herding in a queue: a laboratory experiment

Herding in a Queue: A Laboratory Experiment

Mirko KremerSmeal College of Business, Penn State University, University Park, Pennsylvania 16802, [email protected]

Laurens DeboChicago Booth School of Business, University of Chicago, Chicago, [email protected]

We report results from a set of laboratory experiments designed to understand human queue-joining behavior

when the quality of the good for which the queue develops is uncertain. We confirm that the joining frequency

may increase in the queue length, i.e. human subjects “herd.” Using a combination of choice and judgment

data, we tease apart behavioral deviations from rational behavior to explain the observed queue-joining

behavior. While human queue-joining decisions exhibit random errors compared to the rational strategy,

results show that decisions suffer primarily from judgmental bias: An excessively strong mental mapping

between queue length and quality. As a consequence, short queues are overly strongly associated with a

low-quality firm, reflecting partial neglect of the fact that even high-service-value firms can generate short

queues, while long queues are overly strongly associated with high-quality firms. We demonstrate that due

to the biased quality judgment of short queues, herd behavior in a queue may have a negative impact on a

firm’s throughput (or sales), while, without such bias, a firm would have expected a positive impact.

Key words : Queues, Quality signals, Experiments, Bounded rationality, Quasi-Bayesian judgments

1. Introduction and Motivation

Waiting lines are generally badly perceived by firms as waiting makes consumers waste their valu-

able time. Companies invest a significant amount of money to reduce waiting lines and the perceived

waiting cost (Larson 1987, Koeppen 2007, Barnes 2010). Traditional models in service operations

management almost exclusively focus on such negative externalities. However, folk wisdom reveals

that waiting lines may influence the perception of the value. For example, Waldman (2009) notes

on a blog: “If one walks into a restaurant and there’s nobody eating in the establishment already,

a consumer is less likely to hang around, regardless of the expected food quality. Conversely,

if a consumer walks in and the place is bustling, [he’s] probably going to give it a shot.” The

notion of “empty restaurant syndrome”—a restaurant parlance that describes patrons not joining

a restaurant when it is empty—also supports this observation. In an empirical study of queues at

a deli counter of a grocery store, Lu et al. (2012) suggest that longer queues may actually prompt

(slightly) more consumers to join the queue. In their experimental studies, Giebelhausen et al.

1

Kremer and Debo: Herding in a Queue2

(2011) and Koo and Fischbach (2010) provide evidence for a positive effect of queues on value

perception of the good for which the queue is formed. These examples have in common that a

quality component of the product or service in question is not perfectly known to all consumers.

The important implication is that the queue-joining decisions of better informed consumers may

turn the queue length into an informative signal about quality that influences the value perception

of less informed consumers—queues exert positive externalities.

Our paper studies empirically the endogenous demand formation for a good or service for which

the queue is formed when its quality is uncertain. Rational queue joining with and without positive

externalities is theoretically well understood. If queues exert only negative externalities (through

their impact on waiting), the joining probability of rational agents should be monotonically decreas-

ing in the queue length upon arrival (Naor 1969). In contrast, Debo et al. (2012) predict that the

queue-joining probability of rational agents may in fact be locally increasing when queues exert

negative and positive externalities, i.e., queues can induce “herd” behavior. Whether these the-

oretical accounts describe actual queue-joining behavior is an empirical question, and the main

focus of our study. We present the results from a series of laboratory studies showing that queue

joining is not monotonically decreasing in the observed queue length when the quality is uncertain

for a part of the population. This observed queue-joining pattern is directionally consistent with,

but not fully explained by, the rational model in Debo et al. (2012). We posit, and find strong

support for, the hypotheses that decision makers (a) have excessively strong mappings between

queue length and perceived service value and (b) make random-choice errors (Su 2008, Kremer

et al. 2010, Allon et al. 2012).

Because a better understanding of the demand formation allows firms to make better decisions

(such as those concerning capacity, assortment, price or the availability of information), we place

special emphasis on the question of how human queue-joining behavior impacts the firm’s through-

put. We provide a numerical example for which the firm’s sales (or throughput) increase marginally

due to “rational herding”. However, when taking into account the judgment biases documented in

our empirical studies, “behavioral herding” may cause a significant loss in sales, i.e., the opposite

impact on a firm’s sales than would be expected.

2. Literature Review

Studies related to queue-joining behavior with both positive and negative congestion externalities

are dispersed over different literatures. In the queueing literature, equilibrium joining strategies are

examined by Naor (1969), and the subsequent literature on the economic aspects of queueing—see

Hassin and Haviv (2003) for an excellent overview. The service quality is typically assumed to be

common knowledge. The equilibrium queue-joining strategy is typically characterized by means of


a threshold queue length above which no consumers join. There is a small literature testing in a

laboratory setting the impact of waiting in service environments. These include Oxoby and Bischak

(2005), Carmon et al. (1995), Carmon and Kahneman (1996), Leclerc et al. (1995), Pazgal and

Radas (2008), Kumar and Krishnamurthy (2008), Rapoport et al. (2004) and Seale et al. (2005),

which for brevity, we do not summarize. All these experiments address the negative externalities

(waiting). None of these experiments considers positive externalities.

Related to ours, Giebelhausen et al. (2011) experimentally show that waiting times can indeed be

a signal about quality increasing both purchase intentions and experienced satisfaction. Koo and

Fischbach (2010) experimentally study how consumers’ value perception increases as additional

consumers wait behind them. In both papers, quality is unknown or ambiguous. These papers

provide evidence for the effect of queues on value perception, but are silent on the issue of how queue

length impacts the preceding queue-joining decision. The queue-joining decision is nontrivial as

longer queues may not only indicate higher value; they also imply longer waiting times. The queue-

joining decisions determine, over the long run, the queue-length distribution faced by the consumers

(and hence the occurrence frequency of long queues), and thus the firm’s demand (throughput or

sales). Hence, we rely on theories that explain the joining decision.

The herding literature in economics studies investment decisions of sequentially arriving agents

that observe the investment of their predecessors. Bikhchandani et al. (1991) and Banerjee (1992),

describe situations in which investors observe privately some imperfect information about the

return of an asset, as well as the investment decisions taken by all earlier arriving investors. They

show that, in the long run, rational investors will ignore their private information and follow their

predecessors’ decisions. That is, they “herd.” Chamley (2004) provides an excellent overview of

the theoretical herding literature. Anderson and Holt (1997) and Sgroi (2003) experimentally test

the predictions of herd behavior in an investment-decision context. Put into the queuing context

of our study, the number of previous investors in the typical herding context can be interpreted as

the “queue,” and the investment decision can be considered as the “decision to join the queue.”

Building on herding theory, Debo et al. (2012) consider a Markovian queueing system with

some uninformed consumers who do not know the service quality but do observe the length of

the queue before deciding whether to buy the product; the other consumers are perfectly informed

about the quality. They find that the equilibrium queue-joining frequency of uninformed consumers

nonmonotonic in the queue length and the firm may even want to signal high quality via long lines

by selecting a slow service rate. The equilibrium queue-joining strategy exhibits a local minimum

at some queue length that is (weakly) lower than a threshold above which no consumers join.

Debo et al. (2011) show that a firm may select a low, uninformative price and signal high quality


via long lines instead. Debo and Veeraraghavan (2011) consider a model in which quality and

service rate are correlated, but both are unknown to all consumers. They show also that the

queue-joining frequency is nonmonotonic in the queue length. Veeraraghavan and Debo (2008) and

Veeraraghavan and Debo (2011) extend the single-queue setting to a two-queue setting. With no

waiting costs, consumers join the longest queue (Veeraraghavan and Debo 2008). With waiting

costs (Veeraraghavan and Debo 2011), the equilibrium queue-joining strategy is a complex function

of both queue lengths. In conclusion, theory predicts that queue-joining frequency when there are

both positive and negative externalities is not monotonic in the queue length upon arrival.

3. Theory

We consider a simple queueing model, with a single server, a Poisson arrival rate λ of risk-neutral

consumers and exponentially distributed service times with rate µ. Generally, upon arrival at

the system (server and First Come First Served queue), consumers observe the queue and decide

whether to join or not (i.e., to balk) based on waiting cost and service value considerations. If the

consumer joins, she incurs an outside opportunity cost of c (i.e., misses revenue) every unit of time

spent in the system. Consider a consumer arriving when n other consumers are present in the system

(waiting in line or in service). Due to the memoryless property of the exponential distribution, the

residual service time of the consumer in service is 1/µ. Therefore, the expected time until service

completion is (n+ 1)/µ, resulting in expected opportunity cost of joining, c(n+ 1)/µ. The gross

value obtained upon service completion depends on the service quality, vθ for θ ∈ {ℓ,h} (either

ℓow or high quality), where vℓ < vh. In the context of this paper, we will use the terms “quality”

and “value” interchangeably. The prior that the quality is high is p0 (= Pr(θ = h)). We refer to

v = p0vh + (1− p0)vℓ as the prior service value or as the prior service quality. As in Debo et al.

(2012), we assume that service quality is known by a fraction q of the population. These are the

informed consumers. The remaining fraction 1−q, the uninformed consumers, do not know service

quality realization.

We characterize formally the queue-joining strategy of consumer type τ ∈ {i, u} (either informed

or uninformed) by means of a mapping α = (αi,αu), where αi = (αi,h,αi,ℓ) and αi,h,αi,ℓ,αu ∈

{0,1}N, from the consumer’s information set of queue lengths n∈ {0,1, . . .} (and quality θ ∈ {ℓ,h}

for informed consumers) to a queue-joining decision in {0,1}, where 1 (0) means (not) joining

the queue. We characterize formally the posterior belief of the uninformed consumers that the

quality is high as a mapping p from the set of queue lengths n∈ {0,1, . . .} to the posterior belief in

[0,1]. We focus on pure joining strategies, and impose the following conditions on the consumers’

equilibrium queue-joining strategy α∗ and posterior belief p∗ (Debo et al. 2012): (i) The strategy

needs to maximize the expected utility and (ii) consumers’ posterior beliefs about the quality need


to follow Bayes’ rule, given the queue-joining strategy of all other consumers.

As informed consumers know for sure that the quality is θ (either ℓ or h), they join whenever

the value exceeds the expected opportunity costs: vθ − c(n+1)/µ > 0 (equilibrium condition (i)),

or, n < nθ = ⌊vθµ/c⌋, (where ⌊•⌋ represents the integer part that is lower than •). The threshold

⌊vθµ/c⌋ is referred to as the “Naor threshold.” Informed consumers join thus whenever the queue

is not too long. Of course, consumers informed that the quality is high are willing to join longer

queues than the ones informed that the quality is low (nℓ ≤ nh). To sharply focus on the impact

of uncertainty about the quality, we consider parameter values in our study such that nℓ = 0<nh,

or vℓ < c/µ< vh. That is, if the quality is truly low (high), informed consumers never (always) join

the empty queue. As a direct consequence, high-quality firms generate longer queues on average.

Hence, uninformed consumers can infer quality from queue length, and thus make queue-joining

decisions based on an updated posterior belief about service quality. We let p(n) be the posterior

of a randomly arriving uninformed consumer, and u(n) be the expected utility after observing

queue length n. Rational uninformed consumers join at queue length n⇔ u (n) ≥ 0 (equilibrium

condition (i)), where

u (n) = p(n)vh +(1− p(n))vℓ︸︷︷︸Posterior Value

− c(n+1)/µ︸︷︷︸Opportunity Cost

. (1)

Key to the updating of prior quality beliefs (equilibrium condition (ii)) are the conditional queue

length distributions that are jointly determined by the joining strategies of both informed and unin-

formed customers, α. We denote these long-run distributions as πθ (n,α), which is the probability

that a randomly arriving consumer observes a queue length of n when the quality is θ ∈ {ℓ,h} (via

the PASTA property, Wolff 1982). The likelihood ratio L(n,α) = πh(n,α)/πℓ(n,α) then describes

the statistical strength of evidence of a queue length of n, with higher values providing stronger

evidence in favor of a high-value firm. Further, for convenience of notation, we write posterior and

prior probabilities in odds form, l(n) = p(n)/(1−p(n)), and l0 = p0/(1−p0). Given the queue-joining

strategies, Bayes’ rule requires that the posterior odds that the quality is high, after observing

queue length n, satisfy (equilibrium condition (ii))

l(n) = l0 ×L(n,α). (2)

For convenience of notation in equilibrium, we write u∗ (n), l∗(n), π∗θ (n) and L∗ (n) as shortcuts of

u (n), l(n), πθ (n,α) and L∗ (n,α) evaluated with “decisions” α∗ and “judgments” p∗.

Proposition 1 (Debo et al. 2012). [A. Decision task of informed consumers.]

Informed consumers join according to a threshold strategy with Naor threshold nθ after observing

quality θ ∈ {ℓ,h}. Formally: α∗i,θ(n) = 1 for n∈ {0,1, . . . , nθ − 1}, and α∗

i,θ(n) = 0 otherwise.


[B.] The uninformed consumers’ joining strategy is characterized by the threshold, nh and a “hole,”

n∈ {0, . . . , nh}. Formally: α∗u(n) and l∗(n) are as follows.

[B-i. Judgment task of uninformed consumers.] The posterior odds, l∗(n) can be decomposed

into

l∗(n)︸︷︷︸Posterior

= l0︸︷︷︸Prior

× L∗(0)×Q∗(n),︸︷︷︸Likelihood of queue length n

(3)

with

Q∗(n) =

{(1− q)−n, 0≤ n≤ n+∞, n < n,

(4)

and

L∗(0) =1+

∑n

k=1

(λµ

)k

(1− q)k

1+∑n

k=1

(λµ

)k

+ q∑nh

k=n+1

(λµ

)k< 1.

(From which the posterior probability: p∗(n) = l∗(n)/(l∗(n)+ 1).)

[B-ii. Decision task of uninformed consumers.] Uninformed consumer join at all queue

lengths below nh, except at the hole, n . Formally: α∗u(n) = 1 for n ∈ {0,1, . . . , nh − 1}\{n} and

α∗u(n) = 0 otherwise.

The intuition for the equilibrium results of Proposition 1 is the following. Informed consumers

join according the quality-dependent Naor threshold. Because of this, uninformed consumers know

that queues of high-quality firms are longer than those of low-quality firms. Hence, the longer

the queue, the more likely they think that the quality is high. Both posterior value and expected

waiting cost increase. The likelihood ratio at queue length n below the hole is decomposed into

two factors: the likelihood ratio at the empty queue, L∗(0), and (1− q)−n. The likelihood ratio of

the empty queue, L∗(0) itself is a complicated function of the parameters of the model. Due to the

higher joining rate of the high-quality firm (thanks to the informed consumers), it is intuitive to

see that: L∗(0)< 1 such that an empty queue always requires the decision maker to decrease prior

odds l0; l∗(0)< l0. Hence, an “empty restaurant” is bad news about the quality of the restaurant

as (rational) consumers expect the high-quality restaurant to be more busy than the low-quality

restaurant. The second factor, (1− q)−n, illustrates the intuition that the posterior value increases

in the queue length. Intuitively, the evidence in support of high-quality service (vh) increases in the

queue length n. Once the queue grows long enough, the congestion costs will overtake the posterior

value, making no uninformed consumer willing to join. The queue length at which this happens

is n, the “hole.” As all uninformed consumers follow exactly the same strategy and only informed

consumers, who know that the quality is high, join at n, any queue length that is strictly longer

than n can only be explained by high quality. Hence, upon finding a queue that is strictly longer

than n, uninformed consumers know for sure that the quality is high, making, the posterior odds


jump to infinity above the hole. So, the hole, n, can be thought of as a threshold queue length

above which the quality is fully revealed to uninformed consumers.

4. A Behavioral Model

Based on the assumption that all consumers in the population are rational agents, the prediction

in our model with informed consumers is a sharp one: The queue-joining probability of uninformed

rational Bayesian consumers is 1 below the hole, 0 at the hole, then 1 up to the Naor threshold of

the high-value firm nh and 0 at and above nh. The immediate question is whether queue-joining

strategies of uninformed human are also nonmonotonic in the queue length? Will they identify a

“hole”? We predict systematic behavioral deviations from the benchmark with rational consumers,

and we expect these biases to occur both in the judgment task (how to update prior beliefs) and in

the decision task (how to choose given a belief about service quality). Below, we develop a formal

behavioral model that captures these biases which, in combination, predict a queue-joining pattern

that is distinctly different from the rational benchmark described above. The model provides a

structured intuition for behavioral queue joining in our context, and serves as the basis for our

econometric analyses in Studies 1 and 2.

4.1. Random-Choice Errors

A rational utility maximizer always joins (leaves) the queue if the expected updated utility of

joining is positive (negative)—i.e., u∗(n)>(<) 0. Thus, the model is silent in its prediction about

the fact that net utility from joining u∗(n) is largest at the empty queue n= 0 but decreases when

queue length approaches n. However, decision makers are prone to random-choice errors (Loomes

et al. 2002). Due to limited cognitive abilities, they choose the utility-maximizing alternative most

likely (in particular if |u∗(n)| is large) but not always (in particular if |u∗(n)| is small). Such errors

in the choice process occur for a variety of reasons. For example, the decision maker may make

mistakes when forming the expected utility of each option, when comparing the expected utilities

(i.e., calculating u∗(n) in our context), or when executing her preferences—i.e., when making a

choice. Following Hey and Orme (1994), we model such “bounded” rationality by adding a random

element to the choice rule. In particular, we let the decision maker join when u∗(n)>βϵϵ, where ϵ is

a zero-mean, symmetric random variable with unit standard deviation and distribution Φ(·). The

parameter βϵ is inversely proportional to the decision maker’s “degree of rationality.” We introduce

the -notation to indicate behavioral deviations from the rational strategies, indicated by ∗. The

probability of the decision maker joining a queue length of n is

Pr(a(n) = 1) =Φ(u∗(n)/βϵ) for 0≤ n. (5)


As βϵ grows to infinity, the joining probability approaches 1/2; which implies that the decision

maker randomizes between joining and balking with equal probability, irrespective of the queue

length. On the other hand, βϵ → 0+ models a rational decision maker who makes the decision of

joining the queue strictly based on the sign of u∗(n). This simple model of bounded rationality has

intuitive predictions for a consumer’s queue-joining behavior, as illustrated in Figure 1 for different

values of βϵ. Choice errors introduce a smooth joining pattern with a local minimum joining

probability at the hole, n. A model with βϵ > 0 appears a priori more reasonable as a description

for real queuing behavior than the extreme predictions from the rational model (βϵ → 0+). But can

random-choice errors alone predict queue-joining decisions well? A main proposition of our study

is that deviations from our rational model benchmark are more systematic in nature than caused

by random errors on the choice stage alone. In particular, we hypothesize predictable biases in

the way decision makers update their service value beliefs after observing the queue. The model

of Equation (5) allows for stochastic choice errors, but it assumes that net utility u∗ is calculated

based on the Bayesian posterior belief (p∗) about service value. We next take a quasi-Bayesian

view on the service value updating task.

4.2. Quasi-Bayesian Judgments: Representativeness Heuristic

While our queuing environment is mathematically complex, the key mechanics of the likelihood

ratio in Equation (3) are intuitive: High- (low-)service-value firms tend to generate longer (shorter)

queues. We suggest that human decision makers understand this inference, but are prone to partial

system neglect (Massey and Wu 2005): They respond primarily to the signal (the queue length) and

Figure 1 Posterior value v(n, p(n|βϵ)), cost c(n) = c(n+1)/µ, and joining probabilities α(n|βϵ, βϵ), for

vℓ = 1, vh = 33, p0 = 0.5, λ/µ= 1.19, c/µ= 2 and q = 0.15 with βϵ = 0,1,2.

βε=0,1,2

c(n+1)/µ

n

(a) Expected values and costs (q= 0.15)

βε=1

βε=0

βε=2

n

(b) Joining probabilities (q= 0.15)


secondarily to the system that generated the signal. In particular, we posit an excessively strong

mental mapping between queue length and service quality, resulting from case-based judgments

(Brenner et al. 2005) that rely primarily on evidence regarding the particular case at hand and

tend to neglect relevant aggregate properties of the class to which the case belongs. This essentially

is the representativeness heuristic (Kahneman and Tversky 1973), resulting in the partial neglect

of the relevant base rate (Koehler 1996). We capture the hypothesized bias, formally extending the

Bayesian model in odds form to

l(n|β0

)= l0 × (L∗(0)×Q∗(n))

β0

for 0≤ n. (6)

Our arguments above predict β0 > 1. The implications are most obvious for the empty queue

(n= 0). As L∗(0)< 1 and Q∗(0) = 1 (see Proposition 1), we posit with β0 > 1 that an empty queue

is overly strongly associated with a low-service-value firm, reflecting a partial neglect of the fact

that even a high-service-quality firm can generate an empty queue. Interestingly, this pattern may

reverse at longer queues: Define no as the greatest queue length for which L∗(0)×Q∗(n)< 1—i.e.,

the queue length below (above) which, the decision maker rationally decreases (increases) the prior

judgment.1 While assuming individuals correctly interpret longer queues as increasing evidence in

favor of the hypothesis vh, we posit with β0 > 1 in Equation (6) that, below (above) queue length

no, human queue joiners believe, more than rational consumers, that queues are representative of a

low- (high-)value firm, even though high- (low-)value firms could also generate short (long) queues.

The immediate question is how the hypothesized judgment bias affects queue-joining behavior.

Define the posterior probability of high service value as p(n|β0) = l(n|β0)

l(n|β0)+1. With a slight abuse

of notation, we define the expected utility of joining the queue at length n with a belief, p, that

the quality is high as u(n,p) = vℓ(1− p)+ vhp− (n+1)c/µ. Allowing for random-choice errors, the

predicted joining probability is thus

Pr(a(n) = 1) =Φ(u(n, p(n|β0))/βϵ

)for 0≤ n. (7)

Equation (7) challenges some of the key predictions of Equation (5), which describes a Bayesian

agent prone to random-choice errors alone. As the decision maker switches from lowering the

posterior below no to increasing it at and above no, joining frequencies decrease (increase) below

(above) no (see Figure 2, for which no = 7). Observe that joining rates could be lowest at a queue

length that is strictly smaller than n, while Equation (5) continues to predict a local minimum at

n (see, e.g., β0 = 2 in Figure 2).

1 Of course, when L(0)× (1− q)−n < 1, the decision maker decreases the prior judgment for all queue lengths belowthe hole.


Figure 2 Posterior value v(n, p

(n|β0

)), cost c(n) = c(n+1)/µ, and joining probabilities α

(n|β0, βϵ

), for

vℓ = 1, vh = 33, p0 = 0.5, λ/µ= 1.19, c/µ= 2 and q = 0.15 with β0 = 1/2,1,2 and βϵ = 1.

c(n+1)/µ

n

β0=1/2

β0=1β0=2

(a) Expected values and costs (βϵ = 1)

β0=2

β0=1/2

β0=1

n

(b) Joining probabilities (βϵ = 1)

Recall that above the hole, the posterior odds become infinity (Proposition 1, part B-i, Equation

(4)) as rational uninformed consumers know for sure that the quality is high. Hence, the bias

caused by β0 has no impact on queue lengths above the hole. In §4.3, we introduce another bias

that renders the posterior odds above the hole finite. As a consequence, in the combined model of

§4.4, β0 will have an impact on all queue lengths, below and above the hole.

4.3. Quasi-Bayesian Judgments: The Hole

Recall from our discussion following Proposition 1 that a queue of length of n+ 1 can rationally

be explained only by an informed consumer who has joined at a queue of length n as none of the

uninformed consumers would ever have joined a queue at length n. As a consequence, a rational

uninformed consumer infers a posterior probability of 1 at any queue length above the hole, based on

the reasoning that all other uninformed consumers are identical and act in an identical way. While

we consider it descriptively plausible that decision makers understand the involved equilibrium

directionally, we posit a noisy perception of equilibrium play. Formally, we assume that the posterior

is determined by queue length distributions allowing for noisy queue-joining actions of all other

uninformed (u) and informed (i) consumers:

αu(n|βe) =Φ(u∗ (n)/βe) and αi(n, θ|βe) =Φ((vθ − c(n+1)/µ)/βe), θ ∈ {ℓ,h}, 0≤ n.

These queue-joining probabilities are structurally equivalent to Equation (5), which describes the

focal consumer’s biased queue-joining decision by means of an error parameter βϵ. Here we need

to caution against a self-evident interpretation of βe as a parameter describing a quantal response


equilibrium play among boundedly rational consumers, which would require a mapping βϵ = βe

(McKelvey and Palfrey 1995). Instead, we interpret βe as a judgment parameter that describes

how the focal consumer noisily perceives the rational joining strategies of others, rather than a

choice parameter that describes the degree of rationality of the population. This view is particularly

plausible in light of the particular implementation of our experimental study where human subjects

make decisions in an environment in which all other arriving consumers are rational computer

agents. For convenience of notation, define α(k, θ|βe) = qαi(k, θ|βe)+(1−q)αu(k|βe) for k ∈ {0, . . .}.

Then, using the Birth–Death formulas from queuing theory, the posterior odds are

l(n|βe) =

{l0 × L(0|βe), n= 0

l0 × L(0|βe)× Q(n|βe), 1≤ n,(8)

where

L(0|βe) =1+

∑nhk=1(

λµ)n

∏k−1

l=0 α(k, ℓ|βe)

1+∑nh

k=1(λµ)n

∏k−1

l=0 α(k,h|βe)and Q(n|βe) =

∏n−1

k=0

α(k, ℓ|βe)

α(k,h|βe).

Note that the model of Equation (8) nests the Bayesian updating for βe → 0+, l∗(n), see Equation

(3). To see the impact on queue joining, let again p(n|βe) = l(n|βe)

l(n|βe)+1, such that the queue-joining

probability is given by

Pr(a(n) = 1) =Φ(u(n, p(n|βe))/βϵ) for 0≤ n.

The combined impact of the random-choice parameter βϵ and βe, as well as their interaction, on

queue joining is mathematically complex, but some general intuition can be gleaned from Figure 3:

For different values of βe, the model predicts that the perceived posterior has a smoothed kink at

n+1 (compare with Figure 1(a)), and further that it keeps increasing beyond the hole. Furthermore,

as βe increases, the positive effect of the hole on the posterior value for queue lengths above the

hole diminishes (see, e.g., for βe = 1/2). For large values (for e.g. βe = 2), the queue length becomes

uninformative, as there is too much noise on the perceived joining strategies. Hence, the posterior

value is close to the prior, for any queue length, and the joining strategy is almost monotonically

decreasing in the queue length. We find these predictions plausible.

4.4. Full Behavioral Model

To summarize, we propose a generalized behavioral queue-joining model that will serve as the

basis of our econometric analyses in Studies 1 and 2. Let the behaviorally enhanced posterior

be p(n|βe, β0) =l(n|βe,β0)

l(n|βe,β0)+1, where the parameters βe and β0 characterize the decision maker’s

quasi-Bayesian updating of service-value beliefs, with

l(0|βe, β0

)= l0 ×

(L(0|βe)

)β0

and l(n|βe, β0

)= l0 ×

(L(0|βe)× Q(n|βe)

)β0

for all 1≤ n.


Figure 3 Posterior value v(n, p(n|βϵ)), cost c(n) = c(n+1)/µ, and joining probabilities α(n|βϵ, βe), for

vℓ = 1, vh = 33, p0 = 0.5, λ/µ= 1.19, c/µ= 2 and q = 0.15 with βe = 0,1/2,2, β0 = 1 and βϵ = 1.

βe=0

βe=2βe=1/2

c(n+1)/µ

n

(a) Expected values and costs

βe=1/2

βe=2

βe=0

n

(b) Joining probabilities

Our behavioral queue-joining model for an uninformed consumer is

Pr(a(n) = 1) =Φ(u(n, p

(n|βe, β0

))/βϵ

)for 0≤ n, (9)

where βϵ characterizes the stochastic element in the consumer’s choice whether or not to join the

queue.

Our discussion of the three interacting parameters, (β0, βϵ, βe), points to a queue-joining pattern

distinctly different from the rational benchmark described in Section 3, as well as from the standard

queuing model with no informed consumers (Naor 1969). Our empirical objectives are summarized

in the following hypotheses.

Hypothesis 1 (Decision). In the presence of informed consumers, queue-joining rates of unin-

formed consumers are nonmonotonic in queue length.

While the idea of stochastic choice errors is theoretically sufficient to create a nonmonotonic

joining pattern (even for the fully rational boundary case βϵ = 0), a key objective of our empirical

study is to identify judgment biases that affect queue-joining decisions in a more systematic way.

Hypothesis 2 (Judgment). In the presence of informed consumers, posterior beliefs are too

low at the empty queue, and generally too sensitive to additional customers (β0 > 1).

5. Study 1: The Impact of Informed Customers

5.1. Experimental Design and Implementation

To test our hypotheses, we design experiments in which subjects repeatedly arrive at a facility with

service value that is unknown to them. The basic dynamics of the service system are governed by the


Table 1 Summary of the parameters and equilibrium outcomes

(vℓ = 1, vh = 33, q= 0.15).

Game parameters Strategy

Condition vℓ vh q p0 c µ λ nℓ n nh no

0 1 33 0 0.5 0.5 0.25 0.296 0 8 16 –

1 1 33 0.15 0.5 0.5 0.25 0.296 0 8 16 7

parameters of our theoretical queuing model. Subjects in our experiments are uninformed because

this consumer type’s judgments and choices are the key behavioral ingredient of our theoretical

model.2 In other words, while they can probabilistically infer service quality from the length of the

queue, they learn about the true service value only after completion of service.

As an experimental baseline, we create an environment without informed consumers (q= 0). In

this case, no consumer knows the quality of service before receiving it. Hence, high- and low-quality

firms generate identical queues. As a consequence, the hole reduces to the classical Naor threshold

based on the expected value, v= ⌊vµ/c⌋.3 As there are no informed consumers, the queue will never

grow beyond ⌊vµ/c⌋. This treatment, henceforth labeled Condition 0, is a useful experimental

control: Do subjects make queue-joining decisions that are consistent with the Naor threshold, in

the absence of informed consumers? Do they infer quality from queue length, even in the absence

of informed consumers?

We then replicate Condition 0, the only change being that we inject informed consumers. Specifi-

cally, we set q= 0.15—i.e., 15% of all consumers know the true service value when they arrive at the

service facility. The remaining parameters describing the queueing environment were chosen such

that the hole n is midway between the informed consumers’ thresholds for low- and high-service

value firms. Also, note that we obtain that no = 7, i.e., a Bayesian uninformed consumer would

decrease (increase) her prior p0 at all queue lengths strictly below (at and above) the hole. Table

1 summarizes all parameters and joining benchmarks.

5.1.1. Service value, waiting time, utility, and money. Our queuing model is meant to

represent consumers facing the tradeoff between (anticipated) service quality and (anticipated)

disutility from waiting for service. The hedonic nature of receiving service or enjoying time not

spent in a queue is difficult to control. For example, the same physical measures of service quality—

such as speed, accuracy, or courtesy—may be perceived or valued differently by different people

(Gans et al. 2008). Similarly, waiting in line may be perceived differently depending on a consumer’s

2 In comparison, the task of an informed consumer is less complex.

3 Recall: v= p0vh +(1− p0)vℓ.


outside value of time. Therefore, we operationalize service quality and waiting cost with money,

which is standard practice in experimental economics. The dollar award received after waiting

in queue for service is a proxy for service quality. We face an important design choice regarding

the cost of waiting in line which can be construed either as out-of-pocket cost or opportunity

costs. Queuing theory is silent on this distinction because it does not matter for a rational agent.

Mainly based on the grounds of realism, we decided to implement the disutility from waiting in

an opportunity-cost frame; i.e., subjects do not incur direct costs while waiting for service but

can earn per unit of time they spend not waiting for service.4 Our choice to frame disutility from

waiting as opportunity costs has an important implication: When they choose not to join, subjects

wait (and earn c per second) the same amount of time they would have waited in the queue had

they decided to join the queue, effectively removing the strategic incentive for participants to balk

simply to finish the experimental session early.

5.1.2. Eliciting beliefs. We hypothesize that joining behavior deviates from our model’s

predictions because of systematically biased beliefs about service value. Testing this hypothesis

is empirically challenging because, unlike choices, beliefs are difficult to observe. An empirical

strategy is to infer beliefs from choice data, but this is problematic due to various identification

issues that could be overcome only with additional assumptions (Heath and Tversky 1991, Wang

2011). Because reliable direct measurements of beliefs allow for sharper tests of behavioral theories

than beliefs inferred from choice data (Nyarko and Schotter 2002), we choose to elicit beliefs

directly. After they arrive at the service system and observe queue length, but prior to their joining

decision, we prompt subjects to indicate their subjective (posterior) probability on service value.

On a related methodological note, Croson (2000) show that belief elicitation actually improves

decision making, in particular if it is incentive aligned.

5.1.3. Prior information, sample information, and learning. We provide participants

with full knowledge about the relevant structure and parameters of the environment. For rational

Bayesian agents, this knowledge is sufficient for rational decision making. However, our subjects

are not fully rational, and their environment is complex and noisy. Thus, a mere description of the

relevant parameters is insufficient for good decision making. Rather, we expect subjects to learn

4 There is another reason for our choice of an opportunity-cost-of-waiting frame. If subjects incurred out-of-pocketcosts for waiting in line, then they basically have the choice between a gamble (waiting in line for an uncertaintime with uncertain cost, and then receiving some uncertain revenue) and a zero profit outside option (balking).Further, the gamble of waiting in line includes negative outcomes with a positive probability, potentially triggeringloss aversion (favoring the balking option), which we are not interested in studying here. The opportunity-cost frameavoids some of these issues because subjects have a choice between two gambles, both with strictly positive outcomes:1) waiting in line for an uncertain time, but free of cost, and then receiving service value, or 2) balking and earningper second.


about the options (join or balk) by experiential sampling over time. In order to remove strategic

incentives to join simply for reasons of information acquisition and learning, we provide subjects

with similar (counterfactual) information and experiences regardless of their choices. In terms of

outcomes, subjects observe the service value realizations at the end of each round regardless of

their decision, i.e., even when they do not actually earn the service value because they balked.

In terms of process, subjects observe the complete system dynamics (arrivals, joining, service

completions) even if they balk.5 This is important because observing other consumers’ arrivals

and queue-joining choices might help subjects improve the quality of their own decisions. Because

of our implementation choices, joining (leaving) the queue offers no advantage or disadvantage in

terms of what can be learned about the system and decision making.6

5.1.4. Input data. Because the complexity of our environment requires subjects to learn

by experience, we needed to ensure that what subjects experience throughout the experiment is

representative of their environment. All participants were exposed to the same set of unbiased

random samples from all relevant stochastic populations, sequences of random draws for service

times (governed by µ), arrival times (λ), service value (p0), and type of arriving consumers (informed

with probability q). We also generated a representative history of 30 periods (t∈ {−29, · · · ,0}) for

past queue lengths, total wait times, and service values. Subjects had access to this past information

before arriving at the service system for the first time in period t= 1.

5.1.5. Software implementation, subject recruitment, and payment. The experiment

was implemented in the experimental software zTree (Fischbacher 2007). Throughout the experi-

ment, the top of the computer screen would display all relevant game information such as service-

time and service-value distributions. Further, subjects had access to a history box containing all

relevant information from past periods, such as observed queue lengths, joining decisions, judg-

ments, as well as total wait time and service-value realizations. In each of the 30 independent

rounds played, subjects arrived at the service system and observe the queue. They were then able

to adjust their belief of service value by use of a sliding probability scale with a default value equal

to the base rate p0. They then made a decision whether to join the queue or leave the system.

To help them with some of the key calculations involved in the choice task, subjects could click

a button on the screen that would then display, for the observed queue length n, the expected

wait time as well as the corresponding opportunity cost. After they made their decision to join or

5 In particular, after a decision maker had chosen to balk, the user interface simulated a “what-if version” of thesystem. This version differed from the “real version” (observed after the decision to join) merely in terms of scale ofdisplayed elements (such as arriving and waiting consumers).

6 We note that joining a queue in order to learn about wait times has questionable ecological validity anyway.


balk, the queue dynamics were simulated in real time. In particular, consumers ahead in the queue

were served one after the other, while new consumers arrived at the system. Arriving consumers

themselves observed queue length and then made queue-joining decisions according to their type

(informed or uninformed, randomly determined according to q). After completion of service, sub-

jects learned about the true service value. They then moved on to the next period in which, after a

random period of time, they again arrived at a service system. We provide screenshots of the user

interface in the appendix (in §9).

Participants’ final payoff was based on both their choices and judgments. In each round, partic-

ipants earned laboratory tokens from their decision (join and earn service value, or leave and earn

while waiting). They earned additional laboratory tokens for the accuracy of their probability esti-

mates, according to a incentive-compatible quadratic scoring system that paid 10−(20×(δ−πh)2),

where πh is the submitted probability and δ= 1 if the true service value is high and δ= 0 otherwise.

Such a quadratic payment scheme is proper, and thus encourages honest reporting of subjective

probabilities (Brier 1950, Selten 1998). The two earnings were added for a total earning per round.

At the end of the experiment, the computer calculated average earnings across rounds, which was

then converted into real dollars using a conversion factor of 0.6. In addition, subjects would earn a

show-up fee that was condition-specific to adjust for the time spent in the laboratory.7 The average

total earnings were about $17.

We recruited subjects through a computerized system at a large public university in the United

States. After arriving at the laboratory facilities, participants read written instructions and were

briefed orally. 25 (39) subjects participated in Condition 0 (1). We also removed 6 (16) observations

that clearly indicated a mistake by the subject. For example, we had 5 subjects in Condition 1 that

submitted a posterior of 100% at an empty queue in the early rounds, but then 0% in the same

situations in later rounds, indicating that they did not immediately recognize that the sliding bar

was anchored low (left) to high (right). Our final data set comprises a grand total of 744 (1,154)

recorded choices and probability judgments.

5.2. Results

To describe the structure of our data, let ni denote the length of the queue observed by subject

j in period t, where i is a shortcut for (j, t) (we will omit some, or all, of these subscripts when

convenient). The collected data is then a(ni) ∈ {0,1}, where 1 (0) indicates joining (balking) and

p(ni)∈ [0,1] is the reported posterior probability that the quality is high when subject j is presented

7 Remember that our laboratory implementation did require subjects to actually wait, and that expected wait timescould differ quite dramatically across the different conditions used in our studies.


queue length ni in situation i. Further, define a∗(ni) as the rational decision, and a∗(ni, p(ni)) as

the rational decision conditional on the submitted posterior.8

Figure 4 shows the averaged observed joining a(n), rational joining a∗(n), and averaged condi-

tionally rational joining a∗(n, p(n)) as a function of the queue length n. We can make a number

of observations without formal analysis. For Condition 0, without informed consumers (q= 0), the

observed queue-joining frequencies are close to 100% for low queue length, but they decrease sub-

stantially at n= 5—i.e., in a range where the theoretical model still predicts that every consumer

joins. These observations are directionally consistent with the model of Equation (5) of a decision

maker that joins the queue according to Naor’s threshold, but makes random errors. Not distin-

guishing by queue length, subjects make the rational decision 64.9% of the time (i.e., a∗(ni) = a(ni)

in 64.9% of all queue-joining situations). This percentage does not increase appreciably (67.5%)

when we compare observed choices with conditionally rational choices (i.e., a∗(ni, p(ni)) = a(ni)

in 67.5% of all queue-joining situations). We next look at Condition 1 with informed consumers

(q = 0.15). As expected, the queue-joining frequency is nonmonotonic in queue length, due to the

positive externalities (Hypothesis 1). However, the observed behavior does not fit well with the

predictions of the rational model (see Proposition 1), nor with the smoother version of Equation

(5) which predicts that queue-joining rates dip as queue length approaches the hole (n= 8), see

Figure 1(b). In contrast, we observe that queue-joining rates drop quickly at low queue lengths and

then swiftly increase as queue lengths approaches the hole (at n= 8). One possible explanation for

this choice pattern is that systematic judgment biases, e.g., lead the decision maker to decrease

the posterior probability too heavily at low queue length (Hypothesis 2). We again compare the

proportion of rational choices with the proportion of conditionally rational choices. The difference

(56.2% versus 68.2%) suggests that at least part of the observed choice pattern is explained by

biased service value judgments.

To formally capture the observed choice patterns and explore its underlying behavioral causes,

we structurally estimate our generalized choice model (9) based on our choice data a(ni):

Pr(a(ni) = 1) =Φ(u(ni, p(ni|βϵ, β0))/βϵ

). (Model I)

We consider a series of nested versions of Model I. First, Model Ia constrains βe = 0 and β0 = 1.

This model allows for stochastic errors on the choice stage (through βϵ), while assuming that the

decision maker correctly updates her prior service-value beliefs, p(n|0,1) = p∗(n), including the

prediction that p∗(n) = 1 above the hole (n = 8). Model Ib estimates βϵ as an additional free

parameter which allows for less extreme predictions above the hole, p(n)< 1 for βe > 0. Finally,

8 a∗(n, p(n)) = 1⇔ u(n, p(n))≥ 0


Figure 4 Averaged observed queue-joining frequency a(n) (solid line), averaged conditionally rational joining

a∗(n, p(n)) (dotted line) and rational choices a∗(n) (dashed line) for queue length n∈ {0, . . . , nh} for

Conditions 0 and 1.

a(n)~

a*(n)

a*(n,p(n))~ ~

n

(a) Condition 0 (q= 0)

=n^

a(n)~

a*(n)

a*(n,p(n))~ ~

n

(b) Condition 1 (q= 0.15)

Model Ic relaxes the constraint on β0, which is the key parameter capturing our hypothesis that

decision makers reduce their service-value beliefs too heavily at the empty queue and overall are

too sensitive to additional consumers in the queue (for β0 > 1). Fitting Model I to our choice data

involves the estimation of structural parameters that describe the way decision makers update their

beliefs about service value. Because the attempt to infer beliefs from choice data can potentially

result in biased estimates (Nyarko and Schotter 2002), we also leverage the fact that we elicited

probability estimates for subject i in period t, p(ni). Specifically, we fit a model that conditions a

subject’s observed choices a(ni) on their reported posterior beliefs, p(ni),

Pr(a(ni) = 1) =Φ(u(ni, p(ni))/βϵ). (Model II)

We use maximum likelihood methods implemented in Stata 12 to estimate the parameters of

Models I and II. We assume Φ(z) = exp(z)

exp(z)+1, the logit model. In order to compare model fit, we also

calculate the Akaike information criterion (Akaike 1981) and the Bayesian information criterion

(Schwarz 1978) as likelihood-based measures of the relative goodness of fit of a statistical model.

Table 2 presents the estimation results.

Consider first Condition 0 with no informed consumers, q = 0. Rational consumers should not

learn from the queue length. That is, the posterior is equal to the prior for all queue lengths. Both

information criteria show that the model fit increases when we condition the logit choice model

on submitted posteriors p (Model II) rather than on the assumption that choices are stochastic

but based on the prior p0 (Model Ia). However, the absolute value for the bounded rationality


Table 2 Choices: Estimation results (standard errors in parentheses).

Condition 0 (q= 0), N = 744 Condition 1 (q= 0.15), N = 1,154

Model Ia Model Ib Model Ic Model II Model Ia Model Ib Model Ic Model II

βϵ 6.13∗∗ (0.85) 1.34∗∗ (0.21) 1.43∗∗ (0.42) 6.39∗∗ (0.73) 10.57∗∗ (2.81) 5.67∗∗ (1.34) 2.83∗∗ (0.53) 6.02∗∗ (0.64)

βe – 0.49∗∗ (0.09) 0.45∗ (0.21) – – 1.46∗∗ (0.25) 1.16∗∗ (0.09) –

β0 – – 0.96∗∗ (0.13) – – – 1.40∗∗ (0.14) –

LL −437.13 −404.71 −404.65 −414.64 −782.68 −742.2 −735.76 −648.28

AIC 876.25 813.43 815.30 831.28 1,567.37 1,488.41 1,477.53 1,298.56

BIC 880.86 822.65 829.14 835.89 1,572.42 1,498.51 1,492.68 1,303.62

a ∗∗p= .01, ∗p= .05, †p= .1. The significance level for β0 is based on the test of H0 : β0 = 1.

coefficient βϵ does not change appreciably between the two models. We conclude that, in absence

of informed consumers, random-choice errors explain the observed queue-joining behavior well.

This observation provides empirical support for Allon et al. (2012) who model boundedly rational

consumers in the observable queue model. Thus, without informed consumers, the uninformed ones

are not influenced by the queue length when assessing the quality of the service. The queue exerts

negative externalities only.

Consider next Condition 1, with q = 0.15—i.e., 15% of the population is informed. Estimation

results show that the model fit improves dramatically when we condition choices on submitted

posteriors (Model II, AIC = 1,298.56) rather than rational posteriors (Model Ia, AIC = 1,567.37).

Further, the estimated βϵ decreases notably in magnitude when we condition queue-joining choices

on elicited probabilities (6.02 for Model II versus 10.57 for Model Ia). In other words, Model

Ia absorbs more systematic judgment biases embedded in the elicited p(ni) and simply (and,

quite possibly, incorrectly) attributes them to the bounded rationality coefficient βϵ. To better

understand the mechanisms behind the submitted posteriors p(ni), and test Hypothesis 2 (β0 > 1),

we next compare Models Ib and Ic with the base case of Model Ia. Both information criteria suggest

that the full model Ic explains the choice data best. Importantly, the coefficient β0 = 1.46, which

is significantly larger than 1, in concordance with Hypothesis 2.

Overall, the analysis of our choice data indicates that queue joining in the presence of informed

consumers (Condition 1) is affected significantly by behavioral anomalies on the judgment level.

To sharpen our statistical inferences, we also fitted our judgment model, p(ni|βe, β0), directly to

the judgment data p(ni), rather than inferring its parameters from our choice data.9 Importantly,

the estimated value for β0 (=1.46) was highly significant and near-identical to our choice-based

9 In particular, we assumed that elicited posterior probabilities follow a normal distribution, p(ni) ∼N(p(ni|βe, β0

), σ2

). In this specification, σ is a parameter that must be estimated and that captures, similar to βe

for our choice model, the idea of bounded rationality (see Ho and Zhang 2008 for a similar interpretation).


estimate in Table 2. This result adds further strength to the evidence in favor of our Hypothesis 2.

6. Study 2: Robustness checks

Study 1 established nonmonotonic queue-joining behavior for uninformed customers. Using a com-

bination of choice and judgment data, we find strong support for our hypothesis that queue-joining

decisions are influenced by systematically biased probability judgments on the part of the arriving

customers. In other words, our results suggest that queue joining behavior is not simply a statistical

artifact of a random choice error account, which in other contexts has been shown to be consistent

with seemingly systematic choice and judgment biases (Erev et al. 1994, Budescu et al. 1997, Fox

and Tversky 1998, Su 2008).

To further differentiate our research hypotheses from a pure error explanation, in Study 2 we

take an approach similar to the one adopted in Massey and Wu (2005). We design environments

that differ substantially in terms of their key structural parameters, but are identical in terms

of their Bayesian posteriors and thus predict identical rational queue-joining behavior. A random

error model would predict no systematic differences in choices and judgments across the different

conditions. On the other hand, our queue-sensitivity hypothesis predicts systematic differences in

choices (but not necessarily in judgments), as explained below.

This repetition is valuable also as a robustness check, in an attempt to show that the systematic

judgment biases we posit are not artifacts of the particular parameters that were used in Study

1. Furthermore, the design of Study 2 allows us to explore whether value judgments and joining

decisions are affected in meaningful ways by particular aspects of the queuing environment, such

as service speed or arrival rates.

6.1. Experimental Design and Implementation

We design different environments such that the rational queue-joining strategy is the same as in

Condition 1 of Study 1—i.e., with a hole at n = 8 and balking at nh = 16. While generally a

complicated function of all structural parameters (vℓ, vh, p0, λ, µ, c, q), notice from Proposition 1

that the equilibrium conditions depend crucially on the proportion of informed consumers in the

population, q, as well as the utilization rate ρ= λµ. We keep q constant across conditions. The first

parameter we vary in our experimental design is the prior on service quality, p0 (= 0.5 in Condition

1), and we consider two levels, 0.3 (low prior) and 0.7 (high prior). We also vary arrival rates λ

and service rates µ (i.e., ρ= λµ), with the proper adjustments to c to ensure that we keep constant

cµ, which can be interpreted as the expected waiting cost per consumer in the queue. Intuitively,

relative to the base Condition 1, the resulting changes to ρ= λµdecrease (increase) the expected

waiting cost when the probability of receiving high service value is low (high). To keep the hole at


Table 3 Summary of the parameters and equilibrium outcomes

(vℓ = 1, vh = 33, q = 0.15).

Game parameters Strategy

Condition p0 c µ λ no nℓ n nh Sample Size

1 0.5 0.5 0.25 0.296 7 0 8 16 1,154

2 0.3 0.5 0.25 0.191 2 0 8 16 1,042

3 0.7 0.5 0.25 0.349 8 0 8 16 1,018

4 0.3 0.77 0.387 0.296 2 0 8 16 1,049

5 0.7 0.42 0.212 0.296 8 0 8 16 988

n= 8 with p0 = 0.3, 0.5 and 0.7, the utilization rate must be ρ= 0.77, 1.19 and 1.39, respectively.

The parameters of the four conditions are summarized in Table 3, along with the parameters of

Condition 1.

The five conditions provide a strong test of our hypothesis that queue-joining behavior cannot

be explained by a random-error account alone. Because our design keeps vℓ, vh, and c/µ constant,

it follows immediately that the net expected utility from joining at a given queue length n (u∗(n))

is the same in all conditions, as can be seen from Equation (1). The important implication is

that, for β0 = 1, randomness on the choice stage (through βϵ) and noisy perception of other con-

sumers’ equilibrium joining strategies (through βe) predict the exact same choice and judgment

patterns for all five conditions. Naturally, this prediction presumes that the magnitude of these

parameters (βϵ, βe) does not vary appreciably across conditions, which is an assumption that is

a) theoretically plausible, and b) empirically testable. However, the prediction of invariant choice

and judgment patterns changes under our Hypothesis 2 that decision makers are overly sensitive

to queues (through β0 > 1). Note that the key difference between low-, medium-, and high-prior

environments is the relative location of no—i.e., the queue length below (above) which the decision

maker decreases (increases) her prior (see Table 3). Intuitively, we would mostly expect excessive

upward revisions of service-value probabilities for low-prior Conditions 2 and 4 (where no = 2), but

mostly excessive downward revisions for high-prior Conditions 3 and 5 (where no = 7). The impli-

cation is that service-value judgments and queue-joining patterns differ across conditions, even if

the strength of the bias (magnitude of β0) does not.

6.1.1. Input data. A key principle of our experiments is to expose the subject to signals

(queues, arrivals, service times) that are consistent with the environment the signals are sampled

from. While the experimental macro design of Study 2 controls for the rational predictions regard-

ing choices and judgments, the principle of representativeness poses implementation challenges

because the five conditions of our design are differentiated with respect to the distributions for


priors, arrivals, consumer types, queue lengths, service times, and service values. Although we

draw representative samples from the equilibrium distributions, the implication is that subjects

in different conditions are exposed to different samples throughout the course of the experiment.

Thus, to facilitate comparisons across conditions, it is important to sample intelligently. We take

a number of steps that are based on the idea of common random seeds often used in simulation to

reduce variability. In particular, all stochastic elements in our experiments are based on random

seeds of [0,1]-uniform variables that are common to all conditions. The uniform variates were then

converted into events using the appropriate distributions, such as bivalued for service value and

exponential for service times.

Using common uniform seeds has two implications. First, whenever two conditions are identical

in terms of the distribution of a stochastic element of our model (e.g., service rate for Conditions

1 and 2), then subjects in the two conditions would observe identical sequences of that element

in each round of the experiment. Second, whenever two conditions are not identical regarding a

stochastic element (e.g., service rates for Conditions 3 and 4), then the use of common random

seeds at least controls for order effects. For example, if a high draw from the uniform distribution

would result in a much larger than expected realized service time of the mth consumer in round t

in the fast-service Condition 4 (µ= 0.387), then the same would be true for the mth consumer in

round t in the slow-service Condition 5 (µ= 0.212).

6.1.2. Subject recruitment and payment. The methodology used was identical to that

in Study 2. We recruited a total of 138 (36/34/35/33) subjects from a large public university

in the United States. In each of a total of 30 rounds, subjects arrived at the system, submitted

their subjective belief about the quality of service, and made a queue-joining decision. They then

observed the queue dynamics in real time and, ultimately, the outcome. We used the same incentives

as in Study 1, with a few modifications to the show-up fee and the conversion rates in order to

adjust for the fact that the expected service values and the expected wait times differed by condition

(Table 3). We removed 13 (8/2/1/2) observations, for same the reasons explained in Study 1. This

left us with a total of 5,251 observations, including the 1,154 observations from Condition 1.

6.2. Results

We begin our analysis with a visual inspection of queue-joining frequencies in our five conditions.

Figure 5 shows averaged observed joining a(n), rational joining a∗(n), and averaged conditionally

rational joining a∗(n, p(n)) as a function of the queue length n.10 To save space, we do not repeat

10 We do not have data at for all queue lengths in {0, . . . , nh}. This is because the queue length presented to thehuman subjects was sampled from the equilibrium distribution (π∗

θ (n)), which implies a very low frequency (e.g., lessthan 1%) for certain values of n, especially above the hole (n= 8).


the results of Condition 1 (see Figure 4). For all conditions, the observed queue-joining frequency

is nonmonotonic in queue length, lending broad support for our Hypothesis 1. Further, in all

conditions, the lowest queue-joining frequency appears to be below the hole (and then again, for

the longest observed queues, beyond the hole). Importantly, the joining pattern is much more

pronounced for Conditions 3 and 5 where Bayesian analysis predicts that the high prior (p0 = .7)

should be reduced at all queue lengths below the hole. In both conditions, joining frequencies drop

below 50% below the hole (n = 8), but increase swiftly towards and just above the hole. This

pattern is suggestive of Hypothesis 2, that decision makers decrease their posteriors below the hole

(as they should), but do so too heavily, and then increase their posteriors towards the hole (as they

should), but again do so too heavily.

We again compare the proportion of rational choices with the proportion of conditionally rational

choices. In the low-prior Conditions 2 and 4 (p0 = .3), subjects make a rational decision 63.8%

(61.2%) of the time. These percentages increase (70.5% and 66%, respectively) when we compare

observed choices with conditionally rational choices. Similarly, subjects make the rational decision

51.1% (53.5%) of the time in the high-prior Conditions 5 and 7 (p0 = .7). Again, these percentages

increase (65.6% and 70.2%, respectively) when we compare observed choices with conditionally

rational choices. This observation is suggestive of our Hypothesis 2 that the observed choice pattern

is explained by systematic biases regarding the updating of service value.

We next re-estimate our generalized choice model of Equation (9) for each condition separately.

The estimations for the high-prior Conditions 3 and 5 support our previous findings for Condition 1,

see Table 4. Model fit improves as we add parameters, and all estimated parameters are significant

and of similar magnitude across conditions. Importantly, β0 is significantly larger than 1, supporting

our Hypothesis 2. On the other hand, we do not see a significant effect of β0 for the low-prior

Conditions 2 and 4. Not surprisingly, model fit is generally improved if the utilities entering the

choice rule are conditioned on elicited beliefs (Model II). Figure 5 demonstrates that choice patterns

are moderated by the prior p0, contrary to the rational model that predicts no differences between

the five conditions included in our design. As a consequence, random-choice errors alone (βϵ > 0

and βe > 0) are not sufficient to explain observed queue-joining behavior.

An important objective of Study 2 is to determine if systematic between-condition differences

exist regarding the processes underlying the observed choices—i.e., on the parameter level. To

assess whether the magnitude of random-choice errors varies with environmental conditions, we re-

estimate Model II, which conditions choices on elicited beliefs about the service quality. However,

we now estimate the model on the pooled data and allow the coefficient βϵ to be determined by

condition characteristics. Using Condition 1 (p0 = .5) as the base category, we add a set of nested


Figure 5 Averaged observed queue-joining frequency a(n) (solid line), averaged conditionally rational joining

a∗(n, p(n)) (dotted line) and rational choices a∗(n) (dashed line) for queue length n∈ {0, . . . , nh} for

Conditions 2, 3, 4 and 5.

=n^

a(n)~

a*(n)

a*(n,p(n))~ ~

n

(a) Condition 2

=n^

a(n)~

a*(n)

a*(n,p(n))~ ~

n

(b) Condition 3

=n^

a(n)~

a*(n)

a*(n,p(n))~ ~

n

(c) Condition 4

=n^

a(n)~

a*(n)

a*(n,p(n))~ ~

n

(d) Condition 5

dummy variables. In particular, we create two dummy variables—for the low-prior conditions (I.3 =

1 if p0 = .3) and the high-prior conditions (I.7 = 1 if p0 = .7)—and additional dummy variables for

Conditions 4 (Ic4 = 1 if c= 4) and 5 (Ic5 = 1 if c= 5). We then estimate our structural Model II on

the pooled data (N = 5,251) with βϵ = βϵcons + βϵ

.3I.3 + βϵ.3,c4I.3Ic4 + βϵ

.7I.7 + βϵ.7,c5I.7Ic5. Our results

show that only βϵcons is significant (6.03, p < .01, LL=−3,064.46). This shows a lack of meaningful

between-condition differences in the magnitude of random-choice errors. We find this empirical

result convincing also on theoretical grounds. Because random error enters our choice model after

the (biased) utility calculation, and utilities in the different conditions are approximately on the

same scale, there is little reason to assume that decision makers make more errors in one condition

than in the other.


Table 4 Choices: Estimation results (standard errors in parentheses).

Condition 2 (p0 = 0.3), N = 1,042 Condition 3 (p0 = 0.7), N = 1,018


βϵ 8.37 (2.31) 8.84 (2.65) 8.96∗∗ (2.39) 6.83∗∗ (1.06) 12.24 (3.41) 7.98∗∗ (2.68) 2.16∗∗ (0.41) 7.69∗∗ (0.98)

βe – 2.53ns (1.67) 2.94∗∗ (0.28) – – 0.87∗ (0.35) 1.47∗∗ (0.097) –

β0 – – 0.92ns (0.3) – – – 1.62∗∗ (0.09) –

LL −675.52 −672.1283 −672.06 −588.92 −695.178 −680.51 −639.85 −619.99

AIC 1,353.04 1,348.26 1,350.12 1,179.84 1,392.36 1,365.015 1,285.71 1,241.995

BIC 1,357.99 1,358.15 1,364.97 1,184.78 1,397.28 1,374.866 1,300.48 1,246.92

Condition 4 (p0 = 0.3), N = 1,049 Condition 5 (p0 = 0.7), N = 988


βϵ 10.72∗∗ (3.48) 12.27∗∗ (3.86) 11.03∗∗ (3.67) 6.71∗∗ (0.99) 9.79∗∗ (2.36) 9.60∗∗ (3.47) 2.31∗∗ (0.41) 6.06∗∗ (0.84)

βe – 10.51ns (15.50) 2.00ns (1.52) – – 0.72ns (0.43) 1.45∗∗ (0.11) –

β0 – – 0.90ns (0.50) – – – 1.65∗∗ (0.09) –

LL −697.60 −695.21 −695.11 −626.69 −669.46 −667.723 −629.326 −580.56

AIC 1,397.21 1,394.42 1,396.23 1,255.39 1,340.913 1,339.45 1,264.65 1,163.13

BIC 1,402.16 1,404.33 1,411.09 1,260.34 1,345.809 1,349.24 1,279.34 1,168.02

a ∗∗p= .01, ∗p= .05, †p= .1. The significance level for β0 is based on the test of H0 : β0 = 1.

Finally, as in Study 1, we again fitted our judgment model p(ni|βe, β0) directly to the judgment

data p(ni). Omitting estimation details for the sake of brevity, we highlight here the result that

β0 > 1 in all conditions, although only barely significant for the low-prior Conditions 2 and 4. While

the condition-specific parameter estimates may suggest that decision makers become increasingly

sensitive to queue length when their prior is high (as they do in Table 4), this between-condition

effect is not statistically significant. We conclude from these additional estimations that the strength

of the hypothesized effect (β0 > 1) is fairly robust across substantially different environments.

7. Numerical Illustration and Estimation of Throughput

Recall that our key concern is to understand the impact of behavioral queue joining on the firm’s

queue-length distribution, and hence, on the long-run throughput (or sales). In this section, we show

that systematic behavioral deviations we have identified via our experiments can have a significant

negative impact on a high-quality firm’s sales. We use the parameter settings with a prior of p0 = .7

and 15% informed consumers (as in Conditions 3 or 5). The high-quality firm’s throughput under

rational joining behavior when there are no informed consumers (q= 0) is 0.7020λ. The throughput

(based on both informed and uninformed consumers, q = 0.15) under rational joining behavior is

0.71217λ. Hence, the increase in throughput via a rational “herd effect” is very low. This is because


the rational herd effect occurs at queue lengths above no = 8. These queue lengths occur with very

low frequency.

In Figure 6, we explain qualitatively step-by-step the impact of the different behavioral regular-

ities we have identified. We change the parameters of our behavioral model (βϵ, βe, β0) gradually

from fully rational, (0,0,1), to (2.5,1.5,1.5) which is in the ball-park of the estimations of Condi-

tions 3 and 5 (see Table 4). First, we introduce a random error on the queue-joining decision (left

panel); βϵ = 2.5, keeping the two other parameters the same. The joining frequency of the unin-

formed consumers around n= 8 becomes smoother due to the random errors. Still, the minimum

queue-joining frequency occurs at n= 8. The throughput decreases slightly to 0.7055λ. Second, we

introduce a noisy perception of all other consumers’ joining strategies (middle panel); βe = 1.5.

Interestingly, the notion of the “hole” almost disappears now. The throughput increases slightly

to 0.7066λ.

Notice that this joining pattern is still unlike the empirically observed patterns in Figure 5,

right panel. Therefore, random-choice errors do not explain the observed behavior. Furthermore,

random errors cause only a slight reduction in throughput compared to the rational benchmark

(compare 0.71217λ with 0.7066λ). This is because the random errors matter the most when the

joining decision is less obvious, at long queue lengths, which do not occur frequently. Finally, we

introduce (in the right panel) the bias due to the representativeness heuristic. We selected β0 = 1.5.

The joining frequency (right panel) exhibits a pattern that is qualitatively similar to the empirical

joining frequency of Figure 5, right panel. In addition, the throughput now decreases significantly

to 0.6586λ. Thus, the judgment bias leads to a reduction in throughput because the loss in joining

rate at short queues occurs more frequently than the gain at long queues. Hence, while rational

herding may have a marginally positive impact, “behavioral herding” can have a significantly

negative impact on the high-quality firm’s sales.

Figure 6 Numerical simulation: Queue-joining frequency predicted by the behavioral model.

=n^

(a) (βϵ = 2.5, βe = 0, β0 = 1)

=n^

(b) (βϵ = 2.5, βe = 1.5, β0 = 1)

=n^

(c) (βϵ = 2.5, βe = 1.5, β0 = 1.5)


8. Conclusions

To the best of our knowledge, ours is the first study with observable queue lengths (a la Naor),

that provides experimental and theoretical support to the idea that long queues can attract more

consumers via their a positive impact on the quality perception when the quality is uncertain. The

existing experimental literature (Giebelhausen et al. 2011, Koo and Fischbach 2010) focuses on

how queues impact consumers’ value perception. However, to assess the impact on the throughput

(or sales), the firm must also understand the long-run queue-length distribution. This requires the

analysis of queue-joining decisions, which takes both the value perception and opportunity costs

into account. Our study provides empirical evidence for the hypothesis that queue length exerts

positive externalities on the queue-joining decision.

While our study provides empirical support for the nonmonotonic queue-joining pattern theo-

rized in Debo et al. (2012), we document how systematic choice and judgment biases shape real

queue-joining behavior in ways that are predictably different from the rational benchmark. A main

proposition of our study is that deviations from rational queue-joining decision making are more

systematic in nature than the mere notion of random-choice errors and are, in fact, driven by

systematic biases in arriving consumers’ quality inferences after observing the queue. In particular,

we find empirical support for the idea that consumers’ quality judgments and subsequent queue-

joining decisions suffer from excessive queue-to-quality mappings: They are overly confident that

short queues are indicative of a low-quality firm, but revise their prior quality beliefs in favor of

a high-value firm too readily as queues grow longer. We demonstrate that this excessive queue-to-

quality mapping at short queues can lead to a loss in throughput (or sales) when the firm would

have expected an increase in throughput due to rational herding.

We also view our study as a useful part of a desired empirical–theoretical feedback loop in

Operations Management research. For example, Allon et al. (2012) develop a theoretical model

of boundedly rational consumers (via a stochastic choice rule, as in our study) in the context of

observable queues and focus on situations for which the quality is perfectly known. Our Condition

0 without informed consumers provides some experimental evidence that random-choice errors can

indeed explain the observed queue-joining behavior in absence of quality uncertainty. However,

when service quality is uncertain and in the presence of informed consumers, our study suggests

that random-choice errors alone cannot explain observed queue-joining behavior.

Hopefully, our results can serve as a basis to behaviorally refine future theoretical models of ser-

vice systems in order to improve their descriptive and predictive validity. Further, we believe that

our study contributes to empirical research in the growing field of Behavioral Operations Manage-

ment. In order to establish our hypothesis that (biased) queue-joining decisions are influenced by


(biased) quality judgments, we collect choice and judgment data. This step is useful because of the

difficulty of inferring judgments from choices (Heath and Tversky 1991), which seems important in

light of the plausible hypothesis that decision making in many operations contexts is influenced by

flawed probability judgments. For example, empirically documented biases in newsvendor decisions

are frequently attributed to biased judgments of relevant probability distributions (Schweitzer and

Cachon 2000). Sharing a similar spirit with our study, the experiments in Croson and Ren (2012)

elicit newsvendor decisions as well as confidence judgments regarding the demand distribution in

order to establish their hypothesis that overconfidence causes newsvendor decision bias. Finally, we

believe it is important to distinguish (and, ideally, tease apart) unsystematic and systematic biases

(Kremer et al. 2011). The reason is that effective debiasing strategies are likely to be very different

if the main source of decision bias is random-choice error instead of a systematically biased revision

of service-value beliefs (Arkes 1991).

Our study has natural limitations that point towards interesting opportunities for future research.

For analytical tractability and empirical testability, our simple model of herd behavior uses a

bivalued notion of quality, and all relevant parameters of the queuing environment were given to

our subjects. In real life, however, quality is typically not bivalued, and important parameters as

the fraction of informed consumers, the service rate, the arrival rate and the opportunity costs

can only be learned by experience. Further, subjects in our experiments were making decisions in

an environment in which all other queue joiners are computerized and rational. This experimental

choice controls for many aspects that come with real human interaction (such as fairness) and,

importantly, it gives theory its best shot: subjects had no incentive to deviate from equilibrium

predictions, for example by trying to respond optimally to other boundedly rational consumers.

The use of computerized agents is a common implementation in experimental studies, and we

consider it a useful first step towards a better understanding of system behavior driven by an entire

population of boundedly rational consumers. An immediate extension would be the development

of a theory based on the behavioral biases observed in this paper and testing of endogenous herd

formation in a laboratory setting with congestion. An interesting experiment would analyze human

queue-joining decisions in the presence of other human consumers or, as an intermediate step, in

the presence of computerized consumers that are boundedly rational in the sense of our empirical

results. Obviously, the end goal of this research would be to understand and possibly predict queue

joining and herding phenomena in real businesses (as in Lu et al. 2012). Tightly controlling the

environment in a laboratory setting in order to test behavioral deviations from rational predictions,

we hope our study is a first step in this direction.


Figure 7 (Partial) screenshots of the user interface human subjects faced during our experiments

(a) After arrival.

(b) After joining.

9. Appendix: Screenshots

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