martin a. lariviere (with eren cil) kellogg school of ...reservation customers request service....

Saving seats for strategic customers

Martin A. Lariviere(with Eren Cil)Kellogg School of Management

Copyright 2007 Martin A. Lariviere

The changing nature of restaurant reservations

“It took three years for OpenTable to seat its one-millionth diner. But now, the company seats two million diners every month.” – New York Times, June 18, 2007.

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 when you allow those Web reservations.” Part of that control is saving tables for regulars; you never want to alienate people who 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. However, if all the dining slots were sold online, you'd “have a lot of sweet people you don't know,” and all the familiar faces would be shut out.“Any successful restaurant has to hold back tables,” Gates says. He’s wary of the old Yogi Berra saying: “Nobody goes there anymore because it's too crowded.” – Boston Globe, May 18, 2005.


Research questions

How should the firm allocate capacity between customers demanding reservations and walk-in customers when walk-in customers compete for seats and have a cost to enter?


LiteratureRevenue management

Littlewood (1972); Talluri & van Ryzin (2004); Ovchinnikov and Milner (2005); Cooper,Homem-de-Mello, Kleywegt (2006).

Advance purchasePng (1989); Dana (1998); DeGarba (1995); Xie & Shugan(2001).

Walk-in sales with strategic customersDana and Petruzzi (2001).

Restaurants and barsBecker (1991); Arthur (1994); Kimes et.al.(1998, 1999); Horstmann & Moorthy (2003); Bertsimas & Shioda (2003); Lariviere & Alexandrov (2006).


Model basics

There are two customer segments:Reservation customers

Will only dine if able to secure a reservation.Walk-in customers

Request service immediately.

Customers are atomistic.


Relevant costs

Requesting (or receiving) a reservation is costless as is being denied a reservation.All walk-in customers value the service at V and incur a cost of T to walk in to the restaurant.

A walk-in cannot attempt to acquire a seat without incurring T.


The firm’s problem

The firm is monopoly with fixed capacity K.All customers consume the same amount of capacity.

The firm’s only decision is R – how many seats to make available via reservations.

Reservation holders are guaranteed service.Margin on reservation customers is πr.Margin on walk-in customers is πw.


Timing and uncertainty

Timing:First stage:

Firm announces R.Reservation customers request service.

Second stage:Customers with reservation arrive and are seated.Walk-in customers arrive and seated if possible.

Walk in accepted randomly if insufficient capacity available.

Uncertainty – We have two cases:Reservation demand is uncertain and πr > πw.Walk-in demand is uncertain and πr < πw.


Case 1: Uncertain reservation demand

Suppose there are M walk-in customers and N reservation customers.

M is deterministic and greater than K.N is random with support (Dmin, Dmax) and continuous distribution F(n).

Given K ≥ R ≥ Dmin, reservation sales are πrS(R), where

( ) ( ) ( ).RFRdyyyfRS RDmin∫ +=


The walk-in customer’s problem

Let γ be the probability of getting a seat. A walk-in customer’s expected utility is:

U(γ) = Vγ - T.In equilibrium, we must have γ ≥ T/V.An equilibrium:

Given that R' reservations have been given out, a walk-in customer actually walks in with probability:

( ) .M

RKTV,1R

⎭⎬⎫

⎩⎨⎧

⎟⎠⎞

⎜⎝⎛ ′−=′ minλ


Walk-in analysis

Walk-ins randomize with probability λ*(R) = λ(S(R)) so the number of arriving walk-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 - Dmin.

For R ≤ R, there is sufficient walk-in traffic that all unclaimed capacity is sold.For R > R, there is a positive chance that some capacity goes unused.


Profits

If R ≤ Dmin:

If Dmin < R ≤ R:

If R < R ≤ K:( ) ( )( ) ( )

( )( )( )

( ) ( )( )

( ) .RFRdyyyf

RKFKdyyfRyR

R

RKwr

w

RK

Dwr

min

⎟⎟⎠

⎞⎜⎜⎝

⎛+∫−+

−+∫ +=Π

−

−

μ

μ

ππ

μπμππ

( ) ( ) .RKR wrw πππ −+=Π

( ) ( ) ( ) ( ) .RFRdyyyfKRR

Dwrw

min⎟⎟⎠

⎞⎜⎜⎝

⎛+∫−+=Π πππ


The optimal reservation level

If R > K or R* = K.Likely when K or V/T is large or F(n) is sufficiently tight or πr is large.

Otherwise, R* solves:

F(K - μ(R*)) = P{empty seats}Save more seats when difference in margins is small or walk-in customers have low net utility.

( )( ) .VTRKF

w

wr* ⎟⎠

⎞⎜⎝

⎛ −=−πππμ

( )( )( ),KKF1 TV

wr μππ −+≥


Case 2: Uncertain walk-in demand

The number of reservation customers is now fixed at N > K.

If R are offered, they will all be taken.The number of potential walk-in customers M has continuous distribution G(m) with support (Dmin, Dmax).Assume πw > πr.


Walk-in analysis

Still need the equilibrium probability of getting a seat to be T/V.Walk-in customers dine out with probability λ. The probability of getting a seat when Rreservations have been given out is then:

( ) ( )∫+−

−−max

RK

D

yRKRK dyygG

λ

λλ


Analysis

Equilibrium λ(R) = min{1, (K - R)/Z*}, where Z* is a function of G(m) and T/V.The restaurant’s profit is:

For R < K - Z*, λ'(R) = 0 and

( ) ( ) ( )( ) ( ) ( )( )

.dyygRyGRKRRRRK

minDwR

RKwr ∫+−+=Π

−

−λ

λπππ λ

( ) .RKGw

rwNV π

ππ −=−Copyright 2007 Martin A. Lariviere

More analysis…

Suppose RNV ≥ K - Z*, then profit are:

Linear in R: Either offer everything or nothing via reservations!

( ) ( ) ( ) ( ) .dyyygZGRKRR*

min*ZDZ

1*wr ⎥⎦

⎤⎢⎣⎡ ∫+−+=Π ππ


When are reservations offered…

Reservations are offered if:

Reservations are never offered if:

( ) ( ) .*

min*

* ∫+≥Z

Dw

r dyyygZ1ZG

ππ

.1VT

w

r ≤+ππ


Conclusion

Strategic customers can alter standard policies.When reservation demand is uncertain, may commit to turning away some higher value reservation customers to support walk-in demand.When walk in demand is uncertain, have bang-bang reservation allocation.

Either offer everything or nothing via reservations.


Extensions

Richer model of customer valuations.Suppose walk-in customers draw values from some known continuous distribution.

Competition.Creates a role for strategic behavior by reservation customers.

Spiral down.Ignoring consumer behavior can lead to collapse of walk-in demand.


Strategic customers and spiral down

Suppose that the restaurant starts off knowing G0(m) but fails to recognize the strategic nature of customer behavior.

Choose R0NV and induces λ(R0

NV).Realized demand has distribution G1(m) = G(m/λ(R0

NV)) which is stochastically smaller than G(m).

If the firm revises its estimate of the market, R1

NV > R0NV – i.e., it saves fewer seats.

Leads to lower and lower walk-in demand.


Spiral down example

The optimal policy is R* = 0.G(m) = 1 – (10/m)1.25 for m ≥ 10, K = 25, πr = 1, πw = 1.92, V = 1.3, T = 1.

0

5

10

15

20

25

30

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Seat

s

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

P(A

Wal

k-In

Cus

tom

er E

nter

s)

K-RNV RNV l(RNV)


martin a. lariviere (with eren cil) kellogg school of ...reservation customers request service....

Documents