Joining Longer Queues: Information Externalities in Queue Choice 1

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1 Joinin Loner Queues: Information Externalities in Queue Choice 1 Senthil Veerarahavan Laurens Debo Wharton School, University of Pennsylvania, 3730 Walnut St, Philadelphia, PA 19104, USA Graduate School of Business, University of Chicao, 5807 S. Woodlawn Avenue Chicao, IL senthilv@wharton.upenn.edu ldebo@chicaosb.edu A classic example that illustrates how observed customer behavior impacts other customers decisions is the selection of a restaurant whose quality is uncertain. Customers often choose the busier restaurant, inferrin that other customers in that restaurant know somethin that they do not. In an environment with random arrival and service times, customer behavior is reflected in the lenths of the queues that form at the individual servers. Therefore, queue lenths could sinal two factors - potentially hiher arrivals to the server or potentially slower service at the server. In this paper, we focus on both the factors when customers waitin costs are neliible. This allows us to understand how information externalities due to conestion impact customers service choice behavior. In our model, based on both private information about the service quality and queue lenth information, customers decide which queue to join. When the service rates are the same and known, we confirm that it may be rational to inore private information and purchase from the service provider with the loner queue when only one additional customer is present in the loner queue. We find that, due to the information externalities contained in queue lenths, there exist cycles durin which one service firm is thrivin while the other is not. Which service provider is thrivin depends on luck, i.e. it is determined by the private sinal of the customer arrivin when both service providers are idle. These phenomena continue to hold when each service facility has multiple servers, or when a facility may o out of business when it cannot attract customers for a certain amount of time. Finally, we find that when the service rates are unknown but 1 This paper was previously titled To Join the shortest queue or the lonest queue: Inferrin Service Quality from Queue Lenths. Our sincere thanks to Krishnan Anand, Gerard Cachon, Noah Gans, Seruei Netessine, Alan Scheller-Wolf, Anita Tucker, an anonymous associate editor and three reviewers, and to the seminar participants at Carneie Mellon University, New York University, Washinton University and the Wharton School, whose comments sinificantly improved this paper. An earlier version of the paper was a finalist on the INFORMS Junior Faculty Interest Group paper competition. 1

2 are neatively correlated with service values, our results are strenthened; lon queues are now doubly informative. The market share of the hih quality firm is hiher when there is service rate uncertainty and it increases as the service rate decreases. When the service rates are positively correlated with unknown service values, lon queues become less informative and customers miht even join shorter queues. Keywords: Customer Herdin, Behavioral Operations, Information Externalities, Market Dynamics, Private Sinals, Service Capacity. 1. Introduction In many real life situations, we often have to choose between service providers whose quality is not perfectly known in advance. For example, when selectin a restaurant to dine at, a movie to watch or a sports event to attend, we often do not know the exact valuation of the experience (althouh they miht have an expectation about the service). The examples above have the followin common feature: They create conestion. Waitin lines in front of a restaurant, or queues for a movie or a sports event are commonly observed. Therefore, it is not unreasonable to expect we are influenced by the level of business or buzz at each service facility when makin our selection. In other words, we complement our incomplete private information reardin which service provider to choose with publicly available information such as observed conestion at one service facility. Lon queues at one server but not at another may provide an indication that several customers chose that particular service perhaps because of its perceived superior value compared to the other server. As queues are typically enerated by randomness of either the customer arrival process or the service provision process, lon queues may be created by chance, possibly trierin other customers to join the same queue. What is then the information contained in a queue that forms in front of a service? A lon queue in front of a service provider miht be an indication of slow service. On the other hand, a lon queue in front of the service provider miht be an indication that it provides hiher quality of service and therefore more customers have chosen that service. But if more and more people join a loner queue inorin their own private valuation, then the sinal quality provided by the queue weakens. Therefore, it is unclear what information a loner queue miht contain. Consider a customer who has the flexibility to watch a tourin Broadway show that is playin in town. She has to decide on only one of two shows X and Y (perhaps due to limited budet). 2

3 Since many such shows are experience oods, she does not surely know which show is better. She also does not mind seein a show a few days later if tickets are unavailable immediately. Her private belief is that X miht be a better show, but like every other customer, she is unsure about it. Suppose that the customer could et tickets for show X almost immediately, whereas the show Y is not available for, say, 10 days. Hence, she notices that Y is more popular than X. She can infer that one show is considerably less popular than the other usin the waitin information of 10 days. However, she also knows that other customers would have made different observations and their own rational decisions. She wonders how those choices of other customers would have in turn influenced what she sees. Usin all this information, she determines which show is better and makes her choice. At an areate level, the choices of many customers will be linked with each other and determine the areate demand that each show attracts. In a famous example, Becker (1991) notes a puzzlin implication of how individual decisions that are linked with each other impact the areate demand of two seafood restaurants in Palo Alto; one is always crowded and the other nearly empty, even thouh they have similar prices and amenities, and writes, Suppose that the pleasure from a ood is reater when many people want to consume it, perhaps because a person does not wish to be out of step with what is popular or because confidence in the quality of food, writin or performance is reater when the restaurant, book or theater is more popular. Such an assumption would indicate that all customers have a hiher ex-post utility when they consume the more popular product. Becker s model is static and postulates that demand for a ood is directly, positively influenced by other consumers demand or its popularity by assumin a functional form. Motivated by the examples mentioned, we posit that customers do not make a service selection decision in a vacuum, but they are influenced by what they observe around them, and in particular, by the conestion at the service providers. As queue dynamics play an important role in service context, we develop a simple, stylized, two server queuein system that captures the aforementioned idea. These features have not yet been explored jointly in the previous literature. In our model, service providers are identical ex ante and have an infinite waitin space. By allowin the buffer space to be infinite, we can focus on the information contained by queues. The customers, each carryin some private information about which server provides the best quality, arrive accordin to a Poisson process and observe the queue lenths at both of the servers. Based on this information, 3

4 they decide which queue to join. There is no jockeyin or renein in the system. We assume that all customers are rational Bayesian decision makers that maximize their expected utility. The dynamics introduced in such systems may be intricate. Due to uncertainty about the service value, it may be possible that customers purchase the service that provides lower value. This may even result in an inferior service provider becomin more popular, and hence a loner queue forms at the inferior service provider. It may take a lon time for the less popular service provider (the one with the shortest queue) to attract customers despite offerin the best quality in the market. Our model allows us to answer the followin questions: When do customers inore their private information and make their purchasin decision based on the observed queue lenth information? How do service rates of service providers affect the queue choice behavior? How does information from the queue lenths impact the formation and decay of these queues? How lon do services remain popular (i.e. have lon queues)? Could a server that provides hiher valuation service perform poorly aainst another inferior service provider with the same service rate? Could a slower service provider ain a hiher market share by providin better service? The paper starts by reviewin the related literature in 2, and then outlinin our analytical model in section 3. In section 4, we develop the conditions for pure stratey equilibria to exist, and then discuss the properties of resultin equilibrium strateies for service providers with symmetric service rates. Focusin initially on the equal service rates allows us to study the effect of arrival rates alone on customer choice behavior. We analyze various extensions of the symmetric service model, includin multi-server queues and service providers survival in equilibrium in section 5. We then study the effect of service departures on customer choice behavior in section 6 by modelin asymmetric service rates. Finally, in the concludin section 7, we discuss our insihts and future research directions. 2. Related Literature We are interested in studyin how different aents in a market influence each other s purchasin decisions when the quality of a ood or service is uncertain and there is potential for conestion to impact service-provider selection. Related issues have been addressed separately in the economics, queue choice and operations manaement literature, but the interaction between conestion, service rate and service choice between service providers has not been studied. In the followin pararaphs, we describe the literature in each area and differentiate our model. 4

5 Economics literature: Becker (1991) observed that a popular seafood restaurant (in Palo Alto) had lon queues durin prime hours every day, while the restaurant across the street, with comparable food, had many empty seats most of the time. In explainin the phenomenon, he focuses on monopoly pricin and perfect information. Becker explains why consumer demand is very fickle and why shift of restaurants between in and out cateories occurs usin a pair of equilibria that are unstable under lare demand chanes. Chamley (2004) notes that an analysis of the above interestin problem that employs optimization behavior for the consumers, and a dynamic analysis with imperfect information... remains to be done. We examine how the observation made by Becker (1991) can be explained by modelin the rational behavior of customers in a queuein system. Bikhchandani, Hirshleifer, and Welch (BHW) (1992) study the role that the observed behavior of other actors plays when purchasin an asset whose value is not perfectly known. They explain how informational cascades are created. Informational cascades occur when a series of actors makes a decision that is observed by every subsequent actor, each of whom also makes the same decision independent of his/her private sinal. BHW assume that the entire history of the aents decisions and their sequence is available to every arrivin aent. BHW do not incorporate any supply effects such as conestion or departure of observed customers from the system due to completion of service. BHW find that a commonly observed history of actions can dominate private beliefs. In such cases, aents will choose actions that will not reveal their private information. However, in many cases, arrivin customers may not observe the entire history. For instance, nothin miht be known about the decisions of customers whose service had been already completed. Therefore, the information set of customers in our paper modeled alon the followin ways. First, the customer does not know his arrival sequence number. Secondly, the arrivin customer does not know the full history of prior decisions. Finally, every arrivin customer only sees the number of current customers and their queue choices, but does not see the order those customers came in. To our knowlede, there are only few papers in herdin literature that consider herdin behavior under the limited amount of information. Smith and Sorensen (1998) consider a model where all arrivin aents sample exactly two observations. The actions that each aent observes are exoenously enerated usin a seed population. Smith and Sorensen are interested in the probability of converence to the truth. In Banerjee and Fudenber (2004), in each period a continuum of customers simultaneously choose their actions after observin exoenously chosen N previous actions. The aforementioned papers do not model endoenous censorin of information, and are concerned about converence of public belief. Our paper differs from the analysis in both papers in several 5

6 ways. First, we do not restrict all the customers to see the same number of observations. Moreover, the probabilities of those observations are endoenously enerated in our model. Finally, ours is a stationary model. In this respect, the closest paper to ours is a workin paper by Callander and Horner (2006), who consider a queuin system with a restricted state space. Callander and Horner focus on market heteroeneity, i.e. how the aents in the market are differentially informed, and arue how a minority of informed aents can cause other uninformed aents to follow shorter queues. In contrast, we study a market where all customers are equally informed, and focus on understandin the queue lenths at which herdin occurs. In addition, we also study the effect of service rates that are correlated with service value on customer choice behavior. In contrast with herdin literature, dependin on the service rates of the service providers or strenth of the sinal, we find that customers miht not herd at all. Queue Choice and Operations Manaement Literature: There is a wealth of literature that discusses the decision process of choosin between queues. Hassin and Haviv (2002) provide a comprehensive survey of the literature in equilibrium behavior of customers and service providers in queuin systems. The effect of neative externalities on choice between two queues has been well studied (Whinston 1977, Whitt 1986). Althouh the focus has been joinin shorter queues, simple characterizations of the probability distributions do not exist even for join-the-shortest queue discipline. The steady state probabilities enerated by joinin the loner queues have not been studied, perhaps because of perceived lack of applicability. Consequently, the number of papers that model positive externalities in multiple queues is limited. In a workin paper by Debo et al. (2007) customers arrivin in the market choose between joinin a sinle queue or not. When there are no waitin costs, they find a threshold lenth below which customers with pessimistic information do not join and above which they do join. When there are waitin costs, they show that a non-threshold stratey may determine the equilibrium. In contrast, usin a richer two-queue model, we are able to show that both positive and neative externalities in queues could arise to due to queue lenth information alone. Su and Zenios (2004, 2005) study patient choice in kidney allocation in a queuein context. Gans (2002) studies customers who choose amon various service providers with uncertain quality. Customers learn the true quality of every service provider throuh (expensive) repeated service samplin. There is no conestion externalities in the model (i.e. each customer learns about the service only by experiencin the service and not by observin the choices of other customers). At each service episode a sinle customer determines which service provider to visit. Lariviere 6

7 and van Miehem (2004) model a system in which customers find conestion costly, and therefore plan to arrive when the system is under-utilized. They show that when customers choose arrival times strateically, the equilibrium arrival pattern approaches a Poisson process as the number of customers ets lare. In our paper, customers arrivin accordin to a Poisson process have the option to choose from different service providers after havin observed the queue lenth at each service provider. We show that if the service providers are symmetric in service rates, then lonest queue joinin behavior occurs as soon as the queue difference is one. Our model differs from existin literature in several aspects. We focus on information externalities and show that uncertainty in service valuations alone may either cause customers to join loner queues or to avoid loner queues. We demonstrate that if the service rates are different enouh, customers miht inore the queue lenths completely and may follow their imperfect private sinals. If the service valuations are positively correlated with speed, we find that customers may not necessarily join loner (or slower) queues. We show that in contrast to the existin literature, providers of a better quality service in a market containin uninformed customers miht have a hiher market share by havin lower service capacity (lower service speed). Thus, investin in promotion efforts to indicate quality could be more attractive than investin in hiher service speeds or increasin service value further. 3. The Model In this section, we discuss the players, information sets, and decisions of the customers in the ame we model. Next, we determine the conditions for equilibrium. These conditions allow us to find an equilibrium whose qualitative properties are discussed in the conclusion. Market arrivals and Service rates: We consider a ame in which customers arrive sequentially accordin to a Poisson process with arrival rate λ to a market with two servers, labeled by i = 1 and i = 2. We assume that the service times at both servers are exponentially distributed with mean 1 µ. We allow the servers to have different service rates in Section 6. Also define ρ = λ µ. We assume that ρ < 1, allowin a sinle server to capture the market. The service discipline does not need to be FCFS, as lon as the service rates are exponentially distributed. Service value, public observation and private sinals: The exact service value of the service provided by a server i, v i is unknown to the customers. We assume that (v 1, v 2 ) {(v h, v l ), (v l, v h )} where v h > v l > 0. Thus if v i = v h then v i = v l where i = 1, 2 and i = 2, 1. The initial priors are symmetric, i.e. Pr[v 1 = v h ] = Pr[v 1 = v l ] = 1/2. Without loss of enerality, we can assume 7

8 that service values are fixed, and that server 1 is better. Upon arrival to the market, customers observe the queue lenth in front of each service provider; n = (n 1, n 2 ). We refer to n as the state of the system. We assume that the customers incur no waitin costs. This assumption helps us focus on the information value contained in the lenth of the queue. Our results are sensitive to the presence of waitin costs, and we point out the complexity of analyzin waitin costs in Section 5. Followin the assumption, we note that the value eventually accrued from the service is either v h or v l, reardless of the lenth of the queue. However, the expected value of the service is updated with the queue lenth information. In addition to observin the queue lenth at both servers, each customer receives a private sinal, s S = {1, 2} where S is the set of private sinals. The sinal is an imperfect indicator of which server provides the hihest value; s S is such that: Pr[s = 1 v 1 > v 2 ] = Pr[s = 2 v 1 < v 2 ] =, i.e. if the true state of nature is that server i provides better value than server j, each aent receives sinal s = i with probability and sinal s = j with probability 1. Note that the sinal is imperfect since 1/2 < < 1. When = 1, the sinal completely reveals the better service provider and every customer would then choose the better server, inorin the choices of other customers. When = 1 2, the sinal does not reveal any information about the quality of the service providers. We term such a sinal as bein completely noisy. Customer behavior: Consider any customer to arrive at the market. Let A = {1, 2} be the set of possible actions that the customer can take upon arrival; 1 represents joinin server 1, and 2 represents joinin server 2. A mixed stratey for this customer is then a mappin σ : A S N 2 [0, 1]. Let σ j (a, s, n) be the probability that this customer j joins queue a after observin sinal s and state n (with a A σ j(a, s, n) = 1). As all customers are homoeneous ex ante, we consider only symmetric strateies. For a iven stratey σ, let π i (n, σ) be the lon run probability that the system state is n conditional on v i > v i, with i denotin 2 (1) if i = 1 (2), with σ bein the customer s stratey. With the PASTA property (Wolff 1982), π i (n, σ) is also the probability that the system is in state n for any randomly arrivin customer conditional on v i > v i. Usin Bayes Theorem we have, P r [v 1 > v 2 n, s; σ] = π s(n,σ) π s(n,σ)+(1 )π s (n,σ) v 1 > v 2 (1 )π s (n,σ) π s(n,σ)+(1 )π s (n,σ) otherwise. After observin (n, s), the customer updates her prior expected service value for both service providers: E [v r n, s; σ] = π s (n, σ) E[v r v 1 > v 2 ] + (1 ) π s (n) E[v r v 2 > v 1 ], r, s {1, 2}. π s (n, σ) + (1 ) π s (n, σ) 8 (1)

9 where E[v 1 v 1 > v 2 ] = E[v 2 v 1 < v 2 ] = v h and E[v 1 v 1 < v 2 ] = E[v 2 v 2 < v 1 ] = v l. Notice that only v h and v l are relevant for a risk neutral customer. Consequently, any symmetric distribution F (v 1, v 2 ) over a discrete or continuous domain with the same conditional expectations will lead to the same equilibrium. Note that we have assumed that v l > 0, which alon with the absence of waitin costs, allows us to exclude balkin action from A. The customer equilibrium: Let σ i be customer i s stratey. Fix the stratey of all customers j i at σ. From (1), customer i s belief of the service value upon observin (s, n) is Pr [v 1 > v 2 n, s, σ]. Let BR (σ) be the best response of a customer to σ. Then, σ i BR (σ) if and only if: E [v s n, s, σ] E [v s n, s, σ] σ i (s, s, n) = 1 E [v s n, s, σ] E [v s n, s, σ] σ i ( s, s, n) = 1. (2) σ is a pure stratey symmetric Nash equilibrium if σ BR (σ ). It is apparent from the definition of best responses that the manitude of the difference in service valuations does not play a role in affectin customers decisions and consequently, the equilibrium joinin behavior. This is especially true for customers without waitin costs. We indicate a (s, n) as the equilibrium action in a pure stratey equilibrium upon observin (s, n). A mixed stratey is determined analoously. For convenience of notation, we drop the dependency of the lon-run probabilities and beliefs on σ. In the next section, we characterize σ. 4. Model Analysis for Symmetric Service Rates Road-map of the Analysis: In this section, we first derive conditions for the customers queue joinin stratey to be an equilibrium stratey when the service providers have identical service rates. To achieve this result, we bein with a restricted set of (pure) threshold strateies, which examine fixed queue lenth differences. We show that the followin stratey is an equilibrium stratey: Customers follow their sinals when the queue lenths are equal, and follow the loner queue otherwise. Relaxin the restricted set of strateies to include all possible threshold strateies, any equilibrium threshold stratey is identical to the aforementioned stratey at all recurrent states. We find our equilibrium result is robust when extended to mixed equilibrium strateies, multiple servers, and the possibility that service providers may o out of business if they operate a lon time without customers. We describe the challenes of a full waitin cost analysis in Section 5.3. Finally, we expand the analysis of symmetric service rates to include correlated asymmetric services in Section 6. 9

10 4.1 Equilibrium Customer Strateies We bein by describin the actions of a valuation-maximizin rational customer. We derive the conditions for each action every rational customer would take at each state and each sinal. A stratey is a customer action at each state n and each private sinal s (i.e. Join Queue 1, Join a queue accordin to the sinal, Join Queue 2). Based on the customer s action at each state, we can construct a two-dimensional birth and death process and the correspondin steady state probabilities for all states of the system. A stratey is an equilibrium stratey if it ives rise to stationary probabilities that support the existence of the stratey at each state. To reduce the notational burden, we drop the dependency of π i (n; σ) on σ. As defined before, π 1 (n) is the probability that the system is in state n for any randomly arrivin customer conditional on v 1 > v 2, and π 2 (n) can be defined in a similar way. Let the likelihood ratio 2 l(n) = π 1(n) π 2 (n) be the likelihood that server 1 is better than server 2, iven that the state of the system is n. Then, we can show that: Lemma 1 The equilibrium stratey satisfies: a (1, n) = 1(2) if l(n) ( ) 1 a (2, n) = 1(2) if l(n) ( ) 1 At equality, customers are indifferent between queue 1 and 2. Lemma 1 relates the choice of the customer (whether to choose queue 1 or queue 2) to the observed sinal, state of the system (defined by the number of customers waitin in each queue), and the likelihood ratio at that state. A customer chooses to join queue 1 after seein sinal 1 if the likelihood ratio is reater than 1. If the likelihood ratio is less than 1, the customer chooses queue 2. The above equilibrium conditions combine the imperfect information ained from the sinal about the qualities of the servers, and information about the service provider acquired throuh the choices that other customers have made. Lemma 1 can be understood usin joint probabilities. For example, a customer joins queue 1 upon observin sinal 1 or a(1, n) = 1, when l(n) 1, i.e. when π 1(n) (1 )π 2 (n). In other words, the customer arrivin at some state n on seein sinal 1 follows queue 1 rationally, when the probability of bein at the riht state of nature and seein the true sinal (1 is better than 2 Note l(n) 0 wherever it exists. When π 1(n) > 0, π 2(n) = 0, l(n) = π 1(n) π 2 =. When π1(n) = 0 and π2(n) = 0, (n) l(n) is indeterminate. 10

11 2 and this sinal is true) is hiher than the probability of bein in the wron state (server 2 is better than server 1) and seein a false sinal (sinal 1). Conversely, a(1, n) = 2 when l(n) 1, i.e. when π 1 (n) (1 )π 2 (n). Arrivin at state n, the customer has the followin choices: follow the shorter queue or follow the loner queue independent of the sinal realized; or finally, just follow one s own private sinal. Now, we are ready to define herdin in our model. A customer herds at n when she joins the loner queue at n independent of her sinal 3. The customer is herdin at that state by inorin her private sinal, and joinin the loner queue based on the public information. Now, we can postulate the structure of different strateies and analyze the equilibrium behavior. Clearly there are several possible rational strateies that exist in this ame Fixed Threshold Strateies We initially focus on a class of strateies that are fixed queue difference threshold strateies. Consider the followin stratey for a customer arrivin at some state n: The customer follows her own private sinal when n 1 n 2 B and follows the lonest queue at all other states. A consumer arrives at some state and observes the queue lenths at the servers. If the difference between the queue lenths is strictly reater than B, the customer joins the loner queue. Otherwise, she follows her sinal. We denote such a stratey by A B. B denotes the threshold (queue difference) above which each arrivin customer will join the loner queue. The term fixed refers to the fact that the queue lenth difference thresholds are fixed at B reardless of the lenth of the shorter queue. In other words, reardless of the state at which the customers arrive, if the queue lenth difference is in the set { B, (B 1),..., 0,..., B 1, B}, the customers follow their own private sinal. Since the threshold always refers to the difference in queue lenths, we refer to fixed queue difference threshold strateies as just fixed threshold strateies, whenever it is unambiuous. Fixed threshold strateies miht be a reasonable behavioral assumption, as the assumption reduces the queue selection problem of a customer to a simple comparison between a sinle decision parameter (namely, the queue lenth difference) and her own private sinal. In 4.1.2, we eneralize the fixed threshold strateies. Note that when B =, customers always follow their private sinal and inore the queue lenth information. Notice that in a ame in which customers would not observe the queue lenth, 3 Our definition of herdin is slihtly different from the classical herdin literature (see e.. Chamley, 2003, p ). We consider the decision of an arrivin customer in a steady state, while the classical literature considers a transient reime. 11

12 customers would follow their sinal as this is the only differentiator of the two service providers. We explore now whether this stratey is an equilibrium stratey when queue lenths are observable: Proposition 2 It is never an equilibrium stratey for customers to always follow their private sinal (i.e. to always choose a queue independent of the queue lenths). In other words, when service providers are symmetric in service rates, inorin queue lenths (A ) is never an equilibrium stratey. This result holds independent of the strenth of the private sinal as lon as the sinal is imperfect or noisy ( < 1). If every customer follows his or her sinal, we find that the loner queue forms in front of the hiher valuation provider since more customers et the riht sinal (since > 1 2 ). Hence an arrivin customer has an incentive to deviate from just followin her sinal, and would join the loner queue to improve her expected service valuation. Therefore A fails to be an equilibrium stratey. Proposition 2 captures the arument that customer decisions are influenced by externalities. It is never an equilibrium stratey to completely inore the available public information pertainin to the decisions that other customers have made. This emphasizes the importance of studyin the value of information accrued from the available public information, such as queue lenths. We now examine which fixed queue difference threshold strateies are in equilibrium in the followin proposition. At any state of the system, if followin one s private sinal is an equilibrium stratey, then at that state the customer must have likelihood ratios between 1 and 1. Therefore, for a fixed threshold stratey B to hold in equilibrium, the likelihood ratios in all the states within the queue difference B, should be between 1 and 1. Within the fixed threshold, if these likelihood conditions do not hold, the customers have an incentive to not follow their sinal. The likelihood ratios must be reater than 1 queue outside the fixed queue difference. if the customers inore the sinal and follow the loner Proposition 3 It is never an equilibrium stratey for customers to join the lonest queue only if the queue lenth difference is reater than some positive threshold B 1, and follow their private sinal otherwise. In other words, Proposition 3 implies that A B is never an equilibrium stratey for any B 1. Strateies in which customers follow their private sinal within the fixed queue difference (of size one or more) enerate stationary probabilities (and likelihood ratios) which do not satisfy the condition required for followin the sinal (from Lemma 1) in all states whose queue difference is bounded by the fixed threshold. Hence they fail to hold in equilibrium. For instance, suppose all the customers 12

13 follow the stratey A 1, i.e. each customer follows her private sinal if the queue lenths are equal or differ by 1. Otherwise, she chooses the loner queue. Solvin the two dimensional birth and death process correspondin to the actions taken in the stratey A 1, we obtain the stationary probabilities of bein at each state. Usin these probabilities, we can obtain the likelihood ratios at each state. If A 1 is an equilibrium stratey, the likelihood ratios at each state should in turn satisfy the requirements accordin to Lemma 1 at each state. However, they do not. The intuition for why A 1 is not an equilibrium stratey is the followin: Consider a customer arrivin to the system and observin the state (1, 0) and followin her own private sinal. If the customer would see sinal 1, the customer follows the sinal since his sinal is consistent with the loner queue. If the customer receives sinal 2 at (1, 0), she will follow her sinal rationally only if the probability of bein at the riht state of nature and seein the false sinal (server 1 is better than server 2, but the sinal says 2) is lower than the probability of bein in the wron state (server 2 is better than server 1) and seein a true sinal (sinal 2) (i.e. (1 )π 1 (1, 0) π 2 (1, 0) or simply, l(1, 0) 1 ). Based on the customer actions at each state under the stratey A 1, we derive the probabilities of observin the state (1, 0) when server 1 is better (π 1 (1, 0)) and when server 2 is better (π 2 (1, 0)). Based on these probabilities, we show in proof of Proposition 3 that the likelihood ratio at (1, 0) violates the aforementioned condition (i.e. we show that l(1, 0) > 1 ). Hence, throuh Lemma 1 we find that the best response of a customer arrivin at (1, 0) is to follow the loner queue at (1, 0) if everyone else follows the stratey A 1. Each customer arrivin at the state (1, 0) has an incentive to deviate from the stratey A 1 and improve her payoff, iven that every other customer follows the A 1 stratey. Therefore, the A 1 stratey fails to be an equilibrium stratey. Similar aruments can be made for all other fixed threshold strateies by showin that at the state (1, 0) or some other state, an arrivin customer has an incentive to deviate from A B. Finally note that when all customers follow the fixed threshold stratey with B 1, the customers can queue up in front of both servers with non-zero probability. We consider the sole remainin candidate for the equilibrium stratey in the class of fixed queue difference threshold strateies, A 0 in Proposition 4. Proposition 4 It is an equilibrium stratey for customers to follow their sinal when the queue lenths are equal, and join the loner queue otherwise. The result implies that A 0 stratey is supported in equilibrium. Since > 1/2 whenever the queue lenths are equal, the customers join the better server with hiher probability (since all 13

14 customers follow their own sinal, and the sinal is accurate with probability > 1/2). Once a particular queue becomes loner than the other queue, it keeps rowin with every new arrival until eventual service departures (note that µ > λ) from the loner queue brin the queue lenths back to bein equal. Therefore, on averae the better service provider has a loner queue. The notion that if all the arrivin customers join the loner queue they have a hiher likelihood of ettin better service value is true. It can be shown that l(n, 0) (or l(0, n)) is equal to 1 1 (or ) n under A 0. Hence, the customers weakly prefer the loner queue. This is because under A 0, when the queue is not empty, the private information held by a customer is not reflected in her queue choice. In other words, the herd behavior dilutes the information contained in the queue. Only when a customer arrives at an empty system, the private information she holds is revealed (because she follows her sinal at that state). This naturally raises a question as to whether the customers could randomize their decisions in equilibrium. Mixed Strateies: Our next result asserts that the customers will never mix between the strateies of joinin the loner queue and followin their sinal when they are indifferent between the valuations ained. Let p k be the probability that the customer follows the sinal at state (k, 0) (he inores it and joins the loner queue with probability 1 p k ). Let the actions be symmetric in state (0, k). We show such a mixin stratey will not be an equilibrium stratey. Proposition 5 Under A 0, it is never an equilibrium stratey for customers to mix at all states in which they are indifferent. There exists at least one state at which customers would always follow the loner queue as a best response. The notion that the customers join the loner queue, inorin shorter queue lenths is appealin. We note that in many real life occasions, such as when selectin restaurants, there is some evidence for customers choosin the lonest queue (Becker 1991, Hill 2007) General Threshold Strateies It is certainly a restriction to explore only fixed threshold strateies althouh such a restriction allowed us to reduce the complexity of the equilibrium analysis. Consider the followin two states (2, 0) and (102, 100). Althouh queue lenth differences in those states are the same, the queue lenth difference may be of much more sinificance at the state (2, 0) than at the state (102, 100). Customers miht join the loner queue at (2, 0) and not at (102, 100). We address this issue by considerin eneral threshold strateies where the queue joinin behavior does not only depend on the queue lenth differences alone, but also on the lenth of the individual queues. Under 14

15 the class of eneral threshold strateies, the customers join the loner queue when the lenth of the loner queue is reater than T k + k where k is the lenth of the shorter queue, and T k is a non-neative inteer (no other structure is imposed on T k ). Define the class of threshold stratey where each stratey is defined by S {T 0,T 1, } where T : N N as a threshold stratey. Clearly, fixed queue difference stratey A B is a threshold stratey where T k = B for all k. In other words, A B S {B,B, }. In the followin proposition, we show that the equilibrium threshold strateies are almost surely identical to the A 0 stratey. Proposition 6 Threshold strateies are in equilibrium only if T 0 = 0, i.e. when customers always join the loner queue when the shorter queue is empty. Hence equilibrium threshold strateies are identical to the A 0 stratey at all non-zero probability states. We observe that in equilibrium, reardless of the values of threshold levels that exist at states when the shorter queue k 1, the best response for a customer arrivin at (1, 0) or (0, 1) is to join the loner queue. Thus, just as with A 0, when customers queue in front of one server, the other server is empty. Therefore, the two queues can never develop simultaneously in steady state. 4.2 Market Implications of the Equilibrium Strateies We observe that amon all the strateies where the customers follow the loner queue above a threshold, the unique best response stratey at (1, 0) and (0, 1) is to follow the loner queue. The results we derived in the section imply the followin corollary. Corollary 7 The better server (of the two servers) is busy (and the other server is empty) durin a fraction ρ of time. The worse server is busy durin fraction ρ (1 ) of time. It follows from Corollary 7 that the market share for the hih quality service provider is equal to the strenth of the sinal. The low quality service provider captures the residual market. Somewhat surprisinly, the service rate does not play a role for the market share. It only determines the total traffic ρ to both servers. This is because no customers ever balk from the system. The same market share (and busy times of each service provider) would be obtained if customers would have completely inored the queue lenth information and just have followed their private information 4. However, the total number of customers in the system differs sinificantly: If customers completely inored the queue lenth information, the averae number in the system would be much lower. 4 In that case, the arrival rate to the hih (low) quality service provider would be λ ((1 )λ). 15

16 This is intuitive: the herdin reduces the effective system capacity as only one service provider is active at any point in time. The stationary equilibrium thus has the followin properties: 1. At all times, one of the service providers is necessarily empty. 2. The system is empty with a non-zero probability. A customer that arrives when the queues are empty follows her private sinal. Once she chooses a service, all future customers continue to choose the same service provider until the queue depletes to zero. These cycles then repeat with independent private sinal realizations (for the customer arrivin at the empty system) decidin which service provider s queue rows next. 3. Queues row and decay in cycles. The proportion of time that a hih quality service provider is busy increases with the strenth of the private sinal. A service provider that is busy clears the queue at a rate that decreases in ρ. The lon-run market share of the better firm, however, is equal to the sinal strenth. 4.3 Survival Likelihood A consequence of the analyzed equilibrium stratey is that all the customers line up at the service offered by one of the service providers. Consider the service provider that has an empty queue. We address this service provider as starvin for customers. So far we had assumed that starvation does not impact the service provider in our model. However, starvin service providers do not earn revenues that cover their fixed costs. This may especially be important for start-ups (Archibald et al; 2002) that do not have easy access to capital markets. In this subsection, we model the possibility that a service provider oes out of business because of a lack of revenues from customer arrivals. Let both service providers in the market be described by a 2-tuple (v, τ) where v is the valuation the service provider offers to the consumers from its service and τ [τ, ) is the survival parameter and τ is a lower bound on τ. τ denotes the time a starvin service provider can survive without any revenues from arrivin customers. τ is determined by the firm s ability to cover its fixed costs without earnin revenues. After τ units of time starvin for customers, the service provider oes out of business. Suppose (τ 1, τ 2 ) is distributed with a symmetric density function φ (τ 1, τ 2 ) in the market. In the followin result, we show that A 0 remains an equilibrium stratey althouh starvin service providers miht o out of business. 16

17 Proposition 8 Let a service provider i in the market be described by (v i, τ i ) where v i is the valuation offered to customers, τ i denotes the time it can survive without any revenues from arrivin customers, and τ i [τ, ), i {1, 2}, τ > 0. Conditional on the presence of two servers in the market, stratey A 0 is an equilibrium stratey. When there is one server in the market, all customers join the server. Proposition 8 extends the applicability of the results obtained earlier in this section. The intuition is that the observation by the customers that both service providers are in the market does not contain any additional information about the relative quality of the service providers. Once one service provider oes out of business, the customers in the market are forced to join the other one for the lack of an alternative. Hence, the queue joinin problem becomes trivial when there is only one firm in the market. Note that with probability one, the remainin service provider will eventually o out of business too, as it cannot be uaranteed that the idle period with a sinle service provider is less than τ for sure. 5. Extensions of Modelin Assumptions In the previous section we resorted to simplifyin assumptions primarily in order to obtain sharp analytical insihts. In this section, we explore extensions of the base model. Specifically, we consider multiple servers, allow for the priors to be asymmetric, and discuss the effect of waitin costs on the decision of a sinle customer in the system. 5.1 Multi-server extension We now model each service provider as a service station with N identical servers. In this extension, the customers arrive at restaurants or facilities that have multiple service providers (for example, think of a restaurant with N tables where each table hosts one customer, and all the tables are served by waitin personnel with identical service speeds). Proposition 9 A 0 is an equilibrium stratey when each service provider has N multiple servers. Proposition 9 notes that A 0 continues to hold in equilibrium. We can apply our insihts to two service providers offerin services throuh several identical servers, or sellin products with unknown valuation throuh identical retailin centers. It is also worth notin that customers continue to weakly prefer the loner queue when they are indifferent between valuations ained, just as in the sinle-server case. 17

18 5.2 Asymmetric Priors about Service Valuations We assumed the initial priors of customers were symmetric. i.e. Pr[v 1 = v h ] = Pr[v 1 = v l ] = 1/2. However, customers miht have asymmetric priors. Without loss of enerality, let customers believe that service provider 1 is better with probability q 0. i.e. Pr[v 1 = v h ] = q 0 and Pr[v 1 = v l ] = 1 q 0. We can re-derive the likelihood ratio conditions similar to Lemma 1, now employin asymmetric priors, and show that the equilibrium stratey satisfies the followin condition: a (1, n) = 1(2) if l(n) ( ) 1 1 q 0 q 0 a 1 q 0 (2, n) = 1(2) if l(n) ( ) 1 q 0 At equality, customers are indifferent between queue 1 and 2. Explorin fixed threshold strateies over the conditions, we find that: 1. When q 0, customers always join service provider 1 at all states. 2. When q 0 1, customers always join service provider 2 at all states. 3. When 1 < q 0 < 1/2 and 1/2 < q 0 <, none of the symmetric fixed threshold strateies are in equilibrium. We conjecture that customers always join a loner queue identical to their initial prior at small queue lenth differences, and join a loner queue opposite to their prior when those queues are much loner. The first two results are not surprisin. Suppose the strenth of private sinal that the customers acquire is much weaker than their initial priors. Then the private sinals that customers observe are not stron enouh to overcome the initial priors that customers have aainst or for the service value from a service provider. Hence, all customers follow what the prior dictates. As a result, one service provider attracts all customers and the queue lenths become uninformative. When the priors are strictly asymmetric, but neither too stron nor too weak, we find that none of the symmetric fixed threshold strateies are in equilibrium. In the case with weak initial priors, the equilibrium stratey would have characteristics of the case with symmetric priors and the case with extremely stron priors. In particular, we conjecture that if q 0 > 1/2, customers miht follow loner queue 1 when the queue difference is small. They miht follow queue 2 if it is the loner queue, but only if queue lenth difference is sufficiently lare. 18

19 5.3 Waitin Cost Challenes The presence of waitin costs poses sinificant challenes to analysis. In this subsection, we discuss the underlyin complexity of analyzin waitin costs, and defer a full analysis of the waitin cost problem to future research. The model with waitin cost introduces the customer to a tradeoff she was not facin before: Observin a lon queue indicates that more people chose this service, but it also indicates that the customer will have to wait loner for the service. This presence of additional trade-offs chane the properties of the equilibrium stratey we have derived. An additional issue is that iven some waitin cost, there is no tractable way to theoretically model a two queue system with lare buffer space and analyze the best response at each state. We illustrate this complexity by considerin the best response of a sinle customer in the system. Let c be the waitin cost per unit of time for customers arrivin to a system containin two symmetric servers. Aain let A = {0, 1, 2} be the set of possible actions that the customer can take upon arrival; 1 represents joinin server 1, 2 represents joinin server 2 and 0 represents balkin from the system 5. A mixed stratey for this customer is then a mappin σ : A S N 2 [0, 1]. Let σ j (a, s, n) be the probability that this customer j joins queue a after observin sinal s and state n (with a A σ j(a, s, n) = 1). Let σ i be customer i s stratey. Fix the stratey of all customers j i at σ. Customer i s belief of the service value upon observin (s, n) is Pr [v 1 > v 2 n, s, σ]. The customer balks from the system if the expected valuation on seein the sinal is lower than the waitin cost. We can now characterize the best response of this customer similar to equation (2). Let BR (σ) be the best response of a customer to σ. Then, σ i BR (σ) if and only if: E [V s n, s; σ] c(n s + 1)/µ > max (0, E [V s n, s; σ] c(n s + 1)/µ) σ i (s, s, n) = 1 E [V s n, s; σ] c(n s + 1)/µ > max (0, E [V s n, s; σ] c(n s + 1)/µ) σ i ( s, s, n) = 1 max (E [V s n, s; σ] c(n s + 1)/µ, E [V s n, s; σ] c(n s + 1)/µ) < 0 σ i (0, s, n) = 1 (3) There are two additional complications. Balkin stratey is introduced in the action set, and the difference in expected valuations matters with respect to waitin costs. Given this information, customers miht follow the loner queue, follow their sinal, join the shorter queue, balk on seein a contrary sinal, or balk from the queues completely. However, iven the steady state probabilities π 1 (n) and π 2 (n), their best responses can be immediately determined. 5 Note that the action 0 does not refer to joinin some real queue 0, but instead the action of balkin from queues 1 and 2. We define balkin as joinin queue 0 for notational simplicity. Note that the customer does not observe n 0, the number of customers who have balked. This is commonly true: Customers often do not observe those customers who decided to balk from the queues. 19