Queuing for Expert Services

Laurens G. Debo

Tepper School of Business

Carnegie-Mellon University, Pittsburgh, PA 15213, USA

L. Beril Toktay

Technology Management

INSEAD, 77305 Fontainebleau, France

Luk N. Van Wassenhove

Technology Management

INSEAD, 77305 Fontainebleau, France

Abstract

We consider a monopolist expert offering a service with a ‘credence’ characteristic. A

credence service is one where the customer cannot verify, even after purchase, whether the

amount of prescribed service was appropriate or not; examples include legal, medical or con-

sultancy services and car repair. This creates an incentive for the expert to ‘induce service’,

that is, to provide unnecessary services that add no value to the customer, but that allow

the expert to increase his revenues. We focus on the impact of an operations phenomenon

on service inducement - workload dynamics due to the stochasticity of interarrival and ser-

vice times. To this end, we model the expert’s service operation as a single-server queue.

The expert determines the service price within a fixed- and variable- rate structure and

decides whether to induce service or not. We characterize the expert’s combined optimal

pricing and service inducement strategy as a function of service capacity, potential market

size, value of service and waiting cost. We conclude with design implications of our results

in limiting service inducement.

July 2004

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 is performed. 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 an ‘experience

good’ for which usage reveals quality. Examples of credence goods are medical, legal and

repair services. In each case, if the outcome is satisfactory, the customer is limited in

his ability to detect whether an unnecessary level of service was provided to achieve that

outcome.

In such a setting, 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 an incentive to

perform unnecessary service. We refer to this phenomenon as ‘service inducement.’ Rational

buyers of services with a credence characteristic process ex ante the incentive of the expert

to induce service and calculate their net utility from obtaining the service accordingly. This

affects their decision of whether to purchase the service or not, and impacts the level of

service inducement chosen by the expert.

One key element in the expert’s incentive to induce service is the fee structure. A fee

that is proportional to the level of service provided makes it feasible for the expert to induce

service profitably. Thus, the use of a variable-rate service fee alerts the customer to the

fact that the expert may benefit from service inducement. In contrast, a fixed fee makes it

unprofitable for the expert to induce service. This is because with a fixed fee, the expert is

not compensated for any additional - albeit unnecessary - service.

In their seminal paper, Darby and Karni qualitatively discussed, but did not analyze,

another key element that impacts the expert’s incentive to induce service: the expert’s

workload level. In particular, a customer arriving to a variable-rate service when the expert

has a low workload level may expect that the expert has a high 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. Thus, price structure and workload level both impact the

expert’s choice of service inducement level in equilibrium. Since in many service settings

1

customer interarrival times and service times are stochastic, the workload typically changes

over time. In this paper, we develop a model that allows us to better understand how price

structure and workload dynamics impact the expert-customer interaction.

To this end, we consider a monopolist expert selling a single service. Workload dy-

namics are modelled using a single-server queue with a Poisson arrival process of potential

customers and independently and identically distributed exponential value-adding service

times. Customers are homogeneous in that they place the same value on the service and

have the same waiting cost per unit time. The service price consists of a fixed fee and/or

a variable component that is proportional to the total service time. The expert determines

the fixed and variable fees and chooses a service inducement policy. Customers observe the

queue length upon arrival and decide whether to purchase the service (i.e. join the queue

or not) based on the net utility that they expect from obtaining that service.

Our model allows us to address the following questions: How do characteristics of the

environment, in particular, service capacity, potential market size, value of the service

and waiting cost determine the expert’s incentive to induce services? Which customers

does the expert target for service inducement? Does the customer purchasing strategy

exhibit different characteristics for credence and non-credence services? Will fewer or more

customers visit a service provider with credence good characteristics? How does a credence

service provider set the service fee (fixed and/or variable) differently from a provider of a

non-credence service?

Our analysis is not a normative one: We do not wish to provide advice about when it is

optimal to induce services; our goal is to generate insights for managers of service systems

about drivers that impact choices of service providers. Existing research on credence goods

focuses on the impact of capacity, reputation and competition on the existence of service

inducement. Our research complements this literature by focusing on the role that workload

dynamics - an operations phenomenon - plays in the provision of service inducement. We

also complement the research that uses queuing models to analyze service systems, which,

to the best of our knowledge, ignores the credence good character of certain services.

Elements of our stylized model can be found in different service industries. An example

is legal advice. Many legal services are billed proportional to the time that an attorney puts

into the service, sometimes to the tenth of an hour (Ross 1996). Drawing on his surveys,

the experiences of legal audit firms, and anecdotes, Ross concludes that over-billing is

2

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). Ross also mentions

that the incentives to padding depend on the workload level. In particular, padding occurs

for lawyers who are ‘not busy’ since they have the time to do unnecessary tasks (p. 36-37).

Another example of a credence service 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 (2003) 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. In other words, workload impacts physician-induced demand.

The remainder of the paper is structured as follows: §2 puts our work in the context

of the existing literature and highlights our contributions. Our modelling assumptions are

described in §3. We characterize the expert’s optimal prices and profits with and without

inducement in §4. In §5, we build on the analysis in §4 to derive the expert’s optimal

strategy as a function of the characteristics of the environment. §6 discusses the main

insights from our analysis and the design implications of our results.

2 Related Literature

The model we develop for our analysis draws on 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 Haviv 2003) study the

impact of congestion on the customers’ and service provider’s decisions. In particular,

Hassin (1986) characterizes the equilibrium fixed fee in a single-server observable queue with

a homogeneous customer base, Poisson arrivals and exponential service times. Asymmetric

information models in this context typically assume that it is the expert who does not

observe the customer’s type (e.g. Whang 1989, Radhakrishnan and Balachandran 1996).

With a credence good, it is the customer who does not know his own ‘type.’ To the best of

our knowledge, our paper is the first to model the ‘credence good’ characteristic of services

in this literature. A contribution we make to this literature, described in §4.2, is to identify

3

the existence of the “follow-the-crowd” effect under service inducement; this contrasts with

“avoid-the-crowd” behavior typically observed in queueing models for non-credence goods.

Early papers on credence goods (Darby and Karni 1973, Pitchik and Schotter 1987a,b)

develop simple models of a single firm selling a credence good with an exogenous price,

and identify the existence of demand inducement. Recently, models allowing endogenous

pricing in monopoly (Emons 2001, Fong 2002) and competitive settings (Wolinsky 1995,

Emons 1997, Richardson 1999, Pesendorfer and Wolinksy 2003, Alger and Salanie 2003)

have been developed. Of these papers, only Emons models the expert as being capacity-

constrained, but uses a simple deterministic model. Our contribution to this literature is to

develop a richer model of a capacitated monopoly service system that explores the role of

workload dynamics; this issue has been qualitatively discussed but not analyzed in Darby

and Karni’s seminal paper. In particular, we demonstrate that even with a homogeneous

customer base, which is shown in the literature to eliminate service inducement, workload

dynamics result in the emergence of service inducement under some conditions. Below,

we position our work with respect to papers analyzing the monopoly case and discuss our

contributions in more detail.

Emons (2001) considers a capacity-constrained monopolist serving a homogeneous cus-

tomer base who determines the capacity level and prices of diagnosis and repair. The time

required to serve each customer honestly is identical and deterministic. Emons finds that

charging a flat fee or having a capacity level exactly equal to that required to serve the

whole market honestly (100% utilization) are sufficient to signal credibility. Note that with

stochastic interarrival and service times, 100% capacity utilization is not viable, so the ex-

pert can use only pricing as a mechanism to signal credible service. Fong (2002) shows that

in an uncapacitated system with homogeneous customers, charging a flat-rate regardless of

service type is again optimal and eliminates service inducement.

Note that customer homogeneity is an assumption made by both authors. Dulleck

and Kerschbamer (2003) develop a simple model unifying the literature and delineating

drivers impacting the existence or not of service inducement in equilibrium. They identify

customer homogeneity as one of the necessary conditions to eliminate service inducement.

The logic is the following: With a homogeneous customer base, a single price that leaves

each customer indifferent between purchasing service or not exists. This price extracts all

consumer surplus and maximizes expert profit. Since unnecessary service inducement only

4

“destroys” consumer surplus, the most profitable strategy for the expert is to not induce

service. Only with a heterogeneous customer base may the expert find it profitable to induce

service. In particular, since 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. This outcome

is observed in Fong (2002) who analyzes the heterogeneous customer case.

When workload dynamics are taken into account, a customer base that is homogeneous

with respect to service value and waiting cost becomes effectively heterogeneous upon ar-

rival due to workload dynamics: Customers arriving at different times observe different

workloads, which yields different levels of net utility from service. Our analysis shows that

in line with the literature, a flat fee is sufficient to signal to the customer that no service

inducement takes place. Nevertheless, under some conditions, the expert prefers setting a

fixed and variable fee and selectively inducing service. We also show that surprisingly, more

customers may enter service with demand inducement than without. As our analysis will

demonstrate, this is a combined result of the follow-the-crowd effect and the implicit price

discrimination capability of the fixed- and variable-rate fee.

3 The model

In this section, we outline our assumptions regarding the customer base, the service, the

pricing and service inducement strategies of the expert and the customer strategy. We end

with the specification of the expert-customer game.

The customers. The customer base is homogeneous: All customers place value V on the

service, and incur a disutility of c per unit time spent waiting in queue or in service. We

assume an additive utility structure. If the customer decides to purchase service, his ex-

pected utility is V −E[service cost + waiting cost]. If the customer decides not to purchase

the service, he obtains 0 utility.

Customers arrive at the expert according to a Poisson process of rate Λ. We call Λ

the ‘market potential.’ Since our focus is the impact of workload dynamics 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. For example, Callahan notes that due to the anonymity

of corporate law, “... there is little loyalty between law firms and clients” (p. 35). The

5

Poisson arrival stream generating new customers is particularly appropriate under such

circumstances.

Service Characteristics. The service requirement of each arriving customer is drawn

from the exponential distribution with mean t. We denote the realization of the service

requirement by t. 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 ‘value-adding service time’

and to t− t as the ‘induced service time’. We model the service delivery as follows:

The expert costlessly 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, i.e. the service value V has not been delivered yet. If the expert works t or more, the

service value V is delivered; excess servicing creates no additional value.

We normalize the cost of service to zero. We assume that the time spent by the expert

on service is 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 assume that incomplete servicing can be detected 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. This assumption reflects the fact

that symptoms of the problem the customer wanted solved will persist if the expert does

not provide the appropriate service.

On the other hand, whether service inducement occurred is neither observable by the

customer nor by an outside agency, so the expert is not penalized if he induces service. This

assumption reflects the fact that it is typically very difficult to show that no unnecessary

service has been done (e.g. Ross 1996); the customer only observes that the problem has

been solved.

The expert. We assume that the expert has a monopoly position in the market. He

decides the pricing structure and the service inducement policy. He serves the customers in

a first-come first-served manner.

Pricing structure. The expert uses a price structure (R, r), where R is a flat fee and r is

the service rate per unit of service time. Customers pay for the total reported time by

6

the expert, 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 comple-

tion of the value-adding service time t of the customer in service. We consider the following

service policy of the expert: When the value-adding service time is over, the expert inserts

non-value adding service time of length τz, which is drawn from an exponential distribution

with mean τz. As soon as a new customer enters in the system, the expert stops the service

inducement. This policy is characterized by τ = (τ0, τ1, ...) and is sufficiently flexible to

model an array of situations: When τi = 0 for all i ≥ 0, no service is ever induced. When

τk = +∞, in the long run, a ‘target workload’ of k+1 customers in the system is maintained.

The customer strategy. An arriving customer decides whether to enter service or not

based on his expected net utility from service. 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 service-inducement phase, the expert reports the number

of customers in the queue since he stops service inducement as soon as the new customer

joins the queue. 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(β) = bβc and p(β) = 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, otherwise we have a mixed threshold

strategy. In order to keep the notation simple, we drop the dependence of n and p on β.

Specification of the game. We consider a two-stage game. Since the focus of this paper is

to analyze the impact of information asymmetry concerning the exact service requirement of

the customer, we assume that the price structure and all other parameters (V , c, t and Λ) are

common knowledge. This is a two-stage game with one ‘long-lived’ player (the expert) and

infinitely many ‘short-lived’ players (the customers). In the first stage, the expert chooses

(R, r), which is observable by all customers. In the second stage, the expert determines

his service inducement strategy and the customers determine their joining strategy; the

7

players do not observe each other’s action when making their decisions. 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 threshold strategy.

4 Analysis

The customer strategy is a function of the number of customers he finds in the system upon

arrival. We focus on symmetric threshold equilibria β ∈ R+ as defined by (1).

In what follows, we are able to characterize the equilibrium outcome as a function of

only two fundamental parameters, v.= V

ctand ρ

.= Λt, that capture the four parameters

V, c, t and Λ of our model. The parameter v measures the ratio of the service value to the

expected waiting cost due to the value-adding service time. We assume v ≥ 1, otherwise,

no customer would enter service. We call v the profit potential. Note that the potential

arrival rate Λ may be higher than the service rate 1t. Therefore, ρ > 1 is possible. We call

ρ the base utilization. The actual utilization level of the expert is driven by the effective

arrival rate of customers deciding to visit the expert, and is less than 1; we denote it by ρ.

§4.1 determines the second-stage expert strategy for a given customer entry strategy.

§4.2 determines the second-stage equilibrium customer strategy for a given expert strategy.

Customer and expert strategies that satisfy both conditions are Nash equilibria of the

subgame; these strategies are characterized in §4.3. Finally, we obtain the expert’s optimal

pricing structure and the resulting expert-customer equilibrium under a flat-fee structure

and a variable-fee structure (§4.4). In §5, we compare these two structures and identify the

expert’s optimal strategy.

4.1 The second-stage expert strategy for a given customer entry strategy

We first establish a property of the expert strategy that simplifies the subsequent analysis.

Lemma 1 The expert never induces service if the queue is not empty upon completion of

the value-adding service; τz = 0 for z ≥ 1.

Thus, the expert strategy reduces to the scalar τ0 ∈ [0,∞). This is consistent with Darby

and Karni, who (without formal analysis) focus only on the possibility of service inducement

8

when ‘the length of the queue of customers waiting for service is zero’ (p. 72). According

to Lemma 1, service inducement happens only during the idle time of the underlying queue,

whose dynamics are due to Poisson arrivals and exponential value-adding service times.

Note that when β is integer, the underlying queue is an M/M/1/β queue; β noninteger is

similar except that an arrival finding bβc customers in the system enters with probability

β − bβc.Let us now determine the expert’s expected profit rate for fixed (R, r) as a function of

his inducement strategy τ0 and the customer strategy β. The rate at which customers enter

the system in each state of the system is determined by β. Let δn (β).= P (Sn = join) and

let pn (β) be the limiting probability of state n in the underlying M/M/1/β queue, derived

in Lemma A1 in the Appendix. The expert’s profit rate is

RΛ∞∑

n=0

pn (β) δn (β) + r

(1 − 1

1 + Λτ0p0 (β)

). (2)

To explain this, note that for each joining customer, the expert makes a profit of R. Unless

there is no customer in the system, the expert earns r per unit of time; the probability of

having no customer in the system is 11+Λτ0

p0 (β). Note that if the expert is honest (τ0 = 0),

then the probability of this event reduces to p0 (β), which is the idle time in the underlying

queue. If the expert induces service to the last customer in the queue until a new customer

arrives (τ0 = ∞), the expert earns r per unit of time 100% of the time.

Let us redefine the expert’s inducement policy as α.= 1 − 1

1+Λτ0. τ0 ranges from 0

to ∞, α ranges from 0 to 1. In particular, α = 0 corresponds to an honest policy, and

α = 1 corresponds to inducing service until the next customer joins the system. With this

definition, we can write (2) as: π (β, α;R, r). For a fixed β, we define the expert’s best

response as α (β;R, r) = arg maxα∈[0,1]

π (β, α;R, r) ⊆ [0, 1].

4.2 The second-stage customer equilibrium for a given expert strategy

The analysis proceeds as follows: We derive the expected net utility of an arrival finding

n customers in the system. We determine the best response correspondence of an arriv-

ing customer assuming all others use strategy β, and characterize the resulting customer

equilibria.

Derivation of the Expected Net Utility of an Arriving Customer. For given

prices (R, r), setting the strategy of all other customers to β and the strategy of the expert

9

to α, the expected ex ante net utility of a customer who joins the system in state n is

Un (α, β;R, r) = V −R− cnt− (c+ r) tn (α, β). Here, cnt is the expected queueing cost and

tn (α, β) is the expected service time of a customer who enters when there are n customers

in the system and all other customers adopt the threshold strategy β. The latter expression

includes the expected value-adding service time t and the expected induced service time,

which is a function of n and β. Note that the expected induced service time of an arriving

customer is positive only if no other customer joins the system during the queuing and

value-adding service time of that customer. In this case, the expected induced service time

can be calculated using the memoryless property of the exponential distribution. Lemma 2

uses these properties to derive tn (α, β).

Lemma 2 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.

We illustrate the structure of Un (α, β;R, r) with and without service inducement in

Figure 1. Note that when there is no service inducement, the net expected utility linearly

decreases in the queue length observed by an arriving customer; the more congestion, the

less expected net utility obtained by an arriving customer. When there is service induce-

ment, customers arriving to find fewer customers in the system experience a high service

inducement cost in expectation since their own service time may be inflated. This cost de-

creases at a decreasing rate in the queue length; the longer a customer waits in the queue,

the more likely that another customer will arrive during his wait and queue behind him,

which means no service inducement for the customer in question. The expected waiting cost

in the queue increases linearly in the queue length observed upon arrival. The combination

yields a convex expected waiting cost function and a concave expected net utility function.

Characterizing the Best Response Function. 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, 0 ≤ n′ ≤ n− 1 and Un′ (α, β;R, r) ≤ 0, n′ ≥ n

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

10

α=0

α=1

0

10

20

30

40

50

U

10 20 30 40 50

n

Figure 1: Un (α, β;R, r) as a function of n, the number of customers in the system observed

by an arriving customer when α= 0 and α=1 . V = 100, R = 30, r = 50, c = 2, ρ = 0.5, β =

44.

this, first suppose that for a given β, there exists 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

11

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:

βe (α;R, r) = β ∈ R+ : β ∈ Nc (β;α,R, r) . (3)

The follow-the-crowd effect. It can easily be shown that the best response correspon-

dence is non-decreasing in the threshold strategy of all other customers. This is called the

‘follow-the-crowd’ effect (Hassin and Haviv, p. 6). A nondecreasing best response corre-

spondence results in multiple equilibria. Figure 2 illustrates a typical example. As discussed

in Hassin and Haviv, queuing models typically show an ‘avoid-the-crowd’ effect with a non-

increasing best response correspondence, yielding a unique equilibrium (pp. 8-9). This is

because the more customers arrive, the more congestion there is, and the less attractive

it is for an arriving customer to seek service. The difference in our model comes from the

service inducement effect. This is the first major result of our analysis.

To understand the impact of the service inducement effect better, note that a pure

strategy equilibrium is determined by n satisfying Un−1 (α, n;R, r) ≥ 0 ≥ Un (α, n;R, r)

provided that U0 (α, n;R, r) ≥ 0. Two cost components determine Un−1 (α, n;R, r): waiting

costs due to congestion and costs of service inducement. For high values of n, waiting costs

are high, which normally causes avoid-the-crowd behavior, but service inducement costs are

low, making it more attractive to join the queue. Therefore, there may be an equilibrium

with a high threshold value. For low values of n, the service inducement cost is high, as the

probability that no other customer arrives before the end of the value-adding service of the

entering customer is high. However, the congestion cost is low. Therefore, there may be an

equilibrium with a low threshold value. The lowest equilibrium is 0, which always exists.

Thus, service inducement may lead to a complete market failure: Even though customers

would have been better off by receiving service and the expert would have earned a positive

profit rate, no customers may visit the expert.

4.3 The second stage expert-customer equilibrium

Let B (R, r).= (β, α) ∈ R+ × [0, 1] : β ∈ βe (α;R, r) and α ∈ α (β;R, r). B (R, r) is the

set of all possible second-stage expert-customer equilibria in a subgame characterized by

(R, r). As we have observed in Figure 2, there may be multiple elements in βe (α;R, r).

There may also be multiple elements in α (β;R, r) when r = 0. Therefore, B (R, r) is in

12

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) = 0, 2.901, 3, 3.263, 7 consists of three pure-equilibrium strategies (0, 3 and 7),

and two mixed-equilibrium strategies (2.901 and 3.263).

general not a singleton. We select one equilibrium from this set for our subsequent analysis.

In particular, we select the largest threshold equilibrium for tractability, and denote it by

(β∗, α∗).

Lemma 3 If r = 0, then α∗ (R, 0) = 0 and β∗ (R, 0) =⌊V−Rct

⌋. If r > 0, then α∗ (R, r) = 1

and β∗ (R, r) = n, where n is the largest integer in the set

0 ∪ n ∈ N : n ≥ 1, U0 (1, n;R, r) ≥ 0, Un−1 (1, n;R, r) ≥ 0, Un (1, n;R, r) ≤ 0 .

This result shows that r = 0 precludes service inducement. On the other hand, r > 0

results in α∗ = 1, or, in terms of our policy, the expert induces service to the last customer in

the queue until the next customer arrives: Since there is no cost to the expert of inducing

service, the expert induces service as long as possible. This result also shows that the

largest threshold equilibrium is always a pure strategy threshold equilibrium that can be

characterized using three inequalities that are linear in r and R. This property is what

makes the subsequent analysis tractable.

13

4.4 The first-stage equilibrium

For any pair (ρ, v), α∗ (R, r) and β∗ (R, r) were characterized in Lemma 3. This lemma

allows us to focus on two cases: Either r = 0 and α∗ = 0, or r > 0 and α∗ = 1. We now find

the optimal price structure and the corresponding expert-customer equilibrium strategy.

Rather than attempting to maximize π (β∗ (R, r) , α∗ (R, r) ;R, r) over (R, r) ∈ R2+, we

can take advantage of the structure revealed in Lemma 3 about the largest queue length

equilibrium. We first set α = 0, fix n ∈ N and find the price R ∈ R+ that maximizes the

expert’s profit among all R satisfying β∗ (R, 0) = n. We refer to this profit as π0 (n). This

is the maximum profit the expert can make while inducing no demand and pricing such

that the threshold n will emerge as the largest threshold equilibrium. Second, we set α = 1,

fix n ∈ N and find the price pair (R, r) that maximizes the expert’s profit while satisfying

β∗ (R, r) = n; this is a linear program with three inequalities and non-negativity constraints

on r and R. We refer to this profit as π1 (n); it is interpreted similarly. Finally, we find the

equilibrium threshold that is optimal for the expert in each case. Let ni.= arg max

nπi (n)

and π∗i.= πi (ni) for i = 0 and 1. π∗1 (π∗0) is the maximum profit rate that an expert who

does (not) induce service achieves. If π∗1 > π∗0, then the expert has an incentive to induce

service.

From now on, we introduce explicitly the dependence of all dependent variables on (v, ρ).

The following result characterizes the expert’s optimal fixed-rate pricing decision under no

service inducement.

Proposition 4R∗

0(ρ,v)ct

= v−n0 (v, ρ) maximizes the expert’s profit without inducing service.

The optimal profit π∗0 (ρ, v) satisfiesπ∗

0(ρ,v)c

= (v − n0) ρn0 (ρ), with ρn0 (ρ).=(1 − (1−ρ)ρn0

1−ρn0+1

)ρ.

The profit-maximizing equilibrium threshold n0 (v, ρ) = dxe where x ∈ R solves v = x +

1ρx

(1−ρx+1

1−ρ

)2.

This result identifies a flat-rate contract as optimal if no service inducement is desired

in equilibrium. The resulting equilibrium is then a pure threshold strategy, n. Therefore,

the expert’s queue is of the type M/M/1/n. ρn (ρ) is his corresponding utilization level.

The price, equilibrium threshold policy and profit expressions in this proposition have been

obtained by Hassin (1986) in a paper that does not allow any service inducement and that

assumes a fixed price only. What we show is that within the broader fixed and variable

price structure, fixed pricing is optimal if no service inducement is desired in equilibrium.

14

The following result characterizes the expert’s optimal variable-rate pricing decision

under service inducement. To obtain closed-form solutions, we resort to an asymptotic

approximation for large values of v.

Proposition 5 Let v 1 and ρ > 1+√

2v+1v2

. Then the optimal contract with service induce-

ment 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.

5 The Expert’s Optimal Policy

We now compare the two policies analyzed in §4.4 with respect to the resulting threshold

equilibria and to profit. Let n1 and n0 denote the large-v approximation for n1 and n0.

Proposition 6 Let v 1 and ρ > 1+√

2v+1v2

. Then n1 (v, ρ) > n0 (v, ρ) for ρ 1, with

n1 (v, ρ) = n0 (v, ρ) for ρ > 1.

We have proven that the profit-maximizing equilibrium threshold with service induce-

ment is larger in approximation than that without service inducement for 1v

+ 1+√

2v+1v2

<

ρ 1. A numerical investigation shows that this result holds for a much larger range of

values of ρ < 1, as observed in Figure 3 for v = 500. This is the second major result of

our analysis. At first sight, that more customers seek service when service is induced may

appear surprising. The economic intuition for this result has to do with the difference in

the service prices chosen by the expert in each scenario. We explain this below.

Consider the extreme case where the expert fully price discriminates by charging prices

that are queue length dependent. Then he would be able to extract all surplus from all

customers without resorting to service inducement. In other words, state-dependent pricing

is efficient. In this case, same as a social planner who wishes to maximize total surplus,

the expert would not induce any demand and would set prices so as to maximize total

surplus, all of which he would appropriate. Call the threshold chosen by the social planner

ns. It is well known that ns is larger than the threshold level resulting from the price

selection of a monopolist charging a fixed, state-independent price (Naor; Hassin and Haviv,

Chapter 2). Service inducement in our model has characteristics that are similar to state-

dependent pricing: Even though (R, r) is state-independent, service inducement results in

state-dependent net utility for customers and allows the monopolist to capture consumer

15

surplus in a more efficient way than with a fixed fee only. In other words, service inducement

leads to ‘approximate’ state-dependent pricing enabling a degree of price discrimination.

As a result, it is not surprising that the expert prices his services in such a way that the

equilibrium threshold is larger with service inducement than without service inducement,

but smaller than the socially optimal threshold ns.

Another interesting result is obtained by considering total surplus. With service in-

ducement, the customer waiting time also includes waiting time during non-value adding

service, “destroying” total surplus. Therefore, if n0 = n1, the total surplus with service

inducement will be lower than the total surplus without service inducement. On the other

hand, as discussed above, we also know that at optimality, more customers may visit the

expert with service inducement than without. Thus, 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. Numerical experiments show that this is indeed the case.

We now investigate which of the two strategies will be chosen by the profit-maximizing

expert as a function of ρ and v.

Proposition 7 For v 1, there exists 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) , 1] ,

(iii) π∗1 (ρ, v) ≈ π∗0 (ρ, v) for ρ ∈ (1,√v).

According to Proposition 7, the expert finds service inducement to be the higher profit

choice when the profit potential, v, is high and ρ is high enough and less than 1. This is the

third major result of our analysis. To understand the intuition behind this result, we write

π1 (n; ρ, v)

c=R(ρ, v)

tρn (ρ) + r(ρ, v) ≈ (v − n) ρn (ρ) + (1 − ρn (ρ))

(n− 1

I11 (ρ) − 1

− 1

)(4)

using the price expressions in Proposition 5. The first term is exactly equal toπ∗

0(ρ,v)c

(see

Proposition 4). The second term is the extra profit stream that the expert can capture with

service inducement. Note the term is negative for n < I11 (ρ) and positive for n > I1

1 (ρ).

Thus, there exists a minimum threshold above which service inducement can result in higher

profits than the corresponding system without service inducement (where prices are set such

that the equilibrium n is the same in both systems).

Consider π1(n;ρ,v)c

evaluated at n = n0 (ρ, v). For low values of ρ, n0 (ρ, v) is low and

the inflation factor, I11 (ρ), is high. Therefore, the second term in (4) is negative, and

16

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 threshold n0(ρ, v) (thin line) and n1(ρ, v) (thick line) for v = 500.

the profit π1 (n; ρ, v) evaluated at n0 (ρ, v) is less than π∗0 (ρ, v). Even though π∗1 (ρ, v) is

determined by n1 (v, ρ), which is higher than n0 (ρ, v), this term remains negative due to the

high inflation factor, and service inducement is less profitable than no service inducement.

For higher values of ρ, n0 (ρ, v) increases, while I11 (ρ) decreases. There will be a threshold

value of ρ such that π1 (n; ρ, v) evaluated at n0 (ρ, v) is higher than π∗0 (ρ, v). This implies

π∗1 (ρ, v) ≥ π∗0 (ρ, v). For ρ > 1, the idle period (during which service can be induced) is so

infrequent that service inducement does not have enough potential to generate significantly

more profits for the expert.

The analysis in Proposition 7 is for v 1. Note that v is an upper bound on any

possible threshold strategy because V − cnt ≥ 0 is a necessary condition for the feasibility

17

of n as a threshold strategy. For low values of v, the equilibrium threshold will necessarily

be low and make it impossible to obtain more profits with service inducement; no service

inducement is optimal in this range.

6 Conclusion, Discussion and Further Research

In this paper, we analyze the optimal price structure and service inducement strategy of

a monopolist expert who sells a credence service. In this setting, the expert may have an

incentive to provide unnecessary service since it is difficult for the customer to detect this. In

particular, with a variable-rate contract, the expert’s revenues increase as a function of the

total service time, which makes inducement feasible. If in addition, the expert’s workload

is low, he may have a strong incentive to induce service, referred to as “time-padding”(Ross

1996).

We introduce a simple queuing model that captures the key workload dynamics. The

model incorporates several important elements such as customer value of service, market

potential, waiting cost and service time and is able to capture the important tradeoffs con-

cerning service inducement in a dynamic environment. Within this framework, we determine

the optimal policy for the expert as a function of the characteristics of the environment.

We find that two parameters dictate the optimal policy: (1) the base utilization, 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 expected waiting

cost for the duration of the value-adding service.

The optimal policy for the expert, summarized in Table 1, can be described as follows:

If the profit potential is low, the expert charges a fixed fee and does not induce service

regardless of the base utilization. If the profit potential is high, the optimal strategy depends

on the base utilization. For low levels of base utilization, not inducing demand is optimal.

When the base utilization increases, it becomes profitable to induce unnecessary service.

For levels of base utilization larger than 1, the fraction of idle time during which service

can be induced is so small that service inducement does not result in significantly higher

profits for the expert.

Insights. Our model makes three contributions. First, we show that service inducement

leads to follow-the-crowd behavior: If many customers purchase the service, the expected

18

Potential demand Potential demand < Potential demand >

Service capacity (ρ 1) Service capacity (ρ < 1) Service capacity (ρ > 1)

high profit fixed fee fixed and variable fee fixed and/or variable fee

potential v no inducement inducement indifferent

low profit fixed fee

potential v no inducement

Table 1: The expert’s optimal strategy

service inducement costs are low, which motivates other customers to purchase the service.

On the other hand, when few customers purchase the service, expected costs due to service

inducement are very high, decreasing other customers’ motivation to purchase the service.

In particular, complete market failure is one of the equilibria, that is, nobody purchases

the service even though this would have increased customer surplus and the expert’s profit.

The characterization of this follow-the-crowd phenomenon adds to the literature on strategic

behavior in queues (Hassin and Haviv) where avoid-the-crowd behavior due to congestion

is typically observed.

The second contribution is to demonstrate that in equilibrium, a larger threshold policy

may be observed with service inducement than without. This means that customers who

previously would not have received service now do, and get positive utility instead of 0 utility.

On the other hand, service inducement imposes additional cost without generating value.

In the balance, there are cases where the total surplus increases under service inducement.

This outcome is a consequence of the strategic utilization by the expert of the fixed and

variable-rate fee as a price-discriminating instrument that allows the expert to skim the

surplus of customers arriving at low workload levels. We believe that this interpretation

of service pricing is new within the credence good literature and is obtained by explicitly

modelling the operations aspect of service delivery.

Previous research on the capacitated problem has found that it is optimal for the expert

to not induce service with a homogeneous customer base. Our third contribution is to show

that when workload dynamics due to stochastic arrivals and service times are taken into

account, there are conditions under which the expert will find it optimal to induce service

even when serving a homogeneous consumer base. This result is in fact consistent with

19

other previous research. Dulleck and Kerschbamer (2003) show that inducement occurs

when customers are heterogeneous with respect to their valuation of service. When 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 and makes

inducement desirable for the expert.

Design Levers to Limit Potential Service Inducement. Service inducement can

create negative publicity and result in high losses if detected. From a design perspective,

it is therefore important for management to better understand when service inducement is

likely to emerge within their organization. Certainly, a time-based fee (e.g. a hourly billing

rate) in conjunction with information asymmetry with respect to the service required is

conducive to service inducement. Nevertheless, our model shows that this structure alone

does not result in the expert always inducing services. There is one case in particular where

the expert has a strong incentive to induce services; this is when the potential demand rate

is slightly below the service rate and the profit potential is high.

Our results provide some possibilities to limit inducement in this setting. An obvious

measure would be to restrict the price structure to one with a fixed fee only. However, this

may be perceived as being unfair since customers who need only a low amount of service pay

the same as customers who need a high level of service. Another measure would be to invest

in expert training and an ethical work culture in environments where inducement is deemed

to be more likely. Operational measures can also be considered. Customer heterogeneity, in

this case induced by queue length fluctuations, is what drives service inducement. Steps to

decrease workload fluctuations by decreasing the variability of interarrival or service times

would be effective in limiting the amount of service inducement. Similarly, if customers

are a priori heterogeneous with respect to valuation or waiting cost, segmenting customers

and assigning different servers to homogeneous consumer segments could be considered.

Matching capacity with market demand can also be useful since buffer capacity may be

used to induce services. Using effective workload planning models with the possibility of

redeploying experts in other customer segments could be used to control the incentives to

induce services.

Discussion of Assumptions and Future Research Directions.

Our model makes several simplifying assumptions. We discuss the implications of re-

laxing some of these assumptions below.

20

We assumed a homogeneous customer base and showed that even with a homogeneous

customer base, due to ex-post heterogeneity, service inducement would be observed. If

customer heterogeneity with respect to the valuation of the service and/or with respect to

waiting costs were introduced into our model, we expect the level of service inducement to

only increase.

We assumed that the over-provision of service does not detract value. If the over-

provision of some services can directly harm the consumer, such as with unnecessary medical

intervention, this would destroy total surplus and reduce the ability of the expert to extract

profit via service inducement. Similarly, we assumed that there was no direct cost to the

expert of inducing service. Such a cost would also make is less attractive for the expert to

induce service.

We focused on the largest threshold equilibrium for tractability. In particular, the opti-

mal prices resulting in a given threshold in equilibrium could be characterized as solutions

of a simple linear program; this is what allowed us to obtain closed-form approximations for

optimal threshold policies and profits. For other equilibria, such a characterization is not

possible. Note that the largest threshold equilibrium is the one most favorable to the expert

since the revenue potential is the highest. One may therefore ask whether our results are

driven by the choice of equilibrium. While a closed-form characterization is not possible,

numerical analysis reveals that the insights developed above also apply to other equilibria;

they are not specific to the largest threshold equilibrium.

Finally, we assumed that the workload is observable. If the workload is concealed,

customers remain ex-post homogeneous. As shown in Debo et al. (2004), a fixed fee with

no demand inducement is then optimal, which is consistent with the literature. For an

expert who can choose between revealing or concealing his workload, Debo et al. (2004)

show that workload concealment is optimal only for low values of base utilization. Demand

inducement, in conjunction with workload revelation, is again observed for intermediate

values of base utilization.

We hope that our work generates further research in this area. We developed a basic

model in order to uncover the economics driving the service provider-customer interaction

in a credence good context and to further the insights from the credence good and queuing

economics literatures. More detailed and specific models of expert-customer interactions in

different industries could be developed if the goal is to generate insights into a particular

21

industry. In our model, the expert is owner of the service company and chooses, as a mo-

nopolist, the optimal price structure. Kalra et al. (2003) study compensation schemes in

a principal-agent model in which agent has the opportunity to oversell services. It would

be interesting to investigate design problems as in Kalra et al., but where the agent is a

credence-good expert modelled as a single-server queue. As shown in the economics litera-

ture, competition and reputation are two important factors that limit service inducement.

Analyzing these phenomena with the same expert server model may yield interesting in-

sights into the interaction of workload fluctuation, competition and reputation. Finally, our

findings highlight that it would be interesting to investigate how the credence characteristic

impacts capacity investment decisions.

7 Acknowledgements

Laurens Debo wishes to thank the Sasakawa Young Leaders Fellowship Fund for financial

support during the last two years of his doctoral studies, on which this research is partially

based. The authors also thank the seminar participants at Carnegie Mellon University,

INSEAD, UT Austin, TU Eindhoven and UCLA.

8 References

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Markets. Working Paper, Department of Economics, Boston College, Boston, MA.

Callahan, D. 2004. The Cheating Culture: Why More Americans Are Doing Wrong to Get

Ahead. Harcourt Inc., Orlando, FL.

Darby, M. R. and E. Karni. 1973. Free Competition and the Optimal Amount of Fraud.

Journal of Law and Economics 16: 67-88.

Debo, L. G., L. B. Toktay and L. N. Van Wassenhove. 2004. Queuing for Expert Services.

INSEAD Working Paper 2004/46/TM, Fontainebleau, France.

Delattre, E. and B. Dormont. 2003. Fixed fees and physician-induced demand: A panel

data study on French physicians. Health Economics 12: 741-754.

Dulleck, U. and R. Kerschbamer. 2003. On Doctors, Mechanics and Computer Specialists

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Or Where are the Problems with Credence Goods? Working Paper 0101, University of

Vienna, Vienna, Austria.

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 Orga-

nization 19: 375-389.

Fong, Y. 2002. When Do Experts Cheat and Whom Do They Target? Working Paper,

Kellogg School of Management, Northwestern University, Evanston, IL.

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Case of Queues and Balking. Econometrica 54: 1185–1195.

Hassin, R. and M. Haviv. 2003. To Queue or not to Queue. Kluwer Academic Publishers,

Boston, MA.

Kalra, A., M. Shi, and K. Srinivasan. 2003. Salesforce Compensation Scheme and Consumer

Inferences. Management Science 49(5): 655–672.

Naor, P. 1969. On the Regulation of Queue Size by Levying Tolls. Econometrica 38: 13-24.

Pesendorfer, W. and A. Wolinsky. 2002. Second Opinions and Price Competition: Ineffi-

ciency in the Market for Expert Advice. Review of Economic Studies 70(2): 417-437.

Pitchik, C. and A. Schotter. 1987a. Honesty in a Model of Strategic Information Transmis-

sion. American Economic Review 77: 1032-1036.

Pitchik, C. and A. Schotter. 1987b. Honesty in a Model of Strategic Information Trans-

mission (in Erratum). American Economic Review 77: 1164.

Radhakrishnan, S. and K. R. Balachandran. 1996. Cost of Congestion, Operational Effi-

ciency and Management Accounting. European Journal of Operational Research 89: 237-

245.

Richardson, H. 1999. The Credence Good Problem and the Organization of Health Care

Markets. Working Paper, Private Enterprise Research Center, Texas A&M University,

College Station, TX.

Ross, W. G. 1996. The Honest Hour: The Ethics of Time-Based Billing by Attorneys.

Carolina Academic Press, Durham, NC.

23

Whang, S. 1989. Cost Allocation Revisited: An Optimality Result. Management Science

35(10): 1264-1273.

Wolinksy, A. 1993. Competition in a Market for Informed Experts’ Services. RAND Journal

of Economics 24: 380-398.

9 Appendix

Proof of Lemma 1. Figure 4 describes the Markov process corresponding to the service

inducement policy τ . States 0, 1, ..., n represent the system whenever value-adding service

is being induced on a customer and n customers are in the system. States 0′, 1′, ..., n′ − 1

represent the system whenever service is being induced on a customer and n′ other customers

are waiting (i.e. in total, n′ + 1 customers are in the system).

0

0’

Λ

1/τ0Λ

µ

1

1’

Λ

1/τ1 Λ

µ

2

2’

Λ

1/τ2 Λ

µ

Λn-1

n-1’

1/τn-1

Λ

µ

n

Figure 4: Transition rate diagram corresponding to inducement policy τ .

When the system is in state n, an arrival (which occurs at rate Λ) takes the system

to state n + 1, and a service completion (which occurs at a rate µ = 1t) takes the system

to state n′ − 1, since service is induced on the current customer and n − 1 customers are

waiting.

When the system is in state n′, an arrival (which occurs at a rate Λ) brings the state

to n + 1: The new customer arrival stops the inducement on the customer in service, this

24

customer leaves, and the next customer in line enters the value-adding phase of his service.

When the system is in state n′, a completion of service inducement (which occurs at

a rate 1τn′

), takes the system to state n: The induced customer leaves the system so the

number in the system drops to n, and the next customer in line enters the value-adding

phase of his service.

We now calculate the steady-state probabilities. The policy (τ0, τ1, ...) results in the

following balance equations:

Λp0 =1

τ0p0′

(µ+ Λ) pn =1

τnpn′ + Λ(pn−1 + pn′−1) for 1 ≤ n ≤ n− 1

pn =Λ

µ(pn−1 + pn′−1)

(1

τn+ Λ

)pn′ = µpn+1 for 1 ≤ n′ ≤ n′ − 1.

Using pn′ = µ1τn

+Λpn+1, we can rewrite the second line as

(µ

1 + Λτn−1+ Λ

)pn =

µ

1 + τnΛpn+1 + Λpn−1 for 1 ≤ n ≤ n− 1.

Let µn.= µ

1+Λτn−1. Then, we can rewrite these above expressions as

Λp0 = µ1p1

(µn + Λ) pn = µn+1pn+1 + Λpn−1 for 1 ≤ n ≤ n− 1

pn =Λ

µ(pn−1 + pn′−1)

or,

pn =n∏

l=1

Λ

µlp0 and pn′ = τnΛ

n∏

l=1

Λ

µlp0 for 1 ≤ n ≤ n− 1.

Using∑n−1

k=0 pk +∑n′−1

k′=0 pk′ + pn = 1, we can obtain

p0 =1

1 + Λτ0 +∑n−1

k=1 (1 + Λτk)∏kl=1

Λµl

+ Λµ

(1 + Λτn−1)∏n−1l=1

Λµl

and

pn =

Λµ

(1 + τn−1Λ)∏n−1l=1

Λµl

1 + Λτ0 +∑n−1

k=1 (1 + Λτk)∏kl=1

Λµl

+ Λµ

(1 + Λτn−1)∏n−1l=1

Λµl

.

With µµk+1

= 1 + Λτk, we obtain

p0 =1

µµ1

+∑n−1

k=1µ

µk+1

∏kl=1

Λµl

+ Λµµµn

∏n−1l=1

Λµl

25

and

pn =

Λµ

µµn−1

∏n−1l=1

Λµl

µµ1

+∑n−1

k=1µ

µk+1

∏kl=1

Λµl

+ Λµµµn

∏n−1l=1

Λµl

.

From these expressions, it is straightforward to show the following:

(a) For any 0 ≤ µ1, µ2, . . . , µn ≤ µ,(1 − Λ

µ

)(Λµ

)n

1 −(

Λµ

)n+1≤ pn ≤ 1. (A-5)

(b) If µ2 = µ, ..., µn = µ, then

pn =

(1 − Λ

µ

)(Λµ

)n

1 −(

Λµ

)n+1(A-6)

is independent of µ1 and

p0 =µ1

µ

1 − Λµ

1 −(

Λµ

)n+1 .

For a fixed (R, r) and µ.= (µ1, µ2, ..., µn), π (µ;R, r)

.= RΛ (1 − pn) + r (1 − p0) is the

profit rate of the expert. Above, we obtained closed-form expressions for pn and p0 as a

function of µ. We can now find the optimal service inducement policy µ∗ as follows: (i) If

r > 0, then, using (A-5) and (A-6), the problem

max0≤µ1≤µ,0≤µ2≤µ,...,0≤µn≤µ

π (µ;R, r) (A-7)

is solved by µ∗1 = 0, µ∗2 = µ, ..., µ∗n = µ. This solution results in p0 = 0 with pn given by (A-

6). (ii) If r = 0, then using (A-5), (A-7) is solved by µ∗1 ∈ [0, µ] , µ∗2 = µ, ..., µ∗n = µ. This

solution results in p0 ∈[0,

1−Λµ

1−(

Λµ

)n+1

]with pn given by (A-6). Thus, we have obtained

that the expert never induces non-value adding service if the queue is not empty upon

completion of the value-adding service. This exposition assumed a pure threshold customer

policy n for simplicity; a mixed threshold customer policy results in the same conclusion.

Remark. Recall that we redefined the expert’s inducement policy as α.= 1 − 1

1+Λτ0. This

definition can be interpreted as follows: The expert, with probability α, induces service

on a customer who would leave behind an empty system until the next customer joins the

system; with probability 1 − α, he does not induce any service on such a customer. With

this policy, the expert’s profit rate is

π (β, α;R, r) = RΛ∞∑

n=0

pn (β) δn (β) + r (1 − (1 − α)p0 (β)) . (A-8)

26

It can be shown that the expected inducement duration of an customer is equal under the

two policies τ0 and α if α = 1 − 11+Λτ0

. Since the two policies exhibit this equivalence, we

base our analysis below on service inducement policies determined by α.

Lemma A 1 The limiting probability that the underlying queue is in state n when all cus-

tomers 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 , with n = bβc and (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-9)

Proof. 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+1

n=0 pn = 1, p0 = 1∑nn=0 ρ

n+pρn+1= 1

1−ρn+1

1−ρ+pρn+1

= 1−ρ1−ρn+1(1−p+pρ) = 1−ρ

1−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 .

Following the Remark above, π (β, α;R, r) = RΛ∑∞

n=0 pn (β) δn (β)+r (1 − (1 − α)p0 (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

1 − (1 − α)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 Lemma 2. Please refer to the Remark above. 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.

27

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=

(1

1+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= 1

p.

P3: ξ (0) = ∞ and ξ (n) = n ∈ N+ with n ≥ 1.

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 value-adding service time, that is,

in the event that no other customer enters the system during the value-adding 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 value-adding 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 value-adding 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 inducement is 1Λ . The expected total

service time is then tn (α, β) = t + αPn (β) 1Λ for β ≥ 1, where the first term is the value-

adding 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 from the value-adding service completion of the last customer until the arrival

of the next customer is then 1pΛ . Therefore, the expected length of service inducement is

1pΛ , due to the memoryless property of Poisson arrivals. The expected total service time is

then tn (α, β) = t+ αPn (β) 1pΛ for β ∈ [0, 1).

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 value-adding 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:

28

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= 1

p. Thus, using (P2), we can write αPn (β) 1

pΛ = α(

11+ρ

)ξ−11Λ .

Summarizing cases (i-iii), we obtain tn (α, β) = t

(1 + α

(1

1+ρ

)min(n+1,ξ−1)1ρ

).

Lemma A 2 For any (R, r) ∈ R2+ and α > 0, the set βe (α;R, r) consists of the pure and

mixed threshold strategies satisfying the conditions below:

29

pure strategy equilibria n ∈ N mixed strategy equilibria β ∈ R+ \ N

n = 0 0 < β < 1 : U0 (α, β;R, r) = 0

1 ≤ n :

U0 (α, n;R, r) ≥ 0 (a)

Un−1 (α, n;R, r) ≥ 0 (b)

Un (α, n;R, r) ≤ 0 (c)

1 < β :

U0 (α, β;R, r) ≥ 0 (d)

Ubβc (α, β;R, r) = 0 (e)

For any (R, r) ∈ R2+ and α = 0, the set 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

n =⌊V−R−rt

ct

⌋(f) β ∈ [n− 1, n] if V−R−rt

ct= n ∈ N (g)

Proof. (i) The case with 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 2), 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 > 0, with n ∈ N.

As Un (α, n;R, r) is concave in n, the threshold strategy n is a best response for a new

customer 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 an 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 exactly conditions (a), (b) and (c) in the Lemma.

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, β;R, r) is concave in n, a new customer

will also adopt 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, β;R, r), ensure that the net

expected utility is non-negative in states n ∈ [0, n]. (2) ensures then when arriving in state

30

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

Ubβc (α, β;R, r) = 0

Ubβc+1 (α, β;R, r) < 0.

Since Ubβc+1 (α, β;R, r) < Ubβc (α, β;R, r), the latter condition is always satisfied. There-

fore, the first two conditions are sufficient to characterize mixed strategy equilibria.

(ii) The case with α = 0. Note that Un (0, β;R, r) = V −R−cnt− (c+r)t strictly decreases

in n and is independent of β.

Pure strategy equilibria: If Un−1 (0, n;R, r) ≥ 0, it follows that Un (0, n;R, r) ≥ 0 for

n ∈ [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. The threshold strategy n is thus determined by

V −R− rt− c (n+ 1) t ≤ 0 ≤ V −R− rt− cnt, or, equivalently n ≤ V−R−rtct

≤ n+ 1.

Mixed strategy equilibria: If Un−1 (0, β;R, r) = 0, then the customer is indifferent

in state n − 1 between joining or not. Therefore, any randomization between thresholds

n − 1 (balking at n − 1) and n (joining at n − 1, but balking at n) is an equilibrium, i.e.

all β such that bβc = n − 1 are equilibria. The range [n− 1, n] is thus determined by

V −R− rt− cnt = 0, or, V−R−rtct

= n ∈ N.

Readers can refer to Chapter 3 in Hassin and Haviv (2003) for an extensive treatment of

this case.

Proof of Lemma 3. (i) The case with r > 0. When r > 0, α = 1. This is because

the coefficient of α in the profit function is positive. We prove that the largest threshold

equilibrium 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

U1bβc+k(bβc + k) ≤ 0, that is, bβc + k is a pure strategy equilibrium.

31

Since U1n(β) is nondecreasing in β, U 1

bβc(bβc + 1) ≥ 0. If U 1bβc+1(bβc + 1) ≤ 0, we

are done: bβc + 1 is a pure strategy equilibrium. If not, and U 1bβc+1(bβc + 1) > 0, then

U1bβc+1(bβc + 2) > 0 since U 1

n(β) is nondecreasing in β. If U 1bβc+2(bβc + 2) ≤ 0, we are

done: bβc + 2 is a pure strategy equilibrium. If not, and U 1bβc+2(bβc + 2) > 0, then

U1bβc+2(bβc + 3) > 0 since U 1

n(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→∞ U1

n(n) = −∞.

Thus, a mixed strategy equilibrium can never be the largest threshold equilibrium. With

Lemma A 2, we obtain that the equilibrium conditions are given by (a), (b) and (c).

(ii) The case with r = 0. When r = 0, α = [0, 1], that is, any α ∈ [0, 1] can exist in

equilibrium. From the first part of this lemma, we know that for any α > 0, there exists

a maximum β. It is obvious that the highest threshold will occur for the least amount of

service inducement, α = 0. Therefore, α∗ (R, r) = 0 and β∗ (R, 0) =⌊V−Rct

⌋.

Proof of Proposition 4: With Lemma 3, we obtain that β∗ (R, 0) =⌊V−Rct

⌋= n. From

(A-9) in Lemma A 1 we obtain π = RΛ(1 − (1−ρ)ρn

1−ρn+1

)= 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 show that for a given (v, ρ),

maxn≥0

π0 (n) (A-10)

is solved by the unique value of n satisfying v0n−1 (ρ) < v < v0

n (ρ) with v0n (ρ) = n +

1ρn

(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 + 1

ρn

(1−ρn+1

1−ρ

)2. Let v0

n (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) = dxewhere x ∈ R solves v = v0

x (ρ).

Proof of Proposition 5. We wish to determine the profit maximizing equilibrium under

32

service inducement, n1(v, ρ), by solving maxn≥0

π1 (n; v, ρ). This proof is based on Lemmas T2

to T5 in the Technical Appendix. Lemma T2 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 T2, 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

assume that Case I holds for all feasible n and call the profit maximizing equilibrium that

would result if this were the case nI1(v, ρ). An approximate characterization of nI1(v, ρ) is

given in Lemma T3. We then assume that Case II holds for all feasible n and call the profit

maximizing equilibrium that would result if this were the case nII1 (v, ρ). An approximate

characterization of nII1 (v, ρ) is given in Lemma T4. Lemma T5 delineates the values of ρ

for which Case I and Case II hold at n = nI1(v, ρ).

From Lemma T2, 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+ρρ

≤ 0, 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 T5, 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 T5, we know that Case I applies at

n = nI1(v, ρ). Therefore, with Lemma T 2, 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.

We obtain R∗(v, ρ) and r∗(v, ρ) by substituting (T-20) in

R(n)/ct = v − (n− 1)I11 (ρ)

I11 (ρ) − I1

n−1 (ρ)and r(n)/c =

n− 1

I11 (ρ) − I1

n−1 (ρ)− 1.

Finally, note that n1 (v, ρ) − n0 (v, ρ) ≈ 2 for some range of ρ < 1 and that n0 (v, ρ) −

33

n1 (v, ρ) ≈ 0 for ρ ≈ 1 and ρ > 1.

Proof of Proposition 6. According to the approximation developed in Lemma T 1,

n0 (v, ρ) ≈⌈− ln v

ln ρ

⌉for 0 < ρ 1. According to the approximation developed in Lemma T3,

n1 (v, ρ) ≈⌈− ln v

ln ρ

⌉+2 for 1+

√2v+1v2

< ρ 1 and v 1. We conclude that n0(ρ, v) < n1(ρ, v)

when 1+√

2v+1v2

< ρ 1 and v 1. The result n1(v, ρ) ≈ n0(v, ρ) for ρ > 1 and v 1 also

follows by comparing the approximations.

Proof of Proposition 7. For v 1, we first consider the ρ for which n1 (v, ρ) 1. Note

that for large n, we have that ρn (ρ) ≈ min (1, ρ). Therefore, we can rewrite (4), which gives

the approximate profit under inducement for large n1 (v, ρ):

π∗1c

≈ (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 4, the profit structure is

π∗0c

≈ (v − n0 (v, ρ)) min (1, ρ) .

Comparing these profits for ρ < 1, we see that

π∗1c

− π∗0c

≈ (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 inducement: (1) (n0 (v, ρ) − n1 (v, ρ)) ρ is

due to the difference in thresholds and (2)((n1 (v, ρ) − 1)

(ρ− ρ3

)− (1 − ρ)

)is the extra

profit stream from service inducement. For low ρ, (1) is negative due to Propositions 4 and

5. (2) is also negative.

When increasing ρ, we obtain that n1 (v, ρ) = n0 (v, ρ) + 2 ≈⌈− ln v

ln ρ

⌉+ 2 (see Propositions

4 and 5), and thus

π∗1c

− π∗0c

≈ −2ρ+ (1 − ρ) ((n1 (v, ρ) − 1) ρ (1 + ρ) − 1) .

It follows thatπ∗

1c>

π∗

0c

⇔⌈− ln v

ln ρ

⌉> −1 + 1

ρ(1−ρ) . If v is large enough, this inequality is

satisfied. There exists thus a level ρ′ (v) such that for ρ < ρ′ (v), π∗

0c>

π∗

1c

and for ρ > ρ′ (v),

the opposite holds.

As ρ increases to 1, we know from Proposition 5 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∗

1c→ π∗

0c

). This proves parts (i) and (ii).

From Propositions 4 and 5, we obtain that n0 (v, ρ) ≈ n1 (v, ρ) ≥ 3 for ρ ≤ √v. From

Lemma T6, we know that πI1 (n; v, ρ) ≥ π0 (n; v, ρ) with πI1 (n; v, ρ) ≈ π0 (n; v, ρ) for very

34

high values ρ. 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. This proves part (iii).

35

Queuing for Expert Services: TechnicalAppendix

Laurens G. Debo

Tepper School of Business

Carnegie-Mellon University, Pittsburgh, PA 15213, USA

L. Beril Toktay

Technology Management

INSEAD, 77305 Fontainebleau, France

Luk N. Van Wassenhove

Technology Management

INSEAD, 77305 Fontainebleau, France

Lemma T 1 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

Proof. In Proposition 4, it was shown that n0 (v, ρ) = dxe where x ∈ R solves v = v0x (ρ)

.=

x+ 1ρx

(1−ρx+1

1−ρ

)2. Since 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

)2and 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 −B0

ρx + 1. Solving this equation, we obtain ρx = B02A0

±√(

B02A0

)2− 1

A0, 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: B02A0

−√(

B02A0

)2− 1

A0≈

12ρ+v(1−ρ)2 . Thus, for 0 < ρ 1,

x ≈ln(

12ρ+v(1−ρ)2

)

ln ρ≈ − ln v

ln ρ.

1

If ρ ≈ 1 but less than 1, it can easily be proven that B02A0

−√(

B02A0

)2− 1

A0≈ 1 −

(√v + 1

)(1 − ρ). Therefore, ρn = 1 −

(√v + 1

)(1 − ρ) is solved by

x ≈√v + 1 ≈ √

v.

For large values of ρ, B02A0

+

√(B02A0

)2− 1

A0≈ (v−1)(v+1)

v. Solving for x in ρx = (v−1)(v+1)

vis

approximated by solving for x in ρx = v and we obtain

x ≈ ln v

ln ρ.

Lemma T 2 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)

(T-11)

and the following two cases for n = 1, 2:

Case

II ′ : R12(n) < 0

III ′ : R12(n) ≥ 0(T-12)

The profit maximizing contract (R1(n), r1(n)) among those for which β∗ (R, r) = n has the

following structure:

Case j 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. Remember that α∗(R, r) = 1 when r > 0 and that by Lemma 3, β∗ (R, r) = n ≥ 1

if n is the largest integer that satisfies conditions (d), (e) and (f) in Lemma 2 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. (T-13)

2

Let us impose the additional constraint

Ψ (n;R, r) ≤ n. (T-14)

We now show that if n satisfies (T-13) and (T-14) 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 (T-14) 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

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 T-13. 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 largest threshold 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.

3

Note that Rtρn (ρ) + r is increasing both in R and r (for a fixed n). As Ψ (n− 1;R, r)

is decreasing 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 solution 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 (T-15)

n− 1 ≤ Ψ (n− 1;R, r) (T-16)

Ψ (n;R, r) ≤ n (T-17)

0 ≤ Ψ (1;R, r) (T-18)

The slope of the isoprofit line is − tρn(ρ) , that of the constraint Ψ (n− 1;R, r) = n − 1

is −tI1n−1 (ρ), and that of the constraint Ψ (1;R, r) = 0 is −tI1

1 (ρ). It can easily be shown

that for n ≥ 3, I1n−1 (ρ) < 1

ρn(ρ) < I11 (ρ) for all ρ. Since I1

n (ρ) < I1n−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 (T-16) and (T-18) and below by (T-17). Finally, for n ≥ 3, the R-intercepts of the

three constraints are distinct and ordered with that of (T-17) being the smallest and that of

(T-18) being the largest. Thus, for the feasible region to contain points (R, r) with R ≥ 0

and r > 0, it is sufficient that (T-16) cross the R-axis at a 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 (T-16) and (T-18),

and the objective function is increasing both in R and in r, in the absence of (T-17), the

optimal solution would be either (i) at the intersection of (T-16) and (T-18) if these lines

intersected in the first quadrant, or (ii) at the intersection of (T-16) and the line R = 0

otherwise. With constraint (T-17), we also need to take into account where constraints

(T-16) and (T-17) intersect. Let R12 and R13, respectively, denote the R-intercepts of the

intersection of (T-16) and (T-17), and of (T-16) and (T-18), respectively. We find that the

4

optimal solution to the LP is given by exactly one of the following three cases:

(I) the intersection of (T-16) and (T-18) if R12 ≤ R13 and R13 ≥ 0.

(II) the intersection of (T-16) and R = 0 if R12 < 0 and R13 < 0.

(III) the intersection of (T-16) and (T-17) if R12 > R13 and R12 ≥ 0.

An example for each of these three cases is given in Figure 5.

For n = 1, Ψ (min (1, n− 1) ;R, r) ≥ 0 coincides with (T-16) ; for n = 2, (T-18) is

redundant. Thus in both cases, only (T-16) and (T-17) need be considered. In addition, in

both problems, the slope of the iso-profit function is equal to the slope of (T-16), 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 (T-16) and (T-18), 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 (T-16) and (T-17), 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

Lemma T 3 For a given (v, ρ),

maxn≥3

πI1 (n; v, ρ) (T-19)

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

5

and vI2 (ρ) = 0. For v 1, we can approximate the solution to (T-19) 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

⌉ρ > 1

(T-20)

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 T2 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

))ρ 1−ρn+1

1−ρn+2 −1

1ρ

(1

1+ρ

)− 1ρ

(1

1+ρ

)n − (n− 1)

(1+ 1

ρ

(1

1+ρ

))ρ 1−ρn

1−ρn+1 −1

1ρ

(1

1+ρ

)− 1ρ

(1

1+ρ

)n−1

ρ1−ρn+1

1−ρn+2 − ρ 1−ρn1−ρn+1

.

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

(T-21)

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 (T-21), we use the following approximation:

1

1 −(

11+ρ

)k−1≈

1k−1

(1ρ

+ 12k)

ρ < 2k−2

1 ρ > 2k−2

. (T-22)

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 (T-22) 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

. (T-23)

6

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. (T-24)

Using the approximations in (T-24) in (T-23) 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 2

ρ, which is equivalent to v 1

2ρ− 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, 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 (T-22) 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

=n1−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

. (T-25)

7

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

(T-26)

This gives us three subcases to study.

Subcase (iia): For 2n−3 < ρ 1, we can use (T-26) for (T-25) and obtain

vIn (ρ) ≈n1−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+ρ4

ρ2. 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− 1

1−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) .

Therefore,

v = (n+ 4) (n− 1) ⇒ nI1 (v, ρ) ≈ √v.

8

Subcase (iic): For ρ 1, we obtain

vIn (ρ) ≈n1−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, B22A2

+

√(B22A2

)2− 1

A2≈ v + 1. Therefore, n solves ρn ≈ v + 1 and

nI1 (v, ρ) ≈ ln v

ln ρ.

Lemma T 4 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 T2 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 ≈ 1

ρ+n+1, from which we obtain the approximation nII1 (v, ρ) =

v − 1ρ. Substituting this approximation in the profit expression, we obtain πII1 (v, ρ) ≈

v−(v− 1

ρ−1)

1+ 1ρ

(1

1+ρ

)v− 1ρ−1

−1 ≈1ρ+1

1+ 1ρ(1−(v− 1

ρ−1)ρ)

−1 = 1+ρ

ρ+1−(v− 1ρ−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.

9

Lemma T 5 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 T3, 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 − 1ρ−√

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 T2, 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 T3, 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 = 1

v+ 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.

Lemma T 6 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

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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

)

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

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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

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

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 5: Illustration of cases I, II and III.

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