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Bounded Rationality in Service SystemsTingliang Huang, Gad Allon, Achal Bassamboo,

To cite this article:Tingliang Huang, Gad Allon, Achal Bassamboo, (2013) Bounded Rationality in Service Systems. Manufacturing & ServiceOperations Management 15(2):263-279. https://doi.org/10.1287/msom.1120.0417

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MANUFACTURING & SERVICEOPERATIONS MANAGEMENT

Vol. 15, No. 2, Spring 2013, pp. 263–279ISSN 1523-4614 (print) � ISSN 1526-5498 (online) http://dx.doi.org/10.1287/msom.1120.0417

© 2013 INFORMS

Bounded Rationality in Service SystemsTingliang Huang

Department of Management Science and Innovation, University College London,London WC1E 6BT, United Kingdom, [email protected]

Gad Allon, Achal BassambooKellogg School of Management, Northwestern University, Evanston, Illinois 60208

{[email protected], [email protected]}

The traditional operations management and queueing literature typically assumes that customers are fullyrational. In contrast, in this paper we study canonical service models with boundedly rational customers.

We capture bounded rationality using a model in which customers are incapable of accurately estimating theirexpected waiting time. We investigate the impact of bounded rationality from both a profit-maximizing firm’sperspective and a social planner’s perspective. For visible queues with the optimal price, bounded rationalityresults in revenue and welfare loss; with a fixed price, bounded rationality can lead to strict social welfareimprovement. For invisible queues, bounded rationality benefits the firm when its level is sufficiently high.Ignoring bounded rationality, when present yet small, can result in significant revenue and welfare loss.

Key words : behavioral operations; service operations; bounded rationality; queueing; consumer behaviorHistory : Received: June 21, 2011; accepted: September 1, 2012. Published online in Articles in Advance

March 1, 2013.

1. Introduction and Literature ReviewWhen a customer calls a call center or goes to a fast-food restaurant, a café, or an ATM and has to queuefor service, does he always accurately and perfectly cal-culate the benefits and costs of joining before makinghis decisions? The traditional economics and queue-ing literature has assumed that he does, whereasanecdotal evidence and experimental studies pointto the contrary. In this paper we study queueing orservice systems without making this “perfect ratio-nality” assumption on the part of customers. Ourresearch questions are the following: How should onemodel customer bounded rationality in service sys-tems? What are the implications of bounded rational-ity, for example, on firm revenue, social welfare, andpricing?

Naor (1969) appears to be the first to incorporatecustomer decisions into a queueing model. Naor andsubsequent researchers following his work assumecustomers to be fully rational and able to perfectlyestimate their expected waiting time and thus theexpected utility of joining. Naor (1969) shows that self-interested customers would join a more congested sys-tem than what the social planner prescribes, and heproposes “levying tolls” (i.e., pricing) as a way to max-imize social welfare. In Naor’s model, customers areassumed to be able to compute with great precisionthe expected waiting time and thus the expected utilitythey are about to obtain from making a decision aboutwhether to join or renege. One may ask, are customersfully rational? Specifically, does a customer necessarily

have the capability to perfectly estimate his expectedwaiting time and utility? Ariely (2009) claims that irra-tionality is the real invisible hand that drives humandecision making. Indeed, there is abundant empiri-cal evidence that people are boundedly rational. Inthis work, we study the effects and implications ofbounded rationality in canonical queueing or servicesystems.

Our study is related to several branches of the lit-erature: economics of queues, bounded rationality ineconomics, and behavioral operations.

Economics of Queues. Naor (1969) studies the eco-nomics of queueing systems when customers are fullyrational. Yechiali (1971, 1972) extends Naor’s modelto allow for GI/M/1 queues. Knudsen (1972) extendsNaor’s model to allow for a multiserver queueingsystem in which arriving customers’ net benefits areheterogeneous. Lippman and Stidham (1977) extendthe Naor model to the finite-horizon and discountedcases, showing that, in these settings, the economicnotion of an external effect has a precise quanti-tative interpretation. Hassin (1986) considers a rev-enue maximizing server who has the opportunity tosuppress information on actual queue length, leav-ing customers to decide whether to join the queueon the basis of the known distribution of waitingtimes. See Van Mieghem (2000), Hassin and Haviv(2003), Afèche (2004), and Hsu et al. (2009) for otherextensions and a comprehensive literature review.Although various models along this line are studied,one common theme in this literature is that full ratio-nality is always assumed.

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Huang, Allon, and Bassamboo: Bounded Rationality in Service Systems264 Manufacturing & Service Operations Management 15(2), pp. 263–279, © 2013 INFORMS

Bounded Rationality in Economics. Traditional eco-nomic theory postulates that decision makers are“rational”; i.e., they have sufficient abilities to doperfect optimization in their choices. Simon (1955)seems to be the first to propose an alternative wayto model decision-making behavior: rather than opti-mizing perfectly, agents search over the alternativesuntil they find “satisfactory” solutions. Simon (1957)coins the term “bounded rationality” to describe suchhuman behavior.

Bounded rationality refers to a variety of behav-ioral phenomena in the literature. For a descriptionof systematic errors made by experimental subjects,see Arkes and Hammond (1985), Hogarth (1980),Kahneman et al. (1981), Nisbett and Ross (1980), andthe survey papers by Payne et al. (1992) and Pitz andSachs (1984). Tversky and Kahneman (1974) show thatpeople rely on a limited number of heuristic prin-ciples that in general are useful but sometimes leadto severe and systematic errors. On the basis of theevidence, Conlisk (1996) offers four convincing rea-sons for incorporating bounded rationality in eco-nomic models. Geigerenzer and Selten (2001) adoptheuristics or rules of thumb to model bounded ratio-nality. Thurstone (1927) and Luce (1959) appear tobe the first to develop the framework for stochasticchoice rules capturing that better options are chosenmore often. This approach has attracted consider-able attention and has been adopted in a varietyof settings. For example, following this approach,McKelvey and Palfrey (1995) and Chen et al. (1997)develop a new equilibrium concept quantal responseequilibrium in game theory. Others include, for exam-ple, Bajari and Hortacsu (2001, 2005) in auctions,Cason and Reynolds (2005) in bargaining, Basov(2009) in monopolistic screening, and Waksberg et al.(2009) in natural environments. We also adopt thisapproach in the paper, in the context of expected wait-ing time estimation of service systems.Behavioral Operations. Gino and Pisano (2008) sur-

vey the literature on modeling bounded rationalityin economics, finance, and marketing, and they arguethat operations management scholars should incorpo-rate departures from the rationality assumption intotheir models and theories. There is an emerging liter-ature on behavioral operations management: Lim andHo (2007) and Ho and Zhang (2008) conduct exper-iments on designing pricing contracts for boundedlyrational customers, Davis (2011) investigates pull con-tracts in controlled experiments, and Kremer et al.(2011) analyze how individuals make forecasts basedon time-series data and find that forecasting behaviorsystematically deviates from normative predictions.We refer readers to Bendoly et al. (2006, 2009) forthis stream of research. We point out two papers thatare closely related to our work. The first paper is by

Su (2008), who studies bounded rationality in opera-tional settings. He applies the logit choice frameworkto the classic newsvendor model and characterizesthe ordering decisions made by a boundedly rational(i.e., noisy) decision maker. He identifies systematicbiases and investigates the impact of these biases onseveral operational settings. We apply a similar frame-work in this paper, but we interpret bounded rational-ity as the incapability of estimating expected waitingtime in a service setting. There are several notabledifferences between Su (2008) and our study: First,Su’s (2008) model is static (i.e., one-shot), whereasour model is a dynamic one. Second, the externalityamong boundedly rational decision makers is promi-nent in our setting, whereas it is not in Su (2008).Finally, different from Su (2008), where there are priorempirical and experimental studies of the newsven-dor model that allow for statistical tests, our studyis theoretical and aims to obtain testable theoreticalpredictions that stimulate future empirical and experi-mental work in service systems. Recently, Kremer andDebo (2012) have presented experimental findingsalong this line. The second paper is by Plambeck andWang (2010), who study implications of hyperbolicdiscounting in service systems. Although the researchsetting (i.e., service systems) is similar, the researchfocus and approach are quite different. In their paper,customers lack the self-control to undergo an unpleas-ant experience that would be in their long-run self-interest, which is modeled by psychologists in termsof a hyperbolic discount rate for utility. Our modelof bounded rationality focuses on customer ability tocompute the expected waiting time.

Another stream of research related to ours is exper-imental study in queueing. This literature does notsupport that individuals are fully rational. Rapoportet al. (2004) study a class of queueing problems withendogenous arrival times formulated as noncooper-ative n-person games in normal form. Results fromtheir experimental study cannot be fully explained byrational behavior; see also Bearden et al. (2005) andSeale et al. (2005) for studies along this line.

Traditional models in operations assumed that cus-tomers are rational both in maximizing their util-ity and in their ability to compute the anticipatedexpected waiting time in high precision, regardlessof the complexity of the system. In this paper, ourmodel of bounded rationality focuses on this abil-ity to predict expected waiting times. In particular,we assume that customers may lack the capabilityto accurately estimate their expected waiting time(including the service time) before making their joinor balk decisions and that this capability may varyacross different queue configurations (e.g., visible andinvisible). We thus introduce a random error terminto customer’s expected waiting time estimation that

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Huang, Allon, and Bassamboo: Bounded Rationality in Service SystemsManufacturing & Service Operations Management 15(2), pp. 263–279, © 2013 INFORMS 265

reflects their inability to accurately assess the utilityobtained from each action. From both a revenue-maximizing firm’s perspective and a social planner’sperspective, we study the impact of bounded rational-ity for both visible queues (such as a fast-food restau-rant, a café, or an ATM) and invisible queues (such asa call center).

The three main contributions of our study arethe following: (i) ours is among the first to modelbounded rationality in service systems, for both vis-ible and invisible queues; (ii) our study providesinsights on how service systems should be managedin the presence of boundedly rational customers (e.g.,we show that when the system manager can reducethe ambiguity associated with the process of estimat-ing the expected waiting time, completely eliminatingconsumer bounded rationality may not be the opti-mal strategy and ignoring bounded rationality mayresult in significant revenue and social welfare loss);and (iii) we provide a framework to stimulate futureempirical and experimental work in service systemsand behavioral operations.

The remainder of this paper is organized as follows.Section 2 presents a model of bounded rationality inservice systems. We study revenue maximization andsocial welfare maximization in §§3 and 4. We providea discussion in §5. Proofs of the results are relegatedto the online supplement (available at http://dx.doi.org/10.1287/msom.1120.0417).

2. A Model of Bounded Rationality inService Systems

Consider a customer who has to decide whether tojoin a service system or not. If he joins, he will obtainexpected utility U1 ≡R− p−C Ɛw, where R> 0 is thereward on completion of service, p is the price, Ɛwis the expected waiting and service time, and C > 0is the average waiting and service cost per unit oftime. If he balks, he will get utility U2 = 0 (from anoutside option). The existing queueing literature typ-ically assumes that he is perfectly rational: if U1 ≥ 0,he will join the system; otherwise, he will balk. How-ever, accurately computing expected waiting and ser-vice time is typically not an easy task for customers.We thus depart from this literature by incorporat-ing a more realistic assumption: customers lack thecapability to accurately estimate their expected wait-ing time. As a result, customers cannot guarantee thatthe best choice is always chosen, and they may makemistakes. To formally model this noisy waiting timeestimation, we introduce a random error term, �, intocustomer’s expected waiting time estimation. If V1 ≡

R−p−C4Ɛw+�5≥ 0, he will join the system and balkotherwise. As outside observers, we then obtain thecustomer joining probability � ≡ �4� ≤ U1/C5, which

should be interpreted as the fraction of customers whowill join. For analytical tractability, we assume thatthe error term � follows a logistic distribution F 4x5=

1/41 + e−x/�5 for some � > 0. The logistic distributionprovides a good approximation to the normal distri-bution but has heavier tails (Talluri and van Ryzin2004, pp. 305–306). Following McFadden (1974) andAnderson et al. (1992), we thus obtain the customerjoining probability:

� =eU1/4C�5

1 + eU1/4C�50

It is important to note that in our model customersdo not choose to play mixed strategies. It is the noisyestimation that drives this behavior, and � denotesthe fraction of customers that join the system.

To interpret the meaning of �, note that the stan-dard deviation of � is � ≡

√

Var4�5= 4�/√

35� ≈ 108�.Hence, the parameter � is proportional to the standarddeviation of the error term �. Thus, the parameter �measures the error level of customer expected waitingtime estimation. (See Hey and Orme 1994, p. 1301,for a similar approach and explanations; alternatively,one could normalize the utility value while usingthe variance of the error term to capture the level ofbounded rationality.) Furthermore, the standard devi-ation of customer expected utility V1 is �V1

≡ C� ≈

108C�. For convenience, we define � ≡ C�, whichmeasures the error level of customer expected util-ity estimation. This error level � reflects customerbounded rationality in the sense that customers havelimited computational capability in perfectly estimat-ing their expected waiting time (and as a result, theirexpected utility of joining) in the queueing setting.Hence, we interpret the parameter � as the extentcustomers are incapable of implementing the optimaldecision because of their incapability in estimatingexpected waiting time. As �→ 0, the joining behaviorconverges to full rationality. At the other extreme, as� → �, customers join or balk with equal fractions.Therefore, we can refer to the magnitude of � as thelevel of bounded rationality.

The interpretation of the level of bounded ratio-nality � also follows from the well-known interpre-tation of the coefficients of logit regressions in thatit captures the idea that better options are chosenmore often. One can rewrite the joining probabilityas log4�/41 −�55 = 41/�5U1. The left-hand side is the“log odds” of joining the system, so � is the inverseof the difference in the log odds for any one-unitincrease in the expected utility of joining the system.For example, when � = 005, then the log odds dou-bles for any one-unit increase in the expected util-ity of joining the system; when � = 2, then the logodds decreases by half for any one-unit increase inthe expected utility of joining the system.

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Given that � is the standard deviation of customerutility, it has the same unit as the reward R. We expectthe magnitude of � to depend on the context. Forexample, McKelvey and Palfrey (1995) estimated 1/�in their game settings, which suggests that � rangesfrom 0.3 to 6.7 for a 1982 penny. Bajari and Hortacsu(2005, Table 7) reported the estimates for 1/�: 15.68,17.36, or 11.32 for a 1989 dollar. In another auctionsetting, Goeree et al. (2002) found out that the param-eter � is around 0.09, 0.16, or 0.26 (see Table 4, p. 258)for a 2001 dollar. In the queueing setting, Kremer andDebo (2012) recently ran experiments in laboratoriesto test bounded rationality for visible queues. Theyfound that � is statistically significantly different fromzero: from their estimation, 1/� is 0.296, and the stan-dard deviation is 0.025. They use the expected rewardR= $1002.

Given that � has the same unit as R, one canuse �/R to measure the level of bounded ratio-nality relative to the reward. Note that this ratiois dimensionless and thus allows a unified com-parison across studies without converting monetaryunits. Straightforward computation yields the mag-nitude of �/R in the literature: �/R ∈ 60006150687 inMcKelvey and Palfrey (1995), 600003810000597 in Bajariand Hortacsu (2005), 6000181000527 in Goeree et al.(2002), and around 0.33 in Kremer and Debo (2012).In our numerical study, �/R is in the range of 6010057or 60157 for the low-reward case and high-rewardcase, respectively, which is comparable to the magni-tude observed in the literature.

We believe that people are typically not capableof accurately computing expected waiting time, andwe interpret bounded rationality in terms of inca-pabilities in accurately estimating it in this paper.However, this is not the only possible interpretation.There are a couple of closely related interpretationsin the literature that could be adopted here. First,Chen et al. (1997) propose the boundedly rationalNash equilibrium where agents are not utility max-imizers but instead choose randomly in a fashionthat is influenced by a subconscious utility function. Inour setting, one can interpret U1 as the subconsciousutility because it is not explicitly known to the cus-tomer because of his bounded rationality. Moreover,Chen et al. (1997, p. 34) point out that “any math-ematical structure commonly used with the noiseor random utility interpretations can also be inter-preted as a model of bounded rationality.” Along thisline, one can interpret our model in several ways byrewriting the customer utility function: V1 = R − p −

C Ɛw + �′ = U1 + �′. The noise term does not have tocome from expected waiting time. It can stem fromcustomer noisy perception of the waiting cost C orheterogeneity in customer preference for a particu-lar queueing environment (see Maister 1985, Larson1987): �′ = −�Ɛw.

The second different interpretation comes fromMattsson and Weibull (2002): that agents are utilitymaximizers. However, they have to make some effortin order to implement any desired outcome, and thedisutility of this effort enters their utility function.In our setting, one may think of customers havingto make some effort to implement the optimal join-ing decision, perhaps because of the challenging taskof estimating the expected waiting time. These dif-ferent interpretations provide different justificationsto our model. We follow the interpretation of inca-pability of accurately estimating the expected wait-ing time because we believe that it captures best themain computational limitation customers face in ser-vice systems.

Our model has both usefulness and limitations.The model is useful because (i) it provides a sys-tematic way to capture bounded rationality in servicesystems using a single parameter �; (ii) this singleparameter � is endowed with a concrete meaning thatincludes both conventional full rationality and purelyrandom behavior (i.e., full bounded rationality), andit allows us to investigate the impact of boundedrationality (e.g., in terms of revenue, welfare, andpricing); (iii) this model is flexible to accommodateseveral interpretations; and (iv) it can be readily usedfor further empirical or experimental tests because ofits close connection to commonly used logit regres-sions. However, this model has limitations: flexibilitycomes with both advantages and disadvantages. Thatthe model itself cannot be pinned down to a uniqueinterpretation without lab experiments calls for fur-ther empirical or experimental studies. Indeed, that isone of the goals of the paper: to stimulate a desirabletheoretical–empirical feedback loop in service opera-tions management research. Also, the logit model isa special case of the general class of the “Fechner”approach of modeling the stochastic element in deci-sion making, and we refer the reader to Loomes et al.(2002) for other models that can potentially be useful.

Both the expected waiting and service time Ɛw andthe level of bounded rationality depend on the con-figuration of the service system: whether the queuelength is observable to customers or not. Hereafter,we distinguish the visible and invisible queues, anddenote the levels of bounded rationality � and �I forthem, respectively.

2.1. The Visible QueueConsider a single-server queueing system that isobservable. Customers arrive to the system accord-ing to a Poisson process with rate �. Upon arrival,each customer decides whether or not to join thequeue after observing the queue length and basedon his estimate of the waiting time. Service timesare assumed to be independently, identically, and

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exponentially distributed with mean 1/�. Denote theutilization of the system �≡ �/� if all customers join.Customers are served on a first-come, first-servedbasis. Upon arrival, observing n customers in the sys-tem, the fraction of customers that join the system isgiven by

�n ≡e4R−p−44n+15C5/�5/�

1 + e4R−p−44n+15C5/�5/�1 (1)

for n = 011121 0 0 0 0 To compute the expected waitingtime, each customer needs perfect information whenhe joins about the number of customers and the ser-vice rate and needs the cognitive capability to trans-form this information into an estimate of the expectedwaiting time. Thus, if customers lack either the infor-mation or the cognitive capability, then the waitingtime estimate will be inaccurate. Further, the accu-racy of the expected waiting time is severely impactedbecause they need to do this in real time.

For ease of exposition, we let �n ≡ ��n, n =

011121 0 0 0 , be the state-dependent queue-joining rates.Then, we can treat the number of customers in thesystem as a birth–death process with birth rate �n

and death rate �. Although customers are boundedlyrational, we first show that the stability of the systemis guaranteed as long as � is finite, as stated in thefollowing proposition.

Proposition 1. The visible queueing system withboundedly rational customers is stable for �<�, and theprobability distribution in steady state is as follows:

P0 =1

1 +∑�

k=1 4�0�1 · · ·�k−15/�k

is the probability that the system is in state 0, and

Pn =�0�1 · · ·�n−1

�n41 +∑�

k=1 4�0�1 · · ·�k−15/�k5

is the probability in state n, n≥ 1.

Note that the queue length distribution becomesunbounded from above for � > 0, but the system isalways stable by Proposition 1.

We numerically observe that both utilization (i.e.,�4p1�5 ≡

∑�

n=0 �nPn/�) and expected queue length(i.e., q4p1�5 ≡

∑�

n=0 nPn) have an intricate relation-ship with the level of bounded rationality �. In par-ticular, neither of them is monotonic or unimodalas a function of the level of bounded rationality �.To understand why this happens, we first define ns =

644R− p5�5/C7 as the threshold queue length used byfully rational customers in deciding to join the queueor not. We then divide the states of the system intotwo regions using the threshold ns : Region 1 com-prises the states when the number of customers inthe system is less than ns ; Region 2 comprises all theother states, i.e., those when the number of customers

in the system is greater than ns . When customers arefully rational, customers will join with probability 1in Region 1 and 0 in Region 2. For every strictly posi-tive �, the joining probability will be between 0 and 1.Hence, bounded rationality in Region 1 lowers uti-lization (and expected queue length), and it increasesutilization (and expected queue length) in Region 2.As customers become more boundedly rational, thesetwo effects take place simultaneously. It turns out thatit is not clear which effect dominates.

2.2. The Invisible QueueWe now turn to an invisible queueing system usingthe same model setup as §2.1. The only difference isthat the queue length is invisible to customers. Poten-tial customers arrive to this system according to aPoisson process with rate �. Because customers can-not observe the state of the system, they have tomake a decision a priori whether to arrive to thequeue or not. Different from the visible-queue setting,each customer has to form beliefs about the state ofthe system, which is determined by other customers’strategies. We assume that each customer knows allthe underlying parameters of the system such as R,C, p, �, and �. He knows that he is boundedly ratio-nal, and all the other customers are also boundedlyrational. In other words, he is able to form the correctbelief about the state of the system, which is used toestimate his expected waiting time.

In investigating the system, we are initially inter-ested in the fraction of customers that join the system�4p1�I 5 ∈ 60117 in equilibrium. Again, customers donot choose to play mixed strategies. Only from thepoint of view of an outside observer are customerdecisions probabilistic. A customer’s net benefit orutility of joining is U1 = R − p − C Ɛw = R − p −

C/4�−�4p1�I 5�5+, where we used the fact that the

thinning of a Poisson process with arrival rate �is still a Poisson process with rate �4p1�I 5� andC/4�−�4p1�I 5�5

+ is the customer’s expected wait-ing cost. The arrival rate �4p1�I 5� of the queue willbe referred to as effective demand. According to ourmodel of bounded rationality in §2, each customercannot perfectly estimate his expected waiting timeand hence joins the system with probability �I ≡

eU1/�I /41 + eU1/�I 5. In equilibrium, consistency requiresthat the system effective arrival rate is consistent withcustomer behavior: �4p1�I 5 = �I . Hence, we definethe equilibrium of the invisible queueing system asfollows.

Definition 1 (Equilibrium Joining Fraction).We say that �4p1�I 5 is an equilibrium joining fractionif it satisfies the following:

�4p1�I 5=e4R−p−C/4�−�4p1�I 5�5

+5/�I

1 + e4R−p−C/4�−�4p1�I 5�5+5/�I

1 (2)

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for �I > 0, and

�4p105= min8�01191

where �0 satisfies

R− p−C

�−�0�= 01 (3)

for �I = 0.

When �I > 0, Equation (2) yields a fixed-point prob-lem given that the logit expression in the right-handside includes the equilibrium joining fraction (i.e.,the left-hand side).

When �I = 0, i.e., customers are fully rational, thenthe definition is precisely Hassin’s (1986) equilibriumcondition (Equation (4.1), p. 1189). It is possible thatthere is no �0 ∈ 60117 satisfying Equation (3), and theactual arrival rate then is � because even if all cus-tomers decide to join, each customer’s expected utilityis still strictly positive. According to this definition,we have �4p105= min8�/�−C/4�4R− p55119.

The assumption that customers are boundedlyrational in their strategies but not in beliefs is con-sistent with the economics literature of modelingbounded rationality (see Chen et al. 1997 and refer-ences therein). This approach is certainly restrictivebecause decision makers are capable of correctly cal-culating the expected error-prone actions of the otherplayers, which are certainly nontrivial cognitive tasks(Mallard 2011). But McKelvey and Palfrey (1995) andChen et al. (1997) show that such quantal responseequilibria emerge from learning. Hence, in the short-run or transient states, customers may neglect oth-ers’ bounded rationality, but eventually they wouldbe able to form the correct belief about others’ strate-gies so that the fixed-point outcome according toDefinition 1 would emerge. We relax Definition 1 inAppendix D in our technical report (Huang et al.2012) by allowing customers to have incorrect beliefs.There, the model is closely related to the “level-kthinking” (Stahl and Wilson 1995): the closed-loop,fixed-point type of Equation (2) would become open-loop. The resulting analysis for both revenue andsocial welfare maximization is straightforward. Werefer the reader to Appendix D of Huang et al. (2012)for detailed discussion. In §2 of the online supple-ment, we prove that for a given range of �I , the pathof customer joining decisions over time, when the cus-tomers adaptively learn the expected waiting time,converges to the equilibrium in Definition 1. Also,focusing on bounded rationality as incapable of accu-rately predicting the expected waiting time allowsa fair comparison of the visible queue versus invi-sible queue.

Next, we investigate whether an equilibrium al-ways exists. Proposition 2 shows that there alwaysexists a unique equilibrium.

Proposition 2. There always exists a unique equilib-rium for the invisible queue for any finite price p and levelof bounded rationality �I > 0.

We are now interested in how the (unique) equilib-rium �4p1�I 5 behaves as a function of the price p andthe level of bounded rationality �I . For convenience,we let p̄ ≡ R − 42C5/42�−�5 denote the price underwhich each customer receives exactly zero utility sothat the equilibrium joining fraction is half regardless ofthe level of bounded rationality. The following propo-sition characterizes the equilibrium joining fraction.

Proposition 3. (i) If p < p̄, equilibrium joining frac-tion �4p1�I 5 is strictly decreasing in �I .

(ii) If p > p̄, equilibrium joining fraction �4p1�I 5 isstrictly increasing in �I .

(iii) If p = p̄, equilibrium joining fraction �4p1�I 5 =

1/2 for any �I .(iv) For any fixed �I , equilibrium joining fraction

�4p1�I 5 is strictly decreasing in p.

We offer the following intuition: when the priceis so low that each customer receives strictly posi-tive utility, the initial joining fraction is above half.As the level of bounded rationality increases, betterdecisions are made less often, and thus the joiningfraction decreases as customers are more boundedlyrational. Interestingly, if the price is set so that eachcustomer receives exactly zero utility in equilibrium,then increasing the level of bounded rationality hasno effect on the joining fraction because customersjoin or balk with equal fractions regardless of the levelof bounded rationality.

It is intuitively clear that �4p1�I 5 is strictly decreas-ing in price p by equality (2); i.e., a larger price alwaysresults in a lower joining fraction, regardless of thelevel of bounded rationality, which is the “law ofdemand” in this service setting.

It is useful to note that the invisible queuewith boundedly rational customers is essentially anM/M/1 system with arrival rate �4p1�I 5� and servicerate �. The server utilization is �I 4p1�I 5 ≡ ��4p1�I 5.Thus the utilization behaves the same as the join-ing fraction as a function of p and �I (which isalready characterized in Proposition 3). Using similarlogic, we can characterize the expected queue lengthqI 4p1�I 5≡ �4p1�I 5�/4�−�4p1�I 5�5 as a function of pand �I .

3. Revenue MaximizationIn the previous section, our attention was focused onthe system equilibrium or dynamics. In this section,we focus our attention on the revenue generated fromsuch systems. In this sense, we are looking from arevenue-maximizing firm’s perspective.

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3.1. The Visible QueueThe revenue as a function of price p and level ofbounded rationality �> 0 is

ç4p1�5≡

�∑

n=0

�nPnp =

�∑

n=0

e4R−p−44n+15C5/�5/�

1 + e4R−p−44n+15C5/�5/��Pnp0

Note that we normalize the cost of serving customersto zero without loss of generality.

When customers are fully rational, i.e., � = 0, wenaturally define ç4p105≡ lim�→0 ç4p1�5 for any pricep (one can show that such a limit exists). In the settingwith fully rational customers, Naor (1969) shows thatchoosing the revenue-maximizing price boils down tochoosing the optimal integer n to maximize the rev-enue function çn = �441 −�n5/41 −�n+1554R − Cn/�5so that p4n5=R−Cn/�. Let nr be the maximizer andçnr

be the maximized revenue.We are interested in comparing the optimal revenue

ç4p∗4�51�5 when the revenue-maximizing price p∗4�5is set, if customers are slightly boundedly rational,and the optimal revenue çnr

≡ supp lim�→0 ç4p1�5 ifcustomers are fully rational. For convenience, let p∗ ≡

p∗405 = R − Cnr/� be the revenue-maximizing priceunder full rationality.

Proposition 4. p∗4�5 < p∗ and ç4p∗4�51�5 < çnrwhen � is strictly positive but sufficiently small.

Proposition 4 does not extend to situations whenthe level of bounded rationality becomes high.(As customers become fully boundedly rational, theoptimal revenue goes to infinity.) The intuition behindProposition 4 is as follows: a strictly positive andsmall � forces the revenue-maximizing firm to strictlylower its price compared to full rationality, whichin turn brings strictly lower revenue for the firm.In other words, a strictly positive and small � makesrevenue collecting less profitable. The reason is that astrictly positive and small � strictly reduces the effec-tive demand of the system, ceteris paribus. A strictlylower price is necessary to increase the effectivedemand to maximize revenue.

From our numerical studies, we find that theoptimal revenue is not necessarily monotone withrespect to the level of bounded rationality, and therevenue-maximizing price as a function of the levelof bounded rationality can increase or decrease. Theexplanation is similar to why the utilization is not nec-essarily monotonic in the level of bounded rationalitybecause the revenue directly depends on the utiliza-tion: ç4p1�5= �4p1�5�p.Impact of Ignoring Bounded Rationality. Without tak-

ing into account customer bounded rationality, therevenue-maximizing firm will rationally charge pricep∗405 = R − Cnr/�. We are interested in the revenueloss as a result of bounded rationality and carried out

a numerical study. We approximate the steady-statedistribution by truncating the birth–death process to afinite state space, gradually increasing the number ofstates until the revenue function is no longer sensitiveto the truncation level. To investigate the impact ofthe level of bounded rationality �, the reward-to-costratio R/C, and the traffic intensity �/� on the revenueloss, we normalize C = 1 and �= 1 and vary R and �.Our systematic numerical study is intended to includehigh/low utilization crossed with high/low reward.Figure 1 shows a representative example where R ∈

821209 and � ∈ 8005159. From Figure 1, we have thefollowing observations: (i) The revenue loss can besignificant (more than 200%, for instance), depend-ing on the parameters, and can be arbitrarily largeas � goes to infinity. (ii) The revenue loss is not nec-essarily monotone with respect to �. This is likelybecause the revenue-maximizing price is not neces-sarily monotone in �. To understand this fact, recallthat neither the utilization nor expected queue lengthare necessarily monotone, and our explanation in §2.1applies here given that these basic measures drive therevenue.

3.2. The Invisible QueueThe firm’s objective is to choose a price p to maximizethe expected revenue çI 4p1�I 5 ≡ p�4p1�I 5�, where�4p1�I 5� is the effective demand rate to the system.

To investigate the firm’s revenue maximizationproblem, we first study the behavior of the revenueçI 4p1�I 5 as a function of price p and level of boundedrationality �I . Note that çI 4p1�I 5 is simply a lineartransformation of �4p1�I 5; hence, Proposition 3 char-acterizes how çI 4p1�I 5 behaves as a function of �I forany fixed p.

We next investigate how revenue çI 4p1�I 5 behavesas a function of price p for any fixed level of boundedrationality �I . To state Proposition 5, we denote �0 ≡

R/2 − 2C�/42�−�52, which is the level of boundedrationality at which the optimal price p∗4�05 = p̄.Hence, at the level of bounded rationality �0, eachcustomer receives zero expected utility of joiningunder the revenue-maximizing price.

Proposition 5. (i) For any fixed level of bounded ratio-nality �I , çI 4p1�I 5 is unimodal in p, and thus there existsa unique price p∗4�I 5 that maximizes çI 4p1�I 5.

(ii) The optimal price p∗4�I 5 is strictly increasing in thelevel of bounded rationality �I for �I ∈ 6max8�01091�5.

From this proposition, we obtain that the revenue-maximizing price p∗4�I 5 is monotonically increasingin �I ∈ 601�5 if R is sufficiently small.

When the optimal price induces each customer toreceive strictly negative expected utility in equilib-rium, a higher �I would induce the firm to increaseits price. The reason is that higher �I leads to higher

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Figure 1 Revenue Loss When Revealing the Queue if Bounded Rationality Is Ignored (C = 11 �= 1, Revenue Lossãç4�5≡ 4ç4p∗4�51 �5−ç4p∗4051 �55/ç4p∗4051 �5)

0 5 100

2

4

6

�

Rev

enue

loss

0 5 100

5

10

15

�

� �

Rev

enue

loss

0 5 100

0.02

0.04

0.06

0.08

0.10

Rev

enue

loss

0 5 100

0.2

0.4

0.6

0.8

Rev

enue

loss

R = 2� = 0.5

R = 20

� = 0.5

R = 2� = 5

R = 20

� = 5

joining fractions for a fixed price in this case, accord-ing to Proposition 3(ii). Hence, when the optimal priceis sufficiently high (so that each customer receivesstrictly negative expected utility in equilibrium), thenincreasing �I leads to even higher optimal prices.However, when the optimal price is low so that eachcustomer receives strictly positive utility, then increas-ing �I can lead to lower optimal prices, where thefirm’s trade-off is about the benefit of higher pricesversus the loss of lower effective demand. Proposi-tion 5 above partially characterizes which one of theseeffects dominates.

We are now ready to state the result on theeffect of the level of bounded rationality on the opti-mal revenue çI 4p∗4�I 51�I 5. Using the envelope theo-rem, we obtain the following immediate corollary toProposition 3.

Corollary 1. (i) If p∗4�05 > p̄, then

dçI 4p∗4�I 51�I 5/d�I

∣

∣

�I=�0> 00

(ii) If p∗4�05 < p̄, then dçI 4p∗4�I 51�I 5/d�I ��I=�0< 0.

(iii) If p∗4�05= p̄, then dçI 4p∗4�I 51�I 5/d�I ��I=�0= 0.

By Proposition 5 and Corollary 1, we know thatthe optimal revenue çI 4p∗4�I 51�I 5 strictly increases in�I as �I is sufficiently large. Therefore, the revenue-maximizing firm can exploit the bounded rationalitywhen �I is sufficiently large.

Finally, we are also interested in how the arrivalrate affects revenue because it would later be useful.Recall that in Hassin (1986) where customers are fullyrational, there exists some �0, when � > �0, the rev-enue function is independent of �. Interestingly, inour case with boundedly rational customers, we havethat a higher arrival rate � always leads to strictlyhigher revenue.

Proposition 6. For any fixed price p and level ofbounded rationality �I > 0, the equilibrium joining fractionis strictly decreasing and the revenue is strictly increasingin the arrival rate �.

The result that higher arrival rates lead to lowerequilibrium joining fractions is not surprising becausemore congestion forces each customer to lower hisjoining probability. However, the result that morearrivals always lead to more revenue may appear tobe surprising. The key insight is that the marginalrevenue increment has to be proportional to themarginal joining fraction decrement given the equi-librium condition (2). Hence, the effective demand�4p1�I 5� increases in �. Proposition 6 implies thatfor any price p, not necessarily the optimal price,higher arrival rates lead to higher revenue. In particu-lar, higher arrival rates lead to higher optimal revenue.Such a finding is in stark contrast to Hassin’s (1986)full rationality case.

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Impact of Ignoring Bounded Rationality. Finally,we are interested in the consequence of ignor-ing bounded rationality while customers areactually boundedly rational. Without taking intoaccount customer bounded rationality, the revenue-maximizing firm will rationally charge price p∗405,which is generally different from the revenue-maximizing price p∗4�I 5. Hence, çI 4p∗4051�I 5 ≡

p∗405�4p∗4051�I 5� ≤ çI 4p∗4�I 51�I 5. We are interestedin the revenue loss ãçI 4�I 5 ≡ 4çI 4p∗4�I 51�I 5 −

çI 4p∗4051�I 55/çI 4p∗4051�I 5 as a result of this igno-

rance of bounded rationality. As an example, we usethe same parameters as before. Figure 2 shows thatthe revenue loss can be nontrivial (e.g., more than200%). In general, the revenue loss is not necessarilymonotone with respect to �I . Similar to the visiblequeue, this observation could be driven by the factthat the revenue-maximizing price is not necessarilymonotone in �I (Proposition 5). Hence, when �I

increases, the revenue-maximizing price p∗4�I 5 andp∗405 may become closer, which would result inlower revenue loss; the case when R= 20, � = 005,and �I = 5 in Figure 2 illustrates this point. However,because �I is larger than a certain threshold, the lossis significant. In fact, lim�I→� ãçI 4�I 5 = �; i.e., therevenue loss can be arbitrarily large as �I is suffi-ciently large. This directly follows from the fact thatlim�I→� p∗4�I 5 = � (see the first-order condition inthe proof of Proposition 5) and lim�I→� �4p1�I 5= 005.

Figure 2 Revenue Loss When Hiding the Queue if Bounded Rationality Is Ignored (C = 11 �= 1, Revenue LossãçI 4�I 5≡ 4çI 4p∗4�I 51 �I 5−çI 4p∗4051 �I 55/ç

I 4p∗4051 �I 5)

0 5 100

2

4

6

8

10

�I �I

�I�I

Rev

enue

loss

0 5 1059.4

59.5

59.6

59.7

59.8

Rev

enue

loss

0 5 100

0.05

0.10

0.15

0.20

Rev

enue

loss

0 5 102.230

2.235

2.240

2.245

2.250

2.255

Rev

enue

loss

R = 20

� = 5

R = 2

� = 5

R = 2

� = 0.5

R = 20

� = 0.5

4. Social Welfare MaximizationWe now turn to study the problem from a social plan-ner’s perspective. The social planner is interested inmaximizing social welfare. In this section, we studythe impact of bounded rationality on the socialwelfare, both when the price is exogenously givenand when the social planner charges the welfare-maximizing price.

4.1. The Visible QueueIn many settings, the price is set or influenced byother considerations such as market conditions, com-petition, or a price being set by a third party. Thereare settings where optimizing over the price or evencharging a price may not be feasible—for example,toll charges on public roads, food distribution in nat-ural disasters, etc. We first study how bounded ratio-nality affects social welfare for a given price. Observethat the fixed price p always appears as R−p in Equa-tion (1). For the ease of comparing with the results inthe classic paper by Naor (1969) and for mathemati-cal convenience, we assume p = 0. However, the find-ings extend to the setting where the price is nonzero.We can derive the social welfare function as follows:

W4�5≡W4p1�5�p=0 =

�∑

n=0

�nPnR−

�∑

n=0

nPnC0 (4)

The first term in Equation (4) is the (long-run) averagereward, and the second term is the average waiting

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cost. Notice that if customers are fully rational, i.e.,� = 0, then Pn = 1 for n = 0111 0 0 0 1 6R�/C7 − 1 andPn = 0 for n > 6R�/C7 − 1, in which case our modelreduces to Naor’s (1969) model.

To compare social welfare W4�5 with W405, we firstrecall ns = 6R�/C7 as the threshold queue length usedby fully rational customers in deciding to join thequeue or not and n0 as the equivalent threshold from asocial planner’s point of view. Naor (1969) shows thatns ≥ n0; i.e., self-interested customers typically makethe system more congested than the socially optimallevel.

Intuitively, bounded rationality can create twoeffects for the social welfare: a positive (i.e.,welfare-improving) effect and negative (i.e., welfare-diminishing) effect. To understand how these twoeffects come into play, we divide the states of thesystem into three regions using the two thresholdsns and n0: Region 1 comprises the states when thenumber of customers in the system is less than n0;Region 2 comprises the states when the number of cus-tomers in the system is greater than n0 but less than ns ;and Region 3 comprises all the other states (i.e., thosewhen the number of customers in the system is greaterthan ns). When customers are fully rational, customerswill join with probability 1 in Regions 1 and 2 and 0 inRegion 3. Recall that to maximize social welfare, cus-tomers should join with probability 1 in Region 1 and0 in Regions 2 and 3. However, for any strictly posi-tive level of bounded rationality, the joining fraction isbetween 0 and 1. Hence, social welfare will decreasein Regions 1 and 3 but increase in Region 2 comparedto full rationality. As customers become more bound-edly rational, these effects take place simultaneously,and it seems unclear a priori which effect woulddominate.

Although Equation (4) presents a complete charac-terization of the social welfare in terms of the levelof bounded rationality �, the dependence is quiteintricate. Thus, we begin by analyzing the social wel-fare W4�5 in the neighborhood of zero. We are inter-ested in the relationship between W4�5 and W405when � is sufficiently small. It turns out we are ableto completely characterize the conditions in whichone effect dominates the other. We have the follow-ing simple inequalities showing when the social wel-fare increases or decreases as the customers becomeslightly boundedly rational.

Proposition 7. If any one of the following three con-ditions is satisfied,

(1) ns <R�/C − 1/2,(2) ns = n0, and(3) ns =R�/C − 1/2 and �> 1,

then W4�5 <W405 when �> 0 is sufficiently small. Oth-erwise, W4�5 >W405 when �> 0 is sufficiently small.

According to Proposition 7, if either of the follow-ing two conditions is satisfied,

(a) ns >n0, and ns >R�/C − 1/2, and(b) ns >n0, ns =R�/C − 1/2 and �≤ 1,

then a strictly positive and small � strictly improvesthe social welfare. Finding a simple sufficient andnecessary condition for ns > n0 is difficult. However,Lemma EC.5 in the appendix of Huang et al. (2012)shows that either of the following two conditions issufficient for ns > n0: (1) � > 1 and ns > 1 and (2)√

2 − 1 <�< 1 and ns > 2.In other words, compared with fully rational cus-

tomers, boundedly rational customers err on bothsides, joining a more congested system and balk-ing when congestion is low. Although the former isdetrimental to the social welfare, the latter can bebeneficial. The social welfare can thus be improveddepending on which of these effects dominates.In Proposition 7, we provide a simple characteri-zation of this dichotomy. This result appears to bestriking: although bounded rationality is usually asso-ciated with suboptimal decisions, it might yield bet-ter outcomes for the society overall. This is due tothe externality present among the boundedly rationalcustomers in the system.

As we discussed before, characterizing the socialwelfare as a function of the level of bounded ratio-nality is difficult because of the intricate joint effectscoming from the three regions simultaneously as cus-tomers become more boundedly rational. To under-stand the social welfare as a function of the levelof bounded rationality, we carry out a numericalstudy. To demonstrate that the social welfare func-tion is not necessarily unimodal in a reasonable rangeof bounded rationality, we intentionally use differentsets of parameters. In the first example, the param-eters are R = 140931C = 71� = 31 and � = 5, so thatns = 6 > R�/C − 1/2 = 508986. As shown in thegraph in the upper panel of Figure 3, the social wel-fare strictly increases initially as predicted by Propo-sition 7; however, it decreases and then increasesagain when the customers become more bound-edly rational. In the second example, the parametersare R = 161C = 71� = 31 and � = 206, so that vs =

608571 and ns = 6 < R�/C − 1/2 = 603571. As illus-trated in the graph in the lower panel of Figure 3,the social welfare initially decreases as predicted byProposition 7; however, it increases as the level ofbounded rationality becomes larger and decreasesagain as the level of bounded rationality furtherincreases. Thus, even though the social welfare is wellbehaved for a strictly positive and small �, it does notpossess global properties such as convexity/concavityor even unimodality. This result stands in contrast tothe invisible queue where the social welfare functionis unimodal in the level of bounded rationality.

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Figure 3 Global Behavior of Social Welfare: Example 1(R = 14093, C = 71 �= 31 �= 51 ns = 6> 4R�5/C −

1/2= 508986) and Example 2 (R = 16, C = 7, �= 3,�= 206, vs = 608571, ns = 6< 4R�5/C − 1/2= 603571)

0 1 2 3 4 5 6 7 8 9 1010.4

10.6

10.8

11.0

11.2

11.4

Level of bounded rationality �

Example 2

Example 1

Soci

al w

elfa

re W

(�)

0 1 2 3 4 5 6 7 8 9 1020.0

20.5

21.0

21.5

Level of bounded rationality �

Soci

al w

elfa

re W

(�)

We have analyzed the impact of bounded ratio-nality on social welfare for a given price. However,the social planner may be able to freely charge aprice to maximize the social welfare. We now inves-tigate the implication of bounded rationality for thesocial welfare if the social planner can regulate thesystem by pricing optimally. We are interested inwhether bounded rationality increases or decreasesthe social welfare. We denote the social welfare func-tion W4p1�5 when the social planner charges pricep and the customers’ level of bounded rationalityis �. Obviously, the social welfare W4p1�5 can beexpressed in a similar fashion as Equation (4).

Naor (1969) shows that by levying tolls the socialplanner can achieve the social optimum when cus-tomers are fully rational. In particular, if any price p∗ ∈

4R−C4n0 + 15/�1R−Cn0/�7 is charged by the socialplanner, then the maximum social welfare W ∗405 ≡

suppW4p105 can be achieved. We study whether theoptimal social welfare W ∗405 can be achieved byadding bounded rationality on the part of customers.

We show that when facing boundedly rational cus-tomers, the first-best social welfare can never beachieved, as stated in the following proposition.

Proposition 8. For any price p ∈ � charged to cus-tomers, the social welfare W4p1�5 is strictly lower than thesocial optimum when � is strictly positive; i.e., W4p1�5 <W ∗405 for �> 0.

This proposition proves that bounded rational-ity always results in social welfare losses comparedwith the full rationality case. This is in contrast to(a) Naor (1969), where levying tolls achieves the

socially optimal welfare; (b) the result in Proposi-tion 7 that a strictly positive and small � can increasethe social welfare when an arbitrary price is charged(when the firm charges the optimal price, then onlycase (2) in Proposition 7 arises); and (c) the resultin Proposition 10 of the invisible queue where theremay not be any welfare loss due to bounded rational-ity. This stems from the following: when customersare fully rational, the social planner can always regu-late the service system by charging prices to achievethe social optimality W ∗405. However, each bound-edly rational customer randomizes with nondegen-erate probabilities to join or balk. In this case, thesocial planner loses the precise control over the cus-tomers’ joining decisions, and thus bounded rational-ity dilutes the effectiveness of the price regulation.Of course, if the firm can implement state-dependentpricing, then the social planner can achieve the first-best, even in the presence of bounded rationality.

From our numerical studies, we find that there canbe multiple prices that maximize the welfare for agiven level of bounded rationality. Second, the opti-mal welfare is not necessarily monotone with respectto the level of bounded rationality. The explanationfor the nonmonotonicity behavior is similar to thatfor the global nonmonotonicity behavior of the socialwelfare function with respect to �: bounded rational-ity has the positive and negative effect on welfaresimultaneously.

Impact of Ignoring Bounded Rationality. Without tak-ing into account customer bounded rationality, thesocial planner will pick one price in the range 4R −

C4n0 + 15/�1R−Cn0/�7. We are interested in the wel-fare loss that results from bounded rationality. Usingthe same example as in the revenue maximizationcase, Figure 4 shows the welfare loss when the firmcharges price R−Cn0/�. Again, we observe nontrivialwelfare loss (more than 60% for instance). The intu-ition from the nonmonotonicity behavior is similar tothe explanation we provided for the global nonmono-tonicity behavior of the social welfare function whenp = 0: as � increases, the positive and negative effecton welfare occurs simultaneously.

4.2. The Invisible QueueFor any price p and level of bounded rationality �I ,the social welfare function is denoted as

W I 4�4p1�I 55≡ �4p1�I 5�R−�4p1�I 5�

�−�4p1�I 5�C0 (5)

For mathematical convenience, we may drop thedependence over �4p1�I 5 and write W I 4p1�I 5.

The first term of Equation (5) is the average ben-efit the customers receive from the system, and thesecond term is the average waiting cost incurred bythe customers. Note that price p affects social welfare

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Figure 4 Welfare Loss When Revealing the Queue if Bounded Rationality Is Ignored (C = 11 �= 1, Welfare Loss ãW4�5≡ 4W 4max801 p∗

w 4�591 �5−

W4p∗4051 �55/4W 4p∗4051 �5)

0 5 100

0.2

0.4

0.6

0.8

�

Wel

fare

loss

0 5 100

2

4

6

8

�

� �

Wel

fare

loss

0 5 100

0.05

0.10

0.15

0.20

0.25

Wel

fare

loss

0 5 100

0.5

1.0

1.5

2.0

Wel

fare

loss

R = 2

� = 0.5

R = 2

� = 5

R = 20

� = 5

R = 20

� = 0.5

only indirectly through the equilibrium joining frac-tion �4p1�I 5.

First, observe that the social welfare W I 4�4p1�I 55is strictly concave in �4p1�I 5 (Lemma EC.2 in theappendix of Huang et al. 2012). Combining this factwith the characterization of the equilibrium joiningfraction �4p1�I 5, we can characterize how the socialwelfare behaves as a function of the level of boundedrationality in Proposition 9.

From Proposition 5, one can obtain that p∗405 =

R41 −√

C/4�R55. Note that when customers arefully rational, the welfare-maximizing price and therevenue-maximizing price coincide.

Proposition 9. (i) If p = p̄, then social welfareW I 4�4p1�I 55 is constant for �I ≥ 0.

(ii) If 6p∗405− p̄76p− p̄7≤ 0 and p 6= p̄, then social wel-fare W I 4�4p1�I 55 strictly increases for �I ≥ 0.

(iii) If p ∈ 4min8p∗4051 p̄91max8p∗4051 p̄95 ∪ 8p∗4059,and p∗405 6= p̄, then social welfare W I 4�4p1�I 55 strictlydecreases for �I ≥ 0.

(iv) If 6p∗405 − p̄76p − p∗4057 > 0, then social welfareW I 4�4p1�I 55 strictly increases in 601�w4p57 and strictlydecreases in 4�w4p51�5, where

�w4p5=R− p−

√

RC/�

ln 4�−√

C�/R5/4�−�+√

C�/R50

This proposition fully characterizes the social wel-fare as a function of the level of bounded rationality.

Figure 5 depicts the different scenarios in Proposi-tion 9 based on the relative magnitude of p∗405 and p̄.By comparing the magnitude of p∗405 and p̄, we dis-cuss two cases. For each case, we depict the rangeof price p that falls into scenarios (i)–(iv) in Proposi-tion 9. (The case when p∗405 = p̄ is simple and hencenot illustrated in the figure.) For the first scenario, thejoining fraction at price p = p̄ is precisely half, and it isindependent of the level of bounded rationality. Thusthe social welfare in (i) is not impacted by the levelof bounded rationality. For the second scenario, thefraction 0.5 lies between the joining fraction inducedby the welfare-maximizing price when customers are

Figure 5 Illustration of the Various Scenarios in Proposition 9

2� – � �

2Cp = R–

2� – �2Cp = R–

RCp*(0) = R–

�RCp*(0) = R–

Case I: p*(0) > p

Case II: p*(0) < p

(i) (iii)

(iii)

(iv)

(iv)

(ii)

(ii)(i)

p

p

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fully rational and the joining fraction induced byprice p when customers’ level of bounded rational-ity is �I . In this case, increasing the level of boundedrationality makes their “distance” smaller because theequilibrium joining fraction becomes closer to 0.5 as�I increases, based on Proposition 3, scenarios (ii)and (iii). Thus, the social welfare strictly increases ascustomers are more boundedly rational. For the thirdscenario, the joining fraction induced by the welfare-maximizing price when customers are fully rationalis either too high or too low compared with the join-ing probability induced by price p when customers’level of bounded rationality is �I , so that increasingthe level of bounded rationality can only make their“distance” further apart. Therefore, the social welfarestrictly decreases in the level of bounded rational-ity �I . For the last scenario, the joining probabil-ity induced by the welfare-maximizing price whencustomers are fully rational can be achieved (in theinterior). Hence, as the level of bounded rationalityincreases from zero, the social welfare is “closer” tothe optimal social welfare. In this case, the social wel-fare function is unimodal in the level of boundedrationality, and the first-order condition yields thelevel of bounded rationality �w4p5.

Proposition 9 implies that the social welfare func-tion is unimodal in the level of bounded rationality,as stated in Corollary 2.

Corollary 2. Social welfare W I 4�4p1�I 55 is unimodalin the level of bounded rationality �I for any price p.

We have demonstrated that the impact of boundedrationality on social welfare depends on the magni-tude of the fixed price charged and that the welfarefunction is unimodal in the level of bounded rational-ity. The next question we are interested in is, what isthe welfare-maximizing price and how does the wel-fare behave under such a price? We first prove that thesocial welfare W I 4�4p1�I 55 is unimodal in price p forany level of bounded rationality �I (see Lemma EC.4in the appendix of Huang et al. 2012 for a rigorousjustification). Finding the welfare-maximizing priceboils down to finding the optimal joining probability�∗w. To derive the welfare-maximizing price, we use

the first-order condition

¡W I 4�4p1�I 55/¡�4p1�I 5= 0

and obtain

�∗

w =�−

√

C�/R

�1

which is the optimal equilibrium joining fractionthat induces the optimal social welfare. Suppose thisequilibrium point can be achieved in the interior;then it is required that R ∈ 4C/�1�5 if � < � andR ∈ 4C/�1C�/4�−�525 if � > �. For cases when the

equilibrium point is on the boundary, the problembecomes trivial: if R ≤ C/�, then it is socially opti-mal to keep everybody out of the system; if R ≥

C�/4�−�52 when �>�, then it is socially optimal tolet everyone join the system.

We are now ready to state the welfare-maximizingprice that maximizes the social welfare. To state theresult, we first substitute the joining fraction �∗

w intothe equilibrium condition, i.e., Equation (1), and thenwe obtain the “unconstrained” optimal price by

p∗

w4�I 5 = R−

√

CR

�−�I ln

�−√

C�/R

�−�+√

C�/R

= p∗405−�I ln�∗w

1 −�∗w

1

which can be negative. The welfare-maximizing priceis thus max801 p∗

w4�I 59. Proposition 10 characterizesthis welfare-maximizing price and the correspondingsocial welfare.

Proposition 10. (i) If R>4C�/42�−�52 when �I <�w405, where �w405= 4R−

√

RC/�5/ln64�−√

C�/R5/4�−�+

√

C�/R57, the price p = p∗w4�I 5 is the unique price

that maximizes the social welfare, p∗w4�I 5 strictly decreases

in �I , and the optimal social welfare is W I 4p∗w1�I 5 =

�R + C − 2√

�RC; when �I ≥ �w405, the price p = 0 isthe unique price that yields the maximum social welfareW I 401�I 5.

(ii) If R ≤ 4C�/42�−�52, the price p = p∗w4�I 5 is the

unique price that maximizes the social welfare, p∗w4�I 5

strictly increases in �I , and the optimal social welfare isW I 4p∗

w1�I 5=�R+C − 2√

�RC.

We discuss the implications of Proposition 10as follows. If �∗

w = 4�−√

C�/R5/� > 1/2, thenln4�−

√

C�/R5/4�−�+√

C�/R5 > 0, which impliesthat the price p∗

w4�I 5 is strictly decreasing in level ofbounded rationality �I . This scenario corresponds toCase II depicted in Figure 5, where the equilibriumjoining fraction decreases in �I . In particular, whencustomers are slightly boundedly rational, the opti-mal price strictly decreases. The intuition is that theequilibrium joining fraction decreases as the level ofbounded rationality increases. To achieve the desiredoptimal joining fraction �∗

w, the social planner has tolower the price as the level of bounded rationalityincreases.

Similarly, if �∗w = 4�−

√

C�/R5/� < 1/2, thenln4�−

√

C�/R5/4�−�+√

C�/R5 < 0, which impliesthat the price p∗

w4�I 5 is strictly increasing in level ofbounded rationality �I . This scenario corresponds toCase I depicted in Figure 5, where the equilibriumjoining fraction increases in �I .

The key insight from this proposition is that thefirst-best social welfare (which is independent of the

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Figure 6 Welfare Loss When Hiding the Queue If Bounded Rationality Is Ignored (C = 11 �= 1)

0 5 100

0.005

0.010

0.015

0.020

0.025

�I

Wel

fare

loss

0 5 100.975

0.980

0.985

0.990

�I

�I �I

Wel

fare

loss

0 5 100

0.2

0.4

0.6

0.8

Wel

fare

loss

0 5 100.80

0.82

0.84

0.86

0.88

0.90

Wel

fare

loss

R = 2

� = 0.5 R = 2

� = 5

R = 20

� = 5R = 20

� = 0.5

level of bounded rationality and the arrival rate) canbe achieved when either (i) the optimal joining frac-tion for social welfare maximization is strictly abovehalf and the level of bounded rationality is not toohigh or (ii) the optimal joining fraction for social wel-fare maximization is below half.

The intuition for this key insight is the following:in these settings, the social planner can always cor-rect for the boundedly rationality on the part of cus-tomers; i.e., the social planner still achieves the sameoptimal social welfare by charging appropriate prices.However, when the optimal joining fraction for socialwelfare maximization is strictly above half and thelevel of bounded rationality is sufficiently high, thefirst-best social welfare cannot be achieved. In otherwords, when the desired joining fraction is high, toachieve this, the customers have to join with thisfraction in equilibrium. However, the customers’ join-ing fraction would be much lower if they are tooboundedly rational and too low even if the firm doesnot charge any price. In this case, higher boundedrationality leads to more social welfare losses. Thisresult stands in contrast to (i) the case of revenue max-imization and (ii) the result in Proposition 8 of thevisible queue.Impact of Ignoring Bounded Rationality. Without tak-

ing into account customer bounded rationality, thesocial planner will rationally charge price p∗405, whichis generally different from the welfare-maximizing

price max801 p∗w4�I 59. Similarly, we are interested in

the welfare loss

ãW I 4�I 5 ≡ 4W I 4max801 p∗

w4�I 591�I 5

−W I 4p∗4051�I 55/WI 4p∗4051�I 50

For the same example as the revenue maximizationcase, Figure 6 shows that the welfare loss can besignificant (more than 80%, for instance) and is notnecessarily monotone with respect to the level ofbounded rationality. The nonmonotonicity behaviorand its intuitive explanation stems from Proposition9, where for a fixed price, the social welfare may benonmonotone in �I .

5. DiscussionThe quantal choice paradigm in the behavioral eco-nomics literature posits that people are more likelyto select better choices than worse ones but do notnecessarily succeed in selecting the very best choice.In this paper, we adopted this framework to modelbounded rationality in service systems in the sensethat customers lack the capability to perfectly esti-mate their expected waiting time. We investigatedthe impact of bounded rationality on the revenueof a profit-maximizing firm, social welfare, and pric-ing for both invisible and visible queues. From thefirm’s perspective, higher �I can lead to lower optimalprices, but it leads to higher optimal prices and higher

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revenue when �I is sufficiently large. With the opti-mal price, a strictly positive and sufficiently small �results in revenue losses. From the social planner’sperspective, there may be strictly positive social wel-fare losses when �I is sufficiently large. For visiblequeues with a fixed price, we prove that a strictlypositive and sufficiently small � can lead to strictsocial welfare improvement, and we provide a simpleinequality under which this improvement happens.With the optimal prices, however, bounded rationalitydecreases social welfare. We demonstrate that ignor-ing bounded rationality may result in significant rev-enue and social welfare loss. Our study contributes tothe behavioral operations management literature bydemonstrating the impact of behavioral factors in ser-vice settings.

5.1. Experimental DesignWe hope our theoretical study helps develop a desir-able theoretical–empirical feedback loop in behav-ioral operations management. To capture boundedrationality on the customers’ part, in our model thecustomers have imperfect estimates of the expectedwaiting time. Loosely speaking, the standard devia-tion of the estimate of expected waiting time is usedas a proxy for the level of bounded rationality.

The first step for such a study will be to estimatethe level of bounded rationality � for visible andinvisible queues in carefully controlled experiments.This could be done by assigning the experimentalparameters to subjects and then observing their join-ing fractions. Standard maximum likelihood estima-tion would yield the estimate for �, and we can testwhether it is statistically significantly different fromzero. As we mentioned in §2, McKelvey and Palfrey(1995) actually follow this approach in bimatrix gamesettings and Bajari and Hortacsu (2005) in auction set-tings. Kremer and Debo (2012) have already carriedout experimental studies in service systems and foundthat the model of bounded rationality fits the experi-mental data very well.

There are a variety of interesting conjectures of cus-tomer behavior that could be tested or explored alongthe line of Kremer and Debo (2012). It is of inter-est to estimate � for different queue configurations.For example, Larson (1987) and Maister (1985) dis-cuss the impact of the queue environment on thecustomer waiting time perception. Specifically, theyconjecture that eliminating empty time significantlyreduces customer perception of the length of waitingtime and that explained waits are shorter than unex-plained waits. We conjecture that customers may havedifferent levels of bounded rationality for these dif-ferent queue environments. Some conjectures worthexploring are the following: Are people prone to esti-mation mistakes/errors (and hence bounded rational-ity) depending on the queue structure, for instance,

with one long queue versus many short queues? Aremore educated (or more knowledgeable) people lessboundedly rational? Answering these questions notonly deepens our understanding of customer behav-ior but also helps managers better operate servicesystems.

It is interesting yet challenging to empirically quan-tify the loss that the firm incurs when it disre-gards bounded rationality. Although the arrival rate�, service rate �, and price p are easy to estimate,the reward R, level of bounded rationality �, andcost C are a little more difficult to identify, given thepossible heterogeneity and interaction among theseparameters. These challenges can be overcome usinginstrumental analysis or other structural estimationmethods. Once these parameters are estimated, wecan compute the loss that the firm incurs.

Finally, recall that we have theoretically proventhat bounded rationality can improve social wel-fare. Searching for empirical evidence that in systemswhere bounded rationality exists indeed improves thesocial welfare would be highly valuable.

5.2. Managerial InsightsThere are several implications for how service sys-tems should be managed. First, the study demon-strates the importance of accounting for boundedrationality in pricing the service. In particular,as � increases above a certain threshold, the revenueand social welfare loss of not accounting for it is large.Second, there are settings where the system managercan reduce the ambiguity (or difficulty) associatedwith the process of estimating the waiting time andpotentially reduce the variance in the estimation. It isinteresting to note that because customers are self-interested, when it comes to welfare maximization(e.g., public systems such as the Department of MotorVehicles and traffic on a road network), reducing �to zero might not be the right step because boundedrationality can actually improve social welfare. How-ever, when the service provider can optimize the priceas well as reduce the level of bounded rationality sig-nificantly, then the system performance (both in termsof revenue and social welfare) can be dramaticallyimproved.

Electronic CompanionAn electronic companion to this paper is available aspart of the online version at http://dx.doi.org/10.1287/msom.1120.0417.

AcknowledgmentsThe authors thank the three anonymous referees, the asso-ciate editor, and editor Stephen Graves for many help-ful comments and suggestions that improved the paperenormously.

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manufacturing & service operations management rationality in service... · for visible queues...

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