Strategic Diagnosis and Pricing in Expert Services

Transcription

1 Strategic Diagnosis and Pricing in Expert Services M. Fazıl Paç Wharton School, University of Pennsylvania, Philadelphia, PA Senthil Veeraraghavan Wharton School, University of Pennsylvania, Philadelphia, PA In expert service markets, consumers must rely on experts to identify the type of service they need. Diagnosis is a crucial step of expert services in which the expert identifies the type of problem and prescribes the proper service. Provision of expert services is surrounded by information problems arising at this stage. The expert may not reveal the true diagnosis to the consumer in order to sell more expensive services. This asymmetry of information between the expert and the consumer leads to inefficiencies in the form of over-provision or rationing. Providing extensive treatments yields higher service value; however, it also requires more time and resources, which are valuable for both the expert and the consumer. Furthermore, extensive services result in negative externalities in the form of longer delays for consumers. Pricing serves as an important tool in the provision of expert services. On the one hand, it controls demand for the service and hence congestion in the system, on the other hand, it signals the diagnostic strategy of the service provider to the market. In this study we analyze service providers pricing and diagnosis strategies and consumers procurement decisions in a service environment constrained by capacity and congestion. We find that capacity and congestion concerns mitigate expert cheating. To signal an honest diagnosis an expert has to charge high prices and limit demand for his services. Hence, low prices may serve as a warning for service over-provision. Key words : Expert/Credence Services, Service Operations, Strategic Customers, Queues, Operations Economics. History : December 6, Introduction In a wide variety of services, consumers cannot self-diagnose their problems; hence, they cannot identify the type of service that will address their needs. As a result, they rely on knowledgeable experts, who are also the providers of the service, for the diagnosis of their problem. These services are often referred to as expert services. Medical services, repair services, and consulting services all fall into the broad category of expert services. In these settings, experts are better informed about 1

2 2 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services the potential problem types and the services required to treat these problems. Upon diagnosing the problems, the experts identify consumers problems and gain an informational advantage over consumers. In recommending a service, an expert may not reveal the true diagnosis and may exploit his informational advantage in order to sell more expensive services. As a consequence, even after consumption, consumers are often unsure about the necessity of, and, their true benefit from the recommended service. The information asymmetry between experts and consumers is largely responsible for the inefficiencies in service provision, which appear in the form of over-provision, under-provision or rationing. Economists have long documented the agency problems and the resulting inefficiencies in expert service markets. When the experts responsible for the diagnosis are also the sellers of the service, they have strong incentives to make false recommendations. Quantifying the level of expert cheating is difficult as it requires the analysis of experts diagnostic decisions with and without the presence of information asymmetry. In this manner, Domineghetti et al. (1993) investigate physicians diagnostic decisions for ordinary patients and for the physicians own families and find that ordinary patients had 33% more of the seven major surgical interventions than physicians and their families. Physician-induced demand is a widely used example of over-provision in expert service markets. Gruber and Owings (1996) show that the 13.5% fall in fertility during the period is largely responsible for the 240% rise in cesarean delivery rates from 5.5% to 23.5% of births. The authors argue that declining fertility rates increased the income pressure on obstetricians/gynecologists and led them to substitute the more highly reimbursed cesarean delivery for normal childbirth. Auto repair is another industry causing concern about service over-provision. Mechanics often use their informational advantage over consumers to provide unnecessary services and/or to overcharge consumers. Schneider (2009) provides evidence for under-provision, over-provision and overcharging in the auto-repair market: The field experiment suggests that unnecessary repairs were recommended in 27% of the visits representing 61% of all charges. The estimated welfare loss due to the agency problems in the U.S. auto-repair market is around $8 billion, equalling 22% of industry revenues.

3 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 3 The extant academic research on expert services has not addressed the impact of capacity and service delay on experts decisions. Experts can direct consumers toward more expensive services to extract higher revenues. However, expensive services require more time, a valuable resource for both experts and consumers. Furthermore, longer service times result in long waiting times, which may deter consumers from procuring the service. Clearly, service over-provision (cheating) allows an expert to extract higher revenues from consumers, at the expense of lower demand and longer waiting times. Pricing serves as an important tool in expert service markets. On the one hand it signals the experts diagnostic strategy (honest or fraudulent) to the market; on the other hand, it controls congestion in the service queue by controlling demand for each service type. In this paper, we study an expert s optimal pricing policy and diagnostic decisions in the face of self-interested, delay-sensitive consumers. Congestion costs are an outcome of the experts diagnostic strategy and the aggregate procurement decisions of all consumers in the market, as every consumer procuring the service generates additional waiting time for all other consumers. Service over-provision intensifies the congestion in the system as the expert provides consumers with longer services that they do not need. In addition, signaling an honest diagnosis requires charging sufficiently high prices and limiting the demand for the expensive service. In turn, the expert faces a tradeoff between the cost of signaling an honest diagnosis and the cost of service over-provision. We investigate how capacity and congestion concerns mitigate/aggravate expert cheating (service over-provision) and increase/reduce the efficiency of service provision. We find that capacity and congestion concerns dramatically affect the expert s behavior and lead to outcomes different from those of classical economic models. The expert serves consumers with different service requirements, who impose different levels of delay on others. As a result, price discrimination becomes essential in achieving efficient service provision. This result is in stark contrast with the known results on credence goods/services that prescribe a single admission fee to achieve honest and efficient provision of services. For instance, a single admission fee does not give the expert sufficient control over the congestion generated by consumers with different service

4 4 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services needs. Price discrimination gives the expert control over demand from each consumer type (and the resulting congestion generated) and results in higher revenues. We find that honest price discrimination and efficiency are achieved only under certain conditions. For the expert to achieve efficient service provision, the potential demand for his services has to be large enough. An expert with ample capacity is more likely to cheat, as the cost of congestion diminishes with decreasing utilization and hence over-provision becomes less costly. Also as the value gap between major and minor treatments widens or as the proportion of consumers with major problems increases, signaling an honest diagnosis becomes costlier and service over-provision becomes unavoidable. 2. Related Literature Our work builds on the literature on (i) queuing games and admission control to capacitated systems, and (ii) the economics of expert services and credence goods. The strategic interaction between service providers and consumers in capacitated service settings have long been examined in the queuing games literature pioneered by the seminal work of Naor (1969). This stream of research has shown that admission fees are an important tool to control congestion in service queues. Edelson and Hildebrand (1975) extend Naor s model and study the impact of congestion on service providers and consumers decisions when queue lengths are unobservable to consumers. They show that in this case, revenue maximization coincides with social optimization. For an excellent review of this literature we refer the reader to Hassin and Haviv (2003). Lariviere and Van Mieghem (2004) analyze the effect of congestion on the behavior of consumers strategically seeking service through a timing game. They find that when strategic consumers make decisions based on the long-run average service delay, the arrival strategy approaches a Poisson process. In our model of expert service provision, we consider a setting in which consumers differ with respect to their service requirements and valuations. Price and service discrimination for heterogeneous consumers with private information about their types has received significant attention

5 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 5 after the seminal work of Mendelson and Whang (1990). Van Mieghem (2000) investigates the social and monopoly pricing of heterogeneous consumers with specific service requirements and delay sensitivity and illustrates the incentive compatibility of the generalized cµ scheduling. Along these lines, Afeche (2004) focuses on strategic delays and incentive compatible pricing mechanisms when consumers are heterogeneous with respect to their delay sensitivity, willingness to pay and service requirements. Ha (2001) analyzes the optimal pricing scheme that coordinates queues with customer-chosen service requirements. We model a market for expert service provision; therefore, the information structure in our model differs significantly from those in these papers. In our model the service provider, instead of the consumer, has private information about service valuations and requirements, and he chooses whether to reveal each consumer s true type. The expert s diagnostic strategy and consumers service procurement decisions affect the expected service value and the service rate of the service system. Recommending longer and more expensive services to consumers may induce higher revenues at the expense of slower service and longer delays. Research articles acknowledging the existence of the interaction between service speed and service value in different domains include Kostami and Rajagopalan (2009) (dynamic decisions), de Vericourt and Zhou (2005) (routing unresolved call-backs), Lu, Van Mieghem and Savaskan (2008) (manufacturing rework), Hasija, Pinker and Shumsky (2009) (an empirical study of call centers), de Vericourt and Sun (2009) (judgment accuracy), Wang, Debo, Scheller-Wolf and Smith (2010) (medical diagnostic services) and Anand, Paç and Veeraraghavan (2010) (customerintensive services). All the aforementioned papers illustrate the tradeoff between service speed and service quality without considering the service provider s informational advantage over consumers regarding their service requirements. The research on credence goods/services is pioneered by the seminal work of Darby and Karni (1973), who study the impact of market conditions on the equilibrium level of fraud (i.e., under/over-provision, and overcharging). The authors introduce the term credence goods to refer to goods whose impact on consumers utility is not completely revealed even after consumption. Pitchik and Schotter (1987) find that experts become more honest as the price gap between major

6 6 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services and minor repairs, consumers probability of requiring major repairs and experts competence decrease. Congestion effects reverse this finding. Wolinsky (1993) emphasizes that the information asymmetry between the sellers and consumers might lead to specialization in competitive expert service markets. Pesendorfer and Wolinsky (2003) investigate whether consumer search in a competitive market would give experts an incentive to exert costly but unobservable diagnostic effort. Fong (2005) finds that expert cheating arises as a substitute for price discrimination, and experts cheat on high valuation, high cost customers. Furthermore, the author shows that a fixed fee for all services is optimal and it eliminates service inducement. Wennberg, Barnes and Zubkoff (1982), Dranove (1988), Gruber and Owings (1996) and Delattre and Dormont (2003) investigate the existence of physician-induced demand in different settings and show that physicians increase the level of care in reaction to a fall in the number of consultations (demand). Alger and Salanie (2006) show that in a competitive credence goods market, experts defraud consumers by providing false diagnoses in order to keep them uninformed and charge higher prices. Dulleck and Kerschbamer (2006) emphasize that liability and verifiability are important institutional factors determining experts behavior, while reputation and competition are important market factors. In a follow-up experimental study, Dulleck, Kerschbamer and Sutter (2009) explore the role of liability, verifiability, reputation and competition in credence goods markets. In a different experimental study on the agency problems and reputation effects in the auto repair market, Schneider (2009) finds that mechanics reputation concerns do not improve the service quality or avoid inefficiencies. The above-mentioned literature on the economics of credence goods and expert services does not consider the impact of resource constraints (service capacity) on experts diagnostic decisions. Glazer and Hassin (1983) study the cheating behavior of taxi drivers and find that higher service capacity increases the level of cheating. Emons (1997) models the competition of capacityconstrained experts in a deterministic setting and finds that competition induces an honest diagnosis; however, experts earn positive revenues only if market demand exceeds market capacity. Using a similar deterministic framework, Emons (2001) shows that a monopolist expert can signal an

7 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 7 honest diagnosis either by setting the service capacity exactly equal to the market demand or by charging a fixed admission price. However, when arrivals to the service system are stochastic, it is not possible to achieve 100% capacity utilization. On the other hand, charging fixed admission fees results in revenue loss for the expert. We contribute to the literature by investigating the impact of workload dynamics and service delays on experts diagnostic decisions and consumers service procurement behavior. In such an environment, prices serve two different purposes. On the one hand, they signal the credibility of the expert s diagnosis, and on the other hand, they control congestion in the service system. Debo, Toktay and Van Wassenhove (2009) focus on the relationship between operational dynamics and service inducement in a monopolistic market in which consumers have homogeneous service requirements. They show that the service provider can take advantage of the heterogeneity arising from consumer arrivals at different system states (busy or idle) and induce service to extract higher revenues. Our focus is on the diagnostic strategy of an expert facing a market where consumers differ with respect to their service requirements and their respective valuations. Our analysis leads to outcomes that are in contrast to the findings of the classical economic literature on expert services. We find that under the presence of capacity and congestion, the expert may achieve honest price discrimination, which yields higher revenues and generates higher social welfare compared to a fixed admission fee. 3. Provision of Expert Services in a Monopolistic Market We consider a monopolist providing service to a market of self-interested, risk-neutral and utility maximizing consumers. Each consumer has a problem that requires treatment. We assume that consumers problems fall into two categories: major problems (type M problems), which occur with probability θ (0, 1) in the population, and minor problems (type m), which occur with probability θ = 1 θ. θ is common knowledge. The presence of both types of problems are recognized by similar symptoms. Consumers observe the existence of a problem through symptoms, but they cannot diagnose its type. Hence, they cannot identify the proper service that will address their needs. Therefore, for

8 8 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services both diagnosis and treatment, they rely on a knowledgeable service provider, whom we address as the expert. The resolution of a minor problem provides a consumer a value of v L, while the resolution of a major problem provides a value of v H (> v L ). 1 We assume that consumers are homogeneous in their valuations of the service and their prior beliefs about their problem types. Therefore, for all consumers the ex ante expected value of the treatment of the problem is v θv H + (1 θ)v L. However, upon diagnosis by the expert, they differ in their valuations and updated beliefs based on the service recommendations they receive. We model the monopolist service setting using an unobservable single server queue. Consumers arrive in the market according to a Poisson process at an exogenously specified mean rate of Λ, which we refer to as the potential demand for the service. Each consumer incurs a waiting cost of c per unit of time spent in the system. Upon the consumer s arrival in the system, the expert performs an instantaneous diagnosis and correctly identifies the problem type. While the expert s diagnosis is always accurate, his service recommendation, d, need not be. In other words, the expert does not necessarily reveal the true diagnosis to the consumer when he recommends a treatment. Thus, once the diagnosis is completed, the service provider has an informational advantage over the consumer. Consider a consumer i, following up on his diagnosis, the expert recommends/provides one of the two types of services: a minor treatment (d i = L) and a major treatment (d i = H) 2. For consumers with minor problems (m), the minor treatment is sufficient to treat the problem. However, a major treatment is necessary to resolve major problems (M). Furthermore, a major treatment also resolves minor problems. The average service time for the minor treatment is and the average service time for the major treatment is. Since a major treatment resolves both minor and major problems, its expected duration is longer than that of a minor treatment, i.e., >. For simplicity of exposition, we assume that the service time for each service type is exponentially 1 This setting is equivalent to a setting in which a consumer incurs a higher cost when a major problem is left untreated. 2 There are economies of scope between the diagnosis and the service; ; for instance, providing the service might require equipments used in the diagnosis. Therefore, the same expert provides both the diagnosis and the service.

9 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 9 distributed. However, our analytical conclusions hold for independently and identically distributed general service time distributions. The expert sets the price of the minor treatment at p L and the price of the major treatment at p H, which are both public information. In order to extract higher revenues the expert may provide false recommendations and/or price out certain consumer types (i.e., he may strategically provide unnecessary services or ration services based on capacity and congestion). In particular, the expert may recommend major treatment (H) to a consumer with a minor problem (m), who needs only minor treatment (L). After receiving the diagnosis, consumers update their beliefs about their problem types and decide whether to procure the prescribed service (join the service queue) or quit (balk) based on the price of the recommended service, their beliefs about the service provider s strategy, and their individual expected delay. Note that when making her decision, a consumer knows only her own service recommendation, d i. However, the consumer can deduce the expert s diagnostic strategy and the resulting expected waiting time in equilibrium from the prices set by the provider and the potential demand in the market. The expert s services are verifiable. Upon service completion, consumers know the type of service they received and whether their problems were treated. Consumers pay the service fee only after their problems are treated. However, they still may not know what type of problem they had and whether the provided service procedure was appropriate. Thus, once the problem is fixed, there is no evidence supporting the choice of one treatment over the other. Let α = (α m, α M ) denote the expert s diagnostic strategy. In particular, α m is the probability of prescribing a major treatment (H) to a consumer with a minor problem (m), and α M is the probability of prescribing a major treatment to a consumer with a major problem. Consequently, ᾱ = (1 α m, 1 α M ) denotes the probability of prescribing a minor treatment to consumers with minor and major problems. An honest diagnostic strategy (no overselling) corresponds to α = (0, 1). The information asymmetry arising upon the diagnosis gives the expert an opportunity to oversell services by prescribing a major treatment to consumers with minor problems. Note that under the payment structure we adopt, a rational expert will not recommend minor treatment (L) to

10 10 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services consumers with major problems (M). Minor treatment does not treat major problems, and hence, its provision to consumers with major problems does not yield revenue, i.e., α M < 1 is sub-optimal. This allows us to focus only on diagnostic strategies α = (α m, 1). Consumers receiving the same service recommendation may have different problem types; however, they have identical beliefs prior to their queue joining decisions. To reflect this fact, we refer to the proportion of consumers for whom major treatment is recommended, χ, as the expert s diagnostic decision variable: χ = θ + (1 θ)α m. (1) Note that, χ θ, since all consumers with major problems are recommended for major treatment, i.e., α M = 1. The expert s objective is to maximize his revenues with respect to service prices, (p L, p H ) and the diagnostic strategy χ, while customers make their queue joining decisions in order to maximize their utilities. We model the revenue maximization problem as a two-stage game: First, the expert sets the prices (p L, p H ) for his services and announces it to the market. Then, the expert and arriving consumers play the diagnosis and queue joining game. Following a backward induction approach, we first focus on the diagnosis and queue joining game Diagnosis and Queue Joining Game The diagnosis and queue joining game describes the equilibrium provision of the expert service for a given set of prices. In this stage, the expert sets his diagnostic strategy 3, χ, based on the prices, (p L, p H ), and the potential demand, Λ. In our analysis, we focus on pure diagnostic strategies, i.e., χ {θ, 1} for analytical tractability. In our extensive analysis, we find that pure diagnostic strategies dominate mixed strategies except for the extreme cases when v H >> v L and θ 0. Consumers do not observe the expert s diagnostic strategy χ. However, they can deduce it through 3 The diagnostic strategy, χ, is clearly dependent on θ, and prices (p L, p H). We suppress the dependencies from the notation for convenience.

11 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 11 the prices (p L, p H ) and the potential demand, Λ. Each consumer makes her queue joining decision based on the recommendation, d i, she receives, the prices, (p L, p H ), and the potential demand, Λ. Rational consumers arrive to the service system according to a Poisson process at rate Λ. The potential demand (market size), Λ, mean service times for each service type,,, and waiting cost per unit of time, c, are common knowledge to all arriving consumers. Service prices, (p L, p H ), are observable to all consumers. Upon arrival the expert instantaneously diagnoses the problem and provides a service recommendation, d i : {m, M} {L, H}, to consumer i, based on the diagnostic strategy, χ(p L, p H ). Since consumers pay the service fee only after the resolution of their problem, the expert s services are liable. Due to service liability, the expert always provides a true diagnosis to consumers with a major problem (M). In effect, he recommends the major treatment, i.e., d i (M) = H with probability 1; therefore, χ θ. On the other hand, consumers with minor problems (m) may be subject to over-provision; the expert recommends major treatment (d i (m) = H) with probability χ θ 1 θ, and recommends minor treatment with probability 1 χ 1 θ. After the diagnosis, each consumer updates her belief about the problem type and decides whether to join the (unobservable) service queue, based on the service recommendation, d i, that she receives, and the prices, (p H, p L ). Customer i s queue joining strategy is given by β i (p L, p H ) = (β i L(p L, p H ), β i H(p L, p H )), where β i L(p L, p H ) is the probability of joining the queue upon being recommended for minor treatment and β i H(p L, p H ) is the probability of joining the queue upon being recommended for major treatment, given the prices (p L, p H ). Customers receiving the same service recommendation may differ in their actual problem types; however, they have identical updated beliefs. Therefore, we focus on the symmetric queue joining strategies of consumers receiving the same recommendation; hence, β i (p L, p H ) = β(p L, p H ) = (β L (p L, p H ), β H (p L, p H )) for all i. Prior to their queue joining decisions, consumers do not know the expert s diagnostic strategy χ, the queue s average service rate, and the effective arrival rate. However, through the prices (p L, p H ), they infer the diagnostic strategy, χ, and the resulting equilibrium service, and effective arrival rates.

12 12 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services Given the expert s diagnostic strategy χ, and the consumers symmetric queue joining strategy β = (β L, β H ), the effective demand for the minor treatment is λ L = β L (1 χ)λ, while the effective demand for the major treatment is λ H = β H χλ. We suppress the prices (p L, p H ) from the notation for simplicity of exposition. Service Time Distribution The service rate of the system is determined by the expert s diagnostic strategy, χ, and consumers service procurement decisions β. For a given strategy profile (χ, β L, β H ), the service time, τ, has the following hyper-exponential distribution: ( ) 1 Exp w.p. τ = ( ) 1 Exp w.p. β L (1 χ) β L (1 χ)+β H χ β H χ β L (1 χ)+β H χ (2) with mean, E[τ] = β L(1 χ) +β H χ λ L +λ H, and variance, V ar[τ] = (χβ H (1 χ)β L ) 2 +2χ(1 χ)β L β H (τ 2 L +τ 2 H ) ((1 χ)β L +χβ H ) 2. Expected Waiting Time Consumers joining the service queue receive different service recommendations from the expert; hence, they have different service time distributions. However, the expected waiting time in the queue, W q (χ, β L, β H ), is identical for all consumers joining the queue. Given the potential demand, Λ, and the strategy profile (χ, β L, β H ), a consumer s expected waiting time in the queue can be written as: W q (χ, β L, β H ) = Λ((1 χ)β Lτ 2 L + χβ H τ 2 H) (1 Λ((1 χ)β L + χβ H )), (3) under the stability condition Λ((1 χ)β L + χβ H ) 1, using the Pollaczek-Khintchine formula. If the stability condition is not satisfied, the expected waiting time is infinite. The expected waiting times for consumers joining the service queue upon having, a minor treatment recommended (d = L), and a major treatment recommended (d = H) are given as follows: W L (χ, β L, β H ) = + W q (χ, β L, β H ), (4) W H (χ, β L, β H ) = + W q (χ, β L, β H ). (5)

13 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 13 Consumers Expected Payoff and Expert s Revenues Given the prices (p L, p H ) and the potential demand Λ, under the strategy profile (χ, β L, β H ), the net expected payoff of a consumer from joining the queue upon recommendation d can be written as: v L p L cw L (χ, β L, β H ) V d (χ, β L, β H ) = ( ) ( v H + while the expert s revenues can be written as: θ χ if d = L ) χ θ v χ L p H cw H (χ, β L, β H ) if d = H, (6) R(p L, p H, Λ, χ, β) = Λ (p L (1 χ)β L + p H χβ H ). (7) A consumer will procure the recommended service (join the service queue) only if the net expected payoff, V (d, Λ, χ, β L, β H ), of doing so is non-negative. Equilibrium Conditions Given the potential demand Λ, and the prices (p L, p H ), under symmetric consumer strategies, the set of equilibria E(Λ, p H, p L ) {θ, 1} [0, 1] [0, 1] of the diagnosis and queue joining game consists of strategy profiles (χ e, βl, e βh), e which simultaneously satisfy the following set of conditions: β e L = β e H = max {0 β L 1} max {0 β H 1} {β L β L V d (χ e, β L, β e H) 0}, (8) {β H β H V d (χ e, β e L, β H ) 0}, (9) χ e = arg max {Λ (p L (1 χ)β e L + p H χβh)} e. (10) χ {θ,1} Equations (8) and (9) guarantee that under the strategy profile (χ e, β e L, β e H), a self-interested consumer will join the service queue with positive probability upon receiving recommendation d {L, H}, as long as there is non-negative value in doing so. The diagnosis and queue joining game may have multiple symmetric equilibria satisfying the above conditions, for a given set of prices (p H, p L ). However, as we will show in Section 3.3, under the optimal prices, the equilibrium of the diagnosis and queue joining game is unique in symmetric consumer strategies (i.e., the set E(Λ, p H, p L ) consists of a single strategy profile) in all but one scenario. In that particular scenario we focus on the consumer equilibrium, (β e ), that maximizes the expert s revenues.

14 14 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services Given the potential demand Λ, and the prices (p L, p H ), the equilibrium demand (maximizing the expert s revenues) for the minor and major treatments is λ e L(p H, p L ) = (1 χ e )βlλ, e and λ e H(p H, p L ) = χ e βhλ, e respectively. Before we analyze the expert s optimal pricing strategy and the equilibrium service provision, we discuss the socially optimal (first-best) provision of the service, which will serve as a benchmark Socially Optimal Provision of the Service In this section, we consider the first-best provision of the expert service by a social planner with complete information on each consumer s problem type. The social planner s objective is to maximize the social welfare generated by the capacitated service system (net of congestion costs), by controlling admission to the service queue. The social planner will always provide a true diagnosis (χ = θ), as service over-provision decreases social welfare by increasing waiting costs without increasing the total service value generated. When the potential demand in the market is Λ, the arrival rates of consumers with major and minor problems are θλ, and, (1 θ)λ, respectively. Admitting a consumer with a major (minor) problem into the system generates value v H (v L ), while increasing the expected workload in the queue by ( ) units of time. When the admission rates for consumers with minor and major problems are λ L, and λ H, respectively, a consumer with a minor problem derives the net value ), while a consumer with a major problem derives the net value of of v L cw L (θ, v H cw H (θ, λ L, λ H (1 θ)λ θλ λ L, λ H (1 θ)λ θλ ), from being admitted to the service queue. Using the net value expressions of each consumer type, we can write the social planner s objective function as: ( λ L max {SW (λ L, λ H ) λ L (v L c ) + λ H (v H c ) c(λ L + λ H )W q θ, {0 λ H θλ,0 λ L θλ} (1 θ)λ, λ )} H θλ (11) The objective function in (11) is not quasi-concave. To characterize the optimal policy we initially optimize SW (λ L, λ H ) with respect to a single variable (λ L or λ H ), assuming that the potential demand Λ is unlimited. The following lemma illustrates the socially optimal admission rate for consumers with minor problems, λ L(λ H ), for a given admission rate of λ H ( 1 ) for consumers

15 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 15 with major problems, and, the socially optimal admission rate for consumers with major problems, λ H(λ L ), for a given admission rate of λ L ( 1 ) for consumers with minor problems. Lemma For a given admission rate of λ H for consumers with major problems, the social welfare SW (λ L, λ H ) is maximized by: 1 λ H cγ λ L (λ H ) v L if 0 λ H λ H L(λ H ) = (12) 0 if λ H > λ H where γ L (λ H ) = 1 + (1 λ H )λ H ( ) 2, and λ H is such that dsw (0, λ H ) dλ L = 0. λ L(λ H ) is decreasing in λ H for λ H λ H. 2. For a given admission rate of λ L for consumers with minor problems, the social welfare SW (λ L, λ H ) is maximized by: 1 λ L cγ λ H (λ L ) v H if 0 λ L λ L H(λ L ) = (13) 0 if λ L > λ L where γ H (λ L ) = 1 + (1 λ L )λ L ( ) 2, and λ L is such that dsw ( λ L,0) dλ H = 0. λ H(λ L ) is decreasing in λ L for λ L λ L. 3. a. SW (λ L(λ H ), λ H ) is convex in λ H for λ H λ H and concave in λ H for λ H > λ H. b. SW (λ L, λ H(λ L )) is convex in λ L for λ L λ L and concave in λ L for λ L > λ L. Lemma 1.1 illustrates the optimal admission rate for consumers with minor problems, λ L(λ H ), when the admission rate for consumers with major problems is fixed at λ H. The optimal admission rate for consumers with minor problems, λ L(λ H ), falls as the utilization of the system increases with more consumers with major problems being admitted into the system. If the fixed admission rate for consumers with major problems is higher than λ H, the incremental waiting cost of admitting a consumer with a minor problem exceeds the value of treating the minor problem v L ; hence, λ L(λ H ) for λ H λ H. The results of Lemma 1.2 are analogous to that of Lemma 1.1. Lemma 1.3 shows that for a fixed admission rate for consumers with minor problems, the optimal social welfare function SW (λ L, λ H(λ L )) is convex in λ L for λ L λ L. An immediate consequence

16 16 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services Figure 1: Dis-economies of scale (left) in service time vs. Economies of scale (right). of this result is that when the potential demand Λ is unlimited, the social welfare is maximized by either admitting some of the consumers with major problems and none of the consumers with minor problems, i.e., λ = (0, λ H(0)), or by admitting some of the consumers with minor problems and none of the consumers with major problems, i.e., λ = (λ L(0), 0). The socially optimal admission to the expert s service queue depends on the comparative value per unit of service time provided by the treatment of each problem type (major or minor), v i τ i i {L, H}. We thus analyze the two cases that lead to different socially optimal service provision as follows: (i) Dis-economies of scale in service time: Minor treatment provides higher value per unit of service time, i.e., v L v H. (ii) Economies of scale in service time: Major treatment provides higher value per unit of service time. Figure 1 illustrates the relationship between the service value and the time in the two cases. for Lemma 2. Under dis-economies of scale in service time, if the value of treating a minor problem is low, v H v L < V L = v H + c(v H v H c )( ) 2 cvh then: 1. SW (λ L, λ H ) has a unique saddle point (λ L, λ H ) [0, λ L ] [0, λ H ], such that λ L = λ L(λ H ) and λ H = λ H(λ L ). 2. There exists a unique λ L > λ L such that SW ( λ L, λ H( λ L )) = SW (0, λ H(0)).

17 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services For θλ λ L define Λ H ( θλ) [λ H, ) as, {λ H SW ( θλ, λ H( θλ)) = SW (λ L(λ H ), λ H )} if λ L θλ λ L Λ H ( θλ) = if θλ > λ (14) L a. Λ H ( θλ) is increasing in Λ and decreasing in θ for θλ λ L. b. SW ( θλ, λ H( θλ)) SW (λ L(θΛ), θλ) when Λ H ( θλ) θλ λ H(0). Lemma 2 shows that if the value of treating a minor problem is sufficiently low, i.e., v H v L < V L, prioritizing consumers with major problems over those with minor problems, who derive higher value per service time, generates higher social welfare, SW (λ L, λ H ), in small markets with few consumers with minor problems, θλ < λ L, and sufficiently many consumers with major problems, θλ > Λ H ( θλ). Proposition 1. When there are dis-economies of scale in service time: 1. If the per unit time value of treating minor problems is high, i.e., v H V L v L the socially optimal service policy is: ( min{ θλ, λ L (0)}, 0 ) if θλ > λ L (λ S L, λ S H) = ( θλ, min{θλ, λ H ( θλ)} ) (15) if θλ λ L 2. When the per unit time value of treating minor problems is low, i.e., v H v L < V L the socially optimal service policy is: ( min{ θλ, λ L (θλ)}, min{λ H(0), θλ} ) if θλ < λ L (λ L(θΛ), min{λ H(0), θλ}) if θλ λ L (λ S L, λ S H) = and θλ > Λ H ( θλ) (16) ( min{ θλ, λ L (0)}, λ H( θλ) ) if θλ λ L and θλ Λ H ( θλ) Proposition 1.1 shows that under dis-economies of scale when the value of treating a minor problem is sufficiently large, i.e., V L v L, consumers with minor problems are prioritized over those with major problems, regardless of the potential market size, Λ. Consumers with major problems are admitted to the service queue only if the market is small, Λ < λ L. The expert admits more 1 θ

18 18 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services consumers with major problems into the system as the potential market, Λ, gets smaller, and as the probability of having a major problem, θ, gets higher; i.e., λ H( θλ)) is decreasing in Λ and increasing in θ. On the other hand, when the value per unit of time generated by treating a minor problem is close to that of treating a major problem, i.e., v H v L < V, the consumer type to be prioritized depends on the potential market size, Λ, as shown in Proposition 1.2. In this setting, when the potential market is large, Λ > λ L, only minor problems are admitted into the system as they (1 θ) generate higher value per unit of service time. However, when there are not sufficiently many consumers with minor problems in the market, i.e., (1 θ)λ < λ L, consumers with major problems are prioritized over those with minor problems, and, consumers with minor problems are admitted only if the number of consumers with major problems is low, i.e., θλ < λ H. The underlying reason behind this shift in prioritization from minor problem consumers to major problem consumers is that the expert needs to serve sufficiently many minor problem consumers to generate high value. Although, consumers with minor problems are more valuable per unit of time ( v L v H ), when the market is small, the negative impact of capacity and service delay diminishes. The value per capita effect (v H > v L ) dominates the value per time effect ( v L > v H ). Hence, consumers with major problems are prioritized over those with minor problems. Figure 2 illustrates the socially optimal admission policy as a function of the potential demand, Λ, and the probability of a major problem, θ. The socially optimal admission control policy under economies of scale in service time is converse to that under dis-economies of scale in service time. In this setting, treating customers with major problems generates higher value per capita and per unit of service time. However, due to the longer service time of the major treatment, serving more consumers with major problems also results in lower throughput and longer service delays. Lemma 3. Under economies of scale in service time, if the value of treating a major problem is low, v L v H < V H = v L + c(v L v L c )( ) 2 cvl then:

19 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 19 Figure 2: Socially optimal admission control under dis-economies of scale 1. SW (λ L, λ H ) has a unique saddle point (λ L, λ H ) [0, λ L ] [0, λ H ], such that λ L = λ L(λ H ) and λ H = λ H(λ L ). 2. There exists a unique λ H > λ H such that SW (λ L( λ H ), λ H ) = SW (λ L(0), 0). 3. For θλ λ H define Λ L (θλ) [λ L, ) as, {λ L SW (λ L(θΛ), θλ) = SW (λ L, λ H(λ L ))} if λ H θλ λ H Λ L (θλ) = if θλ > λ (17) H a. Λ L (θλ) is increasing in Λ and θ for θλ λ H. b. SW (λ L(θΛ), θλ) SW ( θλ, λ H( θλ)) for Λ L (θλ) θλ λ L(0)]. Lemma 3 shows that if the value of treating a major problem is sufficiently low, i.e., v L v H < V H, prioritizing consumers with minor problems over those with major problems, who derive higher value per capita and per service time, generates higher social welfare, SW (λ L, λ H ), in small markets with few consumers with major problems, θλ < λ H, and sufficiently many consumers with minor problems, θλ > Λ L (θλ). Proposition 2. When there are economies of scale in service time:

20 20 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 1. If the per unit time value of treating major problems is high, i.e., v L V H v H the socially optimal service policy is: (0, min{θλ, λ H(0)}) if θλ > λ H (λ S L, λ S H) = ( min{ θλ, λ L (θλ)}, θλ ) (18) if θλ λ H 2. When the per unit time value of treating major problems is low, i.e., v L v H < V H the socially optimal service policy is: ( min{λ L (0), θλ}, min{θλ, λ H( θλ)} ) if θλ < λ H ( min{λ L (0), θλ}, λ H( θλ) ) if θλ λ H (λ S L, λ S H) = and θλ > Λ L (θλ) (19) (λ L(θΛ), min{θλ, λ H(0)}) if θλ λ H and θλ Λ L (θλ) Proposition 2.1 shows that under economies of scale in service time, if the value of treating a major problem is sufficiently large v H V H, the socially optimal admission policy prioritizes consumers with major problems over consumers with minor problems, regardless of the potential market size, Λ. It is more desirable to admit consumers with major problems who generate higher value per capita and per unit of service time into the capacitated service system. Figure 3 illustrates the optimal admission policy under economies of scale in service time as a function of the potential demand, Λ, and the probability of a major problem, θ. When the market for the service is large Λ > λ H (0), only a fraction of consumers with major θ problems are admitted into the service system, while consumers with minor problems are not ( ] admitted at all. For Λ λh, λ H (0), all consumers with major problems are admitted into the θ θ system, while consumers with minor problems are not admitted. In this case, the social value generated would increase if there were more consumers with major problems in the market that could be admitted into the system. However, admitting a consumer with a minor problem still does not improve social welfare. Admitting an additional customer with a minor problem, imposes an additional waiting cost externality greater than the incremental value. Thus, when the number of consumers with major problems in the market is small, λ < λ H θ, the social planner admits

21 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 21 Figure 3: Socially optimal admission control under economies of scale. consumers with minor problems into the system; only a fraction of consumers with minor problems are admitted when there is a large number of them in the market (1 θ)λ > λ L(θΛ), while all of them are admitted otherwise. The number of consumers with minor problems admitted to the system increases as the potential demand Λ, and the probability of a major problem θ decrease. If the value of treating a major problem is low, i.e., v L < v H < V H, then consumers with major problems are prioritized as long as there are sufficiently many of them in the market. In markets with few consumers with major problems (θλ < λ H ) and sufficiently many consumers with minor problems, θλ > Λ L (θλ), minor problem consumers, who generate lower value per capita and per unit of service time, are prioritized to utilize the ample service capacity. Even though, there are only few consumers with major problems, only some or none of these are admitted to the service queue to keep the expected waiting costs low. Note that if the consumers in the market have complete information about their problem types, the equilibrium demand under optimal prices will be identical to the socially optimal admission rates (λ S L, λ S H) given in Propositions 1 and 2. In other words, under complete information, the expert can optimally price the services to achieve the social welfare maximizing provision, and fully extract the surplus from consumers joining the unobservable service queue. Thus first best is

22 22 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services achieved if there is no information asymmetry Expert s Revenue Maximization: Optimal Prices and Equilibrium Service Provision Having determined the equilibrium of the diagnosis and queue joining game and the socially optimal provision of the expert service, we focus on the expert s pricing decision. The expert decides on the prices of the minor and major treatments in order to maximize his revenues. Under the presence of information asymmetry between consumers and the expert, prices serve as a tool for controlling demand and congestion as well as for signaling the expert s diagnostic strategy. The expert s objective function can be written as: max {p L λ e L(p H, p L ) + p H λ e H(p H, p L )}, (20) {p L 0,p H 0} We analyze the optimal prices and the equilibrium characteristics for the two distinct cases characterized by Propositions 1 and 2 in the following sections. We first analyze the case of diseconomies of scale in service time ( v L v H ) in Section 3.3.1, and then in Section we focus on the case of economies of scale in service time ( v L < v H ) Dis-economies of Scale in Service Time In this section we consider the case in which the treatment of minor problems provides higher value per unit of service time than the treatment of major problems. Note that the treatment of a major problem is more valuable, v H > v L, but there are diminishing returns to service value from the increasing service time; hence, v L v H. Under dis-economies of scale in service time, consumers with minor problems are prioritized over consumers with major problems when there is no information asymmetry, as presented in Proposition 1. An immediate consequence of this result is that the potential benefit from service over-provision is low for the expert. In fact, in many cases it is lower than the implicit capacity cost and the resulting congestion costs. As a result, the expert is less likely to exploit his information advantage and recommend unnecessary services in this case. Let us define p L (λ L, λ H ) = v L cw L (θ, λ L (1 θ)λ, λ ) H, (21) θλ

23 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services 23 p H (λ L, λ H ) = v H cw H (θ, λ L (1 θ)λ, λ ) H, (22) θλ and ( p C (λ) = v L + θ(v H v L ) cw H 1, 0, λ ). (23) Λ p L and p H are the prices for minor (L) and major (H) treatments under honest diagnosis, χ = θ, and p C is the price for the major (H) treatment when the expert provides a dishonest diagnosis and recommends major treatment to all minor problem consumers χ = 1. For expositional ease, let the maximum number of major problem consumers that can be treated through an honest diagnostic policy be λ H (Λ). Lemma For Λ λ L (0) (1 θ), p H (λ L, λ H (Λ)) λ H (Λ) θλ = p L (λ L, λ H (Λ)). (24) 2. λ H (Λ) is increasing in Λ and θ. 3. For Λ < Λ s the efficient (first best) provision cannot be achieved. For a given potential demand Λ, if all the consumers with minor problems ((1 θ)λ) join the service queue, λ H (Λ) is the largest number of major problem consumers that the expert can treat while still providing honest diagnosis, χ = θ. This upper-bound on admission under honest diagnosis, helps us determine the market size threshold under which the expert benefits from providing false recommendations, χ = 1. The maximum revenues the expert can earn by providing false recommendations (χ = 1), when the potential demand is Λ, is given by: Π c (Λ) = I(Λ > λ c) θv H + (1 θ)v L where I( ) is the indicator function and λ c = 1 is the effective arrival rate maximizing the expert s revenues when χ = 1. 2 c + I(Λ λ c)λ ((θv H + (1 θ)v L ) cw H (1, 0, 1)), c (θv H + θv L ) (25)

24 24 Paç and Veeraraghavan Strategic Diagnosis and Pricing in Expert Services Lemma When θ > c( ) v H v L, there exists a threshold Λ c > 0 such that providing false recommendations, χ = 1, provides higher revenues for the expert than providing honest recommendations, χ = θ for Λ < Λ c. Λ c is the unique solution of the below equation. Π c (Λ c ) = (1 θ)λ c p L ( θλ c, λ H (Λ c )) + λ H (Λ c ) p H ( θλ c, λ H (Λ c )). (26) 2. There exists a threshold θ such that Λ c > λ c when θ > θ. Using the above results, we characterize the optimal prices and the equilibrium outcomes under dis-economies of scale in service time in Proposition 3. Proposition 3. When there are dis-economies of scale in service time, i.e. v H v L, the optimal prices (p L, p H) and the corresponding symmetric equilibrium of the diagnosis and queue joining game are as follows: 1. Minor Treatment Equilibrium: When Λ λ L, the expert provides an honest diagnosis, 1 θ χ = θ, and serves only minor problem consumers. The optimal prices, (p L, p H), and the equilibrium demand, λ e (p L, p H), are as follows: (p L, p H) = (λ e L, λ e H) = ( p L (λ L(0), 0), ) if Λ > λ L (0) 1 θ ( p L ( θλ, 0), ) if λ L (0) 1 θ (λ L(0), 0) if Λ > λ L (0) 1 θ ( θλ, 0) if λ L (0) 1 θ Λ λ L 1 θ Λ λ L 1 θ (27) (28) 2. Honest Price Discrimination: When Λ c Λ < λ L, the expert provides an honest diag- 1 θ nosis, χ = θ, and serves all minor problem consumers and some major problem consumers. The optimal prices, (p L, p H), and the equilibrium demand, λ e (p L, p H), are as follows: ( pl ( θλ, λ H( θλ)), p H ( θλ, λ H( θλ)) ) if λ L > Λ Λ (p L, p 1 θ s H) = ( pl ( θλ, λ H (Λ)), p H ( θλ, λ H (Λ)) ) (29) if Λ s Λ Λ c ( θλ, λ H( θλ)) if λ L > Λ Λ (λ e L, λ e 1 θ s H) = (30) ( θλ, λ H (Λ)) if Λ s Λ Λ c