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8/2/2019 Maxim Afanasyev(2010)Service Provider Competition- Delay Cost Structure-Segmentation and Cost Advantage
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MANUFACTURING & SERVICE
OPERATIONS MANAGEMENT
Vol. 12, No. 2, Spring 2010, pp. 213235issn 1523-4614 eissn 1526-5498 10 1202 0213
informs
doi 10.1287/msom.1090.0266 2010 INFORMS
Service Provider Competition: Delay Cost Structure,Segmentation, and Cost Advantage
Maxim Afanasyev, Haim MendelsonGraduate School of Business, Stanford University, Stanford, California 94305
{[email protected], [email protected]}
We model competition between two providers who serve delay-sensitive customers. We compare a gen-eralized delay cost structure, where a customers delay cost depends on her service valuation, with thetraditional additive delay cost structure, where the delay cost is independent of the customers service valua-tion. Under the additive delay cost structure, service providers offer different prices and expected delays, butcustomers are indifferent between the providers. Under the generalized delay cost structure, when the providershave different capacity or operating costs, we obtain value-based market segmentation, whereby higher-value
customers choose one provider and lower-value customers choose the other. We study how the delay costparameters, the market size, and the service providers costs affect the structure of the equilibrium.
Key words : delay cost structure; value-based market segmentation; service competitionHistory : Received: May 23, 2008; accepted: March 22, 2009. Published online in Articles in Advance
September 14, 2009.
1. IntroductionIn this paper we model competition between service
providers when their customers are sensitive to delay.
Our primary focus is on how the customers delay
cost structure and the asymmetry in the providers
costs affect the market equilibria. In most previousresearch, customers were assumed to have an addi-
tive delay cost structure. In reality, however, we often
observe interdependence between customers service
valuations and their delay costs. We argue that this
interdependence gives rise to major changes both in
the structure of the market equilibrium and in the
levels of arrival rates, prices, and service capacities.
We show how the delay cost structure, market size,
and differences between the providers costs (capacity
costs or variable costs of providing the service) affect
service differentiation and market segmentation.
The assumption of an additive delay cost structure,
which is common in the literature on the economics
of congestion and delay, is reasonable in a variety of
settings. This assumption means that the cost of delay
is independent of the customers service valuation.
This would be the case, for example, if the delay cost
reflects merely productivity losses or the value of lost
time, regardless of the nature of the service provided.
Consider, for example, the case of auto repair. While a
customer is waiting for her car to be repaired, she may
be using a rental car or public transportation. Thus,
her expected delay cost may be approximated by the
product of the expected number of days needed to
repair the car by the daily loss, the latter reflect-ing the value of lost time and the costs of alterna-
tive transportation.1 Assume that this daily loss is
not related to the value of the repair service. 2 Then,
if a repair facility repairs the car a day faster (on
average) than its competitor but charges a premium
equal to the expected daily loss, all customers will be
indifferent between the competing facilities. Although
the repair facilities are differentiated (offering differ-
ent prices and expected delays), there is no market
segmentation.
However, there are numerous situations where the
delay cost and the value of service are interdepen-dent. In electronic brokerage, for example, a delay in
1 Other opportunity costs of time may also be included in the daily
loss.
2 Across customers, the value of the repair service could be pos-
itively correlated with the daily loss, because higher-income cus-
tomers tend to have both higher opportunity costs and more
expensive cars.
213
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trade execution deflates the investors expected profit,
and customers who trade frequently are willing to
pay a larger premium for fast execution. Thus, there is
a positive relationship among an investors frequencyof trading, the total amount traded, and the delay cost
she incurs over, for example, a year, which creates an
interdependence between the value of the brokerage
service to the customer and her delay cost. A similar
relationship holds when the cost of execution delay is
proportional to the trading volume at the transaction
level (see Dewan and Mendelson 1998, 2001). Indeed,
major electronic brokerage firms (e.g., Ameritrade,
E*TRADE, and Scottrade) prominently advertise their
average or median execution speeds, and brokerages
that target frequent traders bear the additional costs
of colocating their servers inside the exchanges toreduce delays (Martin and Malykhina 2007). In hospi-
tals, patients may incur a delay of weeks waiting for
surgery, and the longer the surgical delay, the higher
the mortality rate (Weller et al. 2005). We expect that
for patients in poor health, surgery is particularly
critical and they will suffer from a delay more than
patients whose condition is not as severe, implying
that the value of service and the delay cost are inter-
dependent. In online video streaming, transmission
delays affect the quality of the content viewed by the
customer. The higher the content quality, the more a
customer suffers if there are data transmission delays.
Thus, the delay cost is interdependent with the value
of the content to the customer.
Product development provides another example
of interdependence between service valuations and
delay costs. Adler et al. (1995) show that product
development can be modeled as a queue of projects
sharing common resources. While waiting in the
queue, new products become less attractive because
of changes in the marketplace and the competition.
A longer development lead time may result in a
loss of market leadership, a decrease in potential
revenue, and lower sales, as customers needs change
over time (see Takeuchi and Nonaka 1986, Stalk
1988, Gupta and Wilemon 1990, Mabert et al. 1992,
Millson et al. 1992). Gupta and Wilemon (1990) find
that a six-month delay in the market introduction of
a high-tech product reduces average profit by 33%
over five years. In the auto industry, the larger the
market potential for a new car model, the larger
the revenue loss resulting from model introduction
delays. Some well-known examples of companies
that achieved competitive advantage by reducing
their new product development times includeToyotas introduction of the Prius, Sun Microsystems
in the workstation market, Sony in the CD market,
and Samsung in memory chips and LCD televisions
in the 2000s (Stalk and Hout 1990, Clark et al. 1987,
Smith and Reinertsen 1998, Sun et al. 2004).
In this paper we study how the delay cost structure
affects the competition between service providers and
the resulting market structure. In particular, we dis-
tinguish between service differentiation and market
segmentation (see Smith 1956, Fradera 1986). Whereas
the additive delay cost structure results in differen-
tiated services, interdependence between service val-uations and delay costs leads to value-based market
segmentation. We give the following definitions.
Service differentiation occurs when at least one
firms services are different from those offered by its
competitor(s).
Market segmentation requires that, in addition, the
market is divided into customer segments so that
the customers in at least one segment strictly pre-
fer the services offered by one firm over the services
offered by its competitor(s).
Under service differentiation, (some) firms offer dif-
ferent service variants, typically at different prices,
but customers may be indifferent between these
alternative offerings. Under market segmentation,
different service variants attract different customer
segments. This means that some customers have a
strict preference for one firms offering. The segmen-
tation is value based if the customers are segmented
based on their valuations of the service.
We consider two competing service providers
whose customers are sensitive to delay. We model
interdependent service valuations and delay costs
using the generalized delay cost structure proposed
by Afeche and Mendelson (2004), which allows for
both additive and multiplicative delay cost compo-
nents. Earlier work assumed that customers util-
ity was additive in value and delay cost, namely
uUt = U Dt, where U is the gross utility absent
delay, t is the delay in the system, equal to the sum
of the waiting time in the queue and the service
time, Dt = d t is the expected delay cost, and u is
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the customers expected net utility. Under the gen-
eralized delay cost structure, the expected utility is
uUt = t U Dt, where t is a decreasing
function that deflates the customers value for theservice. Each customer is self-interested and maxi-
mizes her own expected utility, which she forecasts
using the distribution of the steady-state delay; i.e.,
EuUt = Et U Dt. In the linear case,
Dt = d t and t = 1 v t, so the expected utility
equals EuUt = U 1 vW dW, where W is the
expected delay in the system.
We find that the equilibrium is characterized by ser-
vice differentiation. Say one firm provides fast ser-
vice and charges a higher price and the other offers
slow service at a lower price. In addition, when the
service valuations and delay costs are interdependentand the providers costs (capacity costs or operat-
ing costs) are asymmetric, we find value-based mar-
ket segmentation into a high- and low-end customer
segment. We study how the delay sensitivity param-
eters and the total market size affect measures of
service differentiation and market segmentation. We
find that an increase in the multiplicative delay sen-
sitivity affects service differentiation nonmonotoni-
cally and increases the level of market segmentation.
Growth in the total market size decreases both the
degrees of service differentiation and market segmen-
tation until they disappear at the limit.
Our results on value-based market segmentation
are consistent with what we observe in a number of
markets where customers are sensitive to delay. One
example is public transportation in the San Francisco
Bay Area, where Viton (1981) finds that the value of
service and the delay cost are interdependent. Con-
sidering the competition between bus service and the
Bay Area Rapid Transit system (BART), Viton (1981)
finds that the systems differentiate their services in
terms of delay, with BART having a lower average
delay than the bus system and charging a higher
fare. Viton (1981) further finds value-based market
segmentation, with BART focusing on wealthier cus-
tomers and the bus system targeting the lower-income
population.
Mortgage loan origination is another service ex-
hibiting both an interdependence between delay cost
and service value and value-based market segmen-
tation. The longer a customer waits for mortgage
approval, the more the home price and interest rates
fluctuate and the higher the probability that the deal
will fall through before the loan is approved. Also,
the higher the home price, the greater the loss, sothe delay cost and the value of the origination ser-
vice are interdependent. In 1986, Citicorp launched
MortgagePower, a computer-based loan origination
system that dramatically reduced Citicorps mortgage
origination delay. In turn, Citicorp charged customers
higher processing fees, so its service was differenti-
ated by less delay and a higher price. Citicorp tar-
geted customers who borrowed larger amounts and
provided faster service in return for higher fees (Hess
and Kremer 1994, Stalk and Hout 1990). In con-
trast, traditional loan originators charged lower fees
and served lower-end customers whose loans were
smaller. A similar segmentation relating morgtgage
loan amounts, service fees, and average delay is
found by Guttentag (2001).
In the decorative laminates industry, customers
service values and delay costs are interdependent
because the disutility of construction delay is larger
for residential cabinetmakers and commercial speci-
fication customers, who typically work on expensive
projects, than for original equipment manufacturer
(OEM) direct purchasers. Accordingly, the former
are willing to pay a premium for fast delivery.The industry has two major product offeringsfaster
but more expensive and slower but cheaperand is
also characterized by value-based market segmenta-
tion, with Ralph Wilsonart providing faster service to
residential cabinetmakers and commercial specifica-
tion customers, and slower Formica dominating the
price-sensitive OEM segment (Stalk and Hout 1990,
Hamilton 2007, Feldman 2007).
In work under way, Afanasyev and Mendelson
(2009) consider movie production in Hollywood as a
special case of new product development. They esti-
mate the delay costs associated with this process and
show that value and delay cost are interdependent.
They then show how the market for studios is seg-
mentated based on movies market potential, with the
faster studios producing movies with higher expected
box office revenues. They also show that the lead time
in introducing movies into foreign markets exhibits
similar characteristics.
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The remainder of this paper is organized as follows.
We review the literature in 2. Section 3 considers
our benchmark, additive delay cost model. In 4 we
study the effects of the generalized delay cost struc-ture. In 5 and 6 we study the effects of market size
and differences in operating costs, and 7 offers our
concluding remarks.
2. Literature ReviewIn this section, we review the literature on delay sensi-
tivity and service provider (S) competition. Numerous
papers study markets with delay-sensitive customers
(see Hassin and Haviv 2002). Most of these papers
assume a discrete number of classes and consider the
optimal pricing or scheduling of services, or both,
among the classes, sometimes subject to incentivecompatibility constraints. Table 1 summarizes the lit-
erature on the effects of competition on queueing sys-
tems; we focus here on the first four papers, which are
closest to ours. Unlike our model, capacity is exoge-
nous and service valuations are constant in Chen
and Wan (2003) and So (2000), service providers are
symmetric in Chen and Wan (2005), and in all four
models the service providers engage in price compe-
tition. Chen and Wan (2003) study how the market
structure changes as a function of market size. For a
small total arrival rate, the market is dominated by
one of the providers. As the arrival rate increases,
there is a unique equilibrium with both firms in the
market, and then no equilibrium, followed by a con-
tinuum of equilibria and finally, a noncompetitive
market. In a model with endogenous service capaci-
ties, Chen and Wan (2005) find that although some of
the irregular equilibria in Chen and Wan (2003) are
eliminated, for a large market size there is a contin-
uum of equilibria. In contrast, we find a more regular
effect of market size on market structure under our
model.
Cachon and Harker (2002) derive conditions underwhich two competing service providers exist in the
market and study how scale economies affect the mar-
ket equilibrium. They find that the lower-cost S may
have a higher market share as well as a higher price.
They also show that the full price (service fee plus
delay cost) is an increasing function of the operating
and capacity costs. To answer their research questions,
Cachon and Harker (2002) need not explicitly model
customer choice as we do. Unlike Cachon and Harker
(2002), where the firms choose prices and waiting
times and the capacities adjust in equilibrium, in our
model firms choose capacities and arrival rates andprices adjust in equilibrium. This enables us to prove
the existence of an equilibrium.
In So (2000), heterogenous firms with exogenous
capacities compete by announcing their prices and
delivery time guarantees, again unlike our model.
The market has a fixed size and all customers are
served, so each firm serves a portion of the market.
The lower a providers price and delivery time guar-
antee, the higher its market share. So (2000) finds that
providers exploit their relative advantage to differen-
tiate their services; e.g., the high-capacity S will offer
better time guarantees, whereas the lower-operating-cost S charges a lower price.
The equilibria in Chen and Wan (2003) and So
(2000) are characterized by service differentiation but
no market segmentation, with one S offering shorter
delays and charging a price premium and the other
offering a longer delay and a lower price that exactly
offsets each customers delay cost savings. In Chen
and Wan (2005), there is no service differentiation (the
providers are symmetric), and in Cachon and Harker
(2002), there is service differentiation, but the underly-
ing customer choices are not specified in the demand
model.
With the exception of this paper, all the papers in
Table 1 that specify a delay cost structure assume
the additive cost structure, which (as we show) is
an important driver of the equilibria they obtain. To
date, few papers have considered a delay cost struc-
ture that is not additive, and none of them (to our
knowledge) studied the effects of competition. Afeche
and Mendelson (2004) present a generalized delay
cost structure and apply it to a single service facility
that has a fixed capacity. They compare the revenue-
maximizing and socially optimal equilibria for a pri-ority queue and show that some classical results may
not hold under the generalized delay cost structure. 3
They also compare the effects of bidding strategies
to uniform pricing under the generalized delay cost
3 For example, the classical results that the revenue-maximizing
admission price is higher than the socially optimal price, and the
revenue-maximizing use is lower than the socially optimal use.
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Table1
ComparisonofThis
PapertotheRelevantLiterature
Study
Serv
iceproviders
Serviceproviderdecisions
Servicevaluations
Delayco
ststructure
Researchquestions/Results
So(2000)
N
asym
metric
Price,
timeguarantee
Demandbasedonprice,
timeguarantees
Notapplicab
le(time
guarantee
model)
Findsthatprovidersexploitthe
ir
capacityadvantagetodiffere
ntiate
theirpricesandservices,
andstudies
determinantsofequilibrium
prices
CachonandHarker
(2002)
2asym
metric
Price,
delay
Demandfunction
Additive
Findthatscaleeconomiesincrease
incentivestooutsourceandthe
lower-costShashighermar
ketshare
andhigherprice
ChenandWan(2003)
2asym
metric
Price
Constant
Additive
Studyhowmarketsizeaffectsmarket
structure
ChenandWan(2005)
2symm
etric
Price,
capacity
Constant
Additive
Studyhowmarketsizeaffectsmarket
structure
Luski(1976),Levhari
andLuski(1978)
2symm
etric
Price
Constant
Additive/het
erogeneous
Conditionsforservicedifferent
iation
Reitman(1991)
N
symmetric
Price,
capacity
Constant
Additive/het
erogeneous
Conditionsforservicedifferent
iation
Loch(1991)
2symm
etric
Twomodels:(1)price;(2)price
andcapacity
Heterogeneous
Additive
Conditionsforasymmetricequilibria
DeneckereandPeck
(1995)
N
symmetric
Price,
capacity
Constant
Notspecifie
d
Studyexistenceofequilibrium
LedererandLi(1997)
LargeN,
asymmetric
Schedulingpolicy
Demandfunction
Notspecifie
d
Showthatlower-costand
lower-variabilityShavelarge
rmarket
share,
highercapacityutiliza
tion,
and
higherprofits
Gibbensetal.(2000)
2symm
etric
Price,
optiontocreatesubnetwork
Constant
Additive/het
erogeneous
Findthatundercompetitionwithtwo
classes,
eachSfocusesonly
onone
class
ArmonyandHaviv
(2003)
2symm
etric
Price
Constantwithineachof
twocustomerclasses
Additivewithineachclass
Analyzecompetitionandfindservice
differentiation
AllonandFedergruen
(2006)
2,
diffe
rentmarginal
costs
Threemodels:(1)price,
(2)capacity,(3)simultaneo
us
priceandcapacity
Demandfunction
Notspecifie
d
Findthatoutsourcing(servicepooling)
isbeneficial
AllonandFedergruen
(2007)
2asym
metric
Threemodels:(1)delayfollow
ed
byprice,
(2)pricefollowed
by
delay,(3)simultaneousprice
anddelay
Demandfunction
Notspecifie
d
Comparethethreemodels,
sho
wingthat
model1leadstohighestwaitingtime
andlowestprices
Pekgunetal.(2008)
2asym
metric
Price,
leadtime(centralizedvs.
decentralized)
Demandfunction
Additive
Studyeffectsofdecentralizationonprice
andlead-timedecisions
Thispaper
2asym
metric
Arrivalrateandcapacity
Heterogeneous/demand
function
Generalized
Effectsofgeneralizeddelaycos
t
structure;conditionsforand
implicationsofmarketsegm
entation
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structure. Katta and Sethuraman (2005) consider a
queue with priorities and customers with a gener-
alized delay cost structure. They propose an algo-
rithm that sorts customers into priority groups tomaximize the revenues of a monopoly. Kalvenes and
Keon (2007) consider an M/M/m loss system and
customers with a multiplicative delay cost structure
and propose a mechanism to reallocate customer ser-
vice times from periods when the system is highly
congested to periods when it is less congested. Our
paper studies the effects of interdependence between
customers service valuations and delay costs on S
competition. We show that when the delay cost and
service valuations are interdependent and the firms
costs are different, the equilibria must be character-
ized by value-based market segmentation.
3. Basic ModelWe start with a traditional model of competition
between two Ss who serve customers with heteroge-
neous service valuations that are subject to an addi-
tive delay cost. We first present our model of user
behavior and then proceed with a description of the
firms and the competition between them. This model
forms a foundation for the analyses that follow.
3.1. User Behavior
We first present the underlying user behavior model
for a single system with no delays. Potential cus-
tomers arrive following a Poisson process with rate .
The values they derive from receiving the ser-
vice when there is no delay are modeled as an
independent and identically distributed (i.i.d.) sam-
ple from a random variable U with cumulative dis-
tribution function (c.d.f.) F. The probability that an
arriving customer values the service at R or higher isF R = 1 F R. If only customers who value the ser-
vice at R or higher join the system, the effective arrival
rate is = FR, which gives rise to the inverse
demand function R = F1/, or V = F1/
(see Mendelson 1985, Dewan and Mendelson 1990),
where V is the expected total value created for
users per unit of time when the (effective) arrival rate
is , and V is the corresponding marginal value,
which is also equal to the cutoff value R. Given the
c.d.f. F, we can calculate V and V, the demand
curve when there is no delay.
Example. Let the distribution F of consumer values
U be uniform on 0 100 and the total arrival rate be
= 300. Then the expected number of arriving cus-
tomers with service valuations above R per unit oftime equals 300 PrU R, which gives rise to the
linear demand curve
V = 100 /3 (1)
We use this demand curve in our numerical examples
throughout the paper.
We assume that V is strictly decreasing, cor-
responding to the usual assumption of a monotone
decreasing demand curve, i.e.,
V < 0 for 0 (2)
and that
V1 + 2 + 1V1 + 2 < 0 for 1 2 0 (3)
Assumption (3) is needed to ensure that the firms
profit functions are supermodular and is satisfied
by the commonly used linear V = ( > 0,
> 0) and constant elasticity V = k (k > 0,
0 < < 1) demand curves.
Customers are sensitive to delay, and in this sectionwe follow the common assumption that the delay cost
is d per customer per unit of time. We assume that
the queue length is unobservable and that customers
make their decisions based on their expected through-
put time W. Let P be the price charged per job. It is
well known that the analogue of the demand curve
when delay costs are taken into account then becomes
P = V d W (4)
This modeling approach is common in the economics
of queues literature. Note that the model can be spec-
ified in terms of the distribution F of user valuations
or the inverse demand curve V. If the model is
specified in terms of probabilities, the effective arrival
rate is PrU d W P , reflecting the fact that as
the expected delay W increases, customers are willing
to pay less for the service, hence the expected number
of customers decreases.
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3.2. Service Providers
We consider a market with two competing Ss, S1and S2, with respective constant marginal capacity
costs g1 and g2. We assume g1 g2, so S1 is the lower-cost S. Each service provider Sj is modeled as an
M/G/1 queue with arrival rate j. We assume that a
customers service time is B/j, where B is a random
variable with coefficient of variation r (the same for
both firms) and mean 1, and j is the service capac-
ity of Sj. Thus, larger investments in service capacity
stochastically decrease each providers service time.
We further assume the first-come, first-served queue
discipline and unobservable customer service valua-
tions. The expected delay at Sj with capacity j expe-
riencing arrival rate j is given by the Pollaczek-
Khinchin formula for the expected total time in thesystem,4
Wj =
jr
2 + j
j j+ 1
1
2j (5)
The expected delay at each Sj increases in j and r
and decreases in j. We further assume that there is a
direct operating cost c of serving each customer.5 We
assume that each S charges a single price and that the
demand is high enough for both Ss to be profitable at
zero delay cost; i.e.,
V0 > c + maxg1 g2 (6)
To ensure internal solutions, we also assume that for
a high enough arrival rate , it is unprofitable to serve
additional customers; i.e., there is a satisfying
V 0. In this case, there is obviously
a U such that all customers with U > U choose to
receive service, and all customers with U U decline
the service. Because customer choice must satisfy the
individual rationality constraint, Sjs profit maximiza-
tion problem in the first stage is
maxj j
jPj cj gjj
s.t. Vj d Wj = Pj
(8)
Next, consider the customer choice when twoproviders are in the market. Now each customer has
three options to choose from:
(a) join S1 (net utility equals U d W1 P1),
(b) join S2 (net utility equals U d W2 P2), or
(c) do not join (net utility equals 0).
The customer decides among these alternatives by
comparing the net utilities in (a), (b), and (c) above.When both providers serve customers, all cus-
tomers must be indifferent between S1 and S2.7
Indeed, if there exists a customer who strictly prefers
Sj (j= 1 2), namely, for some valuation U,
U d Wj Pj > U d Wj Pj (9)
then all customers will strictly prefer Sj (because
(9) will hold for all U), so Sj will serve no cus-
tomers, which is a contradiction. This means that the
6 If there is no firm in the market, the solution is trivial.7 We will show later that this is the direct result of the assumed
additive delay cost structure.
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customer choice problem is degenerate (as all cus-
tomers are indifferent between the two firms). This
is often modeled by randomizing the choice between
the providers. Thus, there exists a service valua-tion threshold U such that the customers with ser-
vice valuations above U are randomly allocated and
those below U are not served (because their utilities
upon being served by either provider would be nega-
tive). Although customers are indifferent between the
two providers, consistency with the first-stage deci-
sions requires that the probability of joining Sj equals
j/1 + 2 j= 1 2.
To complete the specification of the game, recall
that each Sj, j = 1 2 decides on j and j, building
on the customers decision criteria (which, as shown
above, are degenerate). Hence, a necessary conditionfor an equilibrium is that each Sj solves
maxj j
jPj cj gjj
s.t. V1 + 2 d Wj = Pj
(10)
where Wj is the expected delay given by Equation (5).
If Sj decides to stay out of the market, it sets j =
j = 0, and Sj solves (8). In a mixed-strategy equilib-
rium, customers with valuations above some thresh-
old are randomly allocated between the two firms,
and customers with valuations below this threshold
are not served. It is clear from the above discussionthat under the additive delay cost structure, the only
possible equilibria are mixed-strategy equilibria.
The Cournot game is often presented as a
game with explicit or implicit probabilistic customer
choices. In our game, the probability that a customer
with service valuation U is served by Sj equals
pjU =
j
1 + 2if U > U j = 1 2
0 otherwise.
(11)
This also translates directly to probabilistic choicemodels often used in the marketing literature (see
Currim 1982, Kamakura and Srivastava 1984, Grover
and Srinivasan 1987, Kamakura and Russell 1989).
The marketing literature also allows for more gen-
eral forms than (11), which is implied by our model.
This suggests that a more general model specification
might lead to a generalization of (11), such as that
corresponding to the multinomial logistic model.
3.4. Equilibrium
Intuitively, an increase in the delay sensitivity d in-
creases customers delay costs and lowers the effective
total demand. As a result, for small d, both providersare in the market, but as d increases, only one S sur-
vives. For larger values of d, there is no S in the mar-
ket. Proposition 1 makes this intuition precise. Proofs
of propositions and corollaries are in the appendix.
Proposition 1. There exist constants 0 < d1 d2
d3 d4 such that the following applies.
(1) Duopoly solution: For d < d1, there is a mixed-
strategy equilibrium with both S1 and S2 in the market,
where the capacity j and market size j ofSj are given by
the solution to the equations8
jr2 +jjjj
2+ jr
2
jj+1 1
22j= gj
jd j= 12
V1 +2+jV1 +2dWj
djr2 +1
2jj2
= c j= 12
(12)
where Wj are given by (5).
(2) Single monopoly solution: For d d1 d2 d3 d4,
there is an equilibrium with only the low-cost provider, S1,
in the market. The equilibrium capacity 1 and market size
1 are given by the solution to the equations
1r2 + 111 1
2+
1r2
1 1+ 1
1221
=g1
d1
V1 + 1V1 dW1 d1
r2 + 1
21 12
= c
(13)
where W1 is given by (5) (with j= 1).
(3) Multiple monopoly solutions: For d d2 d3, there
are two equilibria: one with only S1 in the market and one
with only S2 in the market. In the equilibrium with Sm in
the market, the capacity m of Sm and its market size mare given by the solutions to the equations
mr2 + mmm m
2+ mr
2
m m+ 1 1
22m= gm
dm
Vm + mVm dWm dm
r2 + 1
2m m2
= c
(14)
where Wm is given by (5) (with j= m).
8 For d = 0, the equations are j = j and V1 + 2 +
jV1 + 2 = c.
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(4) Market is not served: For d d4, there is no S in
the market, and 1 = 2 = 1 = 2 = 0.
The sequence of market structures in Proposition 1
shows that for small delay sensitivities, both firms
are in the market. As the delay sensitivity increases,
the higher-cost S leaves the market. This happens
because, as a monopoly, S2 can profitably serve fewer
customers than S1; hence the entry barrier for S1 is
lower than for S2. As the delay sensitivity increases
further, only one S can survive. As the delay sensitiv-
ity increases further, S2 cannot obtain positive profits.
Because of its cost advantage, S1 stays in the market
longer but eventually, for larger values of d, leaves the
market as well. Proposition 1 allows for multiple equi-
libria, so the constants d1
through d4
in Proposition 1
need not be unique. Furthermore, the interval 0 d1
consists of up to three subintervals:
(i) 0 dD1 , where only duopoly equilibria exist;
(ii) dD1 dM1 , where duopoly and S1 monopoly
equilibria exist; and
(iii) dM1 d1, where duopoly and two, S1 or S2,
monopoly equilibria exist.
In 5 we study the progression of the different equi-
libria as a function of the market size and discuss
these subintervals in more detail.
Proposition 1 also shows that when the two firms
are in the market, they may offer differentiated ser-vices, but customers are always indifferent between
them; i.e., there is a mixed strategy equilibrium. If the
providers capacity costs are equal, the expected delay
and price are determined, so there is neither service
differentiation nor market segmentation. If g1 < g2,
S1 has a cost advantage: it sets a lower expected delay
than S2 but in equilibrium charges a price premium
that exactly equals the delay cost savings of each cus-
tomer; i.e.,
P1 P2 = d W2 W1 (15)
This means that while the Ss offer differentiated ser-
vices, there is no market segmentation.
4. Generalized Delay Cost StructureIn 3 we found that under the additive delay cost
structure, there is no market segmentation. In this
section we show that under the generalized delay
cost structure, we may obtain value-based market seg-
mentation, where one S serves customers with higher
service valuations (the high-end segment) and the
other S serves customers with lower service valua-
tions (the low-end segment). In this equilibrium, the
high-end S offers a smaller delay but at a higherprice than the low-end S. Unlike in 3, high-value
customers strictly prefer the high-end S and low-
value customers strictly prefer the low-end S. More-
over, if the firms have different capacity costs, only an
equilibrium with value-based market segmentation is
possible.
The generalized delay cost specification allows the
delay cost of a customer to depend on his or her
service valuation. As described in the introduction,
such dependence is common in a variety of settings.
Under the generalized delay cost structure, the utility
derived by a customer with service valuation U anddelay t is t U Dt, where t is a decreasing
function that deflates the customers service valuation
U, and Dt is the additive component of the delay
cost. This means that the customers service valua-
tion and delay cost are interdependent. Each customer
with service valuation U maximizes his or her own
expected utility; i.e., EuUt = Et U Dt.
We assume for the most part that t is linear and
decreasing, t = 1 v t. Under the linear specifi-
cation, a customer with service valuation U has util-
ity EuUt = U 1 vW dW, where W is the
expected delay in the system. Hence, the customerincurs a delay cost of vU + d W, where v > 0 is
a multiplicative coefficient and d 0 is an additive
coefficient of the delay cost. The demand curve (4)
becomes
P = V 1 v W d W (16)
Similar to Equation (8), if only Sj is in the market, it
solves
maxj j
jPj cj gjj
s.t. Vj 1 v Wj d Wj = Pj
(17)
When both firms are in the market, we structure the
game as a Cournot model, as before, where in the first
stage Sj decides on j and j (j = 1 2), and in the
second stage the customers decide which S to join,
if any. In 3.3 we showed that under the additive
delay cost structure, there was only a mixed-strategy
equilibrium; i.e., there was a service valuation U such
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that customers with service valuations above U were
randomly allocated between the two firms, and those
below U were not served. Another type of equilib-
rium is the value-based market segmentation equilibrium,where customers are divided into three groups basedon their service valuations. The lowest value group
receives no service, the highest value group is served
by one S, and the intermediate group is served by theother S. Formally, this means that U1 > U2 exist such
that a customer with service valuation U (a) joins Sj,
if U U1, (b) joins Sj, if U U2 U1, or (c) does not
join either S, if U < U2.
Hence, all customers with service valuations aboveU1 form the high-end segment, served by Sj, and
all customers with service valuations in the interval
U2 U1 form the low-end segment, served by Sj.In our model, the only equilibrium structures pos-
sible are the mixed-strategy and value-based market
segmentation equilibria that we identified. We havealready shown in 3.3 that for v = 0, we obtain only
the mixed-strategy equilibrium under a duopoly. If
for v > 0 some customers with different valuations UAand UB are indifferent between S1 and S2, then
UA vUA +dW1 P1
= UA vUA +dW2 P2 and
UB vUB +dW1 P1 = UB vUB +dW2 P2
(18)
Then, by subtracting the two equations, we obtain that
W1 = W2, which also implies that P1 = P2. Thus, any
customer UC will also be indifferent between the twoproviders. Similarly, if customers with valuations UAand UB strictly prefer Sj, then any customer with valu-
ation UC UA UB will also prefer Sj. Indeed, if UC
UA UB, then there exists an 0 < < 1 such that UC =
UA + 1 UB. Because UA and UB strictly prefer Sj,
UA vUA +dWjPj
> UA vUA +dWjPj and
UB vUB +dWjPj>UB vUB +dWjPj
(19)
Giving the two equations weights and 1 ,
respectively, and summing up, we obtain that
UCvUC+dWjPj>UCvUC+dWjPj (20)
We thus conclude that only our mixed-strategy
or value-based market segmentation equilibria arepossible.
We next derive the structure of the segmented equi-libria for a market with asymmetric providers and a
generalized delay cost structure with v > 0 (symmet-
ric providers are considered later in this section). Weshow that in such a market, there is no equilibriumwithout value-based market segmentation; i.e., among
the customers who are served, those with service val-uations above a threshold U1 choose the fast S, andthose with service valuations below that threshold but
above another threshold U2 choose the slow S. Wefocus on the equilibrium in which the lower-cost S1serves the high-value segment9 and show that for suf-
ficiently small delay sensitivities, both Ss are in themarket and the market is segmented.
Proposition 2. There exist nested sets, with 0 0
1, 1 2 3 4 in vd space such that the fol-lowing applies.
(1) Duopoly solution: For vd 1 and v > 0, thereis an equilibrium with both firms in the market, where S1serves the high-end customers and S2 serves the low-end
customers. The capacity j and the arrival rates j to eachSj are given by the solution to the following equations.
1r2 + 11
1 12
+1r
2
1 1+ 1
1
221
=g1
1V1v + d
V1 + 1V1vW2 W1
+ V1 + 2 + 1V1 + 21 vW2
dW1 1V1v + d
r2 + 1
21 12
= c
(21)
and 2r
2 + 222 2
2+
2r2
2 2+ 1
1
222
=g2
2V1 + 2v + d
V1 + 2 + 2V1 + 21 vW2 dW2
V1 + 2v + dr2 + 1
22 22
= c
(22)
where Wj are given by (5).
9 Under certain conditions, an equilibrium exists in which the
higher-cost S, S2, serves the high-value segment. It can be shown
that whenever such an equilibrium exists, there always exists an
equilibrium in which the lower-cost S serves the high-end segment.
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(2) Single monopoly solution: For vd 2\1
4\3, there is an equilibrium with only S1 in the mar-
ket. The equilibrium capacity 1 and market size 1 of S1
are given by the solution to the following equations.1r
2 + 111 1
2+
1r2
1 1+ 1
1
221
=g1
V1v + d1
V1 + 1V11 vW1 dW1
1V1v + d
r2 + 1
21 12
= c
(23)
where Wm is given by (5).
(3) Multiple monopoly solutions: For vd 3\2,
there are two equilibria: one with only S1 in the market,and one with only S2 in the market. In the equilibrium with
Sm in the market (m = 1 2), the equilibrium capacity mand market size m of Sm are given by the solution to the
following equations.mr
2 + mmm m
2+
mr2
m m+ 1
1
22m
=g2
Vmv + dm
Vm + mVm1 vWm dWm
mVmv + d r2
+ 12m m
2= c
(24)
where Wm is given by (5).
(4) Market is not served: For vd 4, there is no S
in the market and 1 = 2 = 1 = 2 = 0.
Proposition 2 shows that for small values of the
delay sensitivitiesi.e., in the set 1 where we obtain
the duopoly solutionthere is a value-based mar-
ket segmentation equilibrium, where S1 serves the
high-end segment and S2 serves the low-end segment,
and customers are not indifferent between the two
providers. Market segmentation requires delay coststhat are low enough to support two firms in the mar-
ket. As the delay sensitivities increase, so vd is in
the set 2\1 (the single monopoly solution region),
the only possible equilibrium is one with S1 being
a monopoly. In this set, the lower-cost S enters and
crowds out the higher-cost S. As the delay sensitiv-
ities grow furtheri.e., in the set 3\2 (the multi-
ple monopoly solutions region)two equilibria exist
with either of the Ss being a monopoly. This hap-
pens because as the delay sensitivities increase, Sj(j = 1 2) cannot be profitable if it enters the market
with Sj already there. As v and d increase further, S2cannot survive even as a monopoly. Because of its cost
advantage, S1 can survive for larger values of v and d,
but ultimately it cannot be profitable and leaves the
market, too, so the market is not served. Note that
Proposition 2 allows for multiple equilibria, e.g., for
vd 1, there may also be monopoly solutions, and
for vd 3\2 there are two monopoly equilibria.
Corollary 1. For v > 0 and g1 < g2, any duopoly
equilibrium must be a value-based market segmentation
equilibrium.
Corollary 1 shows that market segmentation is
driven by the generalized delay cost structure as
well as by the firms cost asymmetry. Because the
providers capacity costs are different, we expect them
to choose different delays, and, because v > 0, differ-
ent customers value the delays differently.
Similar to what we found in Proposition 1, the set
1 consists of up to three subsets:
(i) D1 , where only duopoly equilibria exist;
(ii) M1 \D
1 , where duopoly and S1 monopoly equi-
libria exist; and
(iii) 1\M1 , where duopoly and two, S1 or S2,monopoly equilibria exist.
We may also have in 1 several duopoly equilibria.
For example, for sufficiently small g2 g1, there is
an equilibrium where S2 serves the high-end segment
and S1 focuses on the low-end segment. It can be
shown that if we have an equilibrium with the higher-
cost S serving the high-end segment, there is always
an equilibrium with the lower-cost S serving the high-
end segment.
Importantly, the structure of the equilibria obtained
under the generalized delay cost structure (Proposi-
tion 2) is substantively different from the structure
we obtained under the additive delay cost struc-
ture. In particular, the framework of an additive
delay cost structure does not adequately explain the
prevalence of value-based segmentation in markets
ranging from transportation to mortgage loan origi-
nation. With generalized delay costs, one of the firms
serves the higher-end segment and the other serves
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Figure 1 Distortions in Total Industry Profits (Solid Line, Left Axis)
and the Difference Between the Capacity of the High-End
and Low-End Service Providers (Broken Line, Right Axis) in
the Duopoly Equilibrium as Functions of the Multiplicative
Delay Sensitivity v
0 1 2 3 4
15
15
10
5
0
Distortionintotalprofits(%)
Distortionincapacitydifference(%)
20
20
25
25
30
35
40
V
Notes. The distortions are caused by assuming that the delay cost is addi-
tive and the average delay cost remains the same. The distortion in total
profits is the change in total profit, 1 + 2, caused by not accounting forthe multiplicative structure of the delay sensitivity relative to the correct total
profits. The distortion in capacity differences is the change in capacity dif-
ference (1 2) caused by not accounting for the multiplicative structure
of the delay sensitivity relative to the correct capacity difference. For both
line graphs, the service valuations U are uniformly distributed over the inter-
val 0 100 and the market size is = 300 (which corresponds to V =100 /3), g1 = 29, g2 = 31, c = 5, and d= 0.
the lower-end segment, resulting in a natural value-
based segmentation. In a market characterized bymultiplicative delay sensitivity, assuming an additive
delay cost structure leads to significant deviations not
only in qualitative structure but also in the quanti-
tative results. Clearly, these deviations are increasing
in v. However, as shown in Figure 1, the distortion in
the capacity differences between the service providers
tends to plateau as v increases, whereas the distor-
tion in total industry profit increases as long as both
service providers are in the market.
Our results link the structure of delay costs to the
existence of service differentiation or market segmen-
tation. Quantitatively, they show how service differ-
entiation and market segmentation depend on the
additive (d) and multiplicative (v) components of
delay cost. Our analysis shows how increasing each
of the two components of delay cost affects market
behavior. To translate this into empirically meaningful
results, we need concrete measures of service differ-
entiation and market segmentation.
In the context of our model, service differentiation
means that customers experience different expected
delays and pay different prices, depending on the
provider that serves them. Thus, two natural mea-sures of service differentiation are the price difference,
P = P1 P2, and the differences in expected delays,
W = W2 W1, between the two providers. Both of
these measures can be empirically estimated in the
marketplace, and each exhibits different behavior as
a function of the delay sensitivities.
Consider first P = P1 P2. Figure 2(a) shows that
P is an increasing function of v. The monotone
relationship between P and v is not surprising: as
v increases, the firms both suffer from lower demand,
but S1 also benefits from higher differentiation, which
gives it greater market power. One might expect P tobe an increasing function of d (as would indeed be the
case when v = 0); however, as shown in Figure 2(b), it
is a quasiconvex. This is the net result of two opposite
effects: on the one hand, P decreases in d because its
component V1vW2 W1 is a decreasing function
of d; on the other hand, its component dW2 W1 is
an increasing function of d.
Next consider W= W2 W1. Although a decreas-
ing relationship between W and d is not surprising
(an increase in d affects customers proportionally to
the expected delay), it is hard to predict the shape of
the relationship between W and v. Numerical exper-
iments show that W is a quasiconvex function of v.
As v increases, all customers are willing to pay less
for the service, so the returns on investments in excess
capacity declinethe services become more similar
and W tends to decrease. In contrast, an increase in
v makes customers more sensitive to the difference in
delay between the two providers, which is beneficial
to the high-end S. It is straightforward to prove that
for sufficiently small v, W is an increasing function
of v, and for sufficiently large v, W is a decreasing
function of v.
Empirically, this means that although W is a
monotone increasing function of d, P is quasicon-
vex, and though P is a monotone increasing function
of v, W is a quasiconcave function of d.
We next measure market segmentation by how
much the average customer in the high-value segment
loses if she switches from S1 to S2. The average con-
sumer of S1 has surplus V1/21 vWj dWj Pj
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Figure 2 The Effects of Increasing the Multiplicative (a) and
Additive (b) Delay Sensitivities on the Difference in
Service Fees
0 0.04 0.08 0.12 0.160
0.1
0.2
0.3
0.4
0.5
0.6
0.7
S1
S2
S1
S2
v
(a) Difference in service feesP
1 P
2as a function ofv for d= 1
0 5 10 150.42
0.43
0.44
0.45
0.46
0.47
d
(b) Difference in service fees P1 P2as a function ofd for v = 0.1
Notes. In these figures, v is the multiplicative delay sensitivity, d is the addi-
tive delay sensitivity, and P1 P2 is the difference in service fees betweenthe providers, which is an alternative measure of service differentiation. In
both figures, the service valuations Uare uniformly distributed over the inter-val 0 100 and the market size is = 300 (which corresponds to V =100 /3), g1 = 29, g2 = 31, and c = 5.
if served by Sj. The reduction in consumer surplusif she switches to S2 equals V
1/2v + dW2 W1
P1 P2. Because the marginal customer (at 1) is
indifferent between the two providers, V1v + d
W2 W1 P1 P2 = 0, which means that the loss
for the average customer is
= v
V
12
V1
W2 W1 (25)
Incentive compatibility means that 0. Indeed,
the loss from switching is commonly used in stud-
ies of incentive compatibility in queueing systems
(see Mendelson and Whang 1990, Yahalom et al.2006), and similar differences are used in the mar-
keting context (e.g., Kalish and Nelson 1991, Werten-
broch and Skiera 2002, Chung and Rao 2003, Jedidi
et al. 2003, Wang et al. 2003). Value-based market seg-
mentation and service differentiation are not indepen-
dent, as is proportional to W. However, when
there is no interdependence between the service val-
uations and delay costs, i.e., when v = 0, we have
W > 0 and = 0. Thus, positive service differenti-
ation is a necessary but not sufficient condition for
positive market segmentation.
The level of market segmentation is an increas-ing function of v. For small v, an increase in v both
increases customers sensitivity to delay and the level
of service differentiation; hence increases as well.
For large v, service differentiation starts declining
but this effect is still weaker than that of the increase
in customers sensitivity to delay (because for large v,
as v increases the number of customers served by S1decreases; hence the service valuation of the average
customer, 1/2, increases).
We next consider symmetric firms and the resulting
market structure for sufficiently small values of the
delay sensitivities.
Proposition 3. When v > 0 and g1 = g2 = g, there
exists a set in (v d) space10 in which there are threeequilibria with both firms in the market.
(1) Mixed strategy: Each Sj chooses capacity j and
market size j satisfyingjr
2 + jj
j j2
+jr
2
j j+ 1
1
22j
=g
jV
1+
2v+
dV2 + V21 vW dW
V2v + dr2 + 1
2 2= c
(26)
where Wj are given by (5).
10 The set is analogous to 1 in Proposition 2.
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(2) Equilibria with value-based market segmentation:
There are two equilibria where Sj serves the high-end seg-
ment and Sj serves the low-end segment, where j and j
satisfy jr
2 + jj
j j2
+jr
2
j j+ 1
1
22j
=g
jVjv + d
Vj + jVjvWj Wj
+ Vj + j + jVj + j1 vWj
dWj jVjv + d
r2 + 1
2j j2
= c
(27)
and jr
2 + jj
j j2
+jr
2
j j+ 1
1
22j
=g
jVj + jv + d
Vj + j + jVj + j1 vWj dWj
Vj + jv + dr2 + 1
2j j2
= c
(28)
where Wj and Wj are given by (5).
Proposition 3 shows that when the capacity costsare equal, there are three equilibria: two equilib-
ria with value-based market segmentation and one
mixed-strategy equilibrium. In the last equilibrium,
the firms have the same capacity and delay, and cus-
tomers are indifferent between them.
5. Effect of Market SizeIn this section we study how the size of the market,measured by the total arrival rate , affects the mar-
ket structure and the level of competition between the
two firms. As discussed in the literature review, Chen
and Wan (2003, 2005) found that the effect of marketsize on the market structure is quite surprising andcounter-intuitive. It is thus interesting to study the
effect of market size in our model vis--vis the mod-
els of Chen and Wan. In addition, we are interested
in the effects of increasing market size on service dif-
ferentiation and market segmentation.The following proposition studies the effects of
increasing on the market structure.
Proposition 4. For any pair vd, there exist con-
stants 0 1 2 3 4 5 such that the follow-
ing applies.
(1) No S in the market: For < 1, no S serves themarket.
(2) S1 monopoly: For 1 2, there exists only a
monopoly equilibrium with S1 in the market.
(3) Two monopoly equilibria: For 2 3, there
exist two monopoly equilibria with either S in the market.
(4) S1 monopoly or duopoly equilibria and two monopoly
equilibria: For 3 4, there exists a threshold value
g such that (i) if g2 g1 > g, there exists only an
equilibrium with S1 being a monopoly; and (ii) ifg2 g1
g, there exist duopoly equilibria with both firms in the
market as well as two monopoly equilibria with either S in
the market.(5) Duopoly equilibria and S1 monopoly: For
4 5, there exist duopoly equilibria with both firms
in the market as well as an equilibrium with S1 being a
monopoly.
(6) Duopoly equilibria: For 5, there exist only
duopoly equilibria with both firms in the market. The
arrival rates and S capacities are given by Proposition 1
( for v = 0) and 2 ( for v > 0).
In Proposition 4 we find an interesting sequence
of market equilibria, which is illustrated in Figure 3.
Potentially, there are seven regions:
Region (1): The market is so small that it cannot
support any S, so the market is not served.
Region (2): is so small that the market can sup-
port only the lower-cost S, S1. Thus, the only equilib-
rium has S1 as a monopoly.
Region (3): The market is large enough to support
either of the firms as a monopoly but too small to
allow another entrant to have nonpositive profits,
so there are two monopoly equilibria.
Region (4a): The market is large enough to support
either of the firms as a monopoly, but it is sufficiently
large that when S2 is in the market, S1 can enter andmake positive profits. However, when S1 enters, it
crowds out the higher-cost S and the only equilibrium
is S1 as a monopoly. Note that this is possible only
when S1s capacity cost is low enough in relation to
S2s.
Region (4b): The market is large enough for both
firms to be in the market. However, if one of the
firms is in the market as a monopoly, the other firm
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Figure 3 The Effect of Increasing the Total Market Size for
Different Levels of the Unit Capacity Cost Differential
on the Market Structure
50 100 150 200 250 3000
5
10
15
g2
g1
(6)(4b)
(5)(3)
(4a)
(1)
(2)
Notes. Region (1): The market is not served. Region (2): Only S1 is in themarket, as a monopoly. Region (3): Two monopoly equilibria, one with S1 in
the market and one with S2 in the market. Region (4a): Only S1 is in the mar-ket, as a monopoly. Region (4b): Duopoly and two monopoly equilibria, one
with S1 and the other with S2 in the market. Region (5): Duopoly equilibriaand a monopoly equilibrium with S1 in the market. Region (6): Only duopolyequilibria. In the figure, the service valuations U are uniformly distributed
over the interval 0 100, which corresponds to V = 100 100/,g1 = 30, c = 5, v= 03, and d= 100.
cannot make positive profit. Hence, this region sup-
ports either a duopoly or two monopoly equilibria.
This region exists only if the capacity cost difference
g2 g1 is small.
Region (5): The market is large enough for the
firms to earn positive profits if both are in the mar-
ket. Only S1 can earn positive profit as a monopoly
and deter S2 from entering the market. However, if S2attempts to operate as a monopoly, S1 enters the mar-
ket. Thus, the only equilibria are duopoly and S1 is a
monopoly.
Region (6): The market is large enough that both
firms can make positive profits. Hence, only duopoly
equilibria exist.Overall, market segmentation occurs when the mar-
ket is large enough and the difference between the
capacity costs of the two firms is not too large. Value-
based market segmentation is the only equilibrium
in Region (6) of Proposition 4, which corresponds to
larger values of and smaller values of g2 g1, and
it is one of the equilibria in Regions (4b) and (5),
where both and g2 g1 have intermediate values.
In summary, larger market sizes and smaller differ-
ences in the firms capacity costs are conducive to
value-based market segmentation.
Next, we show that an increase in the market sizedecreases both service differentiation and value-based
market segmentation, and at the limit both disap-
pear. Figure 4 shows the effects of increasing the
market size on the market share and profit share
of the high-end provider, S1, and on our measures
of service differentiation and market segmentation.
First, an increase in market size benefits the low-
end provider, S2, more than it benefits the high-end
provider, S1, in terms of both market share (Fig-
ure 4(a)) and profit share (Figure 4(b)). This is because
of scale economies in queueing systems: A facility
with few customers benefits from an additional cus-tomer more than a facility with many customers. As
S2 serves fewer customers than S1, an increase in the
total market size decreases the cost of S2 more than it
decreases the cost of S1. Hence, S2 benefits more from
an increase in than S1 does. Figure 4(c) shows that
market segmentation decreases as increases. Ser-
vice differentiation decreases because a given increase
in has a larger incremental effect on S2 than on
S1; hence the difference between the two decreases.
Because market segmentation is increasing in our dif-
ferentiation measure W = W2 W1, our measure of
segmentation (25) decreases. At the limit of a verylarge market, both service differentiation and mar-
ket segmentation disappear. This result is driven by
the fact that as the number of customers increases,
the excess capacity increases at a slower rate than
the equilibrium demand rate j. Because the marginal
cost of excess capacity decreases, the firms reduce the
waiting times to zero at the limit. Formally, we have
as follows.
Proposition 5. lim Wj = 0 for j= 1 2.
Corollary 2. As , the measures of both ser-
vice differentiation and market segmentation tend to zero.
6. Different Unit Operating CostsSo far, we have assumed that the service providers
differ only in their unit capacity costs but not in
their unit operating costs. Do different unit operat-
ing costs lead to value-based market segmentation?
In this section we allow the unit operating costs of the
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Figure 4 Effect of Increasing the Total Market Size on the Market Share of S1 (a), the Profit Share of S1 (b), and Market Segmentation (c)
500 600 700 800 900 1,0000.5258
0.5260
0.5262
0.5264
0.5266
0.5268
0.5270
0.5272
(a) Market share ofS1 as a function of
the market size
500 600 700 800 900 1,0000.5530
0.5535
0.5540
0.5545
0.5550
0.5555
0.5560
0.5565
1/(1
+
2)
1/(1
+
2)
(b) Profit share ofS1 as a function of
the market size
500 600 700 800 900 1,000
0.034
0.036
0.038
0.040
0.042
0.044
0.046
0.048
(c) Market segmentation as a function
of the market size
Notes. In these figures, is the total arrival rate of customers, 1/1 + 2 is the proportion of customers served by S1, 1/1 + 2 is the proportionof profits received by S1, and is our measure of market segmentation. The service valuations U are uniformly distributed over the interval [0,100] (whichcorresponds to V = 100 100/), g1 = 29, g2 = 31, c = 5, v = 01, and d= 1.
two firms, c1 and c2, to be different. We also change
assumptions (6) and (7) so that V0 > maxc1 c2 + g
and there exists a such that V < minc1 c2 + g,
respectively.
First, we show that even if the firms capacity costsare equal, differences in unit operating costs lead to
value-based market segmentation. Then, we numeri-
cally analyze how the differences in the capacity and
operating costs affect the levels of service differentia-
tion and market segmentation.
Proposition 6. Let v > 0, g1 = g2 = g, and c1 < c2,
and let both firms be in the market. Then in equilibrium,
S1 serves the high-end customers and S2 serves the low-end
customers. The capacity j and arrival rates j to each Sjsatisfy
1r2 + 111 1
2+ 1r
2
1 1+ 1 1
221= g
1V1v + d
V1 + 1V1vW2 W1
+ V1 + 2 + 1V1 + 21 vW2
dW1 1V1v + d
r2 + 1
21 12
= c1
(29)
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and 2r
2 + 222 2
2+
2r2
2 2+ 1
1
222
=g
2V1 + 2v + d
V1 + 2 + 2V1 + 21 vW2 dW2
V1 + 2v + dr2 + 1
22 22
= c2
(30)
where Wj are given by Equation (5).
Corollary 3. For v > 0, g1 = g2 = g, and c1 < c2,
there is no duopoly equilibrium without value-based market
segmentation.
Proposition 7. For a 1-unit increase in the capac-ity cost gj of Sj (j = 1 2), to keep profit constant, the
marginal operating cost cj should decrease by / units
for a monopoly and by 1/ units for a duopoly.
Proposition 7 shows that a unit change in capac-
ity costs impacts each service providers profits more
than a unit change in its operating cost. From the
proof of the proposition, for a monopoly /c = ,
i.e., the change in c affects the profits proportionally to
the arrival rate, whereas /g = , i.e., the change
in g affects profits proportionally to capacity, which
is always larger than the arrival rate. Proposition 6implies that other things being equal, a unit reduction
in capacity cost is worth to each S more than a unit
reduction in its operating cost.
7. Concluding RemarksRecent years have seen an increase in the impor-
tance of service industriesand with it, an increased
focus on timeliness. As the value of customers
time increases, prompt service becomes an important
lever for gaining competitive advantage. This calls
for a more detailed examination of the relationship
between the delays and competitive strategies of ser-
vice firms. Previous research on the economics of
queues often took an additive delay cost structure for
granted. We show that when customers service val-
uations and their delay costs are interdependent, and
the service providers have different costs, the market
exhibits value-based segmentation: one of the service
providers focuses on fast service for the high-value
Table 2 How the Equilibrium Market Structures Depend on the
Multiplicative Delay Sensitivity v and Capacity Costsgj j= 1 2
g1 = g2 g1 < g2
v = 0 No service differentiation Service differentiation No market segmentation No market segmentation
v > 0 Multiple equilibria: No service differentiation Service differentiation
No market segmentation Market segmentation
or
Service differentiation
Market segmentation
segment, and the other targets the low-value segment
with a lower price and slower service. Table 2 sum-
marizes our key results: market segmentation (value-based) occurs only under the generalized delay cost
structure, and for service providers with asymmetric
costs this is the only possible equilibrium.
We find an interesting progression of market struc-
tures as a function of market size. In our model, for
a small arrival rate, the market is not served; for a
moderate total arrival rate, the market supports only
the lower-cost S being a monopoly; for a larger mar-
ket size, either S can be a monopoly; and as the mar-
ket size increases further, the only equilibrium has the
lower-cost S as being a monopoly or there are the
duopoly and two monopoly equilibria. As the arrivalrate increases further, the only equilibria possible are
the duopoly equilibria and the lower-cost S being a
monopoly. Finally, for a larger market size, the only
equilibria have both firms competing in the market.
A natural extension of our model is to consider pri-
ority queues. A service provider may assign a higher
priority to customers who choose to pay a higher ser-
vice fee and thus, these customers would be served
faster than those who pay less. We expect our qualita-
tive results (in particular, the existence of value-based
market segmentation) to hold in such a setting. For
a monopoly with two priority classes, all customers
with service valuations above some threshold would
prefer the higher-priority class, and all served cus-
tomers below this threshold would opt for the lower-
priority class. Yet the study of the duopoly case raises
the issue of how the number of priority classes affects
entry and exit decisions, which, to our knowledge,
has not been addressed in the literature.
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Our analytical results were derived for only two
firms with the same coefficient of variation of the ser-
vice times. The duopoly assumption allowed us to use
the properties of supermodular games, and extendingthe results to the oligopoly case requires for numeri-
cal analysis. Numerical explorations suggest that with
multiple firms and different coefficients of variation
of service times, a duopoly market is still character-
ized by value-based market segmentation, with each
of the firms serving an interval of customers with a
range of service valuations.
Our results show that the delay cost structure
makes both a structural and a practical difference in
the analysis of competing congestion-prone service
providers. Markets where the delay cost and the value
of service are interdependent are substantively differ-ent from markets with an additive delay structure,
leading to different competitive strategies, customer
behavior, and industrial structures.
AcknowledgmentsThe authors thank the editor, associate editor, and anony-mous reviewers for their suggestions, which helped toimprove the substance and the exposition of the paper.
Appendix. ProofsProof of Proposition 1. The proof proceeds along the
following steps: (1) We derive necessary and sufficient con-
ditions for the existence of an equilibrium with both firmsin the market for sufficiently small d. (2) We find the largestd (d1) such that there is a duopoly solution in the entireinterval 0 d1. (3) We show that for d slightly above d1,only a monopoly solution is possible, and we derive nec-essary and sufficient conditions for the monopoly equilib-rium. (4) We show that there exist d2, d3, and d4 such thatin equilibrium, any of the firms may be a monopoly for d d2 d3 and only S1 is a monopoly for d d1 d2 d3 d4and that for d d4, none of the firms can earn positiveprofits.
Step 1. By (2) and (6), for d = 0, which reduces our modelto a standard duopoly, both firms earn positive profits inequilibrium. Now consider some d slightly above zero.
We first show that if we suppress the positive profitrequirement, a duopoly equilibrium exists. We then showthat for a small enough d, each duopolist makes posi-tive profit. As shown in 3.3, the customers are indifferent
between the firms and each firm solves the problem:
maxj j
jPj cj gjj
s.t. V1 + 2 d Wj = Pj
(31)
where Wj are given by (5).
Let jj j j = jV1 + 2 d Wj c gjj
be the profit function of player j. From assumption (7)and the nonnegative cross-partial derivatives of the profitfunctions, game (31) is supermodular in 1 1 2 2and a duopoly equilibrium exists (by Theorems 4 and 5 inMilgrom and Roberts 1990).
We next show that for a small enough d, both duopolistsmake positive profits. Theorem 2 in Milgrom and Segal(2002) shows that j (j= 1 2) are continuous functions of d.By the continuity of jd, j0 > 0 implies that for suffi-ciently small d, jd> 0 as well. Summing up, if d is smallenough, there is a duopoly equilibrium with both firmsmaking positive profits. We denote the equilibrium arrivalrate to Sj when the delay sensitivity is d by jd.
11
According to Theorem 6 in Milgrom and Roberts(1990) with exogenous parameter g1, 1 2 and 1 2.This implies that in equilibrium, 1d 2d when
both firms are in the market. By our choice of d, bothfirms are in the market. Because profit goes to asj , there is an equilibrium where j and j satisfythe first-order conditions (FOC) of (31). From Equa-tion (5), Wj/j = r
2 + 1/2j j2 and Wj/j =
jr2 + jj/j j
2 + jr2/j j + 11/2
2j,
which together with the FOC of (31) give the equilibriumcondition (12) for the duopoly.
Step 2. Because of (7), there exists at least one finite dsuch that 2d = 0. Let d1 = infd 2d = 0. Because1d 2d, S1 is still in the market at d = d1 and in 0 d1,we have a duopoly solution with positive profits for bothfirms.
Step 3. Consider d slightly above d1. Then, only one S,
which we call Sm, can survive in the market (the other S willearn negative profits if it enters the market). Then Sm mustset its capacity m and service rate m to maximize its profit:
Mm d = maxm m
mPm cm gmm
s.t. Vm dWm Pm
(32)
By the choice of d and because profit goes to asm , there is an interior solution that satisfies the FOC.From (5), Wm/m = r
2 + 1/2m m2 and Wm/m =
mr2 + mm/m m
2 + mr2/m m + 1
1/22m which together with FOC give the equilibriumcondition (14) for the monopoly.
Step 4. Let jdj (j = 1 2) be the optimal profitin (31) for fixed d and given j. Let d2 be the smallest solu-tion to 1d argmax
M2 d = 0. Because when both
firms are in the market, 1d 2d, d1 d2. Let d3 be thesmallest solution to M2 d = 0, and let d4 be the smallestsolution to M1 d = 0. Because
1d 0 = max
M1 d,
d2 d3. Thus, in the interval d2 d3 there are two equilibria,
11 If there are several equilibria, we choose the equilibrium with the
maximum number of customers for S2.
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with either S being a monopoly, and in the interval d1 d2there is only a monopoly S1 solution. Because g1 < g2, forany value of d, M1 d
M2 d. Hence, d3 d4. Because the
monopoly profit is a decreasing function of d, for d d3 d4only S1 can be a monopoly, and for d d4, the market is notserved.
Proof of Proposition 2. We first show in the Prelimi-nary Step below that when both firms are in the market,only an equilibrium with value-based market segmentationis possible. The proof then follows the same steps as theproof of Proposition 1.
Preliminary Step. Consider a duopoly equilibrium andassume, by contradiction, that it follows our mixed-strategystructure. We show that this assumption leads to the contra-diction that both 1 2 and 1 < 2 hold simultaneously.In our mixed-strategy equilibrium, each Sj, j= 1 2 solves
maxj j
jPj cj gjj
s.t. V1 + 21 v Wj d Wj Pj = 0
(33)
Let jj j j = jV1 + 21 v Wj d Wj c
gjj be the profit function of player j given j. Game (33)is supermodular in 1 1 2 2 by Theorem 4 inMilgrom and Roberts (1990) and the nonnegative cross-partial derivatives of the profit functions. Next, the cross-partial derivatives of j with respect to the strategyvariables of Sj, j = 1 2, and g1 are nonnegative. Hence, byTheorem 6 in Milgrom and Roberts (1990), with exogenousparameter g1,
1 2 (34)
We next show that 1 < 2. As shown in 4, in a mixed-strategy equilibrium, W1 = W2 = W. Let j = j/j. By (5),
112
1r
2 + 1
1 1+ 1
= 2
22
2r
2 + 1
1 2+ 1
= W (35)
By the FOC of Sj (j= 1 2) for (33), we have
V1 + 2v + d
W +
2j
j
r2 + 1
21 j2
=
gj
j (36)
hence for j= 1 2,
j = 3j
V1 + 2v + d
gj jV1 + 2v + dW
r2 + 1
21 j2
(37)
Equation (37) together with (36) leads to
W =
j
V1 + 2v + d
gj V1 + 2v + dWj
2j
r2 + 1
41 j2
j
jr2 + 1
1 j+ j
(38)
W in expression (38) is the product of three functions thatare increasing in j. Thus, g1 < g2 implies 1 < 2. For the
M/G/1 queue, the expected number of customers in thesystem depends only on its utilization and increases in .Therefore, by Littles Law, 2W = 2W2 > 2W1 = 1W, or
2 > 1 (39)
which contradicts (34).Given the value-based market segmentation structure,
the firms determine j and j by solving
max1 1
1P1 c1 g11
s.t. V1v + dW2 W1 P1 P2
(40)
andmax2 2
2P2 c2 g22
s.t. V1 + 21 vW2 dW2 P2 0
(41)
where Wj are given by Equation (5). Note that the constraintin Equation (40) follows from Equations (19) and (20) andthe Preliminary Step, and that Equation (41) is analogousto (10) for v > 0.
We now proceed to prove the core of Proposition 2.Step 1. By Assumptions (2) and (6), for v = d = 0 our
model is reduced to a standard duopoly with both firmsearning positive profits. Now consider some v slightlyabove zero and d slightly above or equal to zero.
We will first show that if we suppress the positive profitrequirement, a duopoly equilibrium exists and we will thenshow that for a small enough d, each duopolist makes a pos-itive profit. Each Sj (j= 1 2) decides on its capacity j andservice rate j, which maximize its profit given the strat-
egy of the other S (Sj). The Preliminary Step shows that aduopoly equilibrium must be characterized by value-basedsegmentationthat is, one of the providers, S1 or S2, servesthe higher-end segment and the other serves the lower-endsegment. The incentive compatibility and individual ratio-nality constraints determine the service fees as
Pj =
VjvWj Wj + Pj if Wj < Wj
V1 + 21 vWj dWj if Wj Wj(42)
From (3), the cross-partial derivatives of the payoff func-tions in (40)(41) are nonnegative. Therefore, by Theorems 4and 5 in Milgrom and Roberts (1990), the game is super-modular in 1 1 2 2 and an equilibrium exists.
We now show that for v and d small enough, bothduopolists earn positive profits. Theorem 2 in Milgrom andSegal (2002) shows that j (j = 1 2) are continuous func-tions of vd. By the continuity of jvd, j0 0 > 0implies that for sufficiently small vd, jvd > 0 aswell. Therefore, an equilibrium with both firms in the mar-ket exists. We denote the equilibrium arrival rate to Sj byjvd.
According to Theorem 6 in Milgrom and Roberts (1990)with exogenous parameter g1, 1 2, and 1 2. This
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implies that in equilibrium, 1vd 2vd when bothfirms are in the market. Indeed, if for some vd, 1vd 0. Because 1vd 2vd, S1 is stillin the market at such vd, and in 1 we have a duopolysolution with positive profits for both firms.
Step 3. Consider vd slightly outside of 1. Then, onlyone S, which we call Sm, can be in the market (because theother S will earn nonpositive profits if it enters the market).Then, Sm must set its capacity m and probabilities m tomaximize its profit:
Mm vd = maxm m
mPm cm gmm
s.t. Vm 1 vWm dWm Pm
(44)
Because we have chosen vd slightly outside of 1 andbecause profit goes to as m , there is an interiorsolution that satisfies the FOC. From Equation (5), Wm/m= r2 + 1/2m m
2 and Wm/m = mr2 + mm/
m m2 + mr
2/m m + 11/22m, which together
with the FOC gives the equilibrium condition (24).Step 4. Let jvdj (j = 1 2) be the optimal profit
determined as a solution to FOC for fixed vd and j. Let2 = vd
1vd = argmax
M2 vd 0. Because
then both firms are in the market, 1vd 2vd,1 2.
Let 3 = vd M2 vd 0. Because
1vd0 =
max M
1
vd, 2 3. Thus, in the set 3\2 there aretwo equilibria with either S being a monopoly. Becauseg1 < g2, for any value of vd,
M1 vd = max
V1 vW dW c g1
max
V1 vW dW c g2
= M2 vd (45)
Hence, 3 4. Because the monopoly profit is a decreas-ing function of v and d, in the set 4\3, only S1 can be amonopoly, and for vd 4, the market is not served.
Proof of Corollary 1. Corollary 1 follows from the Pre-liminary Step in the proof of Proposition 2.
Proof of Proposition 3. We first show that there isa symmetric mixed-strategy equilibrium. Then taking thelimits of the results in Proposition 2 shows that thereare two asymmetric equilibria with value-based marketsegmentation.
To prove the existence of a mixed-strategy equilibrium,it is sufficient to prove the existence of an equilibrium forthe game
maxj j
jPj cj gj