Que
Faculty & Research
uing for Expert Services
by L. Debo
B. Toktay and
L. Van Wassenhove
2004/46/TM
Working Paper Series
Queuing for Expert Services
Laurens G. Debo
Graduate School of Industrial Administration
Carnegie-Mellon University
Pittsburgh, PA 15213
USA
L. Beril Toktay
Technology Management
INSEAD
77305 Fontainebleau Cedex
France
Luk N. Van Wassenhove
Technology Management
INSEAD
77305 Fontainebleau Cedex
France
March 2004
1
Abstract
Many services like car repair, medical, legal and consultant services have the characteristic of a
credence good: The customers cannot verify, even ex post, whether the amount of prescribed service
was appropriate or not. This may create an incentive to provide unnecessary services, that is, the
expert may perform extra service that is of no value to the customer, but that allows the expert
to increase his revenues. This is called demand inducement. Rational customers process ex ante
the incentive of the expert to advise unnecessary service and update their valuation of the service
accordingly. A low workload level (an idling expert), combined with a fee per unit of service time,
may indicate that the expert has a high incentive to prescribe unnecessary service. On the other
hand, a high workload level (a busy expert), or a fixed fee independent of the actual service time
may indicate that the expert has little incentive to perform unnecessary service. When the arrival
rate of potential customers is stochastic and the (true) service time is also stochastic, the workload
level of the expert changes dynamically over time, and impacts an arriving customer’s valuation and
hence his decision to seek service or not.
In this paper, we analyze the optimal service strategy of a monopolist expert offering a credence
good. The monopolist can choose which payment scheme to adopt (with a fixed and/or a variable
component) and whether to reveal the workload information to the customer or not, as well as how
much demand to induce, if at all. We use a simple queuing model with a Poisson arrival process
of potential customers and exponential (true) service times to model the workload dynamics. We
characterize the equilibrium strategies of the customer stream in response to possible strategies
adopted by the expert. We then derive the expert’s optimal pricing and workload information
revelation strategy as a function of the characteristics of the environment (service capacity, potential
market size, value of the service and waiting costs). We find that a monopolist expert has the highest
incentive to induce service when the arrival rate of potential customers is slightly less than the true
service rate. Under these conditions, he reveals the workload level and charges both a fixed and
a variable rate fee. Interestingly, the expert attracts more customers in this case than an expert
that never induces services. When the true service rate is significantly higher than the arrival rate,
the expert prefers not to induce services, conceals the workload level and charges a fixed fee for the
service. When the true service rate is significantly lower than the arrival rate, the expert prefers not
to induce services, but reveals his workload level and charges a fixed fee only. Finally, we find that
service induction may increase the total welfare composed of customer utility and expert profits.
2
1 Introduction
In many service contexts, customers do not know the appropriate level of service required for a
complex product or operation. They rely on the advice of an ‘expert’ who typically also provides
the subsequent service. Furthermore, it is difficult for the customer to verify whether the provided
service was appropriate, even after the service was performed. If selling more services than what is
really required allows the expert to make a higher profit, a moral hazard problem is created: The
expert has then an incentive to advise unnecessary service. In the literature, this is referred to as
‘demand inducement.’
An example is car repair. Car owners typically know when their car needs repair, but cannot
judge the severity of the problem. The mechanic, on the other hand, can. After the repair is
complete, the owner can observe that the car is running without a problem, but he cannot identify
whether unnecessary services have been provided. In this case, the mechanic may have an incentive
to advise unnecessary service in order to capture extra revenues. A noteworthy case occurred in 1992.
Undercover agents of the Californian Bureau of Automotive Repair (BAR) found that Sears, Roebuck
and Co. was systematically defrauding customers by performing unnecessary repairs averaging $233
(Anonymous 1992, Santoro, 1992). At that time, the Sears auto repair chain was the largest in the
US, servicing 20 million cars annually (Callahan, 2004). In September 1992, they agreed to one
of the largest fraud settlements in history. Sear’s sales subsequently fell by 15-20%. The costs of
auto-repair fraud in the US are estimated to be $40 billion a year (Callahan). Ample anecdotal
evidence about moral hazard problems in a car repair context can be found in the press (The Wall
Street Journal, June 12, 1992; Llosa 1996; Koblenz 1999).
Another example of services with a similar moral hazard problem is medical advice. There is an
ongoing debate in the health care literature about the existence of physician induced demand. In
a recent empirical study, Delattre and Dormont (2002) show evidence of physician-induced demand
in France. They find that the number of consultations per doctor only slightly decreases with an
increase in the physician/population ratio. In addition, physicians counterbalance the fall in the
number of customers by an increase in the volume of care delivered in each encounter.
A third example is legal advice. Drawing on his surveys, the experiences of legal audit firms,
and anecdotes, Ross (1996) concludes that over-billing is widespread among attorneys. Much of the
‘padding’ of hours is impossible to detect and “can escape the attention of even the most dedicated
sleuth” (p. 23).
Since it is expensive, if not impossible, to detect service inducement, in the absence of ethical
considerations, can we conclude that experts will always choose to induce demand? Two important
factors affecting the expert’s incentive to induce service are the service price structure and the
workload level of the expert. A flat-rate price structure (independent of the level or type of service
provided) makes it unprofitable for the expert to induce service. A variable-rate structure (composed
3
of a flat fee plus a variable component that is proportional to the level of service provided), on the
other hand, makes it feasible for the expert to consider service inducement. Even with a variable
rate contract, service inducement is only possible if the expert’s workload is less than its capacity
to perform these extra services. Ross, for example, mentions that the incentives to padding and
excessive time only can occur for lawyers who are ‘not busy’. Attorneys that are busy do not have
the time to do unnecessary tasks (p. 36-37).
Rational buyers of services with a credence good characteristic process ex ante the incentive of
the expert to induce services and update their valuation of the service accordingly. This affects
their decision of whether to purchase the service or not and has revenue implications for the expert.
The same factors that influence the expert’s decision concerning service inducement can serve as
indicators to the customer in making this evaluation. For example, the use of a variable-rate contract
alerts the consumer to the fact that the expert may benefit from service inducement. When, in
addition, the customer observes a low workload level, he may expect that the expert has an even
stronger incentive to induce service. If the customer judges the likely cost of service inducement to
be too high, he may decide not to purchase the service. This phenomenon deters service inducement.
As service inducement is quite a widespread phenomenon in different service industries, we want
to better understand its underlying economic motivation. Although in practice there are many
variables deterring service inducement, like the possibility of a second opinion, the concern for
repeat business (reputation) or ethical concerns, we focus here on the role that congestion plays in
such a context. We refer to the literature for discussions of these other variables and the expert
as a monopolist (i.e. no second opinion is possible) who does not have any ethical concerns about
service inducement and has short term interactions with many different customers (i.e. reputation
does not matter). Our analysis is not a normative one: We do not want to provide advice about
when it is optimal to induce services, rather; we want to generate insights for managers of service
systems about the temptations that expert service providers have in similar environments.
We demonstrate that even in the simplest setting with homogenous customers, the expert’s pric-
ing strategy and workload revelation strategy does depend on the characteristics of the environment
(service value, waiting costs, market size and service capacity) in a non-trivial way. We determine
the optimal price structure and workload information revelation strategy for the expert. By setting
a flat rate for service (independent of the amount of work that is necessary to perform the service),
the expert can credibly signal that he has no incentive to induce service. By choosing to make the
workload observable to the customer or not, he can influence each customer’s evaluation of whether
he will experience service inducement or not. Each of these levers affects both the inducement-based
revenue generation potential from an incoming customer and the decision of a customer to purchase
service from the expert.
In the next section, we describe the related literatures in economics and operations management.
4
In the sections following, we describe the models with observable as well as unobservable queue
length. In the sections following, we analyze the models and discuss the results.
2 Related literature
Darby and Karni (1973) coined the name ‘credence good’ for a good whose quality cannot costlessly
be ascertained by the customer even after purchasing it. This is in contrast to ‘experience goods’ for
which usage reveals quality (e.g. whether a car is a ‘lemon’ or not). Using a simple structure where
the probabilities of the customer refusing the current service and refusing future service are known
increasing functions in the level of the service proposed, they show that the higher the anticipated
future profits from a customer, and the higher the ability of the customer to evaluate the nature of
the proposed service, the lower the tendency to induce demand.
Pitchik and Schotter (1987) characterize the Nash equilibrium in a single-customer single-shot
game where the customer may need either ‘minor’ or ‘major’ service. Major service must be provided
truthfully, but the expert can provide honest or dishonest advice (service inducement) to customers
requiring minor service. The customer can either accept or reject the expert’s advice. They find
that a mixed strategy equilibrium (in which both customer and expert randomize) will result.
Wolinsky (1995) studies the market equilibrium with one customer and several competing experts
who also need either minor or major service. Again, major service must be provided truthfully, but
service inducement may take place on the minor service. Experts post prices for these two types of
repair services. Customers visit experts, but may switch to other experts after receiving the expert’s
advice. In markets with a large number of experts, Wolinsky finds that the customer’s search for
multiple opinions and the expert’s concern for repeated sales (his ‘reputation’) induce honest advice.
In markets with sufficiently few experts and without reputation effects, Wolinsky finds specialization
of experts for minor and major services in equilibrium. Pesendorfer and Wolinsky (2000) extend
this model to the case where the diagnosis quality is a function of the expert’s effort, where effort is
costly. They find that price competition may in fact reduce total welfare in this setting.
Both Darby and Karni, and Pitchik and Schotter take prices as given and do not consider the
expert’s price-setting problem. Prices are determined as a consequence of competition in Wolinsky,
and Wolinsky and Pesendorfer. In all four papers, inducing demand is costless for the expert.
Emons (1997) studies the market equilibrium with several competing, capacity-constrained ex-
perts servicing identical customers. The capacity requirements of customers for diagnosis and repair
are deterministic. Diagnosis and repair are verifiable; therefore, all prices charged correspond to
actual services that have indeed used some capacity. If total demand is larger than total industry
capacity, Emons finds that, in equilibrium, the experts charge a flat price equal to the reservation
price of the customers (that make the customers indifferent between buying the experts’ services
5
or not). With this price structure, the experts appropriate the entire consumer surplus while not
inducing any demand. If total demand is lower than industry capacity, competition pushes the
prices to marginal cost, experts are honest even with idle capacity, and all surplus accrues to the
customers. In either case, the service inducement is not observed; the capacitated nature of the
resource and the verifiability of the service provided preclude it.
Emons (2001) considers a monopolist who determines capacity and prices of diagnosis and repair,
and investigates the impact of different types of information available to customers. When customers
can observe services provided (whether or not they observe capacity) or observe only capacity, the
expert invests in the capacity level that is exactly equal to the capacity required to serve the whole
market honestly. When capacity is not observable, he signals his credibility by pricing repair services
such that there is ‘no money in repair’ and setting a flat diagnosis fee in such a way as to extract
all consumer surplus. When capacity is observable, multiple diagnosis-repair prices can exist in
equilibrium, all of them extracting the total surplus of the customers.
While in Emons’ papers, service inducement would consume capacity that could otherwise be used
for other customers, Alger and Salanie (2003) incorporate an explicit cost to the expert of inducing
demand. Customers are again identical. Therefore, in the monopolistic expert case, charging a
flat-rate regardless of service type is optimal and eliminates service inducement. However, price
competition makes flat-rate pricing less sustainable. In particular, the authors show that when
service inducement cost is low enough, competition results in service inducement.
Other literature on credence goods introduces customer heterogeneity with respect to prior infor-
mation, cost of service, or reservation price (Richardson 1999, Fong 2002, Dulleck and Kerschbamer
2003). An interesting result is that in this case, the expert does not find it profitable to charge a flat
price. The intuition is the following: Since with heterogenous customers, the expert is not able to
capture all surplus using a single price, he may find it optimal to selectively induce service to some
customer types.
Although capacity is an important element in the incentives of the expert to induce demand, it
has been modelled only by Emons, who assumes that service requirements are deterministic. Such
models ignore the economics of congestion however. While this may be appropriate for strategic-level
questions about industry capacity, competition etc., it is not particularly appropriate for modelling
service systems. Indeed, the Management Science literature is rich with models that show the
importance of taking the congestion effect into account in the analysis of manufacturing and service
operations.
A key driver in Emons (2001) is the difference between the capacity needed to service all demand
honestly and actual capacity. In particular, any utilization under 100% tells the customer that the
server has an incentive to induce demand. This logic obviously cannot be applied directly to a
classical stochastic service system because 100% utilization is not sustainable.
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In addition, the dynamic impact of the level of workload on the expert’s incentive to induce service
is worth investigating. In particular, a customer arriving when the expert has a low workload level
may expect a high incentive of demand induction. This issue has been qualitatively discussed (but
not analyzed) by Darby and Karni. We note that if the workload level is observable to the customer
upon arrival, then this is a source of effective customer heterogeneity despite the homogeneous nature
of the customer base. Following the recent literature on customer heterogeneity, we can postulate
that the expert may choose to induce demand in this case. In particular, the expert can, by choosing
whether to reveal the queue length or not, affect whether the customer base remains homogeneous
or becomes effectively heterogeneous in the sense described above.
In this paper, we consider a monopolistic capacity-constrained expert and homogeneous cus-
tomers arriving sequentially for service. The arrival time between subsequent customers and the
service requirement of each customer is random. The expert (1) determines the pricing structure
and sets prices; (2) chooses to reveal or conceal the queue length. We solve the expert’s optimization
problem and investigate conditions under which combinations of flat-rate or variable rate contracts
and workload revelation or not would be chosen by the expert.
The queuing model we develop for this analysis fits in the queueing literature that takes into
account the strategic interaction between the server and the customer. Such a strategic interaction
in a queueing context was first studied by Naor (1969). This paper and the subsequent literature (for
an excellent overview, see Hassin and Raviv 2002) study the impact of congestion on the customers’
and service provider’s decisions. The closest models to our ‘credence good’ problem can be found
in papers on service rate decisions made by the expert (Hassin and Raviv, Chapter 8) but in these
models, decreasing the service rate does not correspond to inducing demand, rather it means that the
true service time of each customer increases. Asymmetric information models in this context assume
that it is the expert that does not observe the customer’s type (e.g. Whang 1989, Balachandran and
Radhakrishnan 1996). In another class of models on reneging, Mandelbaum and Shimkin (2000)
consider a model where the customer is not fully informed about service: When a customer decides
to join the queue, he is accepted only with probability q, but he is not informed about whether he
has been accepted or not. As time progresses, the customer updates its assessment of the probability
of having been rejected; eventually, he reneges. To the best of our knowledge, a model equivalent to
the ‘credence good’ problem that we develop here has not been studied in this literature.
3 The model
The customers. The customer base is homogeneous. They have a reservation price V for the
service, and arrive at the expert according to a Poisson process of rate Λ with a service requirement
whose duration is exponential with mean t. Since our focus is the impact of the pricing structure
7
and workload level revelation on the expert’s incentives to induce service, we do not incorporate
the expert’s concern for repeat services: Each customer in the Poisson stream represents a new
customer who does not have a history of transactions with the expert.1 Each customer incurs a
dis-utility of c per unit time spent at the expert (in queue or in service). We assume an additive
utility structure for the expert. If the expert decides to purchase service, his expected utility is
V − E[service cost + waiting cost]. If the customer decides not to purchase the service, he gets
0 utility. Note that by allowing the expert to not purchase service, we are making an implicit
assumption that either the service is not of emergency nature and can be foregone, or the consumer
has some outside option which yields a fixed utility level, normalized to 0, and unaffected by the
expert’s strategy.
Information asymmetry concerning service time. The exact service time required by an
arriving customer is denoted by t. The expert observes t, but the customer cannot. (In fact it’s
sufficient for our analysis to assume that the server can detect when service is complete, but that the
customer cannot). If the expert works less than t, then, the customer is not fully serviced yet. We
assume that incomplete service is observable by the customer and that an institution exists where
the customer can hold the expert liable for incomplete servicing. Therefore, the expert works for at
least t units of time. Let t denote the total service time the customer experiences. We say that the
expert ‘induces service’ if t > t. We refer to t as the ‘true service time’ and to t− t as the ‘induced
service time’. The occurrence of ‘service inducement’ is neither observable by the customer nor by
an outside agency2.
The expert. We assume that the expert has a monopoly position in the market. He decides the
pricing structure, what information to reveal to customers and the service inducement policy. He
serves the customers in a first-come first-served manner.
Information revealed to customers. We assume that the expert has control over the ability of
customers to observe the number in the system.
Pricing structure. Let (R, r) be the flat fee and service rate per unit of service time. Customers
pay for the total reported time by the expert, i.e. R+ rt. We refer to a ‘flat-rate’ contract if r = 0,
otherwise, we refer to a ‘variable-rate’ contract.
Inducement strategy. Let z ∈ N denote the number of customers in the queue upon completion of
the true service time t of the customer in service. We consider the following service policy of the
expert: If z > 0, then the expert does not induce any service time for this customer. If z = 0, then
1According to Gallahan, ‘the anonymity of corporate law makes cutting edges less troubling. Unlike the old days,
there is little loyalty between law firms and clients.’ (p. 35) The Poisson arrival stream generating new customers is
appropriate in that case.2See for example Ross (1996) about litigation for excessive bills for legal advice. When replacing parts for e.g. car
repair, a common practice is to return the broken part to the customer in order to show that no well functioning part
has been replaced. For services not including parts replacement, it is a lot more difficult to show that no unnecessary
service has been done.
8
with probability α, the expert induces service until a new customer arrives. Thus, if α = 0, the
expert is honest, if α = 1, the expert always induces service when it would normally remain idle,
and as a consequence never idles. In that case, the utilization rate of the expert is 100%. Such
a rate cannot be achieved in a classical queuing system with stochastic arrival and independent
service times. In our model 100% utilization rate is only possible because service inducement on a
customer ends as soon as a new customer decides to join. This model allows us to study service
inducement using elements of classical queuing theory. For example, if α = 1, then the expected
demand inducement the expert will do equals the expected idle time in the underlying queue. We
focus on situations where the marginal cost of providing services is equal to zero. In practice, the
expert may be hindered by ethical concerns when inducing services, or, there may be a probability
that he will be caught when inducing services that defers him from inducing services. In order to
have the sharpest insight in the role of the price structure and the workload information to the
customer, we assume that the expert incurs no direct cost when inducing services. The only effect
of inducing services is that it uses the spare capacity of the expert (if available) and makes the
customer wait also during the time of the induced service.
The customer strategy. An arriving customer decides whether to enter service or not. We assume
that a customer who decides to enter does not subsequently renege. The customer strategy depends
on what information is available to an arriving customer.
(i) Unobservable queue length. We use the subscript u to refer to the game with unobservable queue
length. Let Su ∈ join, balk and consider a randomized strategy such that β = P (Su = join).
(ii) Observable queue length. Let n ∈ N denote the number of customers in the queue and in service
at the arrival time of a potential customer. When a customer arrives, the expert informs him about
the number of customers in the system. If there is a customer in service but he is in the demand-
inducement phase, the expert reports n = 0. Each customer makes a decision whether to join or
balk depending on n. Let Sn ∈ join,balk for all n ∈ N be the customer’s strategy profile. A
threshold strategy can be characterized by β ∈ R+, with n = bβc and p = β − bβc such that
Sn =
join if n ∈ [0, n− 1]
join with probability p if n = n
balk if n ∈ [n+ 1,+∞]
(1)
If β is integer, then we have a pure threshold strategy, o/w we have a mixed threshold strategy. In
order to keep the notation simple, we drop the dependence of n and p on β.
Information availability. We assume that the services provided are verifiable, either by the
customer or by some agency. This means that the expert cannot claim to have done work without
actually doing it. This assumption ensures that service inducement has an implicit ‘cost’ to the
expert - it uses up limited capacity.
We also assume that the price structure and all other parameters are common knowledge since the
9
focus of this paper is to analyze the impact of information asymmetry concerning the exact service
requirement of the customer. We nevertheless say a few words about which of these assumptions
are more likely to hold in practice.
It is reasonable to assume that the customer typically is aware of the billing structure. In
particular, since asking for advice is costly, and having spent that money creates a hold-up problem
in which the expert can drive prices up, the customer would want to see detailed information about
the price structure before deciding to purchase the service. The expert would need to commit to
these prices because legal action can ensue if he deviates from posted prices.
In most cases, the expert can judge, with a fair degree of accuracy, the level of demand for its
services. Customers that inquire about prices but do not subsequently choose service can be one
source of information. Information such as the incidence rate of certain medical conditions in the
population, maintenance and repair requirements of various car brands, etc. are types of information
publicly available that the expert can use.
Specification of the game. We consider a two-stage game. In the first stage, the expert chooses
(R, r), which is observable by all customers. In the second stage, the customers determine their join-
ing strategy. Simultaneously, the expert determines his service inducement strategy. This is a game
with one ‘long-lived’ player (the expert) and infinitely many ‘short-lived’ players (the customers).
In the subgame equilibrium, the strategy of each individual player (expert or customers) is optimal
given all other players’ strategies; no player has an incentive to deviate from this equilibrium. We
focus on symmetric equilibria in which all customers follow the same strategy. We first determine the
equilibrium conditions of the customers’ strategies, keeping the expert’s strategy fixed (the second-
stage customer equilibrium for a given α). Next, we determine the equilibrium condition for the
expert’s action given a symmetric customer strategy. Customer and expert strategies that satisfy
both conditions are Nash equilibria of the subgame (the second stage expert-customer equilibrium).
In the first stage, the expert determines the pricing structure that maximizes his profit taking into
account the second-stage equilibrium that will result (the first stage equilibrium).
Below we derive the conditions satisfied by equilibrium strategies in the unobservable and ob-
servable queue length cases, respectively. The detailed characterization and analysis of the game is
left to sections following.
4 Equilibrium specification and analysis under unobservable
queue length
Since the queue length is unobservable, the customer strategy is a randomized strategy where β is
the probability of joining the queue.
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4.1 The second-stage customer equilibrium for a given α
We focus on symmetric randomized strategies where all customers use the same randomization
strategy β. Therefore, the effective arrival process is a Poisson process with rate βΛ. Since the
service times are independent and have an exponential distribution, the system is an M/M/1 queue.
For given prices (R, r), setting the strategy of all other customers to β and the strategy of the
expert to α, the expected ex ante utility of a customer who joins the queue is Uu (α, β;R, r) =
V − R − cWQ (βΛ) − (c+ r) t (α, β). Here, WQ (βΛ) is the expected queueing time in an M/M/1
queue with arrival rate βΛ and expected service time t, and equals βΛt2
1−βΛt . t (α, β) is the expected
service time that an entering customer will experience, and is equal to t + α (1 − βΛt) 1βΛ . The
first term is the true expected service time. The second term is the expected length of the induced
service time. To see this, note that service induction occurs with probability α if a departing
customer would leave the system empty, which occurs with probability 1 − βΛt. The expected
length of service induction is 1βΛ , which is, due to the memoryless property, the expected time until
the next customer arrival of a Poisson process with rate βΛ. This completes the characterization of
the payoff of a customer in the game.
The best reaction of a customer, when all other customers play strategy β, is to join if Uu (α, β;R, r) >
0, to balk if Uu (α, β;R, r) < 0, and to randomize between joining and balking if Uu (α, β;R, r) = 0.
Define Nu (β;α,R, r) : [0, 1] → 2[0,1] (where 2X denotes the set of all subsets of X),
Nu (β;α,R, r) =
[0, 1] if Uu (α, β;R, r) = 0
1 if Uu (α, β;R, r) > 0
0 if Uu (α, β;R, r) < 0.
Let βeu (α;R, r) ⊆ [0, 1] be the set of equilibrium strategies for the customers for a given pair of
prices, (R, r), and expert behavior, α. βeu (α;R, r) can be characterized as follows:
β ∈ [0, 1] : β ∈ Nu (β;α,R, r) , (2)
that is, a symmetric equilibrium strategy β is such that β is in the best response graph of a customer
when all other customers use strategy β. Let A denote V −R− (r+c)t, which is the expected utility
of a customer finding the system empty and not experiencing any demand inducement. A necessary
condition for a customer getting positive expected utility from entering service is that A > 0.
Proposition 1 For any (R, r) ∈ R2+ and α = 0, βeu (0;R, r) = min
(β0u, 1) where β0
u = AA+ct
1Λt
if A > 0 and βeu (0;R, r) = 0 otherwise. For any (R, r) ∈ R2+ and α ∈ (0, 1], βeu (α;R, r) =
0,min(β1u, 1),min(β2
u, 1) where β1u < β2
u, where they exist, are the two distinct real roots of
Uu (βu, α;R, r) on [0, 1Λt ).
This result follows from Equation 2 and the characterization of Uu (β; 0, R, r) in the Appendix.
Possible equilibria are illustrated in Figure 1.
11
1
1
1
1
1
1(i) β β β
1
1 β
1
1 β
1
1 β
(ii) (iii)
(iv) (v) (vi)
Figure 1: These figures show possible best response graphs Nu(β) and the resulting symmetric equi-
libria. (i)-(ii) are for α = 0, and (iv)-(v) are for α > 0. (i) A ≤ 0, βeu = 0; (ii) A > 0, β0u < 1,
βeu = 0, β0u; (ii) A > 0, β0
u > 1, βeu = 0, 1; (iv) A ≤ 0 or A2 ≤ 4αc(c + r)t2, βeu = 0; (v)
A > 0 and A2 > 4αc(c + r)t2, β1u < 1 < β2
u, βeu = 0, β1
u, 1; (vi) A > 0 and A2 > 4αc(c + r)t2,
β1u < β2
u < 1, βeu = 0, β1u, β
2u.
Interpretation of Proposition 1. According to this proposition, whenever service is induced
(α > 0), nobody visiting is an equilibrium. To see this, take a customer who considers visiting the
expert when all other customers’ strategies is to not visit. Then the expected service cost to this
customer will be very high (infinite, in fact), as the expert has an incentive to induce service until
the next customer arrives. Therefore, the new customer will decide not to visit either. Thus, nobody
visiting is an equilibrium.
According to Proposition 1, two other equilibria are also possible: β1u < β2
u. The customer’s
utility is determined by two cost components: queueing (congestion) cost and service cost (including
any service cost due to demand inducement). Fixing the strategy of the other customers at the lower
of the two probabilities, β1u, the expected service cost of a new customer is high (since the expected
time until the next customer arrival is long), but the congestion cost is low. Therefore, the new
customer may be indifferent between joining or not. If so, he will randomize with probability β1u. In
this case, we obtain an equilibrium with high expected cost of service induction and low congestion
cost.
Similarly, fixing the strategy of the other customers at the higher of the two probabilities, β2u,
the expected cost of induced services is low (since the expected time until the next customer arrival
12
is short), but the congestion cost is high. Therefore, the new customer may also be indifferent
between joining or not. If so, he will randomize with probability β2u. We then obtain an equilibrium
with low expected cost of service induction and high congestion cost. These three equilibria are
‘follow-the-crowd’ type of equilibria (Hassin and Aviv).
When α = 0, only congestion cost plays a role since the expected service cost does not depend
on the state of the system upon arrival. There exists a unique equilibrium visiting probability β0u
for which an arriving customer is indifferent between visiting or not.
4.2 The second stage expert-customer equilibrium
Fixing the prices (R, r) and setting the strategy of all customers to β, the expert’s profit rate is
πu (β, α;R, r) = RβΛ + r (βΛt+ α (1 − βΛt)). The expert earns R on each joining customer and,
as long as there are customers in the system, earns r per unit of time that he performs service
(true or induced). The fraction of time that the expert works on true service is βΛt, which is the
utilization of the M/M/1 queue. The rest comes from induced busyness. Let αu (β;R, r).= arg max
α∈[0,1]
πu (β, α;R, r) ⊆ [0, 1]; this is the best response set of the expert fixing the customer strategy at
β. Since πu (β, α;R, r) is linear in α with coefficient (1 − βΛt)r, αu (β;R, r) = 1 if r > 0, and
αu (β;R, 0) = [0, 1] if r = 0.
Let Bu (R, r).=
(β, α) ∈ [0, 1]2
: β ∈ βeu (α;R, r) and α ∈ αu (β;R, r)
. This is the set of all
expert-customer equilibria in the second stage. Finally, let A be a rule that allows to select one
equilibrium from Bu (R, r). Then (β∗u (R, r) , α∗
u (R, r)).= A (Bu (R, r)) is the selected equilibrium
for a given pair of prices, (R, r). For analytical convenience, we select in this paper the equilibrium
for which the probability that a customer enters is the highest (equivalently, for which the expected
queue length is the largest).
Proposition 2 If r > 0, then α∗u (R, r) = 1 and β∗
u (R, r) = min(max(0, β1
u, β2u), 1
). If r = 0, then
α∗u (R, 0) = 0 and β∗
u (R, 0) = min(β0u, 1)
if A > 0 and β∗u (R, 0) = 0 otherwise.
For r > 0, the expert has an incentive to induce services. Since we do not consider any kind of
direct costs of service induction, the expert chooses to induce service when a departing customer
would leave the system empty: α∗u (R, r) = 1. Since we select the equilibrium with the highest prob-
ability of joining, α∗u (R, r) = 1 and the largest of the corresponding customer equilibria determine
the subgame equilibrium.
For r = 0, the expert is indifferent between inducing service or not. Therefore, any α ∈ [0, 1]
may be an equilibrium. The corresponding customer equilibria decrease in the probability of demand
induction, α. Since we wish to select the equilibrium with the highest probability of entry, we select
α∗(R, 0) = 0. For a feasible subgame, min(β0u, 1) determines the subgame equilibrium.
13
4.3 The first-stage equilibrium, or The Expert’s Optimal Pricing Deci-
sion.
Since for all possible sub-games given (R, r), we have characterized an equilibrium with unique
payoff for the expert, we can now derive the optimal (R, r) maximizing the expected profit of the
expert that the expert would select in the first stage. Let Πu (R, r).= πu (β∗
u (R, r) , α∗u (R, r) ;R, r)
be the equilibrium profit rate for the expert for (R, r). Then, the first stage equilibrium (R∗u, r
∗u) is
determined as follows: (R∗u, r
∗u)
.= max
(R,r)∈R2+
Πu (R, r). Finally, let Π∗u = Πu (R∗, r∗).
We define ρ.= Λt and v
.= V
ct. v measures the number of times that one customer’s expected
waiting cost ct during service (excluding waiting in a queue) is contained in the value of the service
(V ) and therefore is the ‘profit potential’ of the system, not taking any effects due to congestion
into account. ρ is the ratio of potential market demand rate (Λ) relative to the true service rate
( 1t). Note that this is not the utilization of the server - the utilization is given by βΛt. Therefore ρ
as defined here, can take values above 1. We refer to ρ as the normalized demand.
Proposition 3 In the case of unobservable queue length, a flat rate contract is optimal, with r∗ = 0,
R∗ = V − cW(min
(1t−√
ctV,Λ))
and Π∗u = c
(v − 1
1−ρ
)ρ, where ρ
.= min
(1 −
√1v, ρ).
The intuition behind this Proposition is the following: Since customers do not observe the queue
length, they are homogeneous and the expected surplus that the expert can extract from each
customer is determined by the service value minus the expected total waiting time for the customer,
V − c (WQ (βΛ) + t (α, β)). Since all customers are homogenous, the expert can extract all the
surplus. This surplus is maximal when there is no service inducement (t (0, β) = t). Therefore, a
fixed rate contract is optimal for the expert. This result is reminiscent of Dulleck and Kerschbamer
(2001) and Emmons (1999, 2001), who identify consumer homogeneity as one of the conditions under
which the monopolist’s credence good problem can be solved by means of a fixed price, extracting
all customer surplus. In our case, customers are ‘ex-ante’ homogenous as the service value, V , and
unit waiting cost, c, are the same for all customers. Since no customer observes the length of the
queue upon arrival, customers are also ‘ex-post’ homogenous. In the next section, we relax ex-post
homogeneity and allow customers to observe the queue length upon arrival.
5 Equilibrium specification and analysis under observable
queue length
Since the queue length is observable, the customer strategy is a function of the number of customers
he finds in the system.
14
5.1 The second stage customer equilibrium for a given α
We focus on symmetric threshold equilibria as defined by (1). For given prices (R, r), setting the
strategy of all other customers to β and the strategy of the expert to α, the expected ex ante utility
of a customer who joins the queue in state n is Un (α, β;R, r) = V − R − cnt − (c+ r) tn (α, β).
Here, cnt is the expected queueing time when n customers are in the system upon arrival and the
expected service time is t. tn (α, β) is the expected service time experienced by a customer who
enters when there are n other customers in the system, and includes the expected true service time
and the expected induced service time given that there are n customers upon arrival.
Lemma 4 tn (α, β) = tIαmin(n+1,ξ(β)−1)(ρ), where Iαz (ρ) = 1+αρ
(1
1+ρ
)zand ξ (β) = n+
ln(p(1−n)++pρ
)
ln(1+ρ) ,
with n = bβc and p = β − bβc.
A complete proof for β ∈ R can be found in the Appendix. Nevertheless, to understand this
result better, let us interpret it for β ∈ N. Consider a potential customer arriving at the system and
finding n other customers. Suppose that all other customers (past and future) adopt a pure threshold
strategy β = n (integer) (join if Sn ≤ n−1). Then p = 0 and ξ = n. If the new customer joins, he will
bring the system state to n+ 1. Two cases are possible: (i) n+ 1 ≤ n− 1. Then, the next arriving
customer will join the queue. If this happens, the current customer will experience no demand
inducement. He can only experience demand inducement if no other customer enters until the end
of his true service time, which happens with probability(
11+ρ
)n+1
(probability of n+ 1 departures
before an arrival). Then service is induced with probability α and the expected inducement time
is 1Λ , due to the memoryless property. Therefore, we obtain tn (α, β) = t + α
Λ
(1
1+ρ
)n+1
. (ii)
n + 1 > n − 1. No arriving customer will join the queue until the system state returns to n − 1 as
a result of customer departures. In states n and below, customers do join upon arrival. Thus, the
probability that no other customer joins before the end of the true service of the current customer
is(
11+ρ
)n−1
. Therefore, we obtain tn (α, β) = t + αΛ
(1
1+ρ
)n−1
. Putting the two cases together
gives tn (α, β) = t + αΛ
(1
1+ρ
)min(n+1,n−1)
= tIαmin(n+1,n−1)(ρ). Iαmin(n+1,n−1)(ρ) is interpreted as
the inflation factor: With no demand inducement this factor would be 1. Lemma 4 completes the
characterization of the payoffs of customers in the game.
The best response of a customer who arrives to find n in the system is to join if Un (α, β;R, r) > 0
and balk if Un (α, β;R, r) < 0. If Un (α, β;R, r) = 0, the customer is indifferent between joining and
balking in state n. Define Ns (β;α,R, r) : R+ → 2N with
Ns (β;α,R, r).= n ∈ N : Un′ (α, β;R, r) ≥ 0 for 0 ≤ n′ ≤ n− 1 and Un′ (α, β;R, r) ≤ 0 for n′ ≥ n .
(3)
Ns (β;α,R, r) is the best response set of pure threshold strategies of an arriving customer when all
other customers adopt a (possibly mixed) threshold strategy β ∈ R+. To see this, first suppose that
15
for a given β, there exist a unique n such that Un′ (α, β;R, r) > 0 for n′ ≤ n−1 and Un′ (α, β;R, r) <
0 for n′ ≥ n. Then Ns (β;α,R, r) = n: n is the customer’s best response pure threshold strategy
to β: He will enter at any state less than or equal to n − 1 and not enter at higher states. Now
suppose that for a given β we have Un′ (α, β;R, r) > 0 for n′ ≤ n − 1, Un (α, β;R, r) = 0, and
Un′ (α, β;R, r) < 0 for n′ ≥ n + 1. Then Ns (β;α,R, r) = n, n + 1: n and n + 1 are both
the customer’s best response pure threshold strategies to β: He is indifferent between the two
strategies and could randomize between them with any probability to specify a mixed threshold
strategy. By allowing for randomization strategies at such points, we can extend Ns (β;α,R, r) to a
correspondence Nc (β;α,R, r) : R+ → 2R+ for which Nc (β;α,R, r) = [n, n+ 1] ⊂ R+. For given α
and (R, r), the set of equilibrium threshold strategies βe (α;R, r) is characterized as follows:
β ∈ R+ : β ∈ Nc (β;α,R, r) . (4)
Figure 2 illustrates one example with five equilibrium symmetric threshold strategies. The next
proposition specifies the conditions that must be satisfied for a mixed or pure threshold strategy
to be an equilibrium strategy for a given inducement probability α ∈ [0, 1] and a pricing structure
(R, r) ∈ R2+.
Proposition 5 For any (R, r) ∈ R2+ and α = 0, βe (0;R, r) consists of the pure and mixed threshold
strategies satisfying the conditions below:
pure strategy equilibria n ∈ N mixed strategy equilibria β ∈ R+ \ N, with bβc = n
0 ≤ n :
Un−1 (0, n;R, r) ≥ 0 (a)
Un (0, n;R, r) ≤ 0 (b)0 < β : Un (0, β;R, r) = 0 (c)
For any (R, r) ∈ R2+ and α ∈ (0, 1], the set βe (α;R, r) consists of the pure and mixed threshold
strategies satisfying the conditions below:
pure strategy equilibria n ∈ N mixed strategy equilibria β ∈ R+ \ N, with bβc = n
n = 0 0 < β < 1 : U0 (α, β;R, r) = 0
1 ≤ n :
U0 (α, n;R, r) ≥ 0 (d)
Un−1 (α, n;R, r) ≥ 0 (e)
Un (α, n;R, r) ≤ 0 (f)
1 < β :
U0 (α, n;R, r) ≥ 0 (g)
Un (α, β;R, r) = 0 (h)
Interpretation of Proposition 5. When α = 0, Un is a strictly decreasing function of n for
any β: As the number in the system increases, the utility a customer derives from joining the queue
decreases. A pure strategy equilibrium β = n is one that leaves a customer finding n − 1 others
indifferent or willing to join (a), and finding n others indifferent or unwilling to join (b). A mixed
equilibrium strategy β is one that leaves a customer finding bβc others indifferent between joining
or not (c).
16
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
6
6.5
7
7.5
8
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8
β
Figure 2: Best response correspondence Nc (β, α,R, r) , β ≥ 0 given R = 6, r = 10 and α = 1 with
Λ = 0.75, V = 25, c = 1 and t = 1. The set of equilibrium threshold strategies βe(1; 6, 10) consists
of three pure-equilibrium strategies (0, 3 and 7), and two mixed-equilibrium strategies (2.901 and
3.263).
It can also be verified that when α > 0, Un is strictly concave and unimodal in n: A low number
of customers means high expected inducement cost and low queueing cost for an arriving customer.
As the number of customers increases, the expected inducement cost goes down, but the queueing
cost increases. The two opposing effects lead to a concave unimodal utility structure. The difference
in utility structure with α = 0 (no inducement) and α > 0 (inducement) is the primary determinant
of the different behavior of these two strategies, as we shall see later.
When α > 0, conditions that determine an equilibrium threshold n should impose (d) a non-
negative utility upon arrival at an empty system (U0 ≥ 0), (e) a non-negative utility upon arrival
at a system with n − 1 customers and (f) a non-positive utility upon arrival at a system with n
customers. (Note that the latter two conditions are sufficient to assure the first when α = 0 since the
utility strictly decreases in n in that case). If customers had strictly negative utility from entering
an empty system, any system that ever empties (which happens with probability 1 for utilization
levels of less than 1) would not attract customers again and would remain empty. Thus, in order to
have customers in the long run, it is necessary that an arriving customer has non-negative expected
utility. This is assured by condition (d). Conditions (e) and (f) assure that if all other customers
17
select a threshold policy n, the best response of a new customer is also the threshold policy n.
Conditions (d), (e) and (f) are formalized in Proposition 5. A similar rationale holds for mixed
strategy equilibria, resulting in conditions (g) and (h).
According to this proposition, whenever service is induced (α > 0), nobody visiting is an equilib-
rium. The rationale is the same as in Proposition 1. Provided that (a) is satisfied, a pure strategy
equilibrium is determined by n satisfying Un−1 (α, n;R, r) ≥ 0 ≥ Un (α, n;R, r). Note that, as in
the unobservable case, two costs components determine Un−1 (α, n;R, r): waiting costs and costs of
service induction. For low values of n, service induction costs are very high, as the probability that
no other customers arrives before the end of the true service is very high. Therefore, there may be
an equilibrium with a low threshold value. For high values of n, service induction costs are low, but,
waiting costs are high. Therefore, there may also be an equilibrium with a high threshold value.
We have thus identified a similar ‘follow the crowd’ effect as in the unobservable case. Due to the
discreteness in the strategy space, there may be more than three equilibria in the observable case.
The economic intuition behind this multiplicity, however, is the same in both cases and is due to
the service induction effect.
5.2 The second stage expert-customer equilibrium
For fixed (R, r), the expert’s expected profit rate depends on the customer’s strategy, β and α. The
arrival rate at each state of the system is determined by β. For each joining customer, the expert
makes a profit of R. As long as there are customers in the system, the expert earns r per unit of
time. At times the system would have been empty with an honest policy, the expert makes r per
unit of time with probability α. Let δn (β).= P (Sn = join) and let pn (β) be the limiting probability
of state n. The expert’s profit rate is then
π (β, α;R, r) = RΛ
∞∑
n=0
pn (β) δn (β) + r
( ∞∑
n=1
pn (β) + αp0 (β)
).
For a fixed β, the expert’s best response is α (β;R, r) = arg maxα∈[0,1]
π (β, α;R, r) ⊆ [0, 1]. Let B (R, r).=
(β, α) ∈ R+ × [0, 1] : β ∈ βe (α;R, r) and α ∈ α (β;R, r). We then select one equilibrium from this
set using rule A: (β∗ (R, r) , α∗ (R, r)) = A(B (R, r)). Similarly as in the unobservable case, we select
the equilibrium for which the expected queue length is the largest.
Proposition 6 If r = 0, then α∗ (R, 0) = 0 and β∗ (R, 0) =⌊v − R
ct
⌋. If r > 0, then α∗ (R, r) = 1
and β∗ (R, r) = n, where n is the highest natural number satisfying conditions (d), (e) and (f) in
Proposition 5.
This is an interesting result: The equilibrium with the longest queue is always a pure strategy
threshold equilibrium. As for the unobservable case, if r = 0, then the equilibrium with the longest
queue occurs for α∗ (R, 0) = 0. With no demand inducement, the model reduces to the one discussed
18
by Naor (1969) or Hassin and Haviv. If r > 0, then α∗ (R, r) = 1. β∗(R, r) is the largest of the
corresponding equilibria. For example, in Figure 2, β∗ (9, 10) = 7.
5.3 The first-stage equilibrium, or, The Expert’s Optimal Pricing Deci-
sion
Having characterized α∗ (R, r) and β∗ (R, r) in Proposition 6, we can now proceed to find the optimal
price structure. Let Π (R, r).= π (β∗ (R, r) , α∗ (R, r) ;R, r) be the corresponding equilibrium profit
rate for the expert. Then, the first stage equilibrium (R∗, r∗) is determined as follows: (R∗, r∗).=
max(R,r)∈R+
2Π(R, r). Finally, let Π∗ = Π(R∗, r∗).
Rather than optimizing Π (R, r) over (R, r) ∈ R2+ and subsequently determining α∗ (R∗, r∗)
and β∗ (R∗, r∗), it is more convenient to first fix (α, n) ∈ 0, 1 × N and find the price pair that
maximizes the expert’s profits over all pairs (R, r) ∈ R2+ satisfying α∗ (R, r) = α and β∗ (R, r) = n.
This is a linear programming problem and can be solved analytically. Given the optimal profits for
a given (α, n), we can optimize the expert’s profits over all pairs (α, n) ∈ 0, 1 × N. Therefore, it
is convenient to partition R2+ into subspaces as follows: Ω0 (n)
.= (R, 0) ∈ R
2+|β∗ (R, 0) = n, for
n ∈ N. Then we can calculate π0 (n).= max Π (R, r)
Rn∈Ω0(n)
, which is the maximum profit that can be
attained without service induction for an equilibrium with threshold n. Similarly, in any subgame
with r > 0, β∗(R, r) is determined by Proposition 6 for a fixed (R, r). Define Ω1 (n).= (R, r) ∈ R
2+|
r > 0 and β∗ (R, r) = n for n ∈ N. Then we can calculate π1 (n).= max Π (R, r)
(R,r)∈Ω1(n)
, which is the
maximum profit that can be attained with service induction for an equilibrium with threshold n.
Let n0.= argmax
n
π0 (n) and n1.= argmax
n
π1 (n). Finally, letting π∗0.= max
nπ0 (n) and π∗
1.= max
n
π1 (n), we obtain that Π∗ = max (π∗0 , π
∗1).
The next two subsections characterize the equilibrium without and with demand inducement.
What is very interesting is that we are able to characterize the equilibrium outcome as a function
of only two fundamental parameters, v and ρ, that capture the four parameters in our model: V, c, t
and Λ. In the following subsections, we explicitly denote the dependence on v and ρ.
5.3.1 Equilibrium without demand inducement
Proposition 7 The optimal contract is of the formR∗
0(ρ,v)ct
= v − n0 (v, ρ). The optimal profit is
π∗
0 (ρ,v)c
= (v − n0) ρn0(ρ), with ρn0
(ρ).=(1 − (1−ρ)ρn0
1−ρn0+1
)ρ. The profit-maximizing equilibrium queue
length n0 (v, ρ) = dxe where x ∈ R solves v = x + 1ρx
(1−ρx+1
1−ρ
)2
. For v 1, n0(ρ, v) can be
approximated as follows:
n0 (v, ρ) ≈
⌈− ln v
ln ρ
⌉for 0 < ρ 1
d√ve for ρ ≈ 1⌈ln vln ρ
⌉for 1 ρ.
19
When customers adopt a threshold strategy n, the expert’s queue is of the type M/M/1/n. ρn (ρ)
is the fraction of time that the expert is busy (his utilization level). The monopolist that does not
induce demand faces the following trade-off: Decreasing the price increases the volume of customers
that are serviced but reduces the profit margin. For very small values of ρ, there is not much
congestion; ρn (ρ) ≈ ρ 1. The service provider occasionally receives a potential customer, who
will likely find the queue empty and experience low or no queueing cost. Therefore, a high price can
be charged. When ρ < 1 but larger, more potential customers arrive and congestion increases. The
service provider can decrease the price to have more customers visit him. For ρ = 1, the minimum
price and the maximum equilibrium queue length are obtained. When ρ > 1 and large, potential
demand largely exceeds ‘capacity’. There is no need to decrease the price in order to attract more
customers. Instead, the expert increases the price while remaining busy most of the time; ρn (ρ) ≈ 1.
This equilibrium behavior has also been described in Hassin and Haviv (which pages? or section?).
The surplus that the customers capture in this case linearly decreases in the state of the system:
Customers that enter an empty system enjoy the highest surplus.
5.3.2 Equilibrium with demand inducement
In the case of demand induction, we find the following approximations for the equilibrium queue
length. The behavior described in this proposition is illustrated in Figure 3.
Proposition 8 For a given (v, ρ) with v 1, the profit-maximizing equilibrium queue length can
be approximated as follows:
n1 (v, ρ) ≈
0 for 0 ≤ ρ < 1v⌈
v − 1ρ
⌉for 1
v< ρ < 1
v+ 1+
√2v+1v2⌈√
2ρ
⌉for 1
v+ 1+
√2v+1v2
< ρ 1 and v < 12ρ
(2−√
2ρ
)
⌈− ln v
ln ρ
⌉+ 2 for 1
v+ 1+
√2v+1v2
< ρ 1 and 12ρ
(2−√
2ρ
)
< v
d√ve for ρ ≈ 1⌈ln vln ρ
⌉for 1 ρ
In the first case, no customer enters and the profit is 0. In the second case, the optimal contract is
of the formr∗1 (ρ,v)
c≈ ρ (v − 1) − 1 > 0 and
R∗
1(ρ,v)ct
= 0. In the last four cases, the optimal contract
is of the formr∗1 (ρ,v)
c≈ n1(v,ρ)−1
I11 (ρ)−1− 1 > 0 and
R∗
1(ρ,v)ct
≈ v − (n1 (v, ρ) − 1)(
1I11 (ρ)−1
+ 1)> 0 for
n1 (v, ρ) 1.
The behavior on 0 ≤ ρ < 1v
is contrary to the case without service induction, where the profits
are non-negative for all ρ > 0. For extremely low values of ρ, no equilibrium with service inducement
is possible. The higher the profit potential, v, of the expert, the lower the threshold for ρ is to obtain
positive profit with service induction. Note that the very steep increase over[
1v, 1v
+ 1+√
2v+1v2
]is
invisible in the figure.
20
For the lowest possible values of ρ that result in positive profit under service induction, the
optimal contract does not contain a fixed component. For v 1, the region over which this
equilibrium is possible is very narrow; it is easy to see that 1v≈ 1
v+ 1+
√2v+1v2
. Nevertheless, in that
region, n1 (ρ, v) increases very steeply and overtakes n0 (ρ, v) in[
1v, 1v
+ 1+√
2v+1v2
].
Comparing Propositions 7 and 8, we see that for ρ ∈ [1,∞], the queue length of with service
induction behaves the same as the queue length without service induction. For ρ < 1, on the
other hand, the queue length with service induction differs from the queue length without service
induction, in particular, it is larger for 1v
+ 1+√
2v+1v2
< ρ 1. The economic intuition is the
following: Remember from Proposition 7 that, when no service is induced, customers arriving at
an empty system (n = 0) enjoy the highest surplus as they do not have to wait. By charging a
fixed fee only, the monopolist cannot extract all utility from these customers. He can only extract
all the utility from customers arriving in state n = n − 1, if n is the threshold level. With service
induction, the expert can set a fixed fee and a variable fee. This allows him to extract more
surplus from the customers. From Proposition 5, we know that the expected utility under service
induction is a concave function of the state of the system upon arrival. By choosing appropriately
the fixed and variable fee, the expert will also extract all surplus from the customers arriving at an
empty system. Consider a high threshold level n 1. Then, the main cost component that the
customers experience when joining at state n− 1 is the waiting costs, as expected service induction
costs are very low (i.e. I1n−1 (ρ) ≈ 1). The expected utility of these customers is approximately
V − R − (n− 1) ct − (c+ r) t. The first terms are the utility for the service, from which the fixed
fee is subtracted. The latter terms are the expected waiting costs in the queue and the expected
waiting plus variable service costs during the service time. The main cost component of customers
joining an empty system is the expected service induction cost, as these customers does not have to
wait in the queue: V − R − (c+ r) tI11 (ρ). The first terms in these two expressions are the same.
The last term is the expected waiting plus variable service costs during the total service time (true
and induced). When extracting all surplus from both customers arriving in state 0 and state n− 1,
the expert will choose to set the variable fee such that (c+ r) tI11 (ρ) ≈ (n− 1) ct + (c+ r) t. The
fixed fee is then determined by R ≈ V − (n− 1) ct − (c+ r) t. The first equation determines the
variable fee:r
c≈ n− 1
I11 (ρ) − 1
− 1. (5)
The fixed fee is then determined by Rct
≈ v − (n− 1)(1 − 1
I11 (ρ)−1
). The expert’s revenues with
service induction are determined by both fixed and variable fees and can be rewritten as
π1 (n) ≈ c
[(v − n) ρn (ρ) + (1 − ρn (ρ))
(n− 1
I11 (ρ) − 1
− 1
)], (6)
where the first term is exactly equal to the profits in case of no service induction. The second term is
the extra profit stream that the expert can capture with demand induction. This term modifies the
21
optimal queue length of the expert when inducing demand. Note the term is negative for n < I11 (ρ)
and positive for n > I11 (ρ). Thus, there exists a minimum queue length above which service induction
can result in higher profits than the corresponding system without service induction (where prices
are set such that the equilibrium n is the same in both systems). This is the ‘follow the crowd’ effect
discussed earlier. For greater values of ρ, the follow the crowd effect is reduced as ρn (ρ) ≈ 1. For
very small but positive values of ρ, the inflation factor, I11 (ρ), is very high. Then, it is necessary
that the threshold, n−1, be very high for the profit to be higher under demand inducement. A high
equilibrium threshold value, in turn, requires a low fixed cost.
0
5
10
15
20
25
30
n
0.2 0.4 0.6 0.8 1 1.2 1.4
rho
Figure 3: Equilibrium queue length for α = 0 (thin line) and for α = 1 (thick line) and v = 500.
22
5.3.3 Comparison of demand inducement vs. no demand inducement equilibria
When determining when service induction is more profitable for the expert than no service induction,
we need to characterize for which region in the (v, ρ) space we have that π∗0 (ρ, v) < π∗
1 (ρ, v) with
π∗i (ρ, v)
.= πi (n
∗i (ρ, v) ; ρ, v). It is very difficult to describe analytically the locus for which π∗
0 (ρ, v) =
π∗1 (ρ, v), as both π∗
i (ρ, v) actually is the solution to an optimization problem over πi (n; ρ, v). The
following Proposition provides some insight:
Proposition 9 For v 1, there exist a ρ′ (v) such that:
(i) π∗0 (ρ, v) > π∗
1 (ρ, v) for ρ ∈ [0, ρ′ (v)]
(ii) π∗1 (ρ, v) > π∗
0 (ρ, v) for ρ ∈ [ρ′ (v) ,√v]
(iii) π∗0 (ρ, v) > π∗
1 (ρ, v) for ρ ∈ [√v,+∞].
According to Proposition 9, experts prefer a variable rate contract when the profit potential
is high and the normalized demand market is neither high nor low. For intermediate normalized
demand, service induction is optimal. Remember from Propositions 8 and 7 that for ρ ∈ [1,√v] and
v 1, we obtain n1 (v, ρ) ≈ n0 (v, ρ). As ρn (ρ) ≈ 1, we obtain with (6) that π∗1 (ρ, v) ≈ π∗
1 (ρ, v).
Thus, the advantage of service induction will be very small in that case. For ρ ∈ [√v,+∞], we
obtain with Propositions 8 and 7 that n1 (v, ρ) ≈ n0 (v, ρ) ∈ 1, 2. Thus, for very high normalized
demand, not inducing services becomes strictly preferable for thresholds less than or equal to 2.
Figure 4 illustrates the conditions under which a variable rate contract is optimal. For low values of
v, service induction is not optimal. The minimum profit potential occurs for ρ = 1 and is v ≈ 9.5.
The region where the variable rate contract is more profitable extends well beyond ρ = 1.
6 The Expert’s Optimal Workload Revelation and Price Struc-
ture
In this section, we compare the expert’s profits when concealing the workload, as characterized by
Proposition 3 to the expert’s profits when revealing the workload, as characterized by Propositions
7 and 8.
Proposition 10 For v 1, there exist a ρ′′ (v) such that:
(i) Πu (ρ, v) > max (π∗1 (ρ, v) , π∗
0 (ρ, v)) for ρ ∈ [0, ρ′′ (v)]
(ii) π∗1 (ρ, v) > max (Πu (ρ, v) , π∗
0 (ρ, v)) for ρ ∈ [ρ′′ (v) ,√v]
(iii) π∗0 (ρ, v) > max (Πu (ρ, v) , π∗
1 (ρ, v)) for ρ ∈ [√v,+∞]
Concealing the queue length allows the expert to extract all surplus as customers are then
homogenous, before arrival as well as after arrival. However, some customers may join when the
queue length is very long (which is inefficient), or may balk when the queue is empty (which is also
23
v
π1<π0
π1>π0
0
10
20
30
40
0.5 1 1.5 2 2.5 3
ρ
Figure 4: Incentives for adopting a billing rate contract in the (ρ, v)-space.
inefficient). Thus, concealing the queue length is advantageous for low and intermediate markets
(see also Hassin and Haviv, p. 53). Remember from the Proposition 9 that the highest gains
can be made from demand induction for intermediate levels of normalized demand. According to
Proposition 10, concealing the queue length is the optimal strategy for low normalized demand. For
a certain level of normalized demand that is still less than 1, revealing the queue length with demand
induction becomes optimal. For higher levels of normalized demand, revealing remains optimal, but,
the optimal price structure will change for very high levels of normalized demand as discussed in
the Proposition 9.
7 The Impact of Service Induction
Finally, we can compare the total surplus and the consumer surplus of the different systems. T0 (n0)
and T1 (n1) denote the total surplus without and with service induction when the threshold is n0
24
and n1 respectively. Then, we obtain:
T0 (n0) =
n0−1∑
n=0
(V − c (n+ 1) t) pn (n0) and T1 (n1) =
n1−1∑
n=0
(V − c
(n+ Imin(n+1,n1−1) (ρ)
)t)pn (n1) .
In the case of no service induction, the term in parentheses is the utility generated by service reduced
by the expected waiting cost (queuing and service) incurred by a customer who enters when the
number in system is n. With service induction, the customer waiting time also includes waiting time
during induced service. Therefore, if n0 = n1, the total surplus without service induction will be
higher than the total surplus with service induction. Compared to the socially optimum, too few
customers will visit the expert that does not induce services, due to his monopoly power (Hassin
and Haviv). From Propositions 7 and 8, we know that more customers visit the expert with demand
induction, due to the follow the crowd effect. Thus, potentially, the total surplus may increase when
inducing services if the gains from having more customers visiting are larger than the extra waiting
costs that are generated. This is indeed the case, as illustrated by Figure 5.
T1/T0 for v=50
0.96
0.98
1
1.02
1.04
0.2 0.4 0.6 0.8 1
ρ
Figure 5: Ratio of the total surplus with and without service induction as a function of ρ
The consumer surplus can easily be derived from the total surplus minus the expert’s profits.
Numerical experiments show that it may be possible that consumer surplus can increase with service
25
induction. These observation open an interesting question from a regulator’s point of view: Should
he encourage service induction? Obviously not in the case that the regulator also can reduce the
monopoly power of the expert. Our model reveals two oppositive effects of service induction on the
social welfare: On one hand, extra unnecessary waiting time during the service induction period
is introduced which reduces the social welfare, while on the other hand, more customers visit the
expert, which increases the social welfare.
8 Conclusions and Further Research
In this paper, we study the choice of price structure and workload information revelation of a
monopolist who sells services to customers that do not know the service time that is appropriate for
them. This occurs when service involves a complex product or system about which the expert has
superior knowledge. In such an environment, the expert may have an incentive to induce unnecessary
services, depending on the price structure and on the expert’s workload level. With a time-variable
contract, the expert’s revenues increase as a function of the total service time. If with such a contract,
the expert’s workload is low, he may have strong incentives to induce services (referred to as “time-
padding”, Ross 1996), which is extremely difficult for the customer to contest. As in many service
environments, customer arrival times are stochastic, as well as customer service times, the workload
level of the expert fluctuates over time. Customers can infer the expert’s incentives from the price
structure and the expert’s workload. The expert has two levers to impact the customer’s decision to
visit the expert: (1) he can make his workload invisible to an arriving customer and (2) he can charge
a price that is independent of the service time (a flat price). In both cases, the expert does not have
an incentive to perform unnecessary services. We introduce a simple queuing model that captures
the key workload dynamics. Within our framework, we determine the best policy for the expert,
as a function of the characteristics of the environment. We find that two parameters dictate the
optimal policy for the expert: (1) the normalized demand, which is the ratio of the potential market
demand (customers per unit of time) over the service rate and (2) the profit potential, which is the
ratio of the service value over the waiting costs during service. Table 8 summarizes our findings.
For low levels of profit potential, the expert never induces services and charges a fixed fee only. The
expert conceals the workload for low levels of normalized demand only. This situation is the same
as discussed in Hassin and Haviv, p. 53.
For high levels of profit potential and low levels of normalized demand, concealing the workload
is optimal. When concealing the workload level, all customers are also ex post (i.e. after arrival)
homogenous and the expert extracts all surplus from the customer. In order to extract the maximum
surplus, the expert does not want to induce unnecessary service and charges a fixed fee. When the
normalized demand increases, concealing the workload becomes less efficient as customers may enter
26
Service value over
waiting costs during
true service
Potential arrival
rate over true
service rate
Expert
Behavior
Workload
informationFee Structure
Consumer surplus
as function of
workload
Low Low No induction Concealed Fixed Flat, Zero
Low High No induction Revealed Fixed Decreasing
High Low No induction Concealed Fixed Flat, Zero
High Medium Induction Revealed Fixed + variable Concave
High Very high No induction Revealed Fixed Decreasing
Table 1: Summary of the expert’s optimal strategy.
when the queue length is long and decide to balk when the queue is empty. There exists a level of
normalized demand less than 1 for which revealing the workload becomes optimal for the expert,
who charges a fixed and a variable fee. In that case, the expert also induces unnecessary service.
When revealing the workload, customers become heterogenous ex post (i.e. after arrival), as they
will decide whether to join the expert’s queue or to balk, depending on the number of customers that
are already in the expert’s system. Due to the possibility of service induction, the expert captures
also a part of the surplus that customers otherwise enjoy in an empty system. The customer surplus
is a concave function of the state of the queue. Furthermore, the equilibrium queue length is longer
than the queue length of an expert that does not induce unnecessary service. This is due to a ‘follow
the crowd’ effect when inducing services: when more other customers are joining the queue, the
expected service induction costs decrease, making it more attractive for customers to join.
For very high levels of normalized demand, with a ratio higher than 1, the expert still reveals his
workload level, but charges a fixed fee only and, consequently, does not induce unnecessary services.
The customers’ expected utility is a linearly decreasing function of the state of the system: When
arriving at an empty queue, the customer enjoys the highest surplus.
Finally, we find that the total surplus may increase with service induction, compared to no service
induction. The reason is that a monopolist that does not induce services restricts too much the queue
length, compared to the social optimal. With service induction, on one hand, extra waiting costs are
incurred during the induced service time. On the other hand, more customers will visit the expert
because of the ‘follow-the-crowd’ effect. It may be that the increase in customers is the dominating
effect.
We believe that our model contributes to both the economic literature on credence goods by
studying explicitly dynamic effects of service induction that have been ignored by economists. We
also believe that our model contributes to the literature in Operations Management as we study
a phenomenon, service induction, in a context where congestion plays a role and for which quite
27
some anecdotal evidence in different service industries has been reported. Furthermore, note that
the model is quite sparse (with only two important parameters; ρ and v), but, in our opinion,
captures the first order effects of service induction in a dynamic environment. Despite being sparse,
be we believe that at the same time is rich as we have identified different phenomena in the (v, ρ)
space. Obviously, in order to obtain such a sparse model that is analyzable, we needed ignored some
aspects that may play a role in real life situations. We hope to relax in further research some of
these assumptions and increase our understanding of the phenomenon further.
9 References
Anonymous (1992). Retailing: Sears Is Accused of Billing Fraud at Auto Centers. Wall Street
Journal June 12.
Callahan, D. (2004). The Cheating Culture, Why More Americans Are Doing Wrong to Get Ahead.
Orlando, Harcourt.
Darby, M. R. and E. Karni (1973). Free Competition and the Optimal Amount of Fraud. Journal
of Law and Economics 16: 67-88.
Dulleck, U. and R. Kerschbamer (2003) “On Doctors, Mechanics and Computer Specialists Or Where
are the Problems with Credence Goods?” Working Paper 0101, University of Vienna.
Emons, W. (1997). Credence Goods and Fraudulent Experts. Rand Journal of Economics 28:
107-119.
Emons, W. (2001). Credence Goods Monopolists. International Journal of Industrial Organization
19: 375-389.
Fong, Y. (2002) “When Do Experts Cheat and Whom Do They Target?” Working Paper, Kellogg
School of Management, Northwestern University.
Hassin, R. and M. Haviv (2003). To Queue or not to Queue. Boston, Kluwer Academic Publishers.
Koblenz, J. (1999). Put your mechanic in check. Black Enterprise April: 135-148.
Llosa (1996). Watch out: Car-repair crooks have some new tricks up their grimy sleeves. Money
June: 172-174.
Paine, L. S. (1992). ”Sears Auto Centers (A).” Harvard Business School Case 9-394-009.
Pitchik, C. and A. Schotter (1987). Honesty in a Model of Strategic Information Transmission.
American Economic Review 77: 1032-1036; Errata.
Richardson, H. (1999) “The Credence Good Problem and the Organization of Health Care Markets.”
Working Paper, Private Enterprise Research Center, Texas A&M University.
28
Wolinksy, A. (1993). Competition in a Market for Informed Experts’ Services. RAND Journal of
Economics 24: 380-398.
10 Appendix
In what follows, we only consider customer strategies that satisfy β < 1Λt since the M/M/1 queue
would not be stable otherwise.
Lemma A 1 Let A.= V −R− (c+ r)t. For α = 0, we have:
(i) If A ≤ 0, then Uu (β; 0, R, r) ≤ 0 ∀ β ∈ [0, 1] ;
(ii) If A > 0, then
Uu (β; 0, R, r)
> 0 for β < β0u,
= 0 if β = β0u,
< 0 if β > β0u,
where β0u is the unique root of Uu (β; 0, R, r) and satisfies β0
u <1Λt .
For α > 0, we have:
(i) If A ≤ 0, or A2 ≤ 4αc(c+ r)t2, then Uu (β; 0α,R, r) ≤ 0 ∀ β ∈ [0, 1] ;
(ii) If A > 0 and A2 > 4αc(c+ r)t2, then
Uu (β;α,R, r)
< 0 for β < β1u,
> 0 if β1 < β < β2u,
< 0 if β > β2u,
where β1u < β2
u are the two distinct roots of Uu (β;α,R, r) and satisfy β1u, β
2u <
1Λt .
Proof. We begin by proving that for any (R, r) ∈ R2+, (i) Uu (β, α;R, r) is concave in β for any
α ∈ [0, 1]. Recall that Uu (β, α;R, r) = V −R− c βΛt2
1−βΛt − (c+ r) t(1 − α
(1 − 1
βΛt
)). Then
dUu (β, α;R, r)
dβ= α (c+ r)
(1 − c
α(c+r)
)(Λtβ)
2 − 2Λtβ + 1
(1 − βΛt)2β2Λ
and
d2Uu (β, α;R, r)
dβ2= 2α (c+ r)
(1 − c
α(c+r)
)(Λtβ)
3 − 3 (Λtβ)2
+ 3Λβt− 1
(1 − βΛt)3β3Λ
.
Since 1 − cα(c+r) < 1, we have
(1 − c
α(c+r)
)(Λtβ)
3 − 3 (Λtβ)2
+ 3Λβt − 1 < (Λtβ)3 − 3 (Λtβ)
2+
3Λβt− 1 = (Λtβ − 1)3 < 0. We conclude that Uu (β, α;R, r) is concave for all α ∈ [0, 1].
Now assume α = 0. Substituting α = 0 in the first derivative shows that Uu (β, 0;R, r) is a
strictly decreasing function of β. Uu (0, 0;R, r) > 0 if and only if V − R − (c + r)t > 0. Thus, if
A ≤ 0, then Uu (β, 0;R, r) ≤ 0 ∀β ∈ [0, 1]. If A > 0, solving for Uu (β, 0;R, r) = 0 yields the root
β0u = A
A+ct1Λt <
1Λt .
29
Now assume α > 0. Then limβ→0 Uu (β, α;R, r) = −∞. Note also that
Uu (β, α;R, r) = 0 ⇔ A =βcΛt2
1 − βΛt+ α (c+ r) t
(1 − βΛt
βΛt
).
(i) A ≤ 0. There exists no β ∈[0, 1
Λt
)that will make the right-hand side non-positive: U has no
root on this interval. Therefore, Uu (β, α;R, r) ≤ 0 for β ∈ [0, 1Λt ].
(ii) A > 0. If βΛt > 1, then, the right hand side is negative, while the left hand side is positive.
Thus, Uu (β, α;R, r) = 0 can never have a root satisfying βΛt > 1. It follows that, any root, if it
exists, belongs to(0, 1
Λt
). Uu (β, α;R, r) = 0 has two distinct solutions, βiu, i ∈ 1, 2 in
(0, 1
Λt
)if
and only if A2 > 4αc (c+ r) t2, with Uu (β, α;R, r) negative on [0, β1) and (β2, 1Λt ], and positive on
(β1, β2). Otherwise, Uu (β, α;R, r) = 0 either has no real root or one double root. In either case,
Uu (β, α;R, r) ≤ 0 for β ∈ [0, 1Λt ].
Proof of Proposition 2. Suppose r > 0. Then, as argued above, αu (β;R, r) = 1 for any
feasible value of β. Then, Bu (R, r) is the set of equilibria with α = 1 and any of the corresponding
equilibria defined in Proposition 1 for α = 1. Recall that we focus on the equilibrium that results in
the highest probability of entry. Therefore α∗u (R, r) = 1 and β∗
u (R, r) is the largest element in the
set βeu (1;R, r). Note that if Uu (βu, 1;R, r) does not have two distinct roots β1u and β2
u on [0, 1Λt ),
then βeu (1;R, r) = 0.Suppose r = 0. If A ≤ 0, then, β∗
u(R, 0) = 0 is the only possible equilibrium. If A > 0, then, as
argued above, αu (β;R, 0) ∈ [0, 1] for any value of β. Then, Bu (R, 0) is the set of equilibria with
α′ ∈ [0, 1] and any of the corresponding equilibria defined in Proposition 1 for α = α′. Note that
βeu (α;R, 0) satisfies
V −R
ct− 1 − βρ
1 − βρ− α
(1
βρ− 1
)= 0 ⇔ β1,2
u Λt =w + 2α±
√w2 − 4α
2 (w + 1 + α),
with w = V−Rct
− 1 > 0. Let β2u denote the highest root. Then
Λtdβ2
u
dα=
1
2
(√w2 − 4α− 2
)(w + 2) + 2 + 2α− w2
√w2 − 4α (w + 1 + α)
2 .
It can be shown that(√w2 − 4α− 2
)(w + 2)+2+2α−w2 = 0 ⇔ w = −1−α, or, V−R
ct= −α < 0.
Thus, ifdβ2u
dαis negative (positive) for one value of α, then it is negative (positive) for all other values
of α ∈ [0, 1]. We find that for α = 1,
Λtdβ2
u
dα=
1
2
(√w2 − 4 − 2
)(w + 2) + 2 + 2 − w2
√w2 − 4α (w + 1 + 1)
2 < 0
for all w > 0. Therefore, we obtain thatdβ2u
dα< 0 for all α ∈ [0, 1] and w > 0. Thus, the largest β2
u
occurs for α = 0. Therefore, we select α∗u (R, 0) = 0. Then Uu (β, 0;R, 0) = V − c
(βΛt2
1−βΛt + t)−R.
Solving U(β) = 0 gives β = V−R−ct(V−R)tΛ . Therefore, β∗
u (R, 0) = min V−R−ct(V−R)tΛ , 1.
Proof of Proposition 3. Let us begin by showing that a flat-rate contract with r∗ = 0 is optimal.
For any feasible (β∗u (R, r) > 0) contract (R, r) with r > 0, we can construct a contract (R′, 0)
30
such that Πu (R′, 0) > Πu (R, r). Let R′ = R + r 1β∗
u(R,r)Λ . β∗u (R, r) satisfies Uu (β, 1;R, r) = 0,
or V − R − r 1β∗
u(R,r)Λ − cβ∗
u(R,r)Λt2
1−β∗
u(R,r)Λt − c 1β∗
u(R,r)Λ = 0. β∗u (R′, 0) solves Uu (β, 0;R′, 0) = 0, or
V −R′ − c
(β∗
u(R′,0)Λt2
1−β∗
u(R′,0)Λt + t
)= 0, or, V −R− r 1
β∗
u(R,r)Λ − cβ∗
u(R′,0)Λt2
1−β∗
u(R′,0)Λt − ct = 0. As 1β∗
u(R,r)Λ > t
is a necessary condition for stability, we conclude that β∗u (R′, 0) > β∗
u (R, r). Substituting into the
expression for Πu, we find that Πu (R′, 0) > Πu (R, r). Thus, we can maximize Πu (R, r) by searching
over contracts of the type (R, 0).
Πu (R, 0) = πu (β∗u (R, 0) , α∗
u (R, 0) ;R, 0) = πu (β∗u (R, 0) , 0;R, 0) = Rβ∗
u (R, 0) Λ. From Propo-
sitions 1 and 2, we have β∗u (R, 0) = minβ0
u, 1 = min V−R−ct(V−R)tΛ , 1 for a feasible contract (R, 0).
Substituting into Rβ∗u (R, 0) Λ, we find that Πu (R, 0) is maximized at R∗ = V − cW (β∗∗Λ) where
β∗∗ =min( 1
t−√
ctV,Λ)
Λ . Finally, Π∗u = Πu(R
∗, 0) = c(v − 1
1−ρ
)ρ, where ρ
.= min
(1 −
√1v, ρ).
Proof of Lemma 4: Step 1: Some useful properties of ξ (β). Before proceeding with the
proof, we derive some properties of ξ (β) that will prove to be useful. Remember that p = β − bβcand n = bβc.
P1: If n ≥ 1, then (1 − n)+
= 0 and ξ = n+ ln(1+pρ)ln(1+ρ) . In this case,
(1
1+ρ
)ξ−1
=(
11+ρ
)n−1+ln(1+pρ)ln(1+ρ)
=(
11+ρ
)n−11
1+pρ .
P2: If n = 0, then p = β, (1 − n)+
= 1 and ξ = ln(p+pρ)ln(1+ρ) . In this case,
(1
1+ρ
)ξ−1
= 1p.
P3: ξ (0) = ∞ and ξ (n) = n ∈ N+ with n ≥ 1.
Step 2: Derivation of tn (α, β). Consider an arriving customer who finds n others in queue.
Under the FCFS discipline, this customer can experience service inducement only in the event that
the queue is empty upon termination of his true service time, that is, in the event that no other
customer enters the system during the true service time of this customer or of the n customers in line
in front of him. The probability of this event depends on the strategy that the other customers follow.
We call this probability Pn(β) to denote the dependence on n and β. Thus, with probability Pn(β),
the queue is empty at the completion of the true service time of the customer under consideration.
At that point, the expert induces service with probability α until the arrival of the next customer.
The length of the service inducement is determined as follows:
Case (a): For any strategy β ≥ 1, the arrival rate to the system in state 0 is Λ. The expected time
between the true service completion of the last customer and the arrival of the first new customer
is then 1Λ , due to the memoryless property of Poisson arrivals. Therefore, the expected length of
service induction is 1Λ . The expected total service time is then tn (α, β) = t+ αPn (β) 1
Λ for β ≥ 1,
where the first term is the true service time and the second term is the expected induced service
time.
Case (b): For β ∈ [0, 1], the arrival rate to the system in state 0 is βΛ = pΛ. The expected time
since true service completion of the last customer until the arrival of the next customer is then 1pΛ .
Therefore, the expected length of service induction is 1pΛ , due to the memoryless property of Poisson
31
arrivals. The expected total service time is then tn (α, β) = t+ αPn (β) 1pΛ for β ∈ [0, 1).
Step 3: Derivation of Pn (β).
Let n be the state of the system when a potential customer arrives. If this customer joins the
queue the state is increased to n + 1. All other customers follow strategy β; n = bβc. The queue
will be empty upon termination of his true service time if the Markov process goes from state n+ 1
through states n, n − 1, . . . , 0 before the next customer arrives and decides to join. Depending on
the value of β, we have the following cases:
Case (i): β ≥ 2 ⇒ n ≥ 2.
Case (ia): 0 ≤ n < n − 1. In this case, n + 1 < n. Since n + 1 < n, any arriving
customer will join. Therefore Pn (β) equals the probability that at each state n′ ∈ [1, n+ 1], a
service completion occurs before a new customer arrival. This is µµ+Λ in each state. Therefore,
Pn (β) =(
µµ+Λ
)n+1
=(
11+ρ
)n+1
.
Case (ib): n ≥ n − 1. In this case n + 1 ≥ n. For all n higher than n, no customer joins
(according to the strategy β). Therefore, with probability 1, the system state will return to n.
Since arriving customers join with probability p in state n, the probability that a service completion
occurs before a new customer joins the queue is µµ+pΛ . For all other states n′ ∈ [1, n− 1], an arriving
customer will enter the queue and the probability that a service completion occurs before a new
customer arrival is, analogous to the previous case, µµ+Λ . Therefore, Pn (β) =
(µ
µ+Λ
)n−1µ
µ+pΛ =(
11+ρ
)n−11
1+pρ . Using (P1), the latter probability can be rewritten as(
11+ρ
)ξ−1
.
Thus, we have obtained that αPn (β) 1Λ = α
(1
1+ρ
)min(n+1,ξ−1)1Λ .
Case (ii): 1 ≤ β < 2 ⇒ n = 1.
Here, any n ≥ 0 satisfies n + 1 ≥ n. Applying case (ib) with n = 1, we obtain Pn (β) =
11+pρ . Using (P1), the latter probability can be rewritten as
(1
1+ρ
)ξ−1
. Note that in this case
min (n+ 1, ξ − 1) = ξ − 1 as ξ < 1 and n ≥ 0. Cases (i) and (ii) can be summarized as follows:
αPn (β) 1Λ = α
(1
1+ρ
)min(n+1,ξ−1)1Λ .
Case (iii): 0 < β < 1 ⇒ n = 0 and p = β.
For all n higher than n = 0, no customer joins (according to the strategy β). In particular, no
customer will join while the customer who last joined is in service. Therefore, with probability 1,
the system state will return to n = 0. This gives Pn (β) = 1. Note that in this case(
11+ρ
)ξ−1
= 1p.
Thus, using (P2), we can write αPn (β) 1pΛ = α
(1
1+ρ
)ξ−11Λ .
Summarizing cases (i-iii), we obtain tn (α, β) = t
(1 + α
(1
1+ρ
)min(n+1,ξ−1)1ρ
).
Proof of Proposition 5.
Case 1: α = 0. Note that Un (0, β;R, r) = V −R− cnt− (c+ r)t decreases in n and is independent
of β.
Pure strategy equilibria: If Un−1 (0, n;R, r) ≥ 0, it follows that Un (0, β;R, r) ≥ 0 for n ∈
32
[0, n− 1]. Thus, if in addition, Un (0, n;R, r) ≤ 0, it is optimal for all arriving customers to follow a
(pure) threshold strategy n.
Mixed strategy equilibria: If Un (0, β;R, r) = 0, then the customer is indifferent in state n
between joining or not. Therefore, any randomization between thresholds n (balking at n) and n+1
(joining at n, but balking at n+ 1) is an equilibrium, i.e. all β such that bβc = n are equilibria.
Case 2: α > 0. Note that for fixed (α, β;R, r), Un (α, β;R, r) has a linear term (−cnt) decreasing
in n and a term (−tn (α, β)) that is concave increasing in n for n ≤ ξ (β) − 1 and constant for
n ≥ ξ (β) − 1. Therefore, Un (α, n;R, r) is concave in n.
Pure strategy equilibria: First, note that n = 0 is an equilibrium for any (R, r): As ξ (0) = +∞(see P3 of Lemma 4), U0 (α, 0;R, r) = −∞ and therefore no customer ever enters the system in
state 0, provided that all other customers adopt the threshold strategy n = 0. Thus, n = 0 is an
equilibrium.
Assume that the threshold strategy of the other customers is β = n ≥ 1, with n ∈ N. As
Un (α, n;R, r) is concave in n, a new customer will also adopt a threshold strategy n if and only
if (1) the net expected utility when entering in state 0 is non-negative (2) the net expected utility
when entering at state n− 1 is non negative and (3) the net expected utility when entering in state
n is non-positive. (1) and (2), together with the concavity of Un (α, n;R, r), ensure that the net
expected utility in states n ∈ [0, n] is non negative. (3) ensures then that n is the optimal threshold
strategy for a new customer, when all other customers adopt the threshold strategy n. Therefore, n
is an equilibrium threshold strategy. Conditions (1), (2) and (3) are thus:
U0 (α, n;R, r) ≥ 0
Un−1 (α, n;R, r) ≥ 0
Un (α, n;R, r) ≤ 0
For n = 1, ξ = 1. We can determine min (n+ 1, ξ (β) − 1) for n = 0, n−1 and n: min (1, n− 1) =
0, min (n, n− 1) = 0 and min (n+ 1, n− 1) = 0 respectively. Substituting this in the expression for
tn(α, β), and subsequently in Un(α, β; r,R) we obtain that the following conditions are equivalent
to (1), (2) and (3):
U0 (α, n;R, r) ≥ 0
Un−1 (α, n;R, r) ≥ 0
Un (α, n;R, r) ≤ 0
Mixed strategy equilibria: Assume that the threshold strategy of all other customers is β > 0
and let n < β < n+1, with n ∈ N. As Un (α, n;R, r) is concave in n, a new customer will adopt also
a (mixed) threshold strategy β if and only if (1) the net expected utility when entering in state 0 is
non-negative, (2) the net expected utility when entering at state n is exactly equal to zero, and (3)
the net expected utility when entering at states n ≥ n+1 is negative. (1) and (2), together with the
concavity of Un (α, n;R, r), ensure that the net expected utility is non-negative in states n ∈ [0, n].
33
(2) ensures then when entering in state n, the new customer is indifferent between joining or not.
In other words, the customer is indifferent between a balking at n or at n + 1. (3) ensures that
balking in states n ≥ n+ 1 is always optimal when all other customers adopt strategy β. Therefore,
any (mixed) strategy in [n, n+ 1] belongs to the best response set of a new customer when all other
customers adopt strategy β. As β ∈ [n, n+ 1], β is an equilibrium threshold strategy. Conditions
(1), (2) and (3) can be written as
U0 (α, β;R, r) ≥ 0
Un (α, β;R, r) = 0
Un+1 (α, β;R, r) < 0.
Since Un+1 (α, β;R, r) < Un (α, β;R, r), the latter condition is always satisfied. Therefore, the first
two conditions are sufficient to characterize mixed strategy equilibria.
Proof of Proposition 6.
When r = 0, any α ∈ [0, 1] is possible in equilibrium. Note, however, that Un (α, β;R, 0) <
Un (0, β;R, 0) for α ∈ (0, 1]. Therefore, for any equilibrium n with α > 0 that satisfies Un−1 (α, n;R, 0) ≥0 and Un (α, n;R, 0) ≤ 0, there exists a larger equilibrium with α = 0. Thus, the equilibrium is de-
termined by Un−1 (0, n;R, 0) ≥ 0 ≥ Un (0, n;R, 0), or, V −R− (n− 1) ct− ct ≥ 0 ≥ V −R−nct− ct,or, n ≤ V−R
ct≤ n+1. When V−R
ctis non-integer, n = bV−R
ctc. When V−R
ctis integer, both n = V−R
ct
and n = V−Rct
− 1 satisfy the two inequalities. Taking the larger of the two, we conclude that for
any value of V−Rct
, β∗ (R, 0) = bV−Rct
c =⌊v − R
ct
⌋.
When r > 0, α = 1 is the unique equilibrium. We prove that the equilibrium with the longest
queue length must necessarily be a pure strategy equilibrium. In the remainder of this proof, we
will suppress α,R and r in the expression Un(α, β;R, r) for simplicity and use Uαn (β) instead. Take
the largest β such that U 1bβc(β) = 0; this is the largest mixed strategy equilibrium. We will show
that there exists a k such that U 1bβc+k−1(bβc + k) ≥ 0 and U1
bβc+k(bβc + k) ≤ 0, that is, bβc + k is
a pure strategy equilibrium.
Since U1n(β) is nondecreasing in β, U 1
bβc(bβc + 1) ≥ 0. If U1bβc+1(bβc + 1) ≤ 0, we are done:
bβc + 1 is a pure strategy equilibrium. If not, and U 1bβc+1(bβc + 1) > 0, then U1
bβc+1(bβc + 2) > 0
since U1n(β) is nondecreasing in β. If U 1
bβc+2(bβc + 2) ≤ 0, we are done: bβc + 2 is a pure strategy
equilibrium. If not, and U 1bβc+2(bβc+2) > 0, then U1
bβc+2(bβc+3) > 0 since U1n(β) is nondecreasing
in β. Repeating the same argument, we will eventually find a k such that U 1bβc+k(bβc + k) ≤ 0.
This is because limk→∞ U1n(n) = −∞. Thus, a mixed strategy equilibrium can never be the longest
queue equilibrium.
Lemma A 2 The limiting probability that the true system is in state n when all customers follow
the threshold strategy profile β ∈ R+is pn (β) = (1−ρ)ρn1−ρn+1+ψ , n = 0 . . . n and pn+1 (β) = (1−ρ)pρn+1
1−ρn+1+ψ , n =
34
0 . . . n, with (1 − p) + pρ = ρψ. The expected steady-state profit rate is
π (β, α;R, r) = RΛ
(1 − (1 − ρ) ρn+ψ
1 − ρn+1+ψ
)+ r
α+ (1 − α) ρ− ρn+1+ψ
1 − ρn+1+ψ. (A-7)
Proof of Lemma A 2: The threshold strategy profile β gives rise to a birth-death Markov process
with the following transition rates: ρi,i+1 = Λ, i = 0 . . . n − 1, ρn,n+1 = pΛ and µi,i−1 = 1/t, i =
1 . . . n+ 1. Recall ρ = Λt. The balance equations for this Markov process are
pn = ρpn−1, n = 1 . . . n
pn+1 = pρpn,
which can be rewritten as
pn = ρnp0, n = 0 . . . n
pn+1 = pρn+1p0
Since∑n+1n=0 pn = 1, p0 = 1∑
nn=0 ρ
n+pρn+1 = 11−ρn+1
1−ρ +pρn+1= 1−ρ
1−ρn+1(1−p+pρ) = 1−ρ1−ρn+1+ψ , where ψ
is defined such that (1 − p) + pρ = ρψ. Then we obtain pn = (1−ρ)ρn1−ρn+1+ψ , n = 0 . . . n and pn+1 =
(1−ρ)pρn+1
1−ρn+1+ψ .
Recall that π (β, α;R, r) = RΛ∑∞n=0 pn (β) δn (β) + r (
∑∞n=1 pn (β) + αp0 (β)) . We have
∞∑
n=0
pn (β) δn (β) = (1 − (1 − p) pn − pn+1)
= 1 − (1 − p)(1 − ρ)ρn − (1 − ρ)pρn+1
1 − ρn+1+ψ
= 1 − (1 − ρ)ρn+ψ
1 − ρn+1+ψ
αp0(β) +
n+1∑
n=1
pn(β) = αp0(β) + 1 − p0(β)
=α+ (1 − α) ρ− ρn+1+ψ
1 − ρn+1+ψ
π (β, α;R, r) is now obtained using the above expressions.
Proof of Proposition 7: Remember that Ω0 (n).= (R, 0) ∈ R
2+ : β∗ (R, 0) = n and with
Proposition 6, β∗ (R, 0) =⌊v − R
ct
⌋= n. From (A-7) in Lemma A 2, we obtain π (n, 0;R, 0) =
RΛ(1 − (1−ρ)ρn
1−ρn+1
)= R
tρn (ρ). Therefore Π∗ (R, 0) = R
tρn (ρ). For n ≥ 1, ρn (ρ) > 0 and π0 (n)
.=
max(R,0)∈Ω0(n)
Rtρn (ρ) is a linear problem in R with a strictly positive coefficient. Let Rn denote the
profit maximizing fixed price as a function of n. The solution to the problem is to set R as high
as possible while satisfying⌊v − R
ct
⌋= n. Therefore, Rn
ct= v − n. For n = 0, ρ0 (ρ) = 0 and also
π0 (0) = 0. Therefore, we obtain π0 (n) = c (v − n) ρn (ρ) for all n ≥ 0. Searching over all n ∈ N
yields the profit maximizing equilibrium n0(ρ, v), which we substitute back to obtain R∗0 and π∗
0 .
We now derive the approximation for n0(ρ, v). We first show that for a given (v, ρ),
maxn≥0
π0 (n) (A-8)
35
is solved by the unique value of n satisfying v0n−1 (ρ) < v < v0
n (ρ) with v0n (ρ) = n+ 1
ρn
(1−ρn+1
1−ρ
)2
.
The profit maximizing value of n satisfies π0(n)−π0(n−1) > 0 and π0(n+1)−π0(n) < 0. Using
π0(n) = c(v−n)ρn(ρ) and simplifying, these two inequalities can be written as n−1+ 1ρn−1
(1−ρn1−ρ
)2
<
v < n + 1ρn
(1−ρn+1
1−ρ
)2
. Let v0n (ρ)
.= n + 1
ρn
(1−ρn+1
1−ρ
)2
. Rewriting the two inequalities, we obtain
v0n−1 (ρ) < v < v0
n (ρ).
It can easily be shown that v0n+1 (ρ)−v0
n (ρ) > 0 for all n ≥ 0. Therefore, for a given (v, ρ), there
exists exactly one n that satisfies v0n−1 (ρ) < v < v0
n (ρ). Thus, n0 (v, ρ) = dxe where x ∈ R solves
v = v0x (ρ). However, as this equation does not have an analytical solution, we approximate the
solution for v 1 by approximating v0x (ρ) by v0
x (ρ) = 1ρx
(1−ρx+1
1−ρ
)2
and solving for x in v = v0x (ρ).
Let A0 = ρ2 and B0 = 2ρ + v (1 − ρ)2. Then v = v0
x (ρ) ⇔ 0 = A0 (ρx)2 − B0ρ
x + 1. Solving this
equation, we obtain ρx = B0
2A0±√(
B0
2A0
)2
− 1A0
, or,
x =
ln
(2ρ+v(1−ρ)2
2ρ2−√(
2ρ+v(1−ρ)2
2ρ2
)2− 1ρ2
)
ln ρ if ρ < 1
ln
(2ρ+v(1−ρ)2
2ρ2+
√(2ρ+v(1−ρ)2
2ρ2
)2− 1ρ2
)
ln ρ if ρ > 1.
For low values of ρ, we can use the following approximation: B0
2A0−√(
B0
2A0
)2
− 1A0
≈ 12ρ+v(1−ρ)2 .
Thus, for 0 < ρ 1,
x ≈ln(
12ρ+v(1−ρ)2
)
ln ρ≈ − ln v
ln ρ.
If ρ ≈ 1 but less than 1, it can easily be proven that B0
2A0−√(
B0
2A0
)2
− 1A0
≈ 1 −(√v + 1
)(1 − ρ).
Therefore, ρn = 1 −(√v + 1
)(1 − ρ) is solved by
x ≈√v + 1 ≈ √
v.
For large values of ρ, B0
2A0+
√(B0
2A0
)2
− 1A0
≈ (v−1)(v+1)v
. Solving for x in ρx = (v−1)(v+1)v
is
approximated by solving for x in ρx = v and we obtain
x ≈ ln v
ln ρ.
Lemma A 3 Assume v − (n − 1) − I1n−1(ρ) > 0. Define R12(n)
.= ct(v − n +
I1n(ρ)
I1n(ρ)−I1n−1(ρ)) and
R13(n).= ct(v − (n− 1)
I11 (ρ)
I11 (ρ)−I1n−1(ρ)). These variables define the following three cases for n ≥ 3:
Case
I : R13(n) ≥ 0 and R12(n) ≤ R13(n)
II : R12(n) < 0 and R13(n) < 0
III : R12(n) ≥ 0 and R13(n) < R12(n)
(A-9)
36
and the following two cases for n = 1, 2:
Case
II ′ : R12(n) < 0
III ′ : R12(n) ≥ 0(A-10)
The profit maximizing contract (R1(n), r1(n)) among those for which β∗ (R, r) = n has the following
structure:
Casej I II, II’ III, III’
Rj1(n)/ct R13(n)/ct 0 R12(n)/ct
rj1(n)/c n−1I11 (ρ)−I1n−1(ρ)
− 1 v−(n−1)I1n−1(ρ)
− 1 1I1n−1(ρ)−I1n(ρ)
− 1
The optimal profit has the form πj1 (n) =Rj1(n)ct
ρn (ρ) +rj1(n)c
. When n = 1, 2, both cases yield profit
π1(n) = v−(n−1)I1n−1(ρ)
− 1. If v− (n− 1)− I1n−1(ρ) ≤ 0, no contract exists for which β∗ (R, r) = n, R ≥ 0
and r > 0.
Proof of Lemma A 3:
Remember that α∗(R, r) = 1 when r > 0 and that by Proposition 6, β∗ (R, r) = n ≥ 1 if n is
the largest integer that satisfies conditions (d), (e) and (f) in Proposition 5 for α = 1. Defining
Ψ (n;R, r).= v − R
ct−(1 + r
c
)I1n (ρ), these conditions can be rewritten as
n− 1 ≤ Ψ(n− 1;R, r) ≤ n and Ψ (min (1, n− 1) ;R, r) ≥ 0. (A-11)
Let us impose the additional constraint
Ψ (n;R, r) ≤ n. (A-12)
We now show that if n satisfies (A-11) and (A-12) with the latter inequality strictly satisfied, then
β∗ (R, r) = n, otherwise, β∗ (R, r) = n+1. By definition, Ψ (n;R, r) strictly increases in n. If (A-12)
holds, we obtain
n− 1 ≤ Ψ(n− 1;R, r) < Ψ(n;R, r) ≤ n⇒ Ψ(n;R, r) − Ψ(n− 1;R, r) ≤ 1.
As Ψ (n;R, r) is strictly concave, it follows that
Ψ (n+ k;R, r) − Ψ(n+ k − 1;R, r) < 1 for all k ≥ 1.
For any k ≥ 2, we obtain:
k−1∑
l=1
[Ψ (n+ l;R, r) − Ψ(n+ l − 1;R, r)] < k − 1 ⇒ Ψ(n+ k − 1;R, r) − Ψ(n;R, r) < k − 1
and, as Ψ (n;R, r) ≤ n, we obtain by adding the latter two inequalities that
Ψ (n+ k − 1;R, r) < n+ k − 1
37
Thus, it is impossible that n + k − 1 ≤ Ψ(n+ k − 1;R, r) for k ≥ 2, which is one of the necessary
conditions for n + k to be an equilibrium. For k = 1, if Ψ (n;R, r) < n, then it is impossible that
n ≤ Ψ(n;R, r) and n is the largest equilibrium. If Ψ (n;R, r) = n, then, in fact n+ 1 is the largest
equilibrium (with n also an equilibrium) since it is the largest value satisfying A-11. Indeed, the
above argument shows that there is no larger equilibrium.
We now write
Ω1 (n).= (R, r) ∈ R
2+ : n − 1 ≤ Ψ(n− 1;R, r) ≤ n, n ≥ Ψ(n;R, r) and Ψ (min (1, n− 1) ;R, r) ≥
0. We would like to find the highest profit contract (r1(n), R1(n)) that results in the pure strategy
equilibrium n as the longest queue equilibrium. To this end, we solve max(R,r)∈Ω1(n)
Rtρn (ρ) + r. If
Ψ (n;R, r) < n at the optimal solution, we’re done. If equality holds, then n and n + 1 both exist.
By imposing Ψ (n;R, r) ≤ n − ε for arbitrarily small ε, we can exclude n + 1. By continuity, the
corresponding profit is arbitrarily close to the profit under the case Ψ (n;R, r) = n and can be
approximated by it. Therefore, for the purposes of making profit comparisons, we work with Ω1 (n)
as defined above.
Note that Rtρn (ρ) + r is increasing both in R and r (for a fixed n). As Ψ (n− 1;R, r) is de-
creasing in R and r and the constraints n− 1 = Ψ (n− 1;R, r) and Ψ (n− 1;R, r) = n are parallel
in the (R, r) space, the constraint Ψ (n− 1;R, r) ≤ n can never be active at the optimal solu-
tion for any n. We therefore redefine Ω1 (n).= (R, r) ∈ R
2+ : n − 1 ≤ Ψ(n− 1;R, r) , n ≥
Ψ(n;R, r) and Ψ (min (1, n− 1) ;R, r) ≥ 0Since this is a two-dimensional linear programming problem with few inequalities, we break the
problem down into subcases according to which corner point will be the optimal solution. This
allows us to characterize the optimal solution in closed form for the three resulting subcases. We
start with n ≥ 3.
For n ≥ 3, we need to solve the following LP:
max(R,r)∈R2
+
R
tρn (ρ) + r (A-13)
n− 1 ≤ Ψ(n− 1;R, r) (A-14)
Ψ (n;R, r) ≤ n (A-15)
0 ≤ Ψ(1;R, r) (A-16)
The slope of the isoprofit line is − tρn(ρ) , that of the constraint Ψ (n− 1;R, r) = n−1 is −tI1
n−1 (ρ),
and that of the constraint Ψ (1;R, r) = 0 is −tI11 (ρ). It can easily be shown that for n ≥ 3, I1
n−1 (ρ) <
1ρn(ρ) < I1
1 (ρ) for all ρ. Since I1n (ρ) < I1
n−1 (ρ) for all n, we obtain I1n (ρ) < I1
n−1 (ρ) < 1ρn(ρ) < I1
1 (ρ)
for n ≥ 3. Moreover, the feasible region is bounded above by (A-14) and (A-16) and below by
(A-15). Finally, for n ≥ 3, the R-intercepts of the three constraints are distinct and ordered with
that of (A-15) being the smallest and that of (A-16) being the largest. Thus, for the feasible region
to contain points (R, r) with R ≥ 0 and r > 0, it is sufficient that (A-14) cross the R-axis at a
38
positive value of R; this can be rewritten as v− (n−1)− I1n−1(ρ) > 0 and will be assumed to hold in
the analysis below. We now use these facts about the problem structure to characterize the optimal
solution.
Since the isoprofit line has a slope between the slopes of constraints (A-14) and (A-16), and the
objective function is increasing both in R and in r, in the absence of (A-15), the optimal solution
would be either (i) at the intersection of (A-14) and (A-16) if these lines intersected in the first
quadrant, or (ii) at the intersection of (A-14) and the line R = 0 otherwise. With constraint (A-15),
we also need to take into account where constraints (A-14) and (A-15) intersect. Let R12 and R13,
respectively, denote the R-intercepts of the intersection of (A-14) and (A-15), and of (A-14) and
(A-16), respectively. We find that the optimal solution to the LP is given by exactly one of the
following three cases:
(I) the intersection of (A-14) and (A-16) if R12 ≤ R13 and R13 ≥ 0. An example (with R12 ≥ 0
as well) is given in Figure 6.
(II) the intersection of (A-14) and R = 0 if R12 < 0 and R13 < 0. An example is given in Figure
7.
(III) the intersection of (A-14) and (A-15) if R12 > R13 and R12 ≥ 0. An example (with R13 ≥ 0
as well) is given in Figure 8.
For n = 1, Ψ (min (1, n− 1) ;R, r) ≥ 0 coincides with (A-14) ; for n = 2, (A-16) is redundant.
Thus in both cases, only (A-14) and (A-15) need be considered. In addition, in both problems, the
slope of the iso-profit function is equal to the slope of (A-14), so any feasible point on this line results
in the optimal profit. The optimal profit expressions are π1 (1) = vI10 (ρ)
− 1 and π1 (2) = v−1I11 (ρ)
− 1.
Note that if R12 ≥ 0, then case III holds, otherwise, case II holds.
From the intersection of (A-14) and (A-16), we obtain
v −(1 + r
c
)I11 (ρ) = R
ct
v − (n− 1) −(1 + r
c
)I1n−1 (ρ) = R
ct
⇒
r13(n)c
= n−1I11 (ρ)−I1n−1(ρ)
− 1
R13(n)ct
= v − (n− 1)I11 (ρ)
I11 (ρ)−I1n−1(ρ)
From the intersection of (A-14) and (A-15), we obtain
v − n−(1 + r
c
)I1n (ρ) = R
ct
v − (n− 1) −(1 + r
c
)I1n−1 (ρ) = R
ct
⇒
r12(n)c
= 1I1n−1(ρ)−I1n(ρ)
− 1
R12(n)ct
= v − n+I1n(ρ)
I1n(ρ)−I1n−1(ρ)
Let rk1 (n) and Rk1(n) for k = I, II, III denote the optimal solution to the LP in the three cases.
Then we have rI1(n) = r13(n), RI1(n) = R13(n), rIII1 (n) = r12(n), RIII1 (n) = R12(n). To determine
the values for k = II, we find the intersection point in case II:
0 = v −R− (n− 1) −(1 + r
c
)I1n−1 (ρ)
R = 0⇒
rII1 (n)c
= v−(n−1)I1n−1(ρ)
− 1
RII1 (n) = 0
39
Ψ(1;R,r)=0
Ψ(n-1;R,r)=n-1
Ψ(n-1;R,r)=n
Ψ(n;R,r)=n
r
R
π(n;R,r)=k
Figure 6: Illustration of case (I).
Lemma A 4 For a given (v, ρ),
maxn≥3
πI1 (n; v, ρ) (A-17)
is solved by n satisfying vIn−1 (ρ) < v < vIn (ρ) with
vIn (ρ) =
n
1−( 11+ρ )
n−11−ρn−1
1−ρn+2 − n−1
1−( 11+ρ )
n−21−ρn−2
1−ρn+1
1−ρn−1
1−ρn+2 − 1−ρn−2
1−ρn+1
for n ≥ 3
and vI2 (ρ) = 0. For v 1, we can approximate the solution to (A-17) by
nI1 (v, ρ) ≈
⌈√2ρ
⌉ρ 1 and v ≤ 1
2ρ2−√
2ρ
⌈− ln v
ln ρ
⌉+ 2 ρ 1 and v > 1
2ρ2−√
2ρ
d√ve ρ ≈ 1⌈ln vln ρ
⌉1 ρ
(A-18)
Proof Step 1. The profit maximizing value of n satisfies πI1(n) − πI1(n − 1) > 0 and πI1(n + 1) −πI1(n) < 0. Using πI1(n) from Lemma 3 and simplifying, these two inequalities can be written as
vIn−1 (ρ) < v < vIn (ρ) with
vIn(ρ) =
n(1+ 1
ρ (1
1+ρ ))ρ1−ρn+1
1−ρn+2 −1
1ρ (
11+ρ )− 1
ρ (1
1+ρ )n − (n− 1)
(1+ 1ρ (
11+ρ ))ρ
1−ρn
1−ρn+1 −1
1ρ (
11+ρ )− 1
ρ (1
1+ρ )n−1
ρ 1−ρn+1
1−ρn+2 − ρ 1−ρn1−ρn+1
.
40
Ψ(1;R,r)=0
Ψ(n-1;R,r)=n-1
Ψ(n;R,r)=n
Ψ(n-1;R,r)=n
r
R
π(n;R,r)=k
Figure 7: Illustration of case (II).
After some algebraic manipulation, we obtain
vIn (ρ) =
n
1−ρn+1−(1−ρ2)
1−ρn+2
1−( 11+ρ )
n−1 − (n− 1)
1−ρn−(1−ρ2)1−ρn+1
1−( 11+ρ )
n−2
1−ρn+1
1−ρn+2 − 1−ρn1−ρn+1
(A-19)
It can be seen (by numerical inspection) that vIn+1 (ρ) − vIn (ρ) > 0 for all n ≥ 0. Therefore, for a
given (v, ρ), there exists exactly one n ≥ 3 that satisfies vIn−1 (ρ) < v < vIn (ρ). πI1 (n; v, ρ) is thus
unimodal for n ≥ 3 and nI1 (v, ρ) = dne can be obtained from solving for n in v = vIn (ρ). Since this
equation does not have an analytical solution, we approximate the solution by approximating vIn (ρ)
by vIn (ρ) and solving for n in v = vIn (ρ) for v 1.
Step 2. In (A-19), we use the following approximation:
1
1 −(
11+ρ
)k−1≈
1k−1
(1ρ
+ 12k)
ρ < 2k−2
1 ρ > 2k−2
. (A-20)
This approximation is obtained by using the first two terms of the Laurent series expansion of the
expression for small ρ, observing that the expression goes to 1 in the limit, and concatenating the
two at the value of ρ for which the expansion equals 1.
Case (i): If 0 < ρ < 2n−2
(< 2
n−3
), then, with (A-20) for k = n− 1 and k = n, we obtain
vIn (ρ) ≈n
1−ρn+1−(1−ρ2)1−ρn+2
1n−1
(1ρ
+ 12n)− (n− 1)
1−ρn−(1−ρ2)1−ρn+1
1n−2
(1ρ
+ 12 (n− 1)
)
1−ρn+1
1−ρn+2 − 1−ρn1−ρn+1
. (A-21)
41
Ψ(1;R,r)=0
Ψ(n-1;R,r)=n-1
Ψ(n-1;R,r)=n
Ψ(n;R,r)=n
r
R
π(n;R,r)=k
Figure 8: Illustration of case (III).
For 0 < ρ 1 we can further approximate
1 − ρn+1
1 − ρn+2− 1 − ρn
1 − ρn+1≈ ρn, 1 − ρn+2 ≈ 1 − ρn+1 ≈ 1 and 1 − ρ− ρ2 ≈ 1. (A-22)
Using the approximations in (A-22) in (A-21) we obtain
vIn (ρ) ≈ gn(ρ).=
12
n2−3n+1− 2ρ
(n−1)(n−2)
ρn−2.
gn (ρ) can be studied analytically: It is unimodal, with limρ→0 = −∞ and limρ→∞ = 0. In addition,
(i) gn(ρ) = 0 for ρ0 (n).= 2
n2−3n+1
(ii) ddρgn(ρ) = 0 for ρm (n)
.= 2
n2−3n+1n−1n−2 .
Thus, vIn (ρ) attains a local maximum in ρ for a fixed n. If v < gn(ρm(n)), vIn (ρ) = v has two
solutions. If v > gn(ρm(n)), vIn (ρ) = v has no solution. One solution falls in [ρ0 (n) , ρm (n)], and
the other in [ρm (n) ,∞]. As the first interval is very small for large values of n, we can approximate
the solution by ρ0(n) ≈ 2n2 or nI1 (v, ρ) ≈
√2ρ. ρ 2
n−2 is satisfied for n =√
2ρ
since√
12ρ 1
ρ+ 1.
Substituting nI1 (v, ρ) ≈√
2ρ
in the condition above, we observe that v = vIn (ρ) has a solution only
if v ≤ 12ρ
(2−√
2ρ
)
.
For the second solution, we use the further approximation vIn (ρ) ≈ 12
1ρn−2 , from which it follows
that nI1 (v, ρ) = 2 − ln(2v)ln ρ . For ρ 2
n−2 to be satisfied for n = 2 − ln(2v)ln ρ , we need − ln(2v)
ln ρ 2ρ,
which is equivalent to v 12ρ
− 2ρ . However, we are interested in large values of v, so an upper bound
on the value of v for which this approximation holds makes it impractical to use. In Case iia below,
42
we develop an approximation for the case 2n−3 < ρ which is arbitrarily close to this approximation
for large v and holds for any v, so we focus on that approximation instead.
Case (ii): If(
2n−2 <
)2
n−3 < ρ, then, with (A-20) for k = n− 1 and k = n then, we obtain
vIn (ρ) ≈n
1−ρn+1−(1−ρ2)1−ρn+2 − (n− 1)
1−ρn−(1−ρ2)1−ρn+1
1−ρn+1
1−ρn+2 − 1−ρn1−ρn+1
=n 1−ρn+1
1−ρn+2 − (n− 1) 1−ρn1−ρn+1 −
(1 − ρ2
) (n
1−ρn+2 − n−11−ρn+1
)
1−ρn+1
1−ρn+2 − 1−ρn1−ρn+1
. (A-23)
We can use the following approximation in the previous expression:
(1 − ρ2
)( n
1 − ρn+2− n− 1
1 − ρn+1
)≈
(1 − ρ2
)ρ 1
4(n+1)(n+2) ρ ≈ 1
0 ρ 1
(A-24)
This gives us three subcases to study.
Subcase (iia): For 2n−3 < ρ 1, we can use (A-24) for (A-23) and obtain
vIn (ρ) ≈n 1−ρn+1
1−ρn+2 − (n− 1) 1−ρn1−ρn+1 −
(1 − ρ2
)
1−ρn+1
1−ρn+2 − 1−ρn1−ρn+1
,
which can be rewritten as
vIn (ρ) =(1 − ρn)
(1 − ρn+2
)
ρn (1 − ρ)2 −
(1 − ρ2
) (1 − ρn+2) (
1 − ρn+1)
ρn (1 − ρ)2 .
Let A = 1 − ρ+ ρ3 and B = (1−ρ)2ρ2
v + 1−ρ+ρ3+ρ4ρ2
. Then v = vIn (ρ) ⇔ 0 = A (ρn)2 − Bρn + 1. We
can solve this equation and obtain
nI1 (v, ρ) ≈ln
(B2A +
√(B2A
)2 − 1A
)
ln ρ
=
ln
(1−ρ+ρ3+ρ4+(1−ρ)2v
2ρ2(1−ρ+ρ3) −√(
1−ρ+ρ3+ρ4+(1−ρ)2v2ρ2(1−ρ+ρ3)
)2
− 11−ρ+ρ3
)
ln ρ.
For small ρ, we further obtain
B
2A 1 and
1
A 1, which means
B
2A−√(
B
2A
)2
− 1
A≈ 1
B.
Thus, for ρ ≈ 0,
nI1 (v, ρ) ≈ln(
ρ2
(1−ρ)2v+1−ρ+ρ3+ρ4
)
ln ρ≈ 2 − ln v
ln ρ.
Subcase (iib): For ρ ≈ 1, we obtain
vIn (ρ) ≈nn+1n+2 − (n− 1) n
n+1 − 4(n+1)(n+2)
n+1n+2 − n
n+1
= (n+ 4) (n− 1) .
43
Therefore,
v = (n+ 4) (n− 1) ⇒ nI1 (v, ρ) ≈ √v.
Subcase (iic): For ρ 1, we obtain
vIn (ρ) ≈n 1−ρn+1
1−ρn+2 − (n− 1) 1−ρn1−ρn+1
1−ρn+1
1−ρn+2 − 1−ρn1−ρn+1
= n+1
ρn(1 − ρn)
(1 − ρn+2
)
(ρ− 1)2 .
We can approximate vIn (ρ) further as follows:
vIn (ρ) =1
ρn(1 − ρn)
(1 − ρn+2
)
(1 − ρ)2 .
Let A2 = ρ2 and B2 = ρ2 + 1 + v (1 − ρ)2. Then v = vIn (ρ) ⇔ 0 = A2 (ρn)
2 − B2ρn + 1. We solve
this equation to obtain
nI1 (v, ρ) =ln
(B22A2
+
√(B22A2
)2− 1A2
)
ln ρ for ρ > 1.
For large values of ρ, B2
2A2+
√(B2
2A2
)2
− 1A2
≈ v + 1. Therefore, n solves ρn ≈ v + 1 and
nI1 (v, ρ) ≈ ln v
ln ρ.
Lemma A 5 For a given (v, ρ), maxn≥0
πII1 (n; v, ρ) is solved by n satisfying vIIn−1 (ρ) < v < vIIn (ρ)
with vIIn (ρ) = n − I1n(ρ)
I1n(ρ)−I1n−1(ρ)and can be approximated by nII1 (v, ρ) =
⌈v − 1
ρ
⌉for low values of
ρ. In addition, πII1 (v, ρ) ≈ ρ (v − 1) − 1 for low values of ρ.
Proof The profit maximizing value of n satisfies πII1 (n)−πII1 (n−1) > 0 and πII1 (n+1)−πII1 (n) < 0.
Using πII1 (n) = v−(n−1)I1n−1(ρ)
− 1 from Lemma 3 and simplifying, these two inequalities can be written
as vIIn−1 (ρ) < v < vIIn (ρ) with
vIIn (ρ) = n− I1n (ρ)
I1n (ρ) − I1
n−1 (ρ)= n−
ρ+(
11+ρ
)n
(1
1+ρ
)n−(
11+ρ
)n−1 = n+ ρ(1 + ρ)n +1
ρ.
For low values of ρ, using (1 + ρ)n ≈ 1 + nρ, we obtain the following approximation: vIIn (ρ) ≈1ρ
+ n (ρ+ 1) + 1 ≈ 1ρ
+ n + 1, from which we obtain the approximation nII1 (v, ρ) = v − 1ρ. Sub-
stituting this approximation in the profit expression, we obtain πII1 (v, ρ) ≈ v−(v− 1ρ−1)
1+ 1ρ (
11+ρ )
v− 1ρ−1
− 1 ≈1ρ+1
1+ 1ρ(1−(v− 1
ρ−1)ρ)
− 1 = 1+ρρ+1−(v− 1
ρ−1)ρ
− 1 ≈ 11−(v− 1
ρ−1)ρ
− 1 ≈ ρ (v − 1)− 1, where we twice used the
approximation(
11+x
)n≈ 1 − nx for x ≈ 0.
44
Lemma A 6 Let v 1 and n = nI1 (v, ρ). For ρ < 1v
+ 1+√
2v+1v2
, R12(n) ≈ R13(n) < 0 and Case
II applies. For 1v
+ 1+√
2v+1v2
< ρ, 0 < R12(n) ≤ R13(n) and Case I applies.
Proof Having v 1 and ρ < 1v
+ 1+√
2v+1v2
means ρ ≈ 0. From Lemma A4, we know nI(v, ρ) =√
2ρ
for ρ ≈ 0. Substituting this into the expression for R13(n) and R12(n) we find R13(n)/ct ≈ R12(n) ≈v − 1
ρ−√
2ρ. R13(n) < 0 if and only if ρ < 1
v+ 1+
√2v+1v2
. This completes the first case.
To complete the case where ρ > 1v
+ 1+√
2v+1v2
, we need to show that R12(n) ≤ R13(n) in this
case. We always have R13(n) ≷ R12(n) ⇔ r13(n) ≶ r12(n). Using the values of r13(n) and r12(n)
from Lemma A3, we find
r12(n)
c≥ r13(n)
c⇔ n− 1 ≤ (1 + ρ)
n−1 − (1 + ρ)
ρ⇔ 1 + nρ ≤ (1 + ρ)
n−1.
This inequality is satisfied for ρ ∈ [0, ρn] where ρn denotes the positive root of 1 + nρ = (1 + ρ)n−1
.
This root can be approximated by solving 1 + nρ = 1 + (n− 1) ρ+ 12 (n− 1) (n− 2) ρ2, which gives
ρn ≈ 2n2−3n+2 . Note that if for a given n, we have that 1 + nρ ≤ (1 + ρ)
n−1, then for all n′ ≥ n,
1 + n′ρ ≤ (1 + ρ)n′−1
. Let n (ρ) denote the solution to ρ = ρn. Since ρn = ρ0(n) in Lemma A4,
it follows that nI1 (v, ρ) ≥ n (ρ). Therefore, the inequality is satisfied, and R12(n) ≤ R13(n), for
n = nI1 (v, ρ).
Note that equating the approximations in Case I and Case II for low values of ρ gives v− 1ρ
=√
2ρ.
Solving this for ρ gives ρ = 1v
+ 1+√
2v+1v2
. This is exactly the boundary point considered in this
lemma, so the approximations preserve the continuity of nI1(v, ρ) across the two cases.
Proof of Proposition 8. We wish to determine the profit maximizing equilibrium under demand
induction, n1(v, ρ), by solving maxn≥0
π1 (n; v, ρ). This proof is based Lemmas A3 to A6 above. Lemma
A3 characterizes the profit maximizing solution (R, r) among those for which β∗(R, r) = n. It is
shown that one of three cases applies and the solution is given in closed form for each case. Suppose
that for a given n, Case III applies. As discussed in the proof of Lemmas A3, the optimal solution
in Case III, (RIII1 (n), rIII1 (n)), in fact yields two successive equilibria n and n+ 1. Since n+ 1 is an
equilibrium, the feasible region corresponding to this equilibrium can also be defined. This region
contains the point (RIII1 (n), rIII1 (n)), so the maximum profit corresponding to equilibrium n+1 will
be at least as much as πIII1 (n). We can therefore focus solely on Cases I and II in our analysis. We
first assumed that Case I held for all feasible n and called the profit maximizing equilibrium that
would result if this were the case nI1(v, ρ). An approximate characterization of nI1(v, ρ) was given in
Lemma A4. We then assumed that Case II held for all feasible n and called the profit maximizing
equilibrium that would result if this were the case nII1 (v, ρ). An approximate characterization of
nII1 (v, ρ) was given in Lemma A5. Lemma 6 delineated the values of ρ for which Case I and Case II
hold at n = nI1(v, ρ).
From Lemma A3, we know that v− (n− 1)− I1n−1(ρ) > 0 is a necessary and sufficient condition
for n to be an equilibrium. Since this expression is decreasing in n, if v − I10 (ρ) = v − 1+ρ
ρ≤ 0,
45
then no equilibrium is possible. Solving v − 1+ρρ
= 0 gives ρ = 1v−1 ≈ 1
vfor large v. Therefore,
n∗1(v, ρ) = 0 if ρ ≤ 1v
for v 1 .
Now consider 1v< ρ < 1
v+ 1+
√2v+1v2
. From Lemma A6, we know that Case II applies. Therefore,
n1(v, ρ) = nII1 (v, ρ) =⌈v − 1
ρ
⌉and π1 (n; v, ρ) = πII1 (n; v, ρ).
Next, consider 1v
+ 1+√
2v+1v2
< ρ. From Lemma A 6, we know that Case I applies at n = nI1(v, ρ).
Therefore, with Lemma A 3, π1
(nI1 (v, ρ) ; v, ρ
)= πI1
(nI1 (v, ρ) ; v, ρ
). From the structure of the
LP, it can be proven that π1 (n; v, ρ) ≤ πj1 (n; v, ρ) for all n. In addition, since nI1 (v, ρ) maximizes
πI1 (n; v, ρ), we have πI1 (n; v, ρ) ≤ πI1(nI1 (v, ρ) ; v, ρ
)for all n. Combining these two inequalities,
we find that π1 (n; v, ρ) ≤ πI1(nI1 (v, ρ) ; v, ρ
)for all n. In other words, nI1 (v, ρ) solves not only
maxn≥3
πI1 (n; v, ρ) but maxn≥3
π1 (n; v, ρ) as well for this range of ρ values.
Thus, nI1 (v, ρ) is the profit maximizing equilibrium when 1v
+ 1+√
2v+1v2
< ρ. This is valid as long
as nI1 (v, ρ) ≥ 3, and we can refer to (A-18) to determine nI1 (v, ρ). Note that in (A-18), nI1 (v, ρ) ≥ 3
for ρ <√v, nI1 (v, ρ) = 2 for
√v ≤ ρ < v and nI1 (v, ρ) = 1 for v ≥ ρ. This can be verified by
evaluating ln vln ρ at ρ =
√v and ρ = v. Thus, we need to consider the ρ ≥ √
v case separately. In this
region, πI1 (n; v, ρ) is maximized by n = 3 since πI1 (n; v, ρ) is unimodal and its maximum occurs at 2
or 1 in this region. We therefore need only compare πII1 (1), πII1 (2) and πI1(3). We find πI1(3) < πII1 (1)
for√v ≤ ρ, πII1 (1) < πII1 (2) for
√v ≤ ρ < v and πII1 (1) > πII1 (2) for v ≤ ρ. We conclude that for
√v ≤ ρ < v, n∗
1 (v, ρ) = 2 and for v ≤ ρ, n∗1 (v, ρ) = 1. Thus, we can use the approximation ln v
ln ρ in
this region as well. This completes the approximate characterization of n1(v, ρ).
Lemma A 7 For any ρ > 1, we have that πI1 (n; v, ρ) ≥ π0 (n; v, ρ) for all n ≥ 3.
Proof Note that πI1 (n; v, ρ) − π0 (n; v, ρ) is independent of v. Calling this difference D (n, ρ), we
have
D (n, ρ) =
(n− (n− 1)
I11 (ρ)
I11 (ρ) − I1
n−1 (ρ)
)ρn (ρ) +
n− 1
I11 (ρ) − I1
n−1 (ρ)− 1
= ρn (ρ) + (n− 1)1 − I1
n−1 (ρ) ρn (ρ)
I11 (ρ) − I1
n−1 (ρ)− 1
Recall ρn (ρ) = ρ(1 − (1−ρ)ρn
1−ρn+1
).
D (n, ρ) > 0 ⇐⇒ ρn (ρ) + (n− 1)1 − I1
n−1 (ρ) ρn (ρ)
I11 (ρ) − I1
n−1 (ρ)− 1 < 0
⇐⇒ 1 − (n− 1)1 − I1
n−1 (ρ) ρn (ρ)
I11 (ρ) − I1
n−1 (ρ)< ρn (ρ)
⇐⇒ 1 − (n− 1)1 − I1
n−1 (ρ)
I11 (ρ) − nI1
n−1 (ρ)< ρ
(1 − (1 − ρ) ρn
1 − ρn+1
)
⇐⇒ 1 − 1n−(1+ρ)n−2
n−1 + ρ (1 + ρ)n−1
< ρ
(1 − (1 − ρ) ρn
1 − ρn+1
)
⇐⇒ n− (1 + ρ)n−2
n− 1+ ρ (1 + ρ)
n−1<
1
(1 − ρ)(1 + ρn+1
1−ρn+1
)
46
It can be shown that D (n, 0) = 0 and limρ→∞D (n, ρ) = 0. We now show that D (n, ρ) = 0
for exactly one ρ0 (n) ∈ (0, 1). Noting that the last term equals 1−ρn+1
1−ρ , we can rewrite the last
inequality as follows:
n− (1 + ρ)n−2
n− 1+ ρ (1 + ρ)
n−1<
1 − ρn+1
1 − ρ
⇐⇒ n
n− 1+
(ρ2 + ρ− 1
n− 1
)(1 + ρ)
n−2<
1 − ρn+1
1 − ρ
⇐⇒ n
n− 1+
(ρ2 + ρ− 1
n− 1
) n−2∑
k=0
(n− 2
k
)ρk <
n∑
k=0
ρk
Note that the ρ0 term cancels the term nn−1 . Then, we can divide by ρ and separate the constant
term and obtain
1 + ρ+
(ρ2 + ρ− 1
n− 1
) n−3∑
k=0
(n− 2
k
)ρk <
n−1∑
k=0
ρk.
Finally, we can solve the latter equation and obtain that ρ0 (3) = 12 , ρ0 (4) = .2953, ρ0 (5) = .2185,
ρ0 (6) = .1762 etc., with limn→∞
ρ0 (n) = 0. D (n, ρ) < 0 on ρ ∈ (0, ρ0(n)) and D (n, ρ) > 0 on
ρ ∈ (ρ0(n),∞). Thus, we have obtained that for any ρ > 1, D (n, ρ) > 0 for all n ≥ 3.
Proof of Proposition 9. For high values of n, ρn (ρ) ≈ min (1, ρ). Rewriting (6), which gives the
approximate profit under induction for large n, we obtain
π∗1
c≈ (v − n1 (v, ρ)) min (1, ρ) + max (1 − ρ, 0) ((n1 (v, ρ) − 1) ρ (1 + ρ) − 1) .
When not inducing service, with Proposition 7, the profit structure is
π∗0
c≈ (v − n0 (v, ρ)) min (1, ρ) .
Comparing these profits for ρ < 1, we see that
π∗1
c− π∗
0
c≈ (n0 (v, ρ) − n1 (v, ρ)) ρ+ (1 − ρ) ((n1 (v, ρ) − 1) ρ (1 + ρ) − 1) .
On this range, we discover two drivers for service induction: (1) (n0 (v, ρ) − n1 (v, ρ)) ρ is due to the
difference in queue length and (2)((n1 (v, ρ) − 1)
(ρ− ρ3
)− (1 − ρ)
)is the extra profit stream from
service induction. For low ρ, (1) is negative due to Propositions 7 and 8. (2) is also negative. For
intermediate values of ρ, for which n1 (v, ρ) = n0 (v, ρ) + 2 ≈⌈− ln v
ln ρ
⌉+ 2 (see Propositions 7 and 8),
we obtain
π∗1
c− π∗
0
c≈ −2ρ+ (1 − ρ) ((n1 (v, ρ) − 1) ρ (1 + ρ) − 1) > 0 ⇔
⌈− ln v
ln ρ
⌉> −1 +
1
ρ (1 − ρ).
If v is large enough, this inequality is satisfied. As ρ increases to 1, we know from Proposition 8 that
n1 (v, ρ) → n0 (v, ρ). The extra revenue term also drops to zero. Thus, the profits of both cases will
become more or less equal (π∗
1
c→ π∗
0
c).
From Propositions 7 and 8, we obtain that n0 (v, ρ) ≈ n1 (v, ρ) ≥ 3 for ρ ≤ √v. From Lemma
A 7, we know that πI1 (n; v, ρ) ≥ π0 (n; v, ρ) with πI1 (n; v, ρ) ≈ π0 (n; v, ρ) for very high values ρ.
47
Since we know that π∗1 (v, ρ) = πI1
(nI1(v, ρ); v, ρ
)in this range, we conclude that π1 (n1(v, ρ); v, ρ) ≥
π0 (n0(v, ρ); v, ρ) with π1 (n1(v, ρ); v, ρ) ≈ π0 (n0(v, ρ); v, ρ) for large ρ.
For ρ ≥ √v, we have that n1 (v, ρ) ≈ n0 (v, ρ) ∈ 1, 2 and π∗
1(n) = πII1 (n) < π∗0 (n).
Proof of Proposition 10. Recall thatΠ∗
u
c=(v − 1
1−ρ
)ρ with ρ
.= min
(1 −
√1v, ρ). For ρ ≈ 0,
Π∗
u
c≈ (v − 1) ρ > 0, while
π∗
1
c= 0 <
π∗
0
c≈ (v − 1) ρ. Thus,
Π∗
u
c≈ π∗
0
c.
For larger values of ρ, we know thatΠ∗
u
c>
π∗
0
c>
π∗
1
c.
From the proof of Proposition 9, we know that for ρ < 1,
π∗1
c≈
(v +
ln v
ln ρ
)ρ− (1 − ρ)
((ln v
ln ρ+ 1
)ρ (1 + ρ) − 1
)
= vρ+ln v
ln ρρ3 + 1 − 2ρ+ ρ3
π∗1
c− Π∗
u
c=
ln v
ln ρρ3 + 1 − 2ρ+ ρ3 +
ρ
1 − ρ> 0
⇔ ln v >−1 + 2ρ− 2ρ2 − ρ3 + ρ4
(1 − ρ) ρ3ln ρ > 0
which will be satisfied for ρ close enough to 1. From similar observations, we obtain:Π∗
u
c>
π∗
0
c.
Finally, for very large values of ρ (and greater than√v), we obtain
π∗
0
c>
π∗
1
c>
Π∗
u
c.
48
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