alexandrov, lariviere - 2012 - are reservations recommended-annotated

MANUFACTURING & SERVICEOPERATIONS MANAGEMENT

Vol. 14, No. 2, Spring 2012, pp. 218–230ISSN 1523-4614 (print) � ISSN 1526-5498 (online) http://dx.doi.org/10.1287/msom.1110.0360

© 2012 INFORMS

Are Reservations Recommended?

Alexei AlexandrovSimon Graduate School of Business, University of Rochester, Rochester, New York 14627,

[email protected]

Martin A. LariviereKellogg School of Management, Northwestern University, Evanston, Illinois 60208,

[email protected]

We examine the role of reservations in capacity-constrained services with a focus on restaurants. Althoughcustomers value reservations, restaurants typically neither charge for them nor impose penalties for failing

to keep them. However, reservations impose costs on firms offering them. We offer a novel motivation foroffering reservations that emphasizes the way in which reservations can alter customer behavior. We focuson a market in which demand is uncertain and the firm has limited capacity. There is a positive chance thatthe firm will not have enough capacity to serve all potential customers. Customers are unable to observehow many potential diners are in the market before incurring a cost to request service. Hence, if reservationsare not offered, some may choose to stay home rather than risk being denied service. This lowers the firm’ssales when realized demand is low. Reservations increase sales on a slow night by guaranteeing reservationsholders service. However, some reservation holders may choose not to use their reservations resulting in no-shows. The firm must then trade off higher sales in a soft market with sales lost to no-shows on busy nights.We consequently evaluate various no-show mitigation strategies, all of which serve to make reservations morelikely in equilibrium. Competition also makes reservations more attractive; when there are many small firms inthe market, reservations are always offered.

Key words : service management; reservations; restaurants; capacity managementHistory : Received: July 10, 2008; accepted: September 9, 2011. Published online in Articles in Advance

February 28, 2012.

1. IntroductionRestaurant reservations are a curious phenomenon.Customers value them, but restaurants give themaway. Indeed, firms such as TableXchange and Today’sEpicure have tried to profit from the resulting arbi-trage opportunity by creating markets to sell reser-vations. What makes offering reservations even moreremarkable is that they are costly to provide. Fischer(2005) identifies three costs to offering reservations.These include additional staff needed to take reserva-tions and added complexity from having to balance theneeds of walk-in customers with commitments madeto reservation holders. Even if technology is substi-tuted for personnel, the restaurant must pay licensingand commission fees to services such as OpenTable.One New York restaurateur estimates his annual costsof offering reservations at $125,000 (Collins 2010).

Fischer’s final consideration is no-shows. Cus-tomers can generally fail to keep reservations withoutpenalty, but restaurants suffer if they hold capacityfor customers that never come. No-shows representa real problem. Bertsimas and Shioda (2003) reporta no-show rate of 3% to 15% for the restaurant theystudied. More generally, rates of 20% are not unusual(Webb Pressler 2003) and special occasions such asNew Year’s Eve can push rates to 40% (Martin 2001).

Why then should restaurants offer reservations?One reason is the operational benefits they provide.Reservations regulate the flow of work. By stagger-ing seatings, a restaurateur can assure that waiters arenot overwhelmed by a rush of customers followedby the bartender and kitchen being swamped withorders. Reservations thus allow fast service withoutexcessive capacity (Fischer 2005). Reservations wouldthen be appealing when either customers are delaysensitive or the firm’s costs increase with arrival vari-ability. Further, reservations may allow a restaurant toestimate demand and improve staffing and sourcingdecisions. This is particularly important when vari-able costs are high. Chicago-based Alinea is knownfor intricate, multicourse menus. It requires that cus-tomers make reservations. To the extent that Alineaincurs significant expense in preparing for its guests,knowing how many are coming on a given night iscrucial.

We abstract from these issues and focus on reser-vations’ ability to influence consumer behavior andthus increase sales. We consider a restaurant with lim-ited capacity whose sole decision is whether to offerreservations. (Our model also applies to other firmsthat serve both reservation and walk-in customersand whose revenues depend on choices customers

218

Alexandrov and Lariviere: Are Reservations Recommended?Manufacturing & Service Operations Management 14(2), pp. 218–230, © 2012 INFORMS 219

make after they arrive for service.) At the time reser-vations are made (assuming they are offered), cus-tomers are uncertain how they will value the serviceat the time of consumption. Customers learn their val-uations before traveling to the service facility. Becausethere is a fixed cost to accessing the facility, thosewith low realized values fail to keep their reserva-tions. If reservations are not offered, customers learntheir value for the service and decide whether to walkin for service. Walking in incurs a travel cost and arisk of an additional penalty if one cannot get a seat.

We assume the market size is uncertain and takesone of two values. The firm can serve all customerson a slow night, but it lacks the capacity to serveeveryone on a busy night. When reservations are notoffered, customers risk not getting a seat despite hav-ing incurred the cost to travel to the firm. This is costlyto the firm on slow nights because some customerswho would have had a positive net utility from din-ing out choose to stay home. Reservations counter-act this by guaranteeing all customers in the marketa seat on a slow night. However, reservations alsoimpose a penalty on the firm because some sales willbe lost to no-shows on busy nights. Hence, the restau-rant faces a trade-off as reservations are valuable ina weak market, but costly in a strong one. We deriveconditions under which this comparison favors offer-ing reservations. That is, reservations may indeed berecommended. Further, we explore various strategiesfor limiting the impact of no-shows. We also show thatcompetition can make offering reservations even moreattractive. In a market with many firms, it is unlikelythat none of the firms will offer reservations.

Section 2 reviews the literature. Section 3 presentsthe basic model, and §4 analyzes the trade-off betweenreservations and relying on walk-in traffic. Section 5evaluates strategies of mitigating no-shows. Section 6examines the impact of competition. Section 7 providesconcluding remarks and comments on possible futureavenues of research. Proofs are in the appendix.

2. Literature ReviewThe existing literature on reservations or advancesales emphasizes pricing or segmentation. In Png(1989), a firm with limited capacity sells to risk-aversecustomers who are uncertain of their need for a ser-vice at some future point. The firm offers reserva-tions and overbooks; bumped customers receive aspecified compensation. Reservations act as insuranceand allow for higher prices than either selling out-right in advance or selling in the spot market. Dana(1998) considers a competitive market with capaci-tated firms. Market segments differ in their willing-ness to pay and the certainty with which they needservice. If those with low valuations are more certain

to need the service, the firms sell some capacity earlyat a discount, but raise the price closer to the time ofservice delivery. In DeGraba (1995), advance sellingallows the firm to capture the same revenue as first-degree price discrimination. Xie and Shugan (2001)and Shugan and Xie (2005) develop similar ideas. (Forreviews, see Courty 2000, Shugan 2002.) None of thiswork addresses the restaurant industry. Restaurantsdo not charge in advance, and menu prices are inde-pendent of holding reservations. Also, they do notpublicize compensation schemes for reservation hold-ers denied service. Furthermore, most of these papersfocus on monopoly settings (Dana 1998 and Shuganand Xie 2005 are exceptions) while we extend ouranalysis to competitive environments.

Dana and Petruzzi (2001) study a newsvendormodel in which customers incur a cost to shop.The stocking level thus affects demand. They focuson determining the stocking level and do not con-sider altering the timing of sales. Deneckere and Peck(1995) examine similar issues in a competitive envi-ronment. We take the restaurant’s capacity as exoge-nous, but allow reservations. See Netessine and Tang(2009) for more on the role of strategic consumers inoperations management.

Appointment systems have been widely studied,particularly in healthcare. See, for example, Robinsonand Chen (2003) or Savin (2006). Gupta and Denton(2008) provide a survey, although this work gener-ally ignores strategic consumer actions. Lariviere andVan Mieghem (2004) consider delay-sensitive con-sumers choosing arrival times. An appointment sys-tem allows them to pick sequentially and results in anarrival pattern that minimizes both period-to-periodvariation in arrivals and total delay costs. Customersare not allowed to decline seeking service. We ignoredelay costs, but explore how reservations can increasesales by encouraging more customers to come in.

There have been some studies applying revenuemanagement to restaurants. Thompson (2010) pro-vides a review and notes that existing research doesnot clearly state under what conditions reservationsshould be offered. We show that reservations areadvantageous when demand is uncertain if the no-show rate is sufficiently low. Bertsimas and Shioda(2003) consider how many reservations a restaurantshould accept and how to accommodate walk-in cus-tomers given reservation commitments. Both demandstreams are exogenous. Here, offering reservationschanges the number of customers patronizing thefirm. Çil and Lariviere (2011) examine a model withtwo segments, one of which will only patronize thefirm if reservations are offered, while the other incurs acost to walk in. They examine how capacity should beallocated between the segments and show that it maybe optimal to save no capacity for late arriving walk-in

Alexandrov and Lariviere: Are Reservations Recommended?220 Manufacturing & Service Operations Management 14(2), pp. 218–230, © 2012 INFORMS

customers even if they are more valuable to the firm.We focus on a single segment who can be served eitherwith or without reservations and focus on the questionof whether reservations should be offered at all.

3. Model FundamentalsWe consider one restaurant serving customers in asingle sales period. Customers are a priori homoge-neous. The net utility from planning to stay homeis normalized to zero. Each customer incurs a cost,T ≥ 0, to travel to the restaurant whether she holdsa reservation. A walk-in customer who fails to get aseat incurs a denial cost, D ≥ 0, which represents theinconvenience of changing plans. A strictly positivedenial cost implies that the opportunities open to acustomer who has been turned away are not as attrac-tive as the customer’s outside option of not visitingthe restaurant. Note that while both T and D are costsincurred by customers, they are not paid to the firm.

While a priori homogeneous, customers are ex postdifferentiated by their value for dining out. Each cus-tomer draws a valuation, V , independently from aknown, continuous distribution, F 4V 5, with densityf 4V 5 and support 401ì5 for ì> T . F 4V 5 = 1 − F 4V 5.One should interpret V as the customer’s valuationnet of expected payment to the firm. This is withoutloss of generality as we take the firm’s pricing pol-icy as fixed. Customers learn their valuations beforeincurring the travel cost. Hence, only those whoserealized valuation V exceeds T will even considerpatronizing the restaurant. We will term V − T thecustomer’s net utility.

The number of customers in the market is uncertainand takes one of two levels, � or 1, with � > 101 Highdemand occurs with probability � for 0 <�< 1. Marketdemand uncertainty does not reflect variations overobservable conditions (e.g., Fridays are busier thanTuesdays), but variations given specific circumstances.One should interpret the model as saying that, giventhat it is Friday night the restaurant will be busy withprobability �0 Customers and the firm cannot learnthe market size until they interact. That is, the firmlearns it is a busy night from the number of customersthat request service, while customers can update theirbelief on whether it is a busy night based on whetherthey can get a reservation or receive service.

Customers are atomistic; they are sufficiently smallrelative to the market that the only aggregate uncer-tainty is the total number of customers in the market.Thus, while it is uncertain whether any specific con-sumer will have a positive net utility from dining out,

1 More generally, we could have demand as taking values �H and�L with �H > K > �L where K is the firm’s capacity. Setting �L = 1represents a normalization of demand and capacity to economizeon notation.

the number of customers interested in dining out iscertain to be �F 4T 5 on a busy night and F 4T 5 on aslow night.

The firm’s sole decision is whether to offer reserva-tions. Its menu and pricing are fixed, and the expectedspending is the same for all customers regardless oftheir realized value for dining out. The restaurant’sobjective is to maximize unit sales. The firm can serveK customers per evening where we assume that everycustomer requires the same amount of capacity. Thatis, we are not differentiating based on party size. Onecan interpret the model as assuming that all arrivalsto the restaurant are parties of the same size.

We assume �F 4T 5 > K > 10 On a slow night, therestaurant can serve everyone who would have a pos-itive net utility to dining out; it cannot do so on abusy night.

Events proceed as follows. The firm announces itspolicy. The number of customers in the market is real-ized, but is not observed by either the firm or indi-vidual customers. If the restaurant offers reservations,customers simultaneously decide whether to ask forone. Requesting customers learn immediately if theyare successful. Subsequently, each customer learns hervaluation V . Reservation holders then decide whetherto use them while nonholders simultaneously decidewhether to walk in. As customers arrive, reserva-tion holders are seated first. If the number of walk-in customers exceeds the available capacity, seatsare rationed randomly. Thus, if M reservations havebeen taken and � > K − M customers walk in, theprobability of any one walk-in customer being seatedis 4K −M5/�. This implicitly assumes the restaurantdoes not realize a given reservation holder is a no-show until it is too late to give her seat to a walk-incustomer. (We will discuss allowing the firm to reof-fer the seats of no-shows in §5.) If reservations are notoffered, customers wait to learn their valuations andthen decide whether to walk in. If the number of walk-ins exceeds capacity, seats are rationed randomly.

4. Analysis: Walk-Ins vs.Reservations

To determine whether reservations are recommendedwe need to contrast firm sales under two policies,offering reservations and relying solely on walk-intraffic. We examine the latter first.

4.1. The No Reservation CaseWhen reservations are not offered, a customer withrealized valuation V faces a lottery. By spending thetravel cost, T , she either receives a benefit of V orincurs a penalty of D0 The probability of getting aseat is thus critical to her decision. Let � denote heranticipated probability of getting a seat. Her expected


utility from walking in is �V − 41 − �5D − T , and shewill walk in if � ≥ 4T + D5/4V + D5. Since all cus-tomers have the same chance of getting a seat, anyonewith realized valuation V ′ ≥ V will also walk in if thecustomer with value V walks in. Thus, the equilib-rium between customers will be in the form of a cut-off strategy in which all customers whose valuationsexceed some value V ∗ ≥ T attempt to receive service.Because all customers are treated equally, the prob-ability of being seated will be given by the fractionof demand that is satisfied from the available inven-tory, i.e., by the fill rate (Cachon and Terwiesch 2009).See Dana and Petruzzi (2001) for a concise proof.Deneckere and Peck (1995) also provide a proof thatuses a customer’s ability to update her belief aboutthe size of the realized market from the fact that she isactively in the market.2 If only those with valuationsexceeding V ∗ walk in, the chance of getting a seat is

�4V ∗5=F 4V ∗541 −�5+K�

F 4V ∗541 −�+��50 (1)

Proposition 1. Suppose the restaurant does not offerreservations.

1. The unique equilibrium has all customer with valua-tions greater than V ∗ walking in to the restaurant, whereV ∗ is found from

F 4V ∗5=�K4V ∗ +D5

��4T +D5− 41 −�54V ∗ − T 50 (2)

The equilibrium probability that an arriving customer getsa seat is 4T +D5/4V ∗ +D5.

2. F 4V ∗5 > K/�1 and V ∗ is increasing in �.3. The firm’s sales are

41 −�5F 4V ∗5+�K0 (3)

In equilibrium, the marginal customer must beindifferent to walking in or staying home. Hence, itmust be the case that there is a positive chance of notbeing seated, i.e., we must have �F 4V ∗5 > K and therestaurant is oversubscribed on busy nights. At thesame time, some customers with a positive net valua-tion stay home so the number of customers walking inon a slow night is just F 4V ∗5. Thus, while the restau-rant fully uses its capacity on a busy night, it losessome potential sales on slow nights. Demand uncer-tainty is essential to this last point. As � goes to zeroand demand is certain to be low, V ∗ goes to T andthe only customers who stay home are those with anegative net utility for dining out.

2 More precisely, suppose there are ä ≥ � potential customers fromwhich nature chooses � or one to be active in the market on agiven night. If the market is small, a customer’s chance of beingselected is 1/ä0 If the market is large, her chance of being chosen is�/ä0 The customer’s ex post belief that the market is large is then�= ��/441 −�5+ ��5; �4V ∗5 then equals 1 − �− �4K/4�F 4V ∗555.

It is straightforward to show that V ∗ increases with�, T , and D, but falls in K. Further, consider two val-uation distributions, F and G, and let V ∗

F and V ∗G be

the corresponding cutoffs. If F 4V 5 ≥ G4V 5 for all V ,V ∗F ≤ V ∗

G. A higher cutoff value does not translate intoa smaller crowd. The chance a walk-in customer getsa seat falls as V ∗ increases (holding T constant and Dconstant). A better restaurant (in the sense of stochas-tically larger valuation distribution) is more crowded.Also, as less capacity is available, fewer customerswalking in does not offset the loss of seats; the chanceof getting a seat falls as K decreases.

4.2. Reservations and the Restaurateur’s ProblemFor simplicity, suppose the restaurant’s entire capacityis offered via reservations. (We relax this assumptionin §5.) It does not overbook, so reservation holdersare guaranteed seats. (We discuss overbooking in §5.)If demand exceeds K, reservations are rationed ran-domly. We do not a priori assume reservations elimi-nate walk-ins; if fewer than K reservations are taken,the restaurant will accept walk-in customers. How-ever, in equilibrium, a restaurant will not serve bothreservation customers and walk-ins. To see why, let� = E6max801V − T 970 � is a representative cus-tomer’s expected value for holding a reservation.Because � ≥ 01 all customers request one, and therestaurant commits all of its capacity to reservations.Recall that the firm is unable to reoffer the seats ofno-show reservation holders. A customer who wasunable to secure a reservation consequently does notwalk in because she knows that she will not get a seat.

If the restaurant offers reservations, it will give out1 on a slow night and K on a busy night. Its sales willthen be

41 −�5F 4T 5+�KF 4T 50 (4)

Comparing (4) and (3) allows us to determine whenreservations are recommended.

Proposition 2. The restaurant offers reservations if

41 −�54F 4T 5− F 4V ∗55≥ �KF 4T 50 (5)

Reservations are never offered if

F 4T 5≥41 −�541 −K/�5

1 −�+�K0 (6)

Reservations alter the behavior of some customerswhose realized valuations fall between T and V ∗.When reservations are not offered, these customersare certain to stay home because their expectedreward is too small relative to the cost of walking in.If, however, reservations are offered and they man-age to secure one, they patronize the firm. On slownights, everyone in the market receives a reservation,


and the firm’s sales are certain to increase. The out-come on busy nights is less sanguine. When reserva-tions are not offered, capacity is fully used on busynights. Hence, it is not possible for reservations toincrease the firm’s sales. Instead, they lower sales.The firm precommits capacity to customers who ulti-mately do not want service (i.e., have realized valua-tions below T ) and thus become no-shows. Note thatno-shows occur on slow nights as well, but are notcostly because capacity does not bind; a no-show ona slow night does not prevent the firm from servinga willing customer. No-shows on busy nights leaveseats empty that would have been full if the firm hadrelied solely on walk-customer traffic.

If the firm opts to offer reservations, it must be thatthe gain on slow nights outweighs the loss on busynights. Reservations are consequently important onevenings in which the firm has plenty of availableseats, but are costly when the place is hopping. Thatis, the restaurant offers reservations not to managedemand when business is good, but to entice morecustomers to come in when demand would otherwisebe slow.

To the extent that a firm’s demand varies system-atically over the week, this view of reservations sug-gests that the firm may want to vary its policy overthe week. For example, it may take reservations onTuesdays but not on Fridays. Intuitively, if weekendnights are almost certain to be busy, reservations aretoo costly because they very frequently idle capacitywhile only rarely boosting slow-night sales. If week-day nights are more likely to be slow, the penaltyof idle capacity is less relevant while the increasedtraffic on slow nights is beneficial. Some restaurantssuch as Audrey Claire in Philadelphia, Café Ibericoin Chicago, and Naam in Vancouver actually followsuch a policy, taking reservations on weekdays, butnot on weekends.

Costly no-shows are essential to our story: Thetrade-off goes away if they are not an issue. If cus-tomers knew their valuations when making reserva-tions, sales in the high state would be K1 and thefirm would always offer them. Alternatively, if theno-show rate, F 4T 5, is too high, reservations are neveroffered, as the bound given in (6) demonstrates. Theright-hand side is decreasing in K and �0 Hence, forany given no-show rate there exists a capacity level orchance of a large market sufficiently high that reser-vations would never be offered.

Proposition 3. Suppose that for a given set of param-eters K1 �1 D1 and T and valuation distribution F 1 (5)holds and the firm prefers offering reservations.

1. The restaurant would offer reservations for any K ′

such that 1 <K ′ ≤K.2. The restaurant would offer reservations for any �′ ≥ �.

3. The restaurant would offer reservations for anyD′ ≥D.

4. The restaurant would offer reservations for any valu-ation distribution G larger than F in the reversed hazardrate order.3

5. The restaurant would offer reservations for any T ′ ≤

T0 if F 4�T 5/F 4T 5 is decreasing in T for � > 1.

Offering reservations becomes more attractive (inthe sense that the gain from offering reservations rel-ative to relying in walk-ins increases) as shrinkingcapacity or an increasing denial penalty further lim-its walk-in demand. Similarly, when a larger valua-tion distribution raises the walk-in cutoff, reservationsbecome more attractive. Together these give the empir-ical predictions that smaller restaurants or better qual-ity restaurants (in the sense of higher valuation distri-butions) are more likely to offer reservations.

The roles of T and � are less clear. A higher travelcost lowers both reservation and walk-in sales. Reser-vation sales fall because no-shows increase in allstates. Walk-in sales fall because a higher T impliesa higher V ∗, lowering sales in the low-demand state.The first effect dominates, and reservations becomeless attractive as T and the resulting no-show rateincreases. The required regularity condition holds formany common distributions.

While reservation sales increase linearly with theprobability of high demand, walk-in sales are not nec-essarily monotone in �. It is consequently difficultto generate comparative statics analytically. However,we know that sales with and without reservationsare equal as � falls to zero, but relying on walk-insis always better if � is sufficiently high. This sug-gests that the gains from offering reservations are notmonotone in �. Figure 1 shows that this can, indeed,be the case. The percentage increase in sales peaksat intermediate values of � while the location of thepeak tends to be at lower values of � for higher travelcosts. Furthermore, as discussed above, the gain fromreservations is higher when capacity is tight.

5. Mitigating No-ShowsIn the preceding analysis, no-shows create a trade-off.The restaurant wants to offer reservations to increasesales on slow nights, but must weigh this gain againstsales loss to no-shows when demand is high. Thefirm then has an incentive to limit the impact ofno-shows. Here, we consider four such strategies:reoffering seats, overbooking, implementing a partialreservation policy, and imposing a no-show penalty.We first consider each mitigation approach in isola-tion, but consider interactions between them at theend of this section.

3 For distributions F and G with respective densities f and g, F issmaller than G in the reversed hazard rate order if f 4V 5/F 4V 5 ≤

g4V 5/G4V 5 for all V ≥ 0 (Shaked and Shanthikumar 1994).


Figure 1 Percent Change in Sales When Moving from Not OfferingReservations to Offering Reservations

0.2 0.4 0.6 0.8 1.0

–20

20

40

60

%ch

ange

in s

ales

(K, T ) = (1.25, 1/10)

(K, T ) = (1.25, 3/10)

(K, T ) = (3, 1/10)

(K, T ) = (3, 3/10)

�

Note. F 4v5= v for 0≤ v ≤ 11 D = 7/41 and � = 5.

5.1. Reoffering SeatsIn our base model, no-shows translate directly intolost sales. Alternatively, one might suppose that atleast some reservation holders call to cancel or thatthe restaurant only briefly holds seats for late-arrivingreservations customers. In either scenario, the firmwould have some capacity which, although initiallycommitted to reservations holders, can be offered towalk-in customers. Here we suppose that a fraction� ∈ 60117 of the capacity held for no-shows is madeavailable to walk-in customers. If K reservations aretaken, �F 4T 5K seats will be available for walk-ins.

We also suppose customers rationally anticipatethat capacity will be available. The availability of re-offered seats is only relevant when customers havebeen denied reservations (i.e., on busy nights). Witha single segment, these customers are the only oneswho might walk in. Thus, we are assuming that cus-tomers are willing to walk in even though they knowthat all capacity had been previously committed toreservations.

Given these assumptions, it is straightforward toshow that customers who were denied reservationsuse a cutoff equilibrium. They know that it is a busynight and walk in if their valuation exceeds some valueV� >T . The resulting number of walk-ins 4�−K5F 4V�5must exceed the available number of seats �F 4T 5K.

In such a setting, the firm’s sales when offeringreservations are

41 −�5F 4T 5+�K(

F 4T 5+�F 4T 5)

1

and the restaurateur prefers relying on reservationsover walk-ins if

41 −�5(

F 4T 5− F 4V ∗5)

≥ �K41 −�5F 4T 50 (7)

Comparing (7) with (5), we see that reoffering seatslowers the costs of no-shows and makes offeringreservations attractive over a larger range of parame-ters. In particular, as the fraction of seats reoffered, �,

goes to one, offering reservations is always the bestpolicy. To the extent that the restaurant can affect �(e.g., limiting how long it will hold tables for latearrivals), it will seek to increase � as much as possible.

5.2. OverbookingReoffering seats copes with no-shows ex post. Therestaurateur could instead compensate for no-showex ante by overbooking. Anticipating that not all cus-tomers will keep their reservations, the restaurateurcould simply give out reservations above the restau-rant’s capacity much as airlines overbook flights. On abusy night, the firm would simply give out K/F 4T 5reservations yielding K actual patrons. The restaurantthen enjoys the best of both worlds, maximizing saleson a slow night (as we have said reservations would)while keeping the restaurant full on a busy night (aswe have said relying on walk-ins would).

Before proclaiming overbooking as the solution tothe restaurateur’s problem, we must acknowledge thatlimitations of our model oversimplify evaluating anoverbooking policy. First, because we have assumedthat customers are atomistic, the firm’s overbook-ing problem is trivial. When K/F 4T 5 reservations aretaken, our model expects exactly K customers to comein and does not allow for excess capacity or (moreimportantly) delayed customers. A more insightfulmodel would have to incorporate discrete customersso there is a possibility that the firm has more than Kreservation customers arriving for service.

Furthermore, a proper model of overbookingwould also have to explicitly consider the sequenceof arrivals over time. Overbooking in service sys-tems with sequential arrivals and processing is a chal-lenging subject because high yields at some pointin the planning horizon can impose delays that per-sist after customers who arrived in an overbookedperiod have left the system. See, for example, Muthu-raman and Lawley (2008) or Liu et al. (2010). Finally,these potential delays introduce an additional diffi-culty in analyzing overbooking. A customer with, say,a 7:30 reservation presumably values being seated at7:30 more than being seated at 8:00. The potentialdelays that overbooking introduces thus systemati-cally lower the expected value of keeping one’s reser-vation. Hence, the no-show rate a firm sees will notbe independent of its overbooking policy.

The complications from overbooking have led oneguide on restaurant management to opine “[B]ut thereis always the chance that everybody will show up.That is a nightmare no maitre d’ wants to experience.It is unwise to play these odds 0 0 0 0 It is better to bookthe dining room to capacity and replace any no showswith last-minute calls or spur of the moment arrivals”(The Culinary Institute of America 2001, p. 58).


5.3. Partial ReservationsReoffering seats limits the cost of no-shows byincreasing the salvage value of capacity while over-booking adds an additional supply of customers.An alternative approach is to minimize the exposureto no-shows by limiting the number of reservationstaken. Thus far, we have assumed that the firm makesits entire capacity available via reservations. Clearly,this is suboptimal. The firm would not offer reserva-tions if it were sure that demand was going to be high.Once reservation requests exceed the low-demandlevel of one, the restaurateur knows demand is highand could do better by refusing any more reservationrequests. Giving out additional reservations increasesthe number of no-shows, but does nothing to increasedemand on slow nights.

We now study allowing the firm to set a reserva-tion level, M , such that 0 ≤ M ≤ 13 K − M seats arethus available for walk-in customers. We suppose thatcapacity unclaimed by no-shows cannot be re-offered(i.e., no-shows result in lost sales). We again assumethat customers are again willing to consider walking ineven if they have been denied a reservation. These cus-tomers follow a cutoff equilibrium with those whoserealized value for the service exceeds VM walking in.

While the structure of the equilibrium is intu-itive, the behavior of VM is not immediately obvious.On the one hand, fewer walk-in customers are in themarket (relative to the no reservation case) becausesome already hold reservations. This suggests that VM

should be decreasing in M0 However, there are twocountervailing forces. First, those customers are com-peting for less capacity since M seats have alreadybeen allocated to reservation holders. Second, beingdenied a reservation is informative because this ismore likely to happen when demand is high. The fol-lowing lemma shows that these latter factors domi-nate and that a smaller fraction of customers will walkin as more capacity is made available via reservations.

Lemma 1. VM is increasing in M .

Given this equilibrium behavior, the restaurant’ssales are

ç4M5 = 41 −�5(

MF 4T 5+ 41 −M5F 4VM 5)

+�(

MF 4T 5+K −M)

= MF 4T 5+ 41 −�541 −M5F 4VM 5+�4K −M50

Note that the first term is increasing in M0 Since VM

is increasing in M1 we then have that the other termsare decreasing.

Proposition 4. Suppose that the firm makes 0 ≤

M ≤ 1 seats available via reservations.1. If

41 −�5(

F 4T 5− F 4V ∗5)

≥ �F 4T 51 (8)then M = 1 is optimal.

2. If (8) fails, then M = 0 is optimal.

3. Reservations are never offered if F 4T 5 ≥ 41 − �5 ·

41 −K/�5.

Out of a continuum of possibilities, only two reser-vations levels need to be considered. M either max-imizes sales on a slow night or is set to zero. Forintuition on why intermediate reservations are neveroptimal, consider the limiting case as D goes toinfinity. This would yield a customer equilibrium ofF 4VM 5= 4K −M5/4�−M5, and firm sales of

MF 4T 5+ 41 −�541 −M5K −M

�−M+�4K −M51

which is convex in M0 The exact shape of the func-tion can vary; it may be monotonically increasing ordecreasing or may first decrease and then increase. Allresult in a corner solution. For finite D1 F 4VM 5 gen-erally mimics the behavior of 4K −M5/4� −M51 andsales are maximized at either zero or one.

Comparing (8) to (5), we again have that limitingthe impact of no-shows increases the range of param-eters over which reservations are preferred. However,unlike the setting in which all seats can be reoffered,it is never the case that reservations are preferredregardless of the no-show rate. Part 3 of the proposi-tion shows that for a given no-show rate there existsa capacity level and probability of a large market suf-ficiently high that reservations are never offered.

5.4. No-Show PenaltiesThe previous options manipulate the impact of no-shows or the firm’s exposure to no-shows but do notalter the root cause of the problem, the customer’spropensity for failing to come in. Our last alterna-tive directly lowers the no-show rate. We now sup-pose that the restaurant charges customers a fee, p > 0,when they fail to keep their reservations. Such chargeshave existed in the U.S. restaurant industry for wellover a decade (Fabricant 1996) but remain controver-sial. No-show penalties currently range from $25 to$175 (Frumkin 2009). Such charges might seem exor-bitant, but restaurateurs insist that the fees are sec-ondary to shaping customer behavior. As one NewYork restaurateur put it, “It’s not really about beingpunitive. It’s about trying to keep the dining roomfull” (Frumkin 2009).

If a customer holds a reservation and has a realizedvaluation less than T 1 she now faces a choice. Shecan keep her reservation and receive a net utility ofV −T < 01 or she can stay home, pay the penalty, andhave a net utility of −p0 She thus keeps the reservationif V > T −p and the restaurant experiences a no-showrate of F 4T − p50

The question then is how the firm should set the no-show fee0 An obvious option is p = T , which wouldeliminate no-shows completely. Because this possibil-ity is open to the firm and will result in higher sales


than relying on walk-ins, we immediately have that thefirm will always prefer reservations to walk-ins if it cancharge a no-show fee. Remarkably, however, the firmmay prefer not to set the no-show fee to the travel cost.

To see this point, note that because no-show feesbring revenue to the restaurant, examining only unitsales is no longer sufficient. We must compare themargin on no-shows with the margin on actually serv-ing customers. We assume that there are no costsincurred for no-shows so the margin on these cus-tomers is p, and we will denote the margin on serv-ing customers as �0 The firm’s expected margin whengiving out a reservation is then

�4p5 = �F 4T − p5+ pF 4T − p5

= � + 4p−�5F 4T − p5 for 0 ≤ p ≤ T 0

There are two possibilities to consider. First, if � ≥ T 1then we must have p ≤� and �4p5 is increasing in p0The optimal solution then is p∗ = T 1 and all no-showsare eliminated. Alternatively, if � < T 1 the firm canchoose a no-show fee such that it makes more moneyon no-shows than on customers who actually showup, while still inducing some no-shows. When � < T 1this is, in fact, optimal.

Proposition 5. Suppose that � < T and F 4V 5 is log-concave.

1. The optimal no-show fee p∗ is less than T and satisfies

p∗=� +

F 4T − p∗5

f 4T − p∗50 (9)

2. Consider two valuations distributions, F 4V 5 andG4V 5, such that G4V 5 is larger than F 4V 5 in the reversedhazard rate order. Let p∗

F and p∗G be the corresponding opti-

mal no-show fees. Then p∗F > p∗

G.

The optimal no-show fee p∗ is greater than �1 andthus the restaurant is better off when customers do notkeep their reservations. The proposition is illustratedby the power function distribution,

F 4V 5=

(

V

ì

)k

for k > 0 and 0 ≤ V ≤ì1

which yields p∗ = 4k/4k+ 155� + 41/4k+ 155T 0 The no-show fee decreases with k, which also correspondswith F 4V 5 increasing in the reverse hazard rate order.

It may seem unreasonable that a restaurant mightset its no-show penalty higher than its margin forserving customers. But consider the case of NewYork restaurant Momofuku Ko. It imposes a $150 perperson penalty for reservations cancelled fewer than24 hours in advance of the seating time or for whichcustomers simply do not show up. Momofuku Ko’sdinner offering is a prix fixe menu at $125. Mostdiners are likely to have wine with their meal, so

the total per-customer revenue exceeds $150. Thus,it is plausible that collecting no-show fees is moreprofitable than serving customers. Similarly, Chicago-based Alinea imposes a $100 no-show fee while offer-ing a prix fixe menu at $210. Its gross margin isreported to range between 8% and 18% (Tanaka 2011).Obviously, the gross margin accounts for variousoverhead costs that would not be included in theper-customer margin. However, it is conceivable thatAlinea earns more when customers fail to show upthan when they do.

Finally, we should highlight an implicit assumptionbehind our analysis. We have assumed throughoutthat customers have a positive expected value to hold-ing a reservation and thus that all customers in themarket would request one. If the firm imposes a no-show fee, this need not be the case because customerswith low realized valuations end up with a nega-tive net utility. We have thus implicitly assumed thatthe valuation distribution puts sufficient weight onlarge values that holding a reservation has a positiveexpected value even if p∗ = T 0 This holds if E6V 7≥ T 0

5.5. Interactions Between Mitigation StrategiesThe various mitigation strategies we have proposedinteract in interesting ways and can be seen as comple-ments or substitutes. For example, if the firm can reof-fer a high percentage of seats, overbooking (and thuspossibly delaying customers) becomes less attractivebecause fewer seats will be idle. The ability to reof-fer seats, however, makes a partial reservation strat-egy more attractive because it effectively increases thesalvage value of unclaimed seats. Conversely, limitingreservations is not consistent with overbooking. Limit-ing reservations to M = 1 generates value for the firmby letting it learn that a given night is, in fact, busy.There is no point in overbooking beyond a level ofM = 1 unless one is willing to offer all capacity viareservations and overbook to the level of K/F 4T 5.

As for no-show fees, if the firm eliminates no-shows,it is indifferent to giving out all or only part of itscapacity via reservations. Furthermore, there is noneed to overbook because no-shows are eliminated. Ifthe firm sets p∗ < T 1 then it strictly prefers giving outas many reservations as possible. If the firm can reof-fer a fraction, �, of unused seats, it can make moneytwice on unkept reservations. The restaurant will noteliminate no-shows as long as T > �41 − �5, and theoptimal no-show fee is decreasing in �. When there aregreater opportunities to reoffer seats, the restaurant ismore tolerant of no-shows. If the firm cannot reofferseats, overbooking to K/F 4T − p5 would be attractivesince the firm can still profit from no-shows who paythe penalty, while having its capacity fully used.


6. CompetitionWe now return to the assumptions of our base modelexcept we suppose there are two restaurants, A and B.Travel and denial costs are the same for both. Restau-rant j has capacity Kj for j = A1B. Let K = KA + KB

and � = 1 − � = KA/K. We assume 1 ≤ KA ≤ ��F 4T 5and 1 ≤KB ≤ ��F 4T 5. Thus either firm could serve theentire market by itself on a slow night, but togetherthey cannot serve all customers on a busy night evenif customers are allocated in proportion to the fractionof industry capacity they control. This structure keepsthe competitive market comparable to our monopolysetting because the industry as a whole could serveall interested customers on a slow night, but not on abusy night. Below we relax the assumption that eitherfirm could serve the entire market on a slow night.

The sequence of events is slightly more involved.Both firms announce their policies, and customersarrive to the market. If at least one offers reservations,customers wanting one make their requests simul-taneously before learning their valuations. If bothoffer reservations, customers seek only one reserva-tion. Once valuations are learned, reservation holdersdetermine whether to use their reservations or stayhome. Customers without reservations wait to learntheir realized valuations and determine whether towalk in, and if so to which firm (i.e., they can onlypatronize A or B). If no one offers reservations, cus-tomers choose whether to walk in (and to which firm)after learning their valuations.

It remains to specify the joint distribution of cus-tomer values. We focus on a simple case. Supposeeach customer draws a valuation V from distributionF and her value for dining at A is the same as hervalue for B, i.e., VA = VB.

We begin with the walk-in equilibrium. Considera customer with realized value V > T 0 Because shevalues dining at the two restaurants the same, shegoes to the firm offering a better chance of get-ting a seat. In equilibrium, the firms must offer thesame probability of successfully securing a seat. Sup-pose customers with valuations greater than some Vattempt to dine out and randomize between the firmsin proportion to their capacities (i.e., a given customergoes to firm A with probability � and to B with prob-ability �). The chance of getting a seat at either firmwould then be

F 4V 541 −�5+ K�

F 4V 541 −�+��50

It remains to find V . Because the chance of a walk-incustomer getting a seat is the same as when there is amonopolist restaurant with capacity K, the cutoff canbe found from (2). We take this monopoly setting asthe base case against which to compare the competi-tive outcome.

Now consider A’s problem. Suppose B does notoffer reservations. If A also forgoes reservations, itssales are 41 − �5�F 4V 5+ �KA0 If it offers reservations,A gets the entire market in the low-demand state.Expected sales are 41 − �5F 4T 5+ �KAF 4T 5, and reser-vations are preferable if

41 −�54F 4T 5−�F 4V 55≥ ��KF 4T 50 (10)

Comparing (10) with (5), it is easy to see that if Bdoes not offer reservations, A would find reservationsattractive over a larger range of parameters than amonopolist with capacity K.

If B offers reservations and A does not, A’s salesare �KA. If both offer reservations, they must offercustomers the same chance of getting a reservation inequilibrium. A’s sales are consequently 41−�5�F 4T 5+�KAF 4T 51 and A offers reservations if

41 −�5�F 4T 5≥ ��KF 4T 50 (11)

Proposition 6. Suppose that firm A is the smallerfirm so that �≤ 1/20

1. The equilibrium has the following structure:(a) If �KF 4T 5≥ 441 −�5/�54F 4T 5−�F 4V 55, neither

restaurant offers reservations.(b) If 441 − �5/�54F 4T 5 − �F 4V 55 ≥ �KF 4T 5 >

max8441 − �5/�54F 4T 5 − �F 4V 551 41 − �5F 4T 59, A offersreservations and B does not.

(c) If 441 − �5/�54F 4T 5 − �F 4V 55 ≥ �KF 4T 5 >41 −�5F 4T 5, only one restaurant offers reservations, and itcan be A or B.

(d) If 41 − �5F 4T 5 > �KF 4T 5, both restaurants offerreservations.

2. If a monopolist with capacity K would offer reserva-tions, both restaurants offer reservations.

3. If F 4T 5 ≥ 41 − �541 − K/�5/41 − �+ ��K5, reserva-tions are never offered.

Competition makes reservations viable over alarger range of market parameters. This is driven bythe smaller firm. The first firm to offer reservationsenjoys a windfall in a soft market if its competitordoes not match its policy. This slow-night windfallis larger for the small firm because each firm’s slownight sales are proportional to the fraction of capac-ity it controls. The amount of sales lost to no-showsis also proportional to capacity implying that thesmaller firm pays a smaller price in the high-demandstate. Indeed, the amount of no-shows the larger firmwould incur may be sufficiently large that it wouldrather forego any low-demand sales than lose sales inthe high-demand state.

Figure 2 provides an example of the prevailingequilibrium for combinations of industry capacity and


Figure 2 Equilibrium Outcome with F 4v5= v for 0≤ v ≤ 1 andParameters �= 002, T = 1/3, D = 1/4, and � = 12

2.5 3.0 3.5 4.0 4.5 5.0

0.30

0.35

0.40

0.45

0.50

K~

NA

A & BA or B

A only

Noreservations

�

Note. Our analysis is not valid in region NA since �< 1/K 0

A’s share of capacity.4 For the given parameters, amonopolist would only offer reservations if K ≤ 1078,but both firms offer reservations for K ≤ 3 (region“A & B” in the figure). As capacity expands, we moveto the region marked “A or B.” Here, only one firmwill offer reservations and it can be A or B. This isthe only region in which there are multiple equilibria.As the split of capacity becomes more skewed, onlyA offers reservations as the larger firm now finds itunprofitable to match offering reservations. If capac-ity expands sufficiently, walk-in business is robust onslow nights and reservations lead to many no-showson busy nights. Thus, no one offers reservations.

To generalize these results to N ≥ 2 firms, weconsider two cases. In the first, the restaurants are“lumpy.” Each has capacity greater than one and thuscan serve the entire market on a slow night. In thesecond case, the firms are arbitrarily small. Any onefirm’s capacity is inadequate for serving the entiremarket on a slow night, and the amount of industrycapacity available via reservations KR can take on anyvalue. In both settings, we assume that in any walk-in equilibrium customers first use a common cutoffvalue and then split in proportion to firm capacities;the cutoff remains V as discussed above.

Proposition 7. Suppose that there are N ≥ 2 firms inthe market and that �F 4T 5 > K > 10

1. Suppose the firms each have capacity K/N > 10 Theequilibrium has the following structure:

(a) If K > N441−�5/�54F 4T 5/F 4T 5− F 4V 55/NF 4T 5,no firm offers reservations.

(b) If 4N/R5441 − �5/�54F 4T 5/F 4T 55 ≥ K > 4N/4R+ 155441 − �5/�544F 4T 5/F 4T 555 for R = 11 0 0 0 1N − 11then R firms offer reservations.

4 Note that we cannot guarantee for a specific set of parameters, thatall of the equilibrium possibilities of the proposition exist. Also, theassumption �K > 1 limits the range of parameters we can consider.In the region marked NA in the figure, industry capacity is too smallrelative to the fraction held by firm A and our results do not apply.

(c) If 441−�5/�544 ¯F 4T 5/F 4T 55≥ K, all N firms offerreservations.

2. Suppose the firms are arbitrarily small and F 4T 5 ≤

41 −�541 − 1/K5. In equilibrium:

KR =1 −�

�

F 4T 5

F 4T 50 (12)

When restaurants are sizeable, the range of indus-try capacities in which everyone offers reservationsis independent of the number of firms in the mar-ket. Yet the region in which no one offers reserva-tions is decreasing in the number of restaurants. Thus,splitting market capacity more finely makes having atleast one firm offer reservations more likely. The intu-ition is similar to the duopoly case. The less capac-ity each firm has, the less is lost to no-shows in thehigh-demand state. However, unless market capac-ity is severely limited, not every restaurant offersreservations. Indeed, the amount of capacity avail-able via reservations RK/N will be very close to441 −�5/�54F 4T 5/F 4T 55 when N is large.

When firms are small, this holds exactly. A marketwith many small restaurants will consequently neverhave all firms following the same policy. Instead,some will offer reservations and have sales levels thatdo not vary with the realized number of customers.Those that do not offer reservations will make do withboom-and-bust sales. On busy nights, they are turn-ing customers away; on slow nights they are empty.

Before closing this section, we return to our strate-gies for mitigating no-shows and consider their effec-tiveness in a duopoly. Reoffering seats, overbooking(given our simplifying assumptions), and partial reser-vation policies will remain attractive to the firm in acompetitive market. None of these strategies affectsthe customer’s decision of whether to ask for a reser-vation or from whom. Thus each strategy serves toreduce the cost of no-shows and will expand the rangeof parameters over which reservations are offered.

The story for no-show penalties is less positive.Suppose that problem parameters are such that with-out no-show penalties, restaurants in a duopolywould both offer reservations. If we now allow no-show penalties, the firms would still offer reserva-tions, but not impose no-show penalties. To see why,consider the customer’s problem. She is indifferent toholding a reservation from either firm if they imposethe same no-show penalty, p∗. However, if one of thefirms drops its penalty from p∗ to p∗ −�, all customerswould prefer this firm. It would get all of the marketon a slow night while incurring only a trivial penaltyon busy nights. Its competitor then has an incentiveto cut its no-show penalty as well. The firms are effec-tively pushed into undifferentiated Bertrand competi-tion that ultimately drops the price to zero.


If instead only one firm offers reservations in equi-librium, it can demand a no-show penalty and enjoyhigher profits. This would expand the range overwhich reservations are part of the market equilibrium.There is a caveat. Relative to the monopoly case, thefirm imposing a no-show penalty faces an additionalconstraint in the duopoly setting. No-show penaltieslower a customer’s expected utility, but now she hasthe option of patronizing the other firm. If the no-show fee is sufficiently high, some customers may optto wait to learn their valuations and then walk in tothe competitor.

7. ConclusionWe have examined whether a restaurant should offerreservations with an emphasis on the ability of reserva-tions to shape customer behavior. Uncertain demandis central to our model. Customers cannot a prioriverify whether demand is high or low. Consequently,slow nights are exceptionally slow when reservationsare not offered because customers with low valua-tions stay home rather than risk being denied ser-vice. Reservations address this problem by guarantee-ing potential diners seats, generating more demandin a slack market. The downside is sales lost to no-shows in a large market. Thus, reservations are valu-able to the restaurant when business is slow, but costlywhen demand is high. In contrast, customers wouldnot value reservations if they knew the market weresmall, but would prize them highly on busy nights.

Hence, no-shows are also crucial to our model,and the restaurateur has an incentive to reduce theirimpact. If the firm can reoffer the seats of no-showsto those initially denied reservations, offering reser-vations becomes more attractive. The firm might alsolimit the number of reservations taken. We show thatit is either optimal to not take reservations or to offerjust enough capacity via reservations to serve a smallmarket completely. Alternatively, the restaurant mayimpose a no-show penalty and completely eliminateno-shows. While it is always possible to eliminate no-shows, the firm may prefer to allow some if its mar-gin on serving customers is sufficiently small. Finally,the firm may choose to overbook, taking additionalreservations on busy nights to offset the anticipatednumber of no-shows. In our simple model, this elim-inates no-shows essentially for free although in prac-tice overbooking introduces significant complications.

Our results carry over to competitive environments.Competition expands the range of parameters overwhich reservations are offered. This is particularly truefor a smaller firm, and a market with many small firmsalmost certainly has some firms offering reservations.

Together, these insights suggest several empricalpredictions. For example, reservations are most attrac-tive to firms facing significant demand uncertainty

(measured by �41 − �5). This follows from two obser-vations. First, reservations are not offered when �approaches one and demand is all but certain tobe high. Second, as � goes to zero, offering reser-vations and relying on walk-ins result in the sameprofit. Hence, if there are any administrative costs toreservations, they are easier to recoup when demanduncertainty is higher. Our model also predicts thatreservations are most attractive to smaller firms. Therate at which customers walk in when no reservationsare offered increases in the firm’s capacity. Smallerfirms thus see a bigger jump in slow-night sales fromoffering reservations while paying a smaller pricein terms of no-shows on busy nights. Further, thiseffect is amplified in competitive markets. Finally, ourmodel suggests that as the number of restaurants in amarket increases, reservations should be more preva-lent. However, in a very large market, it should notbe the case that all firms follow the same strategy.

While we have offered a unique perspective onreservations, it is certainly not the only reason arestaurant may offer free reservations. We have pur-posefully suppressed reservations’ operational bene-fits to emphasize their impact on consumers. To theextent that the firm can use reservations to managethe flow of work or forecast demand, we have sys-tematically underestimated their value. Furthermore,firms may use reservations to segment customers,relying on them to attract customers with differentcharacteristics. For example, one might suppose theexistence of a segment that will only dine out if theyhave a reservation as in Çil and Lariviere (2011).More subtly, in our model, segments could differin their travel or denial costs. If a restaurant doesnot offer reservations, high-cost customers are under-represented among its customers. Reservations movethe firm’s sales mix closer to the underlying marketmix. Tweaking the sale mix would be worthwhile ifthe segments differ in both their costs and spendingproclivities. If high-cost customers are more likely torun up large tabs, reservations would be warranted ifthe gain in the average bill is sufficient to compensatefor the resulting no-shows.

Finally, we should note an alternative mitigationstrategy we have not covered: Simply charge forthe restaurant’s service at the time a reservation isrequested. A restaurant would then be much more likean airline or Broadway theater, and the cost of notshowing up for the reservation would be shifted tothe customer. Chef Grant Achatz has implemented thisscheme for his new Chicago project, Next Restaurant.This model has clear advantages. Besides making thefirm’s revenue independent of no-shows, it allows fordifferential pricing across days of the week and timesin the evening. It also alters the cash flow of the restau-rant, allowing the firm to book its revenue before hav-ing to layout cash for labor and supplies (Wells 2010).


AcknowledgmentsThe authors are grateful for the helpful comments of the ref-erees, the associate editor, Steve Graves, and Gérard Cachon.

Appendix. Proofs

Proof of Proposition 1. In equilibrium, a customer withvaluation V ∗ must be indifferent to walking in and stayinghome, implying that �4V ∗5= 4T +D5/4V ∗ +D5. Equation (2)then follows. For uniqueness, note that the left-hand side of(2) is decreasing in V ′ while the right-hand side is increasing.F 4V ∗5 must exceed K/� because walk-ins would otherwisebe guaranteed a seat regardless of the demand realizationand any customer with a realized valuation V such that T <V < V ∗ would have a positive expected value to walkingin. Given that �F 4V ∗5 > K1 the fill rate (1) is decreasing in �for a fixed V ∗3 V ∗ must increase to compensate. Finally, theexpected number of walk-in customers is F 4V ∗541 − �+ ��5and (3) follows from noting that the restaurant can serve atmost K customers on a busy night. �

Proof of Proposition 2. Reservations are preferable ifthey lead to higher sales:

41 −�5F 4T 5+�KF 4T 5≥ 41 −�5F 4V ∗5+�K

=⇒ 41 −�56F 4T 5− F 4V ∗57≥ �K61 − F 4T 570

For the bound on F 4T 5, F 4V ∗5 ≥ K/� so 41 − �54K/�5 +

�K underestimates walk-in sales. Reservations are neveroffered if F 4T 541 −�+�K5≤ 41 −�54K/�5+�K1 which givesthe bound. �

Proof of Proposition 3. For the first part, as K falls,V ∗ increases. Hence, the left-hand side of (5) increases andthe right-hand side falls as K decreases. Results for � andD follow similarly. For the rest, note that (5) implies thatF 4V ∗5/F 4T 5≥ �K/41−�5+1. Let V ∗

F [V ∗G] be the equilibrium

cutoff for F 6G7. If G is larger than F in the reversed hazardrate order, we have that G4t5 ≤ F 4t5 for all t and F 4t5/G4t5is decreasing in t (Shaked and Shanthikumar 1994). Theformer implies that V ∗

G ≥ V ∗F ; the latter gives F 4T 5/G4T 5 ≥

F 4V ∗F 5/G4V ∗

F 50 We then have

F 4V ∗F 5

F 4T 5≤

G4V ∗F 5

G4T 5≤

G4V ∗G5

G4T 50

For the travel cost, consider T ′ < T and with respectiveequilibrium cutoffs V ′ and V ∗0 Define � = V ∗/T and � =

V ′/T ′0 We first show that �′ ≥ �0 Since T0 > T1, it mustbe that V ∗ > V ′ and 4T + D5/4V ∗ + D5 > 4T ′ + D5/4V ′ + D5because a higher cutoff value means fewer customers walk-ing in which implies a higher probability of getting a seat.If �′ <�, we would have to have

T ′ +D

�′T ′ +D>

T ′ +D

�T ′ +D>

T +D

�T +D1

which implies 4T +D5/4V ∗ +D5 < 4T ′ +D5/4V ′ +D50 Giventhat �′ ≥ �, we then have

F 4V ∗5/F 4T 5 = F 4�T 5/F 4T 5≤ F 4�T ′5/F 4T ′5≤ F 4�′T ′5/F 4T ′5

= F 4V ′5/F 4T ′50 �

Proof of Lemma 1. Let �4V 1M5 denote the fill rate whenM seats are made available for reservations and all cus-tomers with realized valuations greater than V walk in.We have

�4V 1M5 =F 4V 541 −M541 −�5+ 4K −M5�

F 4V 5441 −�541 −M5+�4�−M55

= 1 − �4M5+ �4M5K −M

4�−M5F 4V 51

where �4M5 = 4� − M5��/44� − M5�� + �41 − M541 − �55.Define �4V 1M5 = �4V 1M5 − 4T + D5/4V + D5. In equilib-rium, we must have �4VM1M5 = 0 and can then determine¡VM/¡M through implicit differentiation. We have

¡�4V 1M5

¡V=

4K −M5

4�−M5F 4V 52+

T +D

4VM +D52> 0

and

¡�4V 1M5

¡M=−�4M5

�−K

4�−M52F 4V 5+�′4M5

(

K−M

4�−M5F 4V 5−1

)

0

Note that since � > 1, �′4M5≥ 0 and that in equilibrium wemust have F 4VM 5 ≥ 4K − M5/4� − M5 (otherwise, walk-inswould be guaranteed a seat). We thus have ¡�4V 1M5/¡M <0 and therefore ¡VM/¡M ≥ 00 �

Proof of Proposition 4. ç415 >ç405 if

F 4T 5+�4K − 15≥ 41 −�5F 4V05+�K1

which simplifies to (8). To show that M ∈ 40115 is neveroptimal, we have

ç4M5 = MF 4T 5+ 41 −�541 −M5F 4VM 5+�4K −M5

< MF 4T 5+ 41 −�541 −M5F 4V05+�4K −M5

= M4F 4T 5+�4K − 155+ 41 −M5441 −�5F 4V05+�K5

= Mç415+ 41 −M5ç405

≤ max8ç4051ç41590

The sufficient condition for reservations follows from sub-stituting F 4V ∗5=K/� into (8). �

Proof of Proposition 5. Setting �′4M5 to zero yields (8).The log-concavity of F 4V 5 implies that the reversed hazardrate is decreasing and that the solution is unique. The sec-ond part follows from the definition of the reversed hazardrate order and the log-concavity of F 4V 5. �

Proof of Proposition 6. First, because F 4T 5− �F 4V 5 ≥

�F 4T 51 (11) implies (10). Next, the conditions for B to offerreservations are

41 −�54F 4T 5−�F 4V 55≥ ��KF 4T 51 (13)

41 −�5�F 4T 5≥ ��KF 4T 50 (14)

Comparing (11) and (13), it is evident that the former alwaysholds if the latter does because � ≥ �. Comparing (10) and(14), one sees that they are scalings of each other. Hence,the firms would have same best responses to a competitoroffering reservations.


Now suppose (10) fails (as in part 1 of the proposition).It must also be the case that B does not want to be thefirst to offer reservations. Also, A (and hence B) would notrespond to a competitor offering reservations by also offer-ing reservations. Thus, no one offering reservations is theonly outcome. If, however, (10) holds while (11) and (13) fail(part 1), A is willing to offer reservations if B does not, andB never is willing to offer reservations. If (13) holds but (11)does not, both firms are willing to offer reservations if theother does not do likewise. Because (11) fails, neither willmatch the others offering reservations. Hence, one firm willoffer reservations and it can be either A or B. If (10) and(11) both hold, offering reservations is a dominant strategyfor A and B’s best response is to offer reservations (part 1).

For part 2, (5) immediately implies (11) and both offerreservations. The proof of the final is similar to the corre-sponding part of Proposition 2. �

Proof of Proposition 7. With lumpy firms and no oneoffering reservations, a firm earns �0 = 41 − �54F 4V 5/N 5 +

�4K/N5. If one firm offers reservations, it makes �1 =

41 − �5F 4T 5 + �4K/N5F 4T 50 �0 > �1 requires K > N441 −

�5/�54F 4T 5/F 4T 5 − F 4V 5/NF 4T 55. If R ≥ 1 firms offer reser-vations, a firm offering reservations earns �R = 441 −

�5F 4T 55/R + �4K/N5F 4T 5. A firm without reservationsmakes �NoR = �4K/N ). �R decreases monotonically with Rwhile �NoR is independent of R0 For R firms to offer reser-vations in equilibrium, it must be the case that �R ≥�NoR ≥

�R+11 which leads to (1). For (1), one compares �N and �NoR.For the case of small restaurants, we first show that given

F 4T 5≤ 41 − �541 − 1/K5 the equilibrium must involve reser-vations. Suppose that no firm offered reservations in equi-librium. Letting k be the capacity of a representative firm,this would require that

41 −�5k

KF 4V 5+�k ≥ k41 −�5F 4T 5+�kF 4T 5= kF 4T 51

which implies F 4T 5 ≥ 41 − �541 − F 4V 5/K5 but 41 − �5 ·

41 − F 4V 5/K5 > 41 − �541 − 1/K50 Next, we show that KR

cannot be less than 10 For 1 > KR > 01 the profit of a firmoffering reservations is kF 4T 5 and is constant in KR0 Theprofit of a firm not offering reservations is �4KR5 = 41 −

�541 − KR54k/4K − KR55F 4VR5 + �k1 where VR is the equi-librium appropriate cutoff value for those customers whowere unable to get a reservation. One can easily show that� ′4KR5 < 00 Hence, as more firms offer reservations, but KR

remains less than one, those not offering reservations havean increasing gain from offering reservations. Consequently,KR < 1 cannot be an equilibrium. In equilibrium, the twopolicies must offer the same returns, which leads to (12). �

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