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

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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

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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

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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.

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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

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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.

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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.

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