Saving seats for strategic customers

Transcription

1 Saving seats for strategic customers Martin A. Lariviere (ith Eren Cil) Kellogg School of Management

2 The changing nature of restaurant reservations It took three years for OpenTable to seat its one-millionth diner. But no, the company seats to million diners every month. Ne York Times, June 18, The booking system doesn't mean all seats are up for grabs every night. We have the ability to block out tables at certain times, [Eleanor] Arpino of Davio s says. You can control hen you allo those Web reservations. Part of that control is saving tables for regulars; you never ant to alienate people ho patronize you all the time. [Jeffrey] Gates says it's not difficult to fill Union [Bar & Grill] or any other popular place on a Saturday night since many local restaurants aren t too big. Hoever, if all the dining slots ere sold online, you'd have a lot of seet people you don't kno, and all the familiar faces ould be shut out. Any successful restaurant has to hold back tables, Gates says. He s ary of the old Yogi Berra saying: Nobody goes there anymore because it's too croded. Boston Globe, May 18, 2005.

3 Research questions Ho should the firm allocate capacity beteen customers demanding reservations and alk-in customers hen alk-in customers compete for seats and have a cost to enter?

4 Literature Revenue management Littleood (1972); Talluri & van Ryzin (2004); Ovchinnikov and Milner (2005); Cooper,Homem-de-Mello, Kleyegt (2006). Advance purchase Png (1989); Dana (1998); DeGarba (1995); Xie & Shugan (2001). Walk-in sales ith strategic customers Dana and Petruzzi (2001). Restaurants and bars Becker (1991); Arthur (1994); Kimes et.al.(1998, 1999); Horstmann & Moorthy (2003); Bertsimas & Shioda (2003); Lariviere & Alexandrov (2006).

5 Model basics There are to customer segments: Reservation customers Will only dine if able to secure a reservation. Walk-in customers Request service immediately. Customers are atomistic.

6 Relevant costs Requesting (or receiving) a reservation is costless as is being denied a reservation. All alk-in customers value the service at V and incur a cost of T to alk in to the restaurant. A alk-in cannot attempt to acquire a seat ithout incurring T.

7 The firm s problem The firm is monopoly ith fixed capacity K. All customers consume the same amount of capacity. The firm s only decision is R ho many seats to make available via reservations. Reservation holders are guaranteed service. Margin on reservation customers is π r. Margin on alk-in customers is π.

8 Timing and uncertainty Timing: First stage: Firm announces R. Reservation customers request service. Second stage: Customers ith reservation arrive and are seated. Walk-in customers arrive and seated if possible. Walk in accepted randomly if insufficient capacity available. Uncertainty We have to cases: Reservation demand is uncertain and π r > π. Walk-in demand is uncertain and π r < π.

9 Case 1: Uncertain reservation demand Suppose there are M alk-in customers and N reservation customers. M is deterministic and greater than K. N is random ith support (D min, D max ) and continuous distribution F(n). Given K R D min, reservation sales are π r S(R), here S R ( R) = yf ( y ) dy RF ( R). + D min

10 The alk-in customer s problem Let γ be the probability of getting a seat. A alkin customer s expected utility is: U(γ) = Vγ -T. In equilibrium, e must have γ T/V. An equilibrium: Given that R' reservations have been given out, a alk-in customer actually alks in ith probability: λ V T K R M ( R ) = min 1,.

11 Walk-in analysis Walk-ins randomize ith probability λ * (R) = λ(s(r)) so the number of arriving alk-in customers is μ(r) = M λ * (R). μ(r) is decreasing in R. Let R be the unique reservation level that solves μ(r) = K - D min. For R R, there is sufficient alk-in traffic that all unclaimed capacity is sold. For R > R, there is a positive chance that some capacity goes unused.

12 Profits If R D min : If D min < R R: If R < R K: Π Π K μ ( R) ( R) = ( π y π μ( R) ) f ( y ) D Π + min ( R) = π K + ( π π ) R. r ( R) = π K + ( π π ) yf ( y ) dy + RF ( R). r D R r min dy + π KF ( K μ( R) ) + R ( π π ) yf ( y ) r K μ ( R ) dy + RF ( R).

13 The optimal reservation level π If R > K or r R * = K. Likely hen K or V/T is large or F(n) is sufficiently tight or π r is large. Otherise, R * solves: ( ( * R ) r μ =. F K π ( 1 + V ( K ( K) ) T F μ ), T π V π π F(K - μ(r * )) = P{empty seats} Save more seats hen difference in margins is small or alk-in customers have lo net utility.

14 Case 2: Uncertain alk-in demand The number of reservation customers is no fixed at N > K. If R are offered, they ill all be taken. The number of potential alk-in customers M has continuous distribution G(m) ith support (D min, D max ). Assume π > π r.

15 Walk-in analysis Still need the equilibrium probability of getting a seat to be T/V. Walk-in customers dine out ith probability λ. The probability of getting a seat hen R reservations have been given out is then: G D max ( K R) + K R g( y ) λ K R λ λy dy

16 Analysis Equilibrium λ(r) = min{1, (K - R)/Z * }, here Z * is a function of G(m) and T/V. The restaurant s profit is: Π ( R) = π R ( K R) G K R r + π λ( R) For R < K - Z *, λ'(r) = 0 and K R ( ) ( ) λ R + π yλ( R) g( y ) r ( R ) =. G K NV π π π D min dy.

17 More analysis Suppose R NV K - Z *, then profit are: Π ( ) ( ) ( * ) Z R = π R + π K R G Z + 1 yg( y ) dy. r Linear in R: Either offer everything or nothing via reservations! Z * D * min

18 When are reservations offered Reservations are offered if: π ( ) Z r * 1 G Z + ( ). * yg y dy π Z Dmin Reservations are never offered if: * π r π + T V 1.

19 Conclusion Strategic customers can alter standard policies. When reservation demand is uncertain, may commit to turning aay some higher value reservation customers to support alk-in demand. When alk in demand is uncertain, have bangbang reservation allocation. Either offer everything or nothing via reservations.

20 Extensions Richer model of customer valuations. Suppose alk-in customers dra values from some knon continuous distribution. Competition. Creates a role for strategic behavior by reservation customers. Spiral don. Ignoring consumer behavior can lead to collapse of alk-in demand.

21 Strategic customers and spiral don Suppose that the restaurant starts off knoing G 0 (m) but fails to recognize the strategic nature of customer behavior. Choose R 0 NV and induces λ(r0 NV ). Realized demand has distribution G 1 (m) = G(m/λ(R 0 NV )) hich is stochastically smaller than G(m). If the firm revises its estimate of the market, R 1 NV > R0 NV i.e., it saves feer seats. Leads to loer and loer alk-in demand.

22 Spiral don example % 90% 25 80% Seats % 60% 50% 40% 30% P(A Walk-In Customer Enters) 5 20% 10% K-RNV RNV l(rnv) 0% The optimal policy is R * = 0. G(m) = 1 (10/m) 1.25 for m 10, K = 25, π r = 1, π = 1.92, V = 1.3, T = 1.