Herding and Congestion 1

Transcription

1 Herding and Congestion 1 Laurens G. Debo 2 Tepper School of Business Carnegie Mellon University Pittsburgh, PA Christine A. Parlour 3 Tepper School of Business Carnegie Mellon University Pittsburgh, PA Uday Rajan 4 Ross School of Business University of Michigan Ann Arbor, MI March 1, 2005 PRELIMINARY AND INCOMPLETE DO NOT CIRCULATE 1 The current version of this paper is maintained at 2 Tel: (412) , laurdebo@andrew.cmu.edu 3 Tel: (412) , parlourc@andrew.cmu.edu 4 Tel: (734) , urajan@umich.edu

2 Abstract We develop a model that links the queuing to the herding literature. We enrich a simple two action herding model by introducing a Poisson agent arrival process and exponentially distributed agent processing. Thus, a queue may be formed. Agents incur a per unit time waiting cost. We characterize equilibrium when agents do not know the order in which they arrive, but only observe the queue length. When the service rate is low, we recover a herding result. However, when the service rate is high, only herding-like behavior occurs: Any possible herd formation is broken whenever the queue length is sufficiently small. The latter occurs infinitely often on the long run. Finally, we find that waiting costs may generate an additional set of states that breaks a possible herd.

3 1 Introduction Customers frequently wait before they can consume a good. Lines outside nightclubs, rides at amusement parks and waiting lists for products are part of everyone s experience. If customers dislike waiting, it is natural to suppose that a firm could increase prices and somehow extract benefit from this congestion externality. However, we consider a benefit to letting a queue form; that is, the information it provides to later consumers about earlier arrivals willingness to pay. If a lot of customers are waiting in line to buy a good, then each subsequent arrival must trade off the probability that the product is of high quality (and thus worth waiting for) against possible waiting costs. This suggests that there is a natural tradeoff between congestion or waiting costs and the positive information externality that customers generate when they choose to wait. We model this tradeoff in a simple queueing model in which a firm can be either of high or low quality. Risk neutral consumers receive a signal of the quality of the firm and decide to purchase the good or not. If they decide to purchase the good, they join a first come first served queue and wait. There is a cost associated with waiting. As the firm s service rate is random, the waiting time is also random and governed by a poisson process. To focus on the limiting behavior of the system, we assume that agents do not know when the firm started (i.e., do not know the order in which they arrived), but only see the queue length. We characterize stationary equilibrium with and without waiting costs. If there are no waiting costs, then the queue exerts a pure informational externality. If there are waiting costs, then each arriving agent weighs the information content of the queue against the possible waiting costs. In our model, the firm s stochastic service rate is both a realistic assumption and a natural way of restricting the histories that each agent sees. Thus, we consider social learning in which the history that each agent observes is truncated. However, the history is also endogenous because it is made up of people who have joined a queue. Arriving agents take this into account when the weigh the information content of the queue. They observe the actions of others, only if they were positive. As they do not know the order in which agent s arrive, they do not observe the agents who did not join the queue. In addition, as the service rate is random, they do not know how many long the queue was when the waiting agents made their decision to join. Thus, they do not share a common public history. 1

4 We find that if there are no waiting costs, then agents with bad signals join the queue if it is long enough. Specifically, if an agent has a good signal she will join a shorter queue than one with a bad signal. However, as there is a random service rate, even if all agents have good signals about the value of the good, sometimes the queue will be empty. If there are positive waiting costs, we demonstrate the existence of belief holes: queue lengths for which bad signals will not join but only those with good signals. Thus, some queues are associated with jumps in public beliefs about the quality of the good. These thresholds can only be crossed in equilibrium if an agent received a good signal, which generates a natural mechanism for breaking the link between cascading and herding. Thus, the existence of waiting costs makes it easier for the public to infer an agent s private signal if she joins the queue. We characterize the likelihood ratio as a function of queue length. We find that if there are no waiting costs (thus the queue exerts a pure informational externality), the likelihood is monotonically increasing in queue length. If there are waiting costs, then the likelihood ratio has flat portions, over which the public belief is invariant to queue length. Finally, we note that there are no absorbing states in this model. Thus, while public beliefs may change as a result of queue length, they will never get stuck. The canonical herding models were due to Bikhchandani, Hirschleifer and Welch (1992) and Banerjee (1992). We depart from these frameworks in that agents do not know the order in which they arrive. We make this assumption partly for tractability and partly because when service rates are random it is more natural to consider those who are waiting as opposed to those who have consumed the good in the past. A comprehensive summary of herding and social learning appears in Chamley (2004). Our model is related to the literature on word of mouth communication, and social learning in which agents receive random samples of the history. Banerjee (1993) presents a model in which agents optimally choose whether to pass on rumors. He finds that not all investors with bad signals will invest. Ellison and Fudenberg (1993,1995) find that with boundedly rational agents, can lead all agents to the efficient outcome if each individual receives little social information. Smith and Sorensen (1997) consider a sequential action model in which agents receive a random sample of the history. They find that social learning does not converge to the truth if beliefs are bounded (as they are in our case). By contrast, we do not need the distribution of beliefs to be unbounded to converge to the truth. 2

5 Banerjee and Fudenberg (2002) in a model of sequential moves in large populations find that the system can converge to an informational cascade. Due to waiting costs, the welfare improvement principle of Banerjee and Fudenberg (2002) does not hold in our model. Thus, the value of joining a queue varies according to signal. This leads to belief holes: only agents with a good signal join which leads subsequent arrivals to update the value of the good. While there are many ways to model congestion costs ( e.g., pro rata rationing), waiting costs in first come first served queues seem to accord best with reality. The information externality that we point out provides another natural motivation for this method of rationing. Optimal queue lengths were examined by Naor (1969)and Hassin (1985). Hassin (1986) considers the effect of revealing information about queue length and therefore waiting times. A comprehensive review of the economic aspects of queueing appears in Hassin and Haviv (2003). 2 Model A firm sells a good which can either be of high (h) or low quality (l). The utility an agent gets from the good is ṽ {v l, v h }, where v l < 0 < v h. We also use ṽ to refer to the quality of the firm. While the firm knows the quality of his good, it cannot credibly communicate it to agents. The agents prior belief that the good is high quality is p. The time to deliver the good to the agent is exponentially distributed with parameter µ. We refer to µ as the service rate. If an agent wishes to acquire the product, and another agent is still being serviced, he joins the queue. The queue is serviced on a first-come, first-served basis. Once an agent has joined the queue, he leaves only when he has obtained the good. Each agent incurs a waiting cost c per unit of time waiting to obtain the good. We consider a game with an infinite stream of agents, labeled by i N, that arrive sequentially at the market according to a Poisson process with arrival rate Λ. The arrival order of agent i, t(i) is determined by a random permutation of N. Agents do not know in which order they move (i.e. t(i)) nor do they observe the history of actions taken by previously arriving agents. Instead, each agent receives a private signal s S = {g, b} about the quality of the good, where Pr(s = g v = v h ) = Pr(s = b v = v l ) = q [ 1, 1]. Agent i observes, in addition to his private signal, 2 3

6 s, the number of other agents already in the queue, k, at the time he arrives at the market. The agent takes an action a {0, 1}, where a = 1 indicates a decision to acquire the product, or, in other words, the agent joins the queue. If he does not acquire the good, he obtains a reservation utility of zero. The agent balks. A mixed strategy for an agent is a mapping σ : S N [0, 1]. The agent i s posterior belief that the product is of high quality depends, given the queue length and signal p : S N [0, 1]. Given the signal information and queue length (s, k), agent i s expected utility as a function of his mixing probability at (s, k) is then u : [0, 1] S N [0, 1] R. In a perfect Bayesian equilibrium of this game, each agent plays a best response, σi (s, n), given his belief, p i (s, n): σi (s, n) = max z [0,1] u(z; s, n, p i (s, n)) (s, n), i (1) and p i (s, n) is consistent with Bayes rule on the equilibrium path. We make three assumption on the parameters that we have introduced so far; (v l, v h, c, µ, Λ, q, p): First, we assume that the arrival rate of agents to the market, Λ, is less than the service rate µ. Define ρ = Λ as the expected number of arrivals µ per departure from the system, then, our first assumption can be stated as : ρ < 1. Second, in absence of waiting costs, we assume that an agent will join a queue with a good signal, but not with a bad one in absence of queue information. This is similar to the usual herding assumption that an agent who ignores the information provided in other agents actions will acquire the product if she has a good signal, but not if she has a bad one. We can write this assumption as: where, using Bayes Theorem, p g = p g v h + (1 p g )v l > c µ > p bv h + (1 p g )v l (2) pq pq+(1 q)(1 p) firm is high quality and given signal g, and similarly, p b = is the posterior probability that the p(1 q) p(1 q)+q(1 p) denotes the posterior probability that the firm is high quality, given a bad signal. Third, without loss of generality, we normalize the arrival rate, Λ, to 1. With this assumption, we introduce two new parameters to the BHW model: c and µ. In section 3, we analyze the case where waiting cost, c, is zero but, with µ > 0. In section 4, we analyze the case with c > 0 and µ > 0. 4

7 2.1 Benchmark Cases: A Perfectly Informative and Uninformative Signal about the Quality of the Good In this subsection, we analyze the benchmark cases in which all agents receive a perfect signal about the quality of the good; q = 1 and in which all agents receive an uninformative signal about the quality of the good; q = 1. Letting v = pv 2 h +(1 p)v l be the prior expected service value and denoting n(ν) = νµ -1, the following Lemma c characterizes these equilibria: Lemma 1 When q = 1 and n(v h ) N, the equilibrium profile is: σ (b, k) = 0 for all k {0, 1,...} and σ (g, k) = 1 for k {0, 1,..., n(v h )} and σ (g, k) = 0 otherwise. The equilibrium beliefs are: p (b, k) = 0 and p (g, k) = 1. When q = 1 and n(v) N, the equilibrium profile is: 2 σ (g, k) = σ (b, k) = 1 for k {0, 1,..., n(v)} and σ (g, k) = σ (b, k) = 0 otherwise. p (g, k) = p (b, k) = p for all k {0, 1,...}. In both cases, if n N the equilibrium profile is the same as in the previous case, except for n = n, where any mixing strategy can be sustained. When q = 1, the firm s quality is fully revealed through the signal that agents observe. Upon observing s = b, all agents infer that v = v l < 0. As the waiting costs are non-negative, no agent will ever purchase the good. Upon observing s = g, all agents infer that v = v h. If the expected waiting costs are lower than the value of the good, agents decide buying the good. n(v h ) is the threshold queue length above no agents enter. This threshold increases as the value or service rate increases and decreases in the unit waiting costs increase. When q = 1, the agents do not 2 receive any useful information and decide to purchase the good based on their prior expected utility, v. In both cases, when n(v h ) or n(v) N, an agent observing n other agents in the queue upon arrival is indifferent between joining the queue or not. Any randomization strategy at n is an equilibrium. With a perfectly informative or uninformative signal, the queue length only impacts the waiting costs, not the valuation of the good. In the reminder of this paper, we analyze how the queue length impacts the valuation of the good in the case of an imperfect signal. 5

8 2.2 An Imperfect Signal about the Quality of the Good When the quality of the signal is not perfect, q ( 1, 1), agents incorporate the 2 queue length information when updating the firm s quality. We elaborate the consistency and rationality (see equation (1)) equilibrium conditions. First, we develop consistency conditions for an agent s belief that the firm is of high quality, given the strategies that other agents play: Consistency of agent i s belief. Conditioned on the firm s quality, the random arrival and service processes, in conjunction with the strategies of agents, induce a probability distribution over N. Let π(v h, k, Σ i ) (π(v l, k, Σ i )) denote the probability that there are exactly k agents in the queue when a randomly selected agent, labeled i, arrives, given that the firm s true quality is v h (v l ) and all other agents play Σ i = {σ j,..., j i}. Suppose there are n agents in the queue when a agent j i arrives. Let r(v, n, σ j ) be the probability that a agent j joins a queue with n other agents when the firm s quality is v and σ j is the strategy profile agent j. Conditioning on the quality of the firm, we obtain: r(v h, n, σ j ) = qσ j (g, n) + (1 q)σ j (b, n) and r(v l, n, σ j ) = (1 q)σ j (g, n) + qσ j (b, n). As all agents are homogeneous ex ante, we focus on a symmetric equilibrium: σ j = σ i for all j i. Then, π(v, k, Σ i ) reduces to π(v, k, σ i ). To characterize the agent s decision let us introduce φ(k, σ i ) = π(v h,k,σ i ) π(v l,k,σ i, which is the likelihood that the firm has high quality and that the arriving ) agent sees k agents in the queue, given all agents j i play strategy profile σ i. Every σ i generates a queueing system, of which we can calculate the probability distribution that the queue has length k for the randomly arriving agent i. Define φ s = p s 1 p s as the likelihood ratio of the firm being of high quality, given that signal s is received. Now, we can state the condition for p i (s, n) to be consistent with Bayes rule for any given σ i : Lemma 2 A belief p i (s, n) is consistent with the strategy profile σ i if: where: n 1 φ(n, σ i ) = φ(0, σ i )ρ n k=0 φ s p i (s, n) 1 p i (s, n) = φ(n, σ i) (3) r(v h, k, σ i ) r(v l, k, σ i ) and φ(0, σ i) = n=1 ρn n 1 k=0 r(v l, k, σ i ) n=1 ρn n 1 k=0 r(v h, k, σ i ) 6

9 Next, we develop the rationality conditions of an agent, given his belief about the firm s quality: v l + (k+1)c µ Rationality of agent i s decision. Define γ(s, k) = φ s. Now, we can v h (k+1)c µ determine the best response of agent i: Lemma 3 Let p i (s, n) be the belief of agent i, then, the best response for an agent p i (s,n) is: σ BR (s, n, φ s ) with 1 p i (s,n) σ BR (s, n, φ) = 0 if φ < γ(s, n) [0, 1] if φ = γ(s, n) 1 if φ > γ(s, n). In other words, the agent will join the queue if there is a sufficiently high likelihood that the firm is high quality. γ(s, k) serves as a reservation likelihood; if the likelihood of the firm being high quality exceeds γ(s, k), an agent with signal s will join a queue of k agents. Conditions for a Symmetric Equilibrium. Now, we can combine the consistency (Lemma 2) and rationality (Lemma 3) conditions in order to characterize the perfect Bayesian equilibrium of our game. In a symmetric equilibrium, σi = σ and p i = p for all i. For ease of notation, we will use φ (n) and π (v, n) to denote φ(n, σ ) and π(v, n, σ ) respectively. It will be convenient to introduce for any φ (γ(g, 0), γ(b, 0)) the following functions: Ψ(s, n; φ) = ln(γ(s,n)) ln φ, for k 1: ln( q 1 q ) z(k 1; φ) if Ψ(b, k 1; φ) < z(k 1; φ) z(k; φ) = z(k 1; φ) + 1 if Ψ(g, k 1; φ) < z(k 1; φ) Ψ(b, k 1; φ) (5) if z(k 1; φ) Ψ(g, k 1; φ) with z(0; φ) = 1 and Φ(φ) = 1 + n=1 ρn (1 q) z(n;φ) 1 + n=1 ρn q z(n;φ). (6) Consider z(k; φ) at some value of φ for which all inequalities (5) are strict. Note that for φ + ɛ, with ɛ small enough such that the conditions z(k; φ + ɛ) remain strict for all k, z(k; φ + ɛ) = z(k; φ) and consequently: Φ(φ + ɛ) = Φ(φ). When there exist a ñ such that one constraint in (5) is binding changing φ will result in a discontinuous change of z(ñ; φ) and therefore in a change of Φ(φ). Thus, Φ(φ) is a staircase function with a set of discontinuous points, which we denote as P, over (γ(g, 0), γ(b, 0)). 7

10 Φ(φ) can be interpreted as follows: Assume that all agents believe that the ratio of the probability that the high quality firm has an empty queue over the probability that the low quality firm has an empty queue is φ. The pure strategy best response to this belief is given by σ BR (where agents join in case of a tie). Conditional on the firm s quality, this best response induces a queueing system, of which the ratio of the probability that the high quality firm has an empty queue over the probability that the low quality firm has an empty queue for a randomly arriving agent is Φ(φ). The strategy is an equilibrium strategy when the latter likelihood ratio is equal to φ. The following Proposition characterizes a perfect Bayesian equilibrium strategy profile by means of Φ(φ): Proposition 1 There exist at least one equilibrium, either pure or mixed. There exist a pure strategy equilibrium if there exist a φ such that Φ(φ ) = φ. There exist a mixed strategy equilibrium if there exist a φ P such that either Φ(φ ) < φ < Φ(φ + ɛ) or Φ(φ ) > φ > Φ(φ + ɛ) with ɛ > 0, but, arbitrarily small. In the case that there exist a pure strategy, the strategy profile can be determined ( as follows: φ (n) = φ q 1 q) z(n;φ ) and σ (s, n) = σ BR (s, n, φ (n)) (from Lemma 3) and p (s, n) = φ (n) (from Lemma 2). Otherwise, φ s +φ (n) φ ɛ and φ + ɛ determine two pure strategies that are mixed in equilibrium. Next, we discuss the properties of the equilibrium strategy profiles that are generated by φ. 3 No waiting cost Now, suppose c = 0. In the benchmark cases of Lemma 1, we obtain that n(ν) =, i.e.: With a perfectly informative signal (q = 1), agents observing a bad signal never join, agents observing a good signal always join. With an uninformative signal (q = 1 ) all agents join irrespective of the signal that they receive, as long as the prior 2 service value v is strictly positive. When q ( 1, 1), the queue length serves solely 2 as additional information for the agent of the firm s quality as other agents incur no disutility of waiting in the queue. 8

11 3.1 The Structure of the Equilibrium Policy It follows from Proposition 1 that there exist a φ [γ(g, 0), γ(b, 0)] that characterizes the equilibrium strategy profile (either pure of mixed). When c = 0, we write γ(s) v for γ(s, k) since, for all k, γ(s, k) = φ l s v h. Similarly, we can write Ψ(s; φ). Note that Ψ(g; φ) = Ψ(b; φ) 2 and consequently, for any φ [γ(g, φ), γ(b, φ)]: z(n; φ) = n as long as n Ψ(b; φ). characterization: This specific structure allows us deriving the following Proposition 2 A pure strategy equilibrium, or a strategy profile in a mixed strategy equilibrium is characterized by a threshold queue length below which an agent does not join. Whenever an agent with a bad signal joins, and agent with a good signal joins too. The opposite, does not hold. Proposition 2 sets the stage for herding: When the queue is sufficiently long, all agents join, irrespective of their signal. When the queue is low, only agents with a good signal join. In the following subsection, we fully characterize the equilibrium profile. 3.2 The Impact of Market Congestion on Herding The special structure of z(n; φ) results in a simple expression of Φ(φ) for φ [γ(g, 0), γ(b, 0)]: Φ(φ) = Ψ(b;φ) 1 ((1 q)ρ) 1 + (1 q) ρ + ((1 q)ρ) Ψ(b;φ) +1 1 (1 q)ρ 1 ρ 1 + qρ 1 (qρ) Ψ(b;φ) 1 qρ + (qρ) Ψ(b;φ) +1 1 ρ (7) which characterizes the equilibrium (Proposition 1). Recall that ρ is the expected number of service departures during an inter-arrival time and, by assumption ρ < 1. We can refer to ρ as a measure of market congestion. In the following subsections, we study the impact of market congestion on the equilibrium strategy profile. We provide insight in the impact of market congestion on the equilibrium strategy profile by solving Equation (7) in the special cases where the market is either not congested (ρ ( ) Ψ(φ) +1 0) or highly congested (ρ 1). It follows from (7) that lim ρ 1 Φ(φ) = and lim ρ 0+ Φ(φ) = 1. In the following Propositions, we discuss qualitative properties of the perfect Bayesian equilibrium as a function of ρ. 1 q q 9

12 Proposition 3 If ρ is close to, but less than, 1, then, agents observing a bad signal do not enter queues of length 2 or less and always enter longer queues. Agents observing a good signal mix upon arrival at an empty queue and always enter non-empty queues. Proposition 3, introduces essential herding features: First, agents receiving a bad signal join the queue only when the queue is long enough. The threshold is reminiscent of the classical BHW result that two subsequent agents purchasing the asset generate a positive herd. In our case, as the service process is faster than the arrival process (ρ < 1), whenever the queue length is above 3, it will drop into the set {0, 1, 2} with probability 1. The positive herding phenomenon cannot continue indefinitely. Second, agents receiving a good signal randomize upon arrival at an empty queue. As agents receiving a bad signal do not enter an empty queue, it may remain empty for a while, until an agent with a good signal arrives and the outcome of his randomization process is to enter. This is similar to the classical BHW result that two subsequent agents not purchasing the asset generate a negative herd. In our case, due to the randomization process, there will be with probability 1 an agent observing a good signal that enters an empty queue. The negative herding phenomenon cannot continue indefinitely. The herding phenomena are present when the market is highly congested (even though this does not create any disutility for the agents). The queue length contains information about the recent history as, due to the service process (with rate µ) any agent who is in the queue will leave for sure and will not be included anymore in the queue length observed by a later arriving agent. In highly congested markets, the service process is slow compared to the arrival process. Therefore, the queue length contains information about a long history. It is thus intuitive that the BHW herding features are observed in these markets. Next, we study the equilibrium in markets with low levels of congestion or, equivalently, a high service rate compared to arrival rate: Proposition 4 If ρ 0, then agents follow a threshold strategy: join the queue when observing a good signal, irrespective of the queue length upon arrival, balk when observing a bad signal when the queue length below than a threshold, that is strictly less than 2. If the equilibrium contains mixing, the equilibrium strategy is similar except that mixing occurs occurs when observing a bad signal when the queue length is exactly equal 10

13 to the threshold. In markets with few congestion effects, a short queue is not necessarily an indication of low product quality, revealed indirectly through the choice of previous agents. Short queues may be the result of fast service. Agents take both possibilities into account. Only agents observing a bad signal and low queue lengths do not join. Propositions 3 and 4 allow understanding better the role that congestion (which does not generate any disutility to agents) plays in herd formation: For a high quality firm, herding in a congested environment results in the loss of agents with the correct signal, but, arriving at an empty queue and in the loss of agents with a wrong signal, who hold on to their signal when arriving at a short queue. The less congested the market, the more beneficial for the high quality firm: For markets with only a little congestion, all agents observing the correct signal joint the queue, while agents with a wrong signal hold on to their signal only for very short queues. On Figure 1, we illustrate Φ(φ) for three different levels of market congestion: ρ = 0 (Proposition 4), an intermediate level, ρ = 0.9 and ρ = 1 (Proposition 3). For each ρ > 0, Φ(φ) is a staircase function with one discontinuity in (γ(g, 0), γ(b, 0)) (i.e. P is a singleton). Each φ on the continuous part of Φ(φ) determines an unique best response (with σ BR (s, n, φ)). The equilibrium is determined by the intersection of Φ(φ) and φ. For low values of φ (below γ(b, 0)), the best response is to never enter the queue. This response is determined by an agent observing a good signal and an empty queue, whose best response is not to join the queue. Obviously, agents observing a bad signal will not join either. As no agent joins the empty queue, the queue remains empty. Thus, Φ(φ) = 1. However, γ(g, 0) < 1 no φ on the lowest segment on the Figure can characterize an equilibrium. For higher values of φ (on the second segment of the staircase function), the best response is to join a queue that is longer than 2 upon observing a bad signal and always join the queue upon observing a good signal. For higher values of φ, the best response is to join a queue that is longer than 1 upon observing a bad signal and always join the queue upon observing a good signal. As the market congestion decreases, the equilibrium φ increases and the equilibrium policy will follow the above described pattern when the strategy profile is pure. Finally, note that there is a range of market congestion levels with a mixed strategy profile in which agents observing a bad signal mix at a queue length of 1. 11

14 1 ρ=0 0.8 Φ 0.6 ρ= ρ= φ Figure 1: Φ(φ) for three different values of ρ: 0, 0.9 and 1. The other parameters are: v l = 0.9, v h = 1.1, p = 0.6, q = The Impact of the Prior Value on Herding in a Congested Environment Proposition 5 In non-congested markets, the threshold value for agents observing a bad signal decreases from 1 when the expected prior value of the good is negative to 0 when the expected prior value of the good is positive. In highly congested markets, the threshold value for agents observing a bad signal remains 2, when the expected prior whether the expected prior is positive or negative. The joining probability of agents observing a good signal and an empty queue is independent of the expected prior. Proposition 5 highlights the differential impact of the prior expected value on the threshold for agents observing a bad signal. In non-congested markets, an increase in prior expected value increases the positive herding of agents observing a bad signal. In highly congested markets, the prior expected value has less impact on the positive herding of agents observing a bad signal. Finally, note that Proposition 5 is illustrated on Figure 1, where v is positive and therefore in un-congested markets (ρ low), the 12

15 threshold queue length for agents observing a bad signal is 1. 4 Positive Waiting Cost In the case that agents incur some disutility due to waiting, we need to introduce again γ(s, k) (rather than γ(s) as in the previous section), for all k. The queue length has two effects here. In a traditional queuing model, a longer queue implies higher waiting costs, and thus a lower incentive to join the queue. Here, the queue length is also potentially informative about the quality of the product, as agents participation decisions depend on their signals. 4.1 The Structure of the Equilibrium Policy Building on Proposition 1, we can obtain insight in the equilibrium strategy: Proposition 6 A pure strategy equilibrium, or a strategy profile in a mixed strategy equilibrium is characterized by means of the following strategies: For an agent observing a bad signal: A threshold below which no agent joins the queue, a threshold above which no agent joins the queue and a set of isolated queue lengths ( holes ) in between the two thresholds at which the agent does not join the queue. For an agent observing a good singal: A threshold above which no agent joins the queue. Similarly as for the case without waiting costs, there exist a threshold below which an agent observing a bad signal does not join the queue. From Proposition 6, note that waiting costs make the long run equilibrium strategies significantly more complex. We observe two phenomena: (1) as waiting is expensive, there exist a maximum queue length above which no agent joins. This maximum queue length is higher for an agent observing a good signal than for an agent observing a bad signal. (2) The long run equilibrium strategy is complicated by a set of isolated queue lengths ( holes ) at which of an agent observing a bad signal does not join. The holes in the equilibrium strategy profile are explained as follows: Agents observing a bad signal have a lower updated expected value, than agents observing a good signal. If only agents observing a good signal join, information will be revealed 13

16 to agents arriving at a queue length that is just one higher: As only an agent with a good signal can have brought the queue in that position, the likelihood that the firm s quality is high increases. If for a certain queue length both agents join, no information will be revealed to agents arriving at a queue length that is just one higher. Thus, as agents join the queue irrespective of their signal, the updated expected value does not increase. As agents observing a bad signal have the lowest updated and waiting cost are increasing, there will exist a queue length for which the agent observing a bad signal does not join. Interestingly, if at that queue length the agent observing a good signal does join, information will be generated for the agent arriving at a queue length that is exactly one higher: Only an agent with a good signal can have brought the system in that state. The increase in expected value may be sufficient to have an agent observing a bad signal join again. Intuitively, the waiting costs generate some critical queue lengths that can only be overcome by agents observing a good signal. Arriving at a queue above such a critical length reinforces the confidence in the firm s quality. 4.2 The Impact of Waiting Costs on Herding The equilibrium profile characterized by (6) contains in the case of positive waiting costs more points of discontinuity than case without waiting costs. This is illustrated on Figure 2. As the structure of the equilibrium is more complex (i.e. with holes and upper threshold level), there are more points of discontinuity in the set P. This is illustrated on Figure 2. Observe that with more waiting cost friction, the equilibrium φ is greater than in the no waiting cost friction case. Next, we study the impact of market congestion on herding, in when waiting costs are positive. Therefore, we vary ρ, or, equivalently µ. With positive waiting costs, however, we need to make sure that condition (2) is satisfied. A lower bound on µ is thus determined by: (p g v h + (1 p g )v l )µ = c. Proposition 7 In non-congested markets (i.e. µ ), the equilibrium strategy profile with positive waiting costs is equal to the equilibrium strategy profile without waiting costs. In highly congested markets, i.e. µ µ, the equilibrium strategy profile is... 14

17 1 0.8 c=0.1 Φ 0.6 c= φ Figure 2: Φ(φ) for two different values of c: 0 and 0.1. The other parameters are: v l = 0.9, v h = 1.1, p = 0.6, q = 0.6, ρ = 0.9. The first part of Proposition 7 is intuitive: In non-congested markets, no significant queues can ever develop. The second part of Proposition 7 is needs to be explained. Therefore, even with positive waiting costs, the equilibrium behavior is similar as in Proposition 4. From Proposition 7, it follows thus that the combination of congestion and waiting costs may result in a natural separation of agents with bad signals (who do not join) from agents with good signals (who do join). This phenomenon prevents positive herding. In the next Proposition, we provide sufficient conditions for when positive herding cannot occur: Proposition 8 For v h = v l = v and p = 1, if the condition: ξ ξ ln q 1 q ξ ln ln 1 1 ξ q 1 q ln q 1 q < 1 + ln q 1 q (8) with ξ = 2 c is satisfied, only agents observing a good signal join the queue. Agents vµ observing a bad signal always balk. 15

18 No Herding q ξ Figure 3: Condition for no herding (8), as a function of ξ and q. Proposition 8 illustrates the impact of waiting cost frictions on the herding behavior. Remember from Proposition 2 that for queues that are sufficiently long, in absence of waiting costs, positive herding occurs: agent s receiving a bad signal ignore their signal and join the queue. Waiting costs may prevent this positive herding. Condition (8) is illustrated on Figure 3: For signals with either low or high information content, the herding phenomenon disappears for either high waiting costs, low service rate or low service value. The straight line on the Figure indicates when condition (2) is binding. 5 Discussion and Conclusion In this paper, we studied herding behavior in an environment in which queues may be formed. We explain the effect that queue length information and waiting cost has on an arriving agent s decision. We believe to contribute to the literature on queueing, which typically ignores information asymmetry on the value of the good/service and to the herding literature, which typically ignores queueing phenomena. 16

19 References [1] Banerjee A. V. (1992) A simple model of herd behaviour, Quarterly Journal of Economics 107, [2], Banerjee, Abhijit (1993), The Economics of Rumours, Review of Economic Studies Vol 60 No 2, p [3] Banerjee, A. and D.Fudenberg, (2002) Word of Mouth Learning, Harvard working paper. [4] Bikhchandani, S., D.Hirschleifer and I. Welch (1992) A Theory of Fads, Fashion, Custom and Cultural change as Information Cascades Journal of Political Economy 100, [5] Chamley, Chrisophe P. (2004), Rational Herds, Cambridge University Press, Cambridge UK. [6], Ellison, G. and D. Fudenberg (1995) Word-of-mouth communication and Social Learning, Quarterly Journal of Economics Vol 110, No 1 p [7] Ellison, G. and D. Fudenberg (1993) Rules of Thumb for Social Learning, Journal of Political Economy Vol 101, No 4, p [8] Hassin, R. (1986) Consumer information in markets with random products quality: The case of queues and balking, Econometrica 54, [9] Hassin, R. and M. Haviv, (2003) To Queue or not to queue: Equilibrium behavior in queueing systems. Kluwer Academic Publishers. [10] Noar, P. (1969), The regulation of queue size by levying tolls, Econometrica 37, p [11] Hassin, R. (1985) On the optimality of first come last served queues, Econometrica 53, [12] Smith, Lones and Peter Sorensen (2000), Pathological Outcomes of Observational Learning, Econometrica Vol 68, No 2 p

20 [13] Smith, Lones and P. Sorensen (1997), Rational Social Learning with Random Sampling, University of Michigan Working paper. [14] Wolff, R.W. (1982) Poisson arrivals see time averages, Opns. Res., 30, p Appendix PROOF of Lemma 1. Due to the memoryless property of the exponential distribution, the remaining time of the agent currently being serviced is exponentially distributed with parameter µ. Thus, if k agents are ahead of him, the total expected waiting cost for an arriving agent is (k+1)c µ (which includes the own service time). The agent s expected utility is then u(z; s, k, p) = (pv h + (1 p)v l c(k + 1)c/µ)z. For q = 1, the signal reveals fully ( the firm s ) quality: p(g, n) = 1 and p(b, n) = 0 for all n. Then, u(z;, g, k, p) = v h c(k+1) z. Maximizing u(z; g, n, p) results µ ( ) in the equilibrium strategy. Similarly: u(z; b, k, p) = v l c(k+1) z. Maximizing µ u(z; g, n, p) results in the equilibrium strategy. For q = 1, the signal does not reveal 2 any information about the firm s quality: p(s, n) = p for s = g or s = b and all n. Maximizing u(σ, g, n) results in the equilibrium strategy. PROOF of Lemma 2. As all agents are homogeneous ex ante, we can characterize Σ i by means of σ i. Then, conditioned on the quality of the firm, any profile Σ i induces a birth-death process with the queue length, n, as state. With the PASTA property (Wolff, 1982). PROOF of Lemma 3. Let φ(k, σ i ) be an agent s likelihood of the firm being high quality when the queue length is k. The agent should join the queue if u(1, s, k, p(s, k)) > 0 or, φ > γ(s, k). It follows that if φ < γ(s, k) the best response is to not join the queue, and if φ = γ(s, k), the agent is indifferent between joining and not. 18

21 PROOF of Proposition 1. (Sketch) The function Φ(0) = 1 > 0 and Φ(γ(b, 0)) < 1 < γ(b, 0). The inclusion of mixed strategy profiles at the points of discontinuity corresponds to an extension the function Φ(φ) to a correspondence with the vertical segment in between the left and right limit as set value for a discontinuous point. This correspondence has thus at least one intersection point with φ. PROOF of Proposition 2. Follows from the special structure of Ψ(s, n, φ), which results in z(n; φ) = n until Ψ(s; φ) is reached. For these values of n, the best response is for agents observing a bad signal not to enter, while agents observing a good signal do enter. If z(n 1; φ) = n 1 > Ψ(s; φ), then, agents observing a bad signal also enter. As Ψ(s; φ) is independent of n and z(n 1; φ) remains constant, the policy remains the same for all higher queue lengths. PROOF of Proposition 3. (Sketch) Follows immediately from solving Φ(φ) = φ in the limiting case. PROOF of Proposition 4. (Sketch) Follows immediately from solving Φ(φ) = φ in the limiting case. PROOF of Proposition 5. (Sketch) Follows immediately from the threshold solution of the Propositions 3 and 4. PROOF of Proposition 6. It will be convenient to define s n (φ) = 0 k n 1 ( 1 σ k b (φ) ), i.e. the number of states strictly below n in which the agent observing b does not enter. For convenience of notation, we drop φ from all arguments. Given the recursive structure of s n, we can easily see that s n s n 1 = 1 only if s n 1 < Ψ b (n 1) = ln γ b(n 1) ln φ ln( 1 q) q and s n s n 1 = 0 otherwise. Note that γ b (n) = (1 p)q v l +c(n+1)t, p(1 q) v h c(n+1)t As Ψ b (n) Ψ b (n 1) = ln γ b (n) ln γ b (n 1) ( ) = ln v l+c(n+1)t v h c(n+1)t ( ) ln ln ( ) d vl + c (n + 1) t v h cnt dn v h c (n + 1) t v l + cnt q 1 q = q 1 q v h cnt v l +cnt ( v l + ) ( v h ct ct v l v h ct ct + 2n + 1 ) ( v l + n) 2 ( vh ct ct 1 n ) 2 which is positive and increasing in n [CHECK] for n v h ct 1. Therefore, γb (n) is a convex function. Furthermore, and d Ψ dn b (n) = 1 1 ln( 1 q) q γ b (n) ( v l ct + v h ct )( v l ct v h ct +2n+1) ( v l ct +n)2 ( v h ct 1 n) 2 is also positive and increasing in n [CHECK]. Thus, Ψ b (n) is also a convex function. Note that lim v n h ct 1 d Ψ dn b (n) = +. As φ < γ b (0), we obtain that, s 1 = 1. As long as s n < Ψ b (n), s n = n. As s n increases with a slope of 1, while the slope of Ψ b (n) 19

22 increases faster [BE MORE PRECISE]. Therefore, there should exist a n 0 such that n 0 < Ψ b (n 0 ) and n > Ψ b (n 0 + 1). (*) Then, it follows that s n0 +1 = n and s n = s n0 +1 for n n As s n increases with a slope of 0, while the slope of Ψ b (n) increases faster, there should exist a n 1 such that s n1 > Ψ b (n 1 1) and s n1 < Ψ b (n 1 ). Let for now j = 1. Consider now two possibilities: (i) Ψ b (n j + 1) Ψ b (n j ) < 1. Then, s nj +1 = s nj + 1 > Ψ b (n j + 1) and s n = s nj +1 for n n j + 1 and we are at the same situation as (*) and j increases by 1. (ii) Ψ b (n j + 1) Ψ b (n j ) > 1. Then, s nj +1 = s nj +1 < Ψ b (n j + 1). As Ψ b (n j + 1) Ψ b (n j ) > 1 and, due to convexity of Ψ b (n), Ψ b (n + 1) Ψ b (n) > 1 for all n n j. In that case, s n = s nj + n n j < Ψ b (n). As s n increases with a slope of 1, while the slope of Ψ g (n) increases faster [BE MORE PRECISE]. Therefore, there should exist a n such that n > Ψ g (n) and n + 1 < Ψ b (n + 1). d As Ψ dn b (n) = +, in a finite number of iterations, we obtain Ψ b (n + 1) Ψ b (n) > 1 in a finite number, say J, of iterations. Proceeding in this way, we construct an interval [0,..., n 0 ], J holes [n j ] and an interval [n 0,..., n] where Ψ g (n) < s n < Ψ b (n) PROOF of Proposition 8. Let n solve d ln(γ(b,n)) = 1. If n < ln(γ(b, n)), then the curves dn ln q ln q 1 q 1 q n and Ψ(b, n, 1) are tangent. Also, n < Ψ(b, n, 1) for all φ < γ(g, 0). Therefore, it follows from (5) that z(k; φ) < k as long as z(k 1; φ) < Ψ(g, k 1; φ). Algebraic manipulation of the above condition leads to the stated result in the Proposition. 20