). We also calculate

the expected value of the contrarian queue, longest queue and the empty queue (Table 7). While the rational

model suggests that contrarian queues provide the most value on average, it is the longest queues that provide

the highest average value in the data. The estimated QRE model is consistent with this pattern, suggesting

indeed that contrarian queues lose their attractiveness when accounting for possible random choice behavior.

Observation 5: QRE provides a plausible explanation for why uninformed subjects flock to the longest queue

rather than choosing a contrarian queue.11

11For the single-queue state, recall that in the experiment, when the single queue has only one subject, only 42% of the subjectschoose to join that single queue and the rational model predicts that 100% rational customers should join that queue. Under QREmodel, it predicts that 41% of the subjects should join that single queue.

31

Table 7: Average Value in Contrarian State

Qui

Observed Rational QRE

Longest Queue 558.1 405.0 599.0

Contrarian Queue 413.3 573.0 379.4

Empty Queue 314.3 245.0 287.2

6.3 Results: Aggregate Performance Level

How does individual level behavior translate into aggregate level performance? In this section, we first compare

the average value, welfare and queue length at locations across all four experimental conditions. Then we

compare these performance measures with the rational model and the QRE model.

6.3.1 Comparison Across Experimental Conditions

Average Value and Welfare. Recall from Section 5 that the presence of informed customers will increase

(decrease) the average value and welfare of uninformed customers when the congestion cost is negligible (is

high) (Proposition 10 (i) and 11 (i)). But the presence of informed customers will always increase the average

value and welfare of all customers in the system (Proposition 10 (ii) and 11 (ii)). To test these predictions, and

letting Vk(c)t denote the value received by subject k (nested in cohort c) in round t, we develop the following

regression model:

Vk(c)t = ↵0

+ ↵�I� + ↵qIq + ↵�,qI�Iq + ↵�I� + ↵TTt + ↵HHt + ↵RRt

+ ↵THTtHt + ↵TRTtRt + ↵RHRtHt + �c + ✏kt,

(11)

which includes a cohort-level random e↵ect, �c ⇠ Normal(0,�2

v). In order to test for between-condition di↵er-

ences, we include dummies Iq , {q=0.3} and I� , {�=0.3}, as well as their interaction terms. Furthermore,

the dummy I� = {�=i} allows to distinguish between subject types in those conditions with both informed

and uninformed subjects. As each subject in a given cohort will play multiple independent rounds (Tt), with

di↵erent arrival rank in each round (Rt) and with di↵erent number of high quality locations in each round (Ht),

we include Tt, Ht and Rt together with their pairwise interaction terms to control for these factors. Specifically,

Tt is the mean-centered round variable, defined as Tt , t� 17, where t = 1, 2, 3, ..., 33, that captures the possi-

bility that subject may learn to make better decisions over time; Ht (mean-centered) is the total number of high

quality locations in round t that controls the e↵ect of di↵erent number of high quality locations in each round

on the value gained by subjects; Rt (mean-centered) is the rank of each subject in round t. We also estimate

model (11) with welfare Wk(c)t as the dependent variable.12

12Note that, unlike Wk(c)t, Vk(c)t is a discrete variable that has only two possible outcomes (high or low), and thus lends itselfto the estimation of a discrete choice model. We provide the results of a logistic regression on Vk(c)t in Appendix C, but present

32

Table 8: Regression Results for Average Value and Welfare

Value (Vct) Welfare (Wct)constant ↵

0

385.50⇤⇤ 382.33⇤⇤

I� ↵� 258.98⇤⇤ 226.28⇤⇤

I� ↵� 8.08 � 77.82⇤⇤

Iq ↵q 66.64⇤⇤ 67.35⇤⇤

I�Iq ↵�,q �46.06 � 116.31⇤⇤

Tt ↵T 0.41 0.45Ht ↵H 223.30⇤⇤ 188.85⇤⇤

Rt ↵R 1.67 � 12.82⇤⇤

TtHt ↵TH 0.43 0.50TtRt ↵TR 0.26 0.18RtHt ↵RH 1.42 � 6.51⇤⇤

�v 16.27 11.27LL -28,286 -27,811N 3,960 3,960

Notes: ⇤⇤p < .01, ⇤

p < .05.

Table 8 presents the results. Not surprisingly, informed subjects reap higher average value (↵� = 258.98, p <

0.01) and welfare (↵� = 226.28, p < 0.01) than uninformed subjects. Besides, value and welfare is larger in

rounds with more high quality locations (↵H = 223.3, p < 0.01 for value and ↵H = 188.85, p < 0.01 for welfare).

With vanishing congestion cost (� = 0.01), the presence of informed subjects increases the average value of all

subjects in the system (↵q + 0.3↵� = 144.33, p < 0.01) and increases the average welfare of all subjects as well

(↵q +0.3↵� = 135.23, p < 0.01). In the experimental condition with � = 0.3, the presence of informed subjects

also increases the average value of all subjects (↵q +0.3↵� +↵�,q = 98.27, p < 0.01). However, it does not have

statistically significant impact on the average welfare of all subjects (↵q + 0.3↵� + ↵�,q = 18.92, p = 0.22).

When congestion costs are vanishing (� = 0.01), uninformed subjects’ average value increases in the presence

of informed subjects (↵q = 66.64, p < 0.01), which corresponds to a welfare e↵ect of similar magnitude (↵q =

67.35, p < 0.01). In contrast, when congestion costs are not vanishing (� = 0.3), the presence of informed

subjects does not have a statistically significant impact on the uninformed subjects’ average value (↵q + ↵�,q =

20.58, p = 0.29); however, the presence of informed subjects decreases the average welfare of uninformed subjects

(↵q+↵�,q = �48.96, p < 0.01). Together, these observations are related to the “reversal e↵ect” from Proposition

10 (i) and Proposition 11 (i), which posit that the presence of informed subjects generally benefits uninformed

subjects when congestion costs are vanishing, but decreases their average welfare when congestion costs are

high. Notably, while the propositions hold for high congestion cost � > 1�vL/vH , we observe the reversal e↵ect

even when the congestion cost is much lower (� = 0.3 < 0.8 = 1� vL/vH).

Observation 6: When the congestion cost is high, the presence of informed subjects reduces the average welfare

in Table 8 instead the results from simple linear random-e↵ect regressions, for the sake of a more direct comparison between thecoe�cient estimates for both models. Given this model, the constant ↵0 provides the baseline e↵ect for the average value ofuninformed subjects in the absence of informed subjects and with vanishing congestion cost (i.e., condition Qu).

33

of uninformed subjects.

Finally, in terms of average value, we find no evidence that subjects learn over time (i.e., round), or by

arriving late (i.e., rank) and observe more subjects’ choices. Specifically, as the Table 8 shows, ↵T , ↵R and ↵TR

are not statistically significant. For average welfare, experience does not play a role (↵T = 0.45, p = 0.32), but

the arrival rank does matter (↵R = �12.82, p < 0.01).

Average Queue Length. We next test the prediction that the average queue length at a high (low)

quality location will increase (decrease) with the presence of informed customers, regardless of the congestion

level (Propositions 12 and 13). Consistent with the definition in Section 5, we use cohort level averages as the

unit of analysis. Specifically, for each round, we calculate the average queue length at high and low quality

locations after all 10 subjects have made choices. We run the following regression model:

QHct = ↵

0

+ ↵�I� + ↵qIq + ↵�,qI�Iq + ↵TTt + ↵HHt + ↵THTtHt + �c + ✏ct

where QHct is the queue length at high quality location by the end of round t in cohort c. We run the same

regression for QLct as well. The results are shown in Table 9. When the congestion cost is vanishing (� = 0.01), the

Table 9: Regression Results for Average Queue Length

QHct QL

ct

constant ↵0

2.32⇤⇤ 2.56⇤⇤

I� ↵� 0.10 �0.03Iq ↵q 1.79⇤⇤ � 0.60⇤⇤

I�Iq ↵�,q �0.58 0.19Tt ↵T 0.00 0.00Ht ↵H 2.80⇤⇤ � 0.93⇤⇤

TtHt ↵TH 0.00 0.00�v 0.00 0.00LL -739 -304N 396 396

Notes: ⇤⇤p < .01, ⇤

p < .05.

presence of informed subjects increases the average queue length at high quality location (↵q = 1.79, p < 0.01)

and decreases the average queue length at low quality location (↵q = �0.60, p < 0.01). Similarly, with non-

vanishing congestion cost (� = 0.3), the presence of informed subjects increases the average queue length at

high quality location (↵q+↵�,q = 1.21, p < 0.01) and decreases the average queue length at low quality location

(↵q + ↵�,q = �0.41, p < 0.01), although the e↵ects are smaller in magnitude than for � = 0.

Observation 7. With the presence of informed subjects, the average queue length at high (low) quality location

increases (decrease) regardless of the congestion cost.

34

6.3.2 Comparison with Model Predictions

We next compare the observed system performance (experimental data) with the predictions from our rational

model, as well as with the predictions from the QRE model. In particular, the performance measures of interest

are the average value (welfare) of all subjects, VA (WA), the average value (welfare) of uninformed subjects,

V u

A (W u

A), and the average number of subjects at a high (low) quality location, QHA (QL

A). We use cohort-level

averages (by round t) as unit of analysis, e.g., we let Vct denote the average value of all subjects in cohort

c in period t.13 Similarly, let V ⇤ct and Vct be the average value from applying the rational model and QRE

model, respectively, to the scenarios from the experiment. To compare predictions with data, let �Yct denote

the di↵erence between the predictions from the rational model and the data (e.g., V ⇤ct � Vct), or between the

predictions from the QRE model and the data (e.g., Vct � Vct). For each experimental condition and each

performance measure, we run the following regression model:

�Yct = ↵0

+ ↵TTt + ↵HHt + �c + ✏ct (12)

where ↵0

captures the di↵erence between data and predictions (rational or QRE), after controlling for round (Tt)

and number of high quality locations (Ht). Table 10 summarizes the means of the six performance measures,

for all experimental conditions under the “rational” model, the QRE model, and in the experimental data

(“observed”). For example, V ⇤A = 620.71⇤⇤ for condition Q

ui

indicates that the di↵erence between the rational

model (620.71) and data (528.89) on the measure VA is statistically significant at the 1% level.

In the absence of informed subjects (Qu

,Qccu

), the predictions from both equilibrium and QRE models fall in

line with the data. This is not surprising, as the simplicity of the task in these conditions (i.e., always joining the

shortest queue) does not leave much room for bias. However, when the presence of informed subjects (Qui

,Qccui

)

creates an environment where observational learning should take place, QRE model predictions are much better

aligned with the data than rational predictions.

Observation 8. Without informed customers, our data is consistent with the rational model (as well as the

QRE model) in terms of value, welfare, and queue lengths. With informed customers, our data aligns better

with the QRE model than with the rational model.

13Note that our theoretical results are based on Q

HA (QL

A) defined at the end of each round, i.e., as a cohort-level average. Forconsistency, we also use this unit of analysis for VA and WA.

35

Table 10: Performance Metrics under QRE, Equilibrium Model and Data

VA WA V u

A W u

A QHA QL

A

Qu

QRE 389.80 386.66 389.80 386.66 2.37 2.54

observed 385.86 382.74 385.86 382.74 2.32 2.56

rational 389.77 386.66 389.77 386.66 2.37 2.54

Qccu

QRE 389.81 302.48 389.81 302.48 2.37 2.54

observed 393.94 304.92 393.94 304.92 2.42 2.53

rational 389.77 303.28 389.77 303.28 2.37 2.54

Qui

QRE 538.81 526.77 462.56 450.93 4.24 1.92

observed 528.89 516.84 455.41 443.30 4.11 1.96

rational 620.71⇤⇤ 603.26⇤⇤ 573.89⇤⇤ 555.03⇤⇤ 5.26⇤⇤ 1.58⇤⇤

Qccui

QRE 475.03 324.15 376.44⇤⇤ 260.78 3.44 2.19

observed 490.91 322.70 415.01 262.91 3.64 2.12

rational 511.82⇤ 332.37⇤⇤ 442.87 280.73⇤⇤ 3.90⇤ 2.03⇤

Notes: ⇤⇤p < .01, ⇤

p < .05.

7 Conclusion

The nature, and management, of social learning processes has received increasing attention by academia and

practitioners alike, thanks to the rise of large online market places. In this paper, we study customer behavior

when the quality of di↵erent choice options is uncertain to some, but, not all customers. Under these conditions,

informed customers’ choices exert positive information externalities, in the sense that less informed customers

can learn something about the quality of the di↵erent options from their popularity. On the other hand,

in many economically significant environments, customers’ choices exert negative congestion externalities on

subsequent customers, due to crowding, or long wait times. When popular choice options exhibit both positive

and negative externalities, uninformed customers’ choice problem is non-trivial. Our theoretical model analysis

provides a sharp contrast with the join the shortest queue strategy that typically arises from the focus on

negative congestion externalities in the Operations Research literature (e.g., Winston 1977), as well as with

the join the longest queue strategy that typically arises from the focus on positive information externalitites

in the economics literature on observational learning (e.g., Bikhchandani et al. 1992). In particular, while the

equilibrium strategy is generally complicated, we are able to fully characterize the equilibrium when there are

many more locations than customers (Propositions 6 and 9). In such a setting, the equilibrium strategy is to

join the “smallest crowd”, i.e., join the shortest non-empty queue. The stochastic process that results in such

choice behavior has some similarity with the Chinese Restaurant Process (CRP, see e.g. Pitman et al. 2002).

The CRP also assumes that infinitely many locations (or tables in CRP parlance) labeled by k = 1, 2, ... are

available to sequentially arriving (homogeneous) customers with rank r = 1, 2, .... The CRP is characterized by

a single parameter, ↵ > 0. The customer with rank r > 1 joins an empty location with probability proportional

36

to ↵ or joins an non-empty location, k, with a probability proportional to the size of that location, nk (> 0). The

CRP is closely related to Dirichlet processes and Polya’s urn scheme, however, does not have micro-foundations

based in decision theory. Our model is slightly di↵erent as we have heterogeneous customers (informed vs.

uninformed), but, provides decision theoretic micro-foundations for “location selection”. We find that, even in

the absence of congestion costs, rational customers that join an existing location select the one with the smallest

number of occupants, instead of selecting one with a probability proportional to the number of occupants as in

the CRP. In other words, while the CRP captures intuitively the notion of “popularity”, ignoring observational

learning amounts to ignoring the wisdom of the minority.

We test the rational location selection strategies in a laboratory setting with human subjects. Interestingly,

subjects seem to under-value the appeal of minority queues, and instead tend to be more attracted to majority

queues. This observation is consistent with the notion of “random choice”—the possibility that minority queues

may have been formed only by mistake, significantly diminishes their signalling value in favor of longer queues.

To better understand how decision noise drives (or, hinders) the observational learning process, we develop

a quantal response equilibrium (QRE) version of our basic theoretical model. Our paper is a first step in

understanding how demand shares for products (or services) with uncertain quality is formed. Theoretically

and empirically, we find that a growing fraction of informed customers increases the overall welfare and via

better allocation of customers to the high quality options, at the expense of some imbalance in the queues.

However, we also find that a growing fraction of informed customers actually reduces the average welfare of the

uninformed customers, when congestion costs are high.

Our QRE demand model can be used as a basis to determine the optimal number of locations factoring in

real human choice behavior in presence of quality uncertainty and congestion. Assuming a fixed cost per location

o↵ered, a manager faces the following trade-o↵: O↵ering more locations will alleviate the customers’ congestion,

at the expense of higher fixed costs. However, as discussed above, the number of locations also impacts the

observational learning. We leave exploration of the optimal number of choices in presence of observational

learning for future research.

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