European Journal of Operational Research

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1 Euroean Journal of Oerational Research 15 (011) Contents lists available at ScienceDirect Euroean Journal of Oerational Research journal homeage: Interfaces with Other Discilines Otimizing referral reward rograms under imression management considerations Ping Xiao a, Christoher S. Tang b, Jochen Wirtz a, a NUS Business School, National University of Singaore, Singaore b UCLA Anderson School, UCLA, United States article info abstract Article history: Received 1 October 010 Acceted May 011 Available online 30 May 011 Keywords: Referral reward rograms Stackelberg game Imression management We examine referral reward rograms (RRP) that are intended for a service firm to encourage its current customers (inductors) to entice their friends (inductees) to urchase the firm s service. By considering the interlay among the firm, the inductor, and the inductee, we solve a nested Stackelberg game so as to determine the otimal RRP in equilibrium. We determine the conditions under which it is otimal for the firm to reward the inductor only, reward the inductee only, or reward both. Also, our results suggest that RRP dominates direct marketing when the firm s current market enetration or the inductor s referral effectiveness is sufficiently high. We then extend our model to incororate certain key imression management factors: the inductor s intrinsic reward of making a ositive imression by being seen as heling a friend, the inductor s concerns about creating a negative imression when making an incentivized referral, and the inductee s imression of the inductor s credibility when an incentive is involved. In the resence of these imression management factors, we show that the firm should reward the inductee more and the inductor less. Under certain conditions, it is otimal for the firm to reward neither the inductor nor the inductee so that the otimal RRP relies urely on unincentivized word of mouth. Ó 011 Elsevier B.V. All rights reserved. 1. Introduction To gain market share, many service firms have launched various referral reward rograms (RRPs) that are intended to encourage their existing customers (inductors) to recommend the firms service to their friends (inductees). Because RRPs utilize existing customers (who are familiar with the firm s service) as inductors to reach out to the same affinity grou, their testimonies tend to be more credible and effective in communicating and tailoring the value roosition than direct marketing (Brown and Reinigen, 1987; Tuk et al., 009). Consequently, RRPs are generally seen as articularly effective in acquiring new customers in the service industry. At the same time, RRPs can be highly cost effective because the rewards for the inductor and the inductee are contingent on the inductee s successful urchase. RRPs have become ubiquitous: a recent search on Bing using the keywords recommend-a-friend rogram yielded over 3 million hits. Most of the to hits originated from service firms offering rewards to their existing customers for referring their services to their friends, and, in many instances, their friends also get rewards for signing u as new customers. Examles of various RRPs include: (1) Entertainment services: DirectTV offers $100 credit to the inductor and the inductee, Time Warner (cable TV) offers the Corresonding author. addresses: bizx@nus.edu.sg (P. Xiao), ctang@anderson.ucla.edu (C.S. Tang), jochen@nus.edu.sg (J. Wirtz). inductor one month free service and offers the inductee a reduced service fee for the first year, and Netflix (an online video rental/ streaming comany) offers the inductor $16 credit and offers the inductee a reduced service fee for the first year, resectively; () Wireless services: AT&T wireless, Verizon wireless, and Vonage (a VOIP hone service rovider) offer different credits to both the inductor and the inductee; (3) Financial services: Scott Trade (an online stock brokerage firm) offers 3 free trades and Hutchinson Credit Union (a financial service cooerative) offers $5 rewards to the inductor; however, neither firm offers rewards to the inductee; () Fitness center: hours Fitness (a oular fitness center in the US) offers a $0 couon to the inductors and offers the inductee a reduced service fee for the first year; and (5) Professional societies: the Institute of Oerations Research and Management Science (INFORMS) launched a Member-Get-A-Member rogram by offering a $10 Amazon voucher but offers no reward to the inductee. Desite the oularity of RRPs, little academic research has focused on the otimal design of RRPs or when to use referral rewards. By noting that traditional sales effort models examine ways for a firm to choose a comensation lan to maximize its rofit by taking into account the salesersons effort levels under different comensation lans, our base model of RRPs resented in Section 3 is related to the sales effort model in the following sense. First, the inductor in our model is essentially the saleserson in the sales effort model. Second, the inductor s reward in our model can be viewed as a secific comensation lan. (The reader is referred to Basu et al. (1985), Lal and Staelin (1986), /$ - see front matter Ó 011 Elsevier B.V. All rights reserved. doi: /j.ejor

2 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) Rao (1990), and Lal and Srinivasan (1993) for various sales effort models in the marketing literature, and Chen (000) and Chen (005) in the oerations literature). Desite the aforementioned similarities between our base model of RRPs and the traditional sales effort models, there are two subtle differences that deserve more clarifications. First, in our base model of RRPs, the firm chooses the reward (r) for the inductor and the discount incentive (d 1 ) for the inductee. In traditional sales effort models, the discount incentive for the consumers (inductees in our RRP context) is not considered. One excetion is the model resented in Bhardwarj (001) that deals with a situation in which the firm chooses a comensation lan for the saleserson and the selling rice for the customers. Unlike our base model his sales resonse function is a linear function that is given exogenously (Bhardwarj (001,. 16)). Secifically, Bhardwarj (001) examines a situation in which cometing firms sell through sales res; however, each firm may delegate the ricing decision to its sales res. Each firm offers a comensation lan that takes the form of W = a + b(q i ), where a is the fixed salary and b is the commission on the unit sold for firm i = 1,. The comensation lan is intended to entice the sales res to exert effort e i. For any given retail rice i and effort level e i, the demand for firm i s roduct is given by: q i ( i,e i, j,e j )=h i + h j + e i h s e j + d, where h reresents rice cometition intensity, h s reresents sales res effort cometition intensity, and d reresents demand uncertainty. By examining the non-cooerative game layed by these two cometing firms, Bhardwarj (001) determines the Nash equilibrium (Proosition 3 on age 151) that exhibits the following roerties: both firms will delegate their ricing decisions to their sales res in equilibrium when rice cometition is more intense (i.e., when h > h s ); and both firms will not delegate in equilibrium when effort cometition is more intense (i.e., when h s > h ). Observe that our base model is related to Bhardwaj s model when the firm does not delegate and that the inductor s reward r is related to the sales res commission b in Bhardwaj s model. However, we obtain different results because our model is monoolistic while his model is duoolistic; our sales resonse function (i.e., the demand function) is endogenous while his demand function is exogenous; and our inductor s reward r is a decision variable while the sales res commission b is given exogenously. Secifically, Bhardwarj (001) shows that the commission b has no imact on the selling rice in equilibrium (see the no delegation case in Table 1 on age 15), while we show that the inductor s reward r is directly related to the discount incentive d 1 for the inductees in equilibrium (Theorems 1 and in our aer). Second, in the sales effort model, customers (i.e., inductees in the RRP context) are assive articiants so that more customers will buy the roduct if the sales agents (i.e., inductors in the RRP context) exert more efforts. Consequently, the sales resonse function considered in most sales effort models is usually given exogenously so that the urchasing behavior of each customer (or inductee) is not modeled exlicitly. However, in order to develo a base model that would enable us to cature the issue of imression management that would affect the inductor s effort and the inductee s urchasing behavior, our base model of RRPs catures the urchasing behavior of each inductee in an exlicit manner. Secifically, we consider each inductee is an active articiant and each inductee s urchasing decision deends on the inductee s valuation and the discounted rice offered by the firm that are not considered in the sales effort model. Hence, the inductee s resonse in our model is determined endogenously and it is derived from the rational urchasing behavior of each inductee. In the traditional sales effort models, it is assumed that the sales agent (i.e., the inductor) can increase the number of customers (i.e., inductees) to sign u the firm s service (in exectation) by increasing his efforts. However, as we extend our base model to cature the issue of imression management, the extended model resented in Section is different from the traditional sales effort models in the following sense. First, each inductee s urchasing decision deends on the incentive for the inductee to sign u the firms service as well as the inductee s imression about the inductor s referral reward and his effort. Second, due to the otential negative imression about the inductor s intention due to his referral reward and his effort, the inductees may refuse to sign u the firm s service when the inductor s referral reward and his effort create strong negative imression. To our knowledge, the interaction between the incentives given to the agents (inductors in our RRP context) and the consumers (inductees in our RRP context) has not been considered in the sales effort literature. The sales effort model focuses on the dynamics between two arties: the firm and the sales agent (i.e., the inductor in the RRP context); however, because the inductee in the RRP model is an active articiant, one needs to examine the inter-lays among three arties: the firm, the inductor, and the inductee. Our research sheds light on a number of imortant questions related to RRPs. First, by considering the interlay among the firm, the inductor, and the inductee, we solve a nested Stackelberg game so as to determine the otimal RRP in equilibrium. We determine the conditions under which it is otimal for the firm to reward the inductor only, reward the inductee only, or reward both. Second, we study under what conditions RRPs would dominate direct marketing. Third, we then extend our model by incororating imression management factors that are imortant asects of incentivized referrals but have hitherto not been modeled in the literature. They are the inductor s intrinsic reward of making a ositive imression by being seen as heling a friend, the inductor s concerns about creating a negative imression when making an incentivized referral, and the inductee s imression of the inductor s credibility when an incentive is involved. Secifically, we examine how imression management factors affect the reward structure of the otimal RRP, and we show that strong imression management factors can make referral rewards subotimal. Our aer contributes to the extant literature by develoing a comrehensive model to examine a firm s otimal RRP design and by determining how this design is affected by the aforementioned imression management factors. In this aer, we resent a model to cature the inductor s and the inductee s behavior as well as the environmental factors. Secifically, to exlicate the mathematical structure of our model, we first resent the base model that is intended to analyze the underlying dynamics among the firm, the inductor, and the inductee without considering imression management factors. Our base model uses a nested Stackelberg game in which the firm acts as the suer leader of an outer game who decides on the reward mechanism to be given to the inductor and the inductee. For any given reward mechanism, the inductor and the inductee enter an inner game in which the inductor is the leader who chooses his referring effort in romoting the service (on behalf of the firm), and the inductee is the follower who makes her urchasing decision. By solving this nested game using backward induction, we show how the regular selling rice and the inductor s referral effectiveness affect the structure of the otimal RRP that the firm should adot in equilibrium. By comaring the firm s rofits under the otimal RRP and under direct marketing, we identify conditions under which RRP dominates direct marketing. We then extend our base model of RRP to incororate imression management factors and show how they significantly change the otimal reward structure of a RRP. From our analysis we obtain the following results. First, the regular rice and the inductor s referral effectiveness affect the reward structure of an otimal RRP. Second, RRP dominates direct marketing when the firm s number of current customers (i.e., its market

3 73 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) enetration) or their referral effectiveness is sufficiently high. Third, when we extend our model to incororate imression management factors, our analysis reveals that the otimal reward structure shifts from rewarding the inductor only towards rewarding both or rewarding the inductee only. Also, there are cases in which the firm should reward none so that the firm should rely on unincentivized word of mouth (WOM) via the inductors social network to acquire new customers. The remainder of this aer is organized as follows. We rovide a brief review of the related literature in Section. Section 3 resents our base model that is intended to exlicate the mathematical structure of the nested Stackelberg game without considering imression management factors. We determine the otimal RRP reward structure and show when a RRP dominates direct marketing. In Section, we extend our base model so as to examine the imact of imression management factors on the otimal reward structure. The aer ends with some concluding remarks and future research directions.. Literature review Desite the oularity of RRPs in ractice, only a few studies have exlored the otimal design of a RRP. In the marketing literature, exerimental work focused largely on the inductor s resonse to referral rewards associated with RRPs. Ryu and Feick (007) examined the imact of tie-strength (the relationshi between the inductor and the inductee), brand strength and the RRP s reward structure (reward only the inductor, only the inductee, or reward both) on the inductor s likelihood to make referrals. By considering the extent to which the inductor is satisfied with the firms roducts, Wirtz and Chew (00) showed that the referral reward is an effective mechanism to increase the inductor s likelihood to make referrals, esecially when the inductor is highly satisfied. In contrast to Ryu and Feick (007) and Wirtz and Chew (00) who focused on the inductor s resonse, Tuk et al. (009) examined the inductee s resonses to the RRP reward structure. They showed that the inductor s reward can reduce the inductee s likelihood to urchase because the inductor s reward can be ill-erceived by the inductee and reduces the erceived sincerity of the inductor. In this aer, we cature the key findings of this stream of exerimental work as follows: for any given reward structure (r, d) (Ryu and Feick, 007), the inductor s likelihood to make referrals and the inductee s likelihood to urchase ossess the same roerties in our model as discovered in Tuk et al. (009) and Wirtz and Chew (00). As such, this resent study comlements the extant stream of exerimental work. There are two recent modeling aers that deal with RRPs in the following manner. To our knowledge, Biyalogorski et al. (001) are the first to analyze a RRP in which the reward is given to the inductor only: either through a referral reward that is contingent uon a successful referral, or through a lower rice for the inductor that is intended to delight him so as to imrove his willingness to make referrals. They show that, because the referral reward is contingent uon successful referral, the referral reward can be more effective because it avoids the free-rider roblem that is resent in the lower rice strategy. Our model exands the issues examined by Biyalogorski et al. (001) as follows. In additional to the inductor s referral reward, we consider an RRP that also offers a reward to the inductee via a rice discount. Because there are rewards for the inductor and for the inductee, we develo a nested Stackelberg game that enables us to cature the behavior among three major layers (the firm, the inductor, and the inductee). In addition to Biyalogorski et al. (001), and Kornish and Li (010) are the first who incororated a sychological factor: the inductor will take the exected inductee s satisfaction level with a roduct into consideration when deciding whether to recommend that roduct. Kornish and Li consider the situation in which the inductor is more knowledgeable about a roduct s erformance than his friend, and he knows his friend s needs and wants, which means that the inductor can then make an informed recommendation to the inductee. They investigate the imact of the inductor s concern over the inductee s satisfaction with the roduct on the otimal inductor s referral reward and the inductee s rice. Secifically, inductors were modeled to be concerned about otentially negative effects of inductee dissatisfaction on the inductor s reutation, which led to incentives having to be increased at first to comensate for that risk until it became too high and inductors referred inductee discounts (and no reward to the inductor) to mitigate that risk. Our aer comlements Kornish and Li s work as follows. While Kornish and Li (010) address the inductor s concern about the inductee s satisfaction level with the roduct, we examine imression management factors that directly relate to the reward structure of the RRP: (a) the inductor s concern of making a negative imression when otentially being seen to be motivated by a reward and by benefiting from a friend s urchase; (b) the fli side of the inductor s negative imression management factors is the inductee s concern about the inductor s objectivity when an incentivized referral is involved; and (c) the inductor s intrinsic reward though making a ositive imression of being helful to a friend, esecially if the friend can receive an attractive discount as art of the RRP. In summary, our aer contributes to the existing literature in the following ways: First, we develo a model about RRPs that catures the major findings of a stream of exerimental work about the inductor s referral and inductee s urchasing behaviors. Hence, our work comlements the extant stream of exerimental work. Second, relative to the existing game theoretical models of RRPs, our model deals with additional decision variables (c.f., Biyalogorski et al., 001), and the hitherto unexlored imression management factors that are directly related to inductor s and inductee s rewards (c.f., Kornish and Li, 010). Third, we obtain closed form exressions for the otimal RRP reward structure, which enable us to examine the imact of different imression management and environmental factors. Fourth, besides heling us to formalize our understanding of RRPs, our model can also serve as a building block for determining otimal reward structures in more general settings. 3. The base model Consider an existing service firm (e.g., AT&T or Verizon) who offers a service in a market that is comrised of m + N customers. Currently, the firm has m existing customers and the firm s market enetration is /, where / = m/(m + N). Hence, the otential number of new customers is N ¼ m Currently, the firm is offering / its service at a regular rice to all existing customers. In our model, we assume is given exogenously for two reasons. First, this assumtion is consistent with actual ractice: the regular rice remains the same in the examles mentioned in Section 1. Second, in the context of RRPs, a firm should not lower its regular rice because it would allow those existing customers who do not serve as 1 Our model does not deal with a start-u comany that offers RRP so that each consumer can decide to be an inductor (i.e., an early adoter who signs u the firm s service without referral and then refers to his friends) or an inductee (i.e., a late adoter who signs u later and obtains some rewards). To cature the consumers strategic behavior over time for a start u firm, one would need to develo a twoeriod dynamic model which is beyond the scoe of this aer. The authors thank one reviewer for this insightful comment. Even though is exogenously given, one can always search for the otimal rice numerically.

4 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) inductors to ay a lower rice and would cause the firm to lose revenue. Instead of lowering the rice to all existing customers by leaving money on the table, we focus on the usage of referral reward to inductors and the rice discount to inductees. Moreover, we focus on analyzing the equilibrium behavior of the firm, the inductor, and the inductee. To obtain tractable results, we assume each new customer has a service valuation V, where V U[0,1]. To solicit new customers, the firm can use either (1) direct marketing (e.g., advertising and romotions) or () a RRP The benchmark model: direct marketing Under direct marketing, the firm offers a discount d 0 (a decision variable that satisfies d 0 (0,1]) so that each new customer can urchase the service at a discounted rice d 0. The direct marketing rogram has two cost comonents: a fixed cost for advertising such as the design for TV and rint advertisements K, and a variable cost for mass communication to otential customers k. This direct marketing rogram will enable the firm to reach an otential new customers, where a [0,1] is the enetration rate of direct marketing rogram. Under direct marketing, the interaction between the firm and the new customer can be modeled as a simle Stackelberg game in which the firm acts as the leader who selects the discount rate d 0, and the new customer acts as the follower who decides whether to urchase (Fig. 1). Given d 0, each customer will buy the service if her valuation V P d 0 + h 0, where h 0 is the hurdle rate which catures the imlicit costs associated with the customer s erceived risk due to unfamiliarity with the service and/or lack of knowledge and skills of using the service (Su, 009). In this case, the robability of a otential customer to urchase the service is P{V P d 0 + h 0 }. Knowing the customer s urchase robability, the firm can determine the otimal discount d 0 ð0; 1Š by solving the following roblem: max ½Na PfV P d 0 þ h 0 gðd 0 cþ K knš; d 0 ð0;1š where c is the unit cost for serving a customer. Notice that Na is the effective number of otential customers generated by the direct marketing rogram, and kn is the cost associated with mass communication to all otential customers N. By using the fact that V U[0,1], the firm s exected rofit can be rewritten as: [Na(1 (d 0 + h 0 )) (d 0 c) K kn]. By considering the first-order condition, we obtain the following result: Lemma 1. Under n o direct marketing, the otimal discount d 0 ¼ min 1 h 0þc ; 1, and the firm s otimal exected rofit 0 ðn; h 0Þ satisfies 0 ðn; h 0Þ¼m 1 / 1 a 1 h 0 þ c K km 1 / 1 : ð1þ Lemma suggests that the firm should offer a deeer discount d 0 if the regular rice or the hurdle rate h 0 is high. By noting that d 0 ¼ 1 when 6 1 h 0þc, there is no incentive for the firm to set < 1 h 0þc esecially when the firm anticiates rice discount exante. As such, we focus on the case when P 1 h 0þc in the remainder of this aer. 3.. The base model: referral reward rogram Note that our model is a one-eriod model which catures the context in which the firm needs to increase revenue by leveraging existing consumers. Therefore the underlying assumtion is the firm has already existing consumers when it makes the decision Firm d 0 New customer s urchasing behavior Customer s surlus discount offered to new customers Fig. 1. Game structure for direct marketing. regarding the structure of the referral reward rogram. Under RRP, the firm offers a reward structure with two comonents: (a) a referral reward r (a decision variable that has r P 0) to be given to each existing customer (an inductor) who refers a friend (an inductee) successfully; and (b) a rice discount d 1 (a decision variable that satisfies d 1 (0,1]) to be given to each inductee so that the inductee can urchase the service at a discounted rice d 1. (We shall refer to the term (1 d 1 ) as the discount deth under RRP throughout this aer.) Observe that the reward structure (d 1,r) can be classified into four basic tyes: (1) Reward the inductor only; i.e., r >0,d 1 = 1; () Reward both; i.e., r >0,d 1 (0,1); (3) Reward the inductee only; i.e., r =0, d 1 (0,1); and () Reward none; i.e., r =0,d 1 =1. Given the reward structure, the interlay among the firm, the inductor and the inductee can be modeled as a nested Stackelberg game that integrates an inner game with an outer game (Fig. ). We assume the inductor s reward r and inductee s urchase discount d 1 are common knowledge. In most instances, the reward structure is common knowledge through the firm s marketing communication or the communication between the inductor and the inductee. Given any reward (r,d 1 ), the inductor and the inductee enter a Stackelberg game in which the inductor acts as the leader who chooses his referral effort e, and the inductee acts as the follower who makes her urchase decision. Throughout this aer, we shall refer to the inductor and the inductee as he and she, resectively. Anticiating the inductor s and the inductee s equilibrium resonses in the inner game, the firm acts as the suer leader who determines its reward structure ðr ; d 1Þ, that maximizes its equilibrium rofit in the outer game. Once the equilibrium reward structure ðr ; d 1Þ is determined, one can retrieve other equilibrium outcomes accordingly Analysis of the inner game For any reward structure (r,d 1 ) and for any referral effort e exerted by the inductor, the inductee will urchase the service if her valuation V P d 1 + h(e), where h(e) is the inductee s hurdle rate associated with inductor s effort e (e.g., communication, exlanation, demonstration, and ersuasion). We assume the inductee s hurdle rate h(e) is decreasing and convex in the inductor s effort e. For ease of exosition, we consider the case when hðeþ ¼h 0 h ffiffi e. Notice that the arameter h reresents the inductor s referral effectiveness. Managerially seaking, the inductor s referral effectiveness h reflects the inductors overall satisfaction with the service (Wirtz and Chew, 00). For instance, when the inductor is highly satisfied with the service, he is more effective in articulating the value of the service to the inductee (e.g., Jones and Sasser, 1995; Rust et al., 1995). Therefore, the inductee will urchase the service with robability bðeþ ¼PfV P d 1 þðh 0 h ffiffi e Þg. Given the inductee s

5 73 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) Firm r : Referral reward for the Inductor Inductor Inner game Outer game e : Inductor s referral effort Inductee d 1 : Purchase discount for Inductee s surlus the Inductee Fig.. The inner and outer games of a referral reward rogram. urchase robability b(e) and the referral reward r, the inductor s exected ayoff for his referral effort e is equal to rbðeþ e ¼ r PfV P d 1 þðh 0 h ffiffi e Þg e. By using the fact that V U[0, 1], the inductor s roblem can be written as: max ep0 ½rð1 d 1 h 0 þ h ffiffi e Þ eš: By considering the first-order condition, the inductor s equilibrium referral rate satisfies: e ¼ rh : ðþ Observe from () that the equilibrium referral effort e in the inner game is increasing in the inductor s referral reward r and the inductor s referral effectiveness h. This result is consistent with the exerimental findings reorted by Ryu and Feick (007). Therefore, for any given referral reward r, the inductor will articiate the RRP if his net surlus from referring is non-negative, i.e., if (rb(e ) e ) P 0. It follows from () that bðe Þ¼1 d 1 h 0 þ rh : ð3þ Hence, the inductor s articiation constraint can be rewritten as:! ðrbðe Þ e Þ¼r 1 d 1 h 0 þ rh P 0: ðþ Observe from (3) that the inductee s urchase robability b(e ) and from () that the inductor s surlus are increasing with the inductor s referral effectiveness h and the inductee s discount deth (1 d 1 ). Hence, these results are consistent with our intuition Analysis of the outer game By anticiating the inductor s and the inductee s resonse as described in Section 3..1, we now analyze the outer game. For tractability, let us assume that each inductor will refer one inductee (when the inductor articiates in the RRP). 3 Hence, for any given (d 1,r), the firm s exected rofit is equal to 1 ðd 1 ; rþ ¼½m bðe Þðd 1 cþ m r bðe ÞŠ; 3 We make this assumtion for tractability. However, the same aroach can be used to analyze the case when each inductor refers multile inductees. ð5þ where the first term reresents the gross rofit obtained from the inductees and the second term reresents the referral cost aid to inductors. By consider b(e ) given in (3), we can determine the firm s equilibrium reward structure by solving the following otimization roblem: "! # 1 ¼ max m 1 d 1 h 0 þ rh ðd d 1 ð0;1š;rp0 1 c rþ ; subject to ðþ: By relaxing all constraints and by considering the Langrangean associated with the above otimization roblem, we can identify the set of active constraints in the otimal solution. As it turns out, the otimal reward structure deends on the regular rice. For ease of exosition, we shall resent our results according to different rice levels. First, the following theorem deals with the case when the firm s regular rice is low. Theorem 1 (Low Regular Price). Suose < 3 3h 0 c. Then the firm should never reward both the inductor and the inductee; i.e., either d 1 ¼ 1 or r = 0. Secifically, the firm s otimal reward structure satisfies 8 >< ðd 1 ; r Þ¼ >: h ; 0 ; if < 0:5; ; otherwise: 1 h 0 þc 1; h þ ch þh 0 h Theorem 1 can be interreted as follows. When the firm s regular rice is low, the firm cannot afford to reward both the inductor and the inductee without sreading its rewards too thin and rendering them ineffective. Hence, the firm should not reward both. To reward either the inductor or the inductee but not both, the firm has to choose the layer who offers the highest return for its reward. Consequently, it is intuitive to reward the inductor only when his referral effectiveness is sufficiently high (i.e., when h P 0:5). Otherwise, the firm should offer the incentive to the inductee only. Substituting the otimal reward structure given in (6) into the firm s exected rofit given in (5), we can determine the firm s otimal exected rofit in equilibrium 1 ¼ 1ðd 1 ; r Þ accordingly. By considering the case when = 0.7, c = 0., h 0 = 0. so that < 3 3h 0 c, the to chart in Fig. 3 illustrates the imact of the inductor s referral effectiveness (measured in terms of h ) on the firm s otimal rofit and its otimal reward structure (measured ð6þ

6 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) Base Model: Low Regular Price in terms of the inductor s referral reward r and the inductee s discount deth ð1 d 1 Þ). By using the same aroach as before, we can establish the following result for the case when the firm s regular rice is high: Theorem (High Regular Price). Suose P 3 3h 0 c. Then the firm s otimal reward structure satisfies 8 1 h 0 þc ; 0 ; if h 6 0:5; 1 h 0 þ h =ð1 h 0 cþ 1 h ; 0 c >< ; if 0:5;< h 6 ðþh 0 1Þ ðd 1 ; r ð1 h Þ¼ =Þ ð1 h =Þ 1; ðþh 0 1Þ ; if h >: ; if h > 3ðþh 0 1Þ : ð cþ 1; h þ ch þh 0 h Inductor's Referral Effectiveness (theta^/) Base Model: High Regular Price Inductor's Referral Effectiveness (theta^/) Otimal rofit Otimal reward to inductor (r*) Otimal discount to inductee (1-d1*) Otimal Profit Otimal reward to inductor (r*) Otimal discount deth to inductee (1-d1*) Fig. 3. Otimal RRP under low and high regular rice. ð cþþðþh 0 1Þ ; ðþh 0 1Þ < h 6 3ðþh 0 1Þ ð cþþðþh 0 ; 1Þ ð cþ Contrary to Theorems 1, asserts that it can be otimal for the firm to reward both the inductor and the inductee when is sufficiently high. For instance, when the inductor s referral effectiveness h is in the medium range; i.e., when 0:5 < h 6 ðþh 0 1Þ, ð cþþðþh 0 1Þ the firm should reward both the inductor and the inductee. Also, as the inductor s referral effectiveness h increases, the inductee s hurdle rate h(e )=h 0 rh / decreases and becomes more eager to urchase the service. In this case, the firm can discount less (i.e., a higher d 1 ). Finally, when the inductor is very effective in ðþh referring, say, when 0 1Þ < h ð cþþðþh 0, the third statement reveals 1Þ that it is otimal for the firm to reward the inductor only (i.e., d 1 ¼ 1Þ. Also, by examining the inductor s referral reward r as stated in Theorem, it can be shown that r is decreasing in the inductor s referral effectiveness h when h > 3ðþh 0 1Þ. This is because, as h ð cþ is sufficiently large, the inductee s hurdle rate h(e )=h 0 rh / is sufficiently low. As such, the firm can afford to lower the reward for the inductor r without affecting the inductee s urchase robability significantly. Substituting the otimal reward structure resented in Theorem into the firm s exected rofit given in (5), we can determine the firm s otimal exected rofit in equilibrium 1 ¼ 1ðd 1 ; r Þ accordingly. By considering the case when = 1., c = 0., h 0 = 0. so that P 3 3h 0 c, the bottom chart in Fig. 3 illustrates the imact of the inductor s referral effectiveness level (measured in terms of h /) on the firm s otimal rofit and its otimal reward structure (measured in terms of the inductor s referral reward r and the inductee s discount deth ð1 d 1 Þ) Comaring direct marketing and referral reward rogram Comaring the firm s otimal rofit under direct marketing rogram 0 ðm; h 0Þ given in (1) in Lemma and the firm s otimal rofit under RRP 1 ðd 1 ; r Þ by using the results stated in Theorems 1 and, we can establish the following intuitive result formally: Corollary 1 (Direct Marketing versus RRP). There exists two thresholds s 1 and s so that the Referral Reward Program dominates Direct Marketing when the firm s current market enetration is sufficiently high (/ P s 1 ) or when the inductor s referral effectiveness is sufficiently high (h P s ).. Extension: the issue of imression management In this section, we extend the base model resented in Section 3. by incororating imression management factors associated with referral reward (d 1,r). First, from the inductee s ersective, the credibility of the inductor s recommendation would decrease if the inductor received a referral reward r. Hence, as the inductor s referral reward r increases, the inductee s hurdle rate increases because the inductee may have a susicion about the inductor s motive. To cature this effect, we modify the inductee s hurdle rate h(e) defined in Section 3..1 by including an additional term Dr so that hðeþ ¼h 0 h ffiffi e þ Dr; ð7þ where D >0. Next, when incororating the inductor s imression management factors associated with any RRP reward (d 1,r), the inductor s effective reward equals to r + s(r,d 1 ), where r is the inductor s direct benefit and s(r,d 1 ) catures the imression management costs and benefits associated with the following factors. First, regardless of the inductor s referral reward r, the inductor may obtain an intrinsic reward s 0 P 0 that is associated with heling a friend make a good urchase decision, and make an imression of being helful and knowledgeable when making a recommendation (e.g., Arndt, 1967; Dichter, 1966; Ryan and Deci, 000). We shall refer s 0 as the intrinsic reward in the remainder of this aer. Second, when r > 0, the inductor may be concerned about being seen as wanting to benefit from his friend s urchase decision (c.f., Tuk et al., 009). Hence, there is a negative benefit s 1 r that is associated with this concern, which we shall refer to as the negative imression management in the remainder of this aer. Third, when r = 0 and d 1 > 0, the inductor may erceive a reward from managing a ositive imression s (1 d 1 ) I {r=0} (with s >0 and I {r=0} as the indicator function) that is associated with making the imression of being genuinely interested in heling the inductee to receive a discount without receiving any referral reward himself. In view of these three imression management factors, the inductor s effective reward under RRP is equal to r + s(r,d 1 ), where sðr; d 1 Þ¼s 0 s 1 r þ s ð1 d 1 ÞI fr¼0g : Observe from (8) that the inductor s effective reward r + s(r,d 1 ) involves a ste function s (1 d 1 )I {r=0}. For this reason, we now first determine the firm s otimal rofit for the case when r =0 in ð8þ

7 736 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) Section.1. Then, in Section., we determine the firm s otimal rofit for the case when r > 0. Then, by comaring the firm s otimal rofits associated with these two cases, we can obtain the firm s otimal reward structure that yields the highest rofit for the firm numerically in Section.3. bðeþ ¼P V P d 1 þðh 0 h ffiffi e þ DrÞ : ð13þ Anticiating the inductee s urchase robability, one can determine the inductor s referral effort in equilibrium by solving the following roblem:.1. Analysis of the inner game and outer game when the referral reward r = 0 When the inductor s referral reward r = 0, we can use the hurdle rate h(e) given in (7) to show that the inductee s urchase robability b(e) satisfies: bðeþ ¼PfV P d 1 þðh 0 h ffiffi e Þg: ð9þ By considering the inductor s effective reward given in (8) for the case when r = 0 and by anticiating the inductee s urchase robability b(e) given in (9), one can determine the inductor s otimal referral effort in equilibrium by solving the following roblem: max ½ðs 0 þ s ð1 d 1 ÞÞ bðeþ eš: ð10þ ep0 By using the fact that the valuation V U[0,1] and by considering the first-order condition, we obtain the following result: Lemma. When r = 0, the inductor s referral effort in equilibrium satisfies: e ¼ ðs 0 þ s ð1 d 1 ÞÞh : ð11þ Substituting e into (10), the inductor s articiation constraint can be written as: 1 d 1 h 0 þ ðs 0 þ s ð1 d 1 ÞÞh P 0: ð1þ Hence, the firm s roblem associated with the outer game can be written as: "! # ¼ max m 1 d 1 h 0 þ ðs 0 þ s ð1 d 1 ÞÞh d 1 ð0;1š ðd 1 cþ s:t: 1 d 1 h 0 þ ðs 0 þ s ð1 d 1 ÞÞh P 0: By considering the Kuhn Tucker conditions associated with the Langrangean function, we can establish the following result: Theorem 3 (No Inductor Referral Reward). When r = 0, the firm should offer the inductee a discount d 1 in equilibrium that satisfies: ( d 1 ¼ min 1; 1 h 0 þðs 0 þ s Þh = ; þ s h = ) ð þ s h =Þc þð1 h 0 þðs 0 þ s Þh =Þ : ð þ s h =Þ By noting that d 1 is decreasing in the inductor s referral effectiveness h /, it is easy to check that d 1 ¼ 1 (i.e., no discount for the inductee) when the inductor s referral effectiveness h / is sufficiently high. This result reveals the ossibility of having the firm to reward none ðr ¼ 0; d 1 ¼ 1Þ in the resence of imression management factors. We examine this ossibility in Section.3... Analysis of the inner game and outer game when the referral reward r > 0 When the inductor s referral reward r > 0, we can use the hurdle rate h(e) given in (7) to show that the inductee s urchase robability b(e) satisfies: max ep0 ½ðr þ sðr; d 1ÞÞ bðeþ eš; ð1þ where (8) yields s(r,d 1 )=s 0 s 1 r. By using the fact that the valuation V U[0, 1] and by considering the first-order condition, we obtain the following result: Lemma 3. When r > 0, the inductor s referral effort in equilibrium satisfies: e ¼ ðs 0 þð1 s 1 ÞrÞh : ð15þ Because s 1 [0,1], it is easy to check that the inductor s referral effort e is increasing in the inductor s referral reward r, the inductor s intrinsic benefit s 0, and the inductor s referral effectiveness h. Also, observe that the inductor s referral effort e given in (15) reduces to the base case as given in () when s 0 = s 1 =0. By using (15) and (1) along with the fact that s(r,d 1 )=s 0 s 1 r, the inductor s articiation constraint ((r + s(r,d 1 ))b(e ) e P 0) can be written as: 1 d 1 h 0 þ ðs 0 þð1 s 1 ÞrÞh Dr P 0: ð16þ Hence, the firm s roblem associated with the outer game can be written as: "! # 3 ¼ max m 1 d 1 h 0 þ ðs 0 þð1 s 1 ÞrÞh Dr ðd d 1 ð0;1š 1 c rþ s:t: 1 d 1 h 0 þ ðs 0 þð1 s 1 ÞrÞh Dr P 0: ð17þ By considering the Kuhn Tucker conditions associated with Lagrangean functions, we can determine the otimal reward structure ðd 1 ; r Þ that involves the solutions associated with different regions within the feasible set. To simlify our exosition, let us introduce the following quantities. Let: a ¼ 1 h 0 þ s 0h ; a0 ¼ 1 h 0 þ s 0h ; k ¼ 1 D þ ð1 s 1Þh ; and k 0 ¼ 1 D þ ð1 s 1Þh. Also, we define: ðd a ;r a Þ¼ a0 þðk 0 þ 1Þr a ;min a0 k 0 ;max ðk k0 Þða 0 cþþk 0 ða a 0 Þ ðk k 0 Þk 0 ðd b ;r b Þ¼ a þðk þ Þr þ c b ;min a0 k 0 ; 1 h0 c 1 þ D ðd c ;r c Þ¼ð1;0Þ ðd d ;r d Þ¼ a0 þðk 0 þ 1Þr d ; 1 h0 c 1 þ D if 3 d 1 ¼ a0 þðk 0 þ 1Þr ;r ¼ 1 h0 c 1 þ D ðd d ;r d Þ¼ a0 þðk 0 þ 1Þr d ; a0 k 0 Otherwise ðd e ;r e Þ¼ a þðk þ Þr e þ c ;min ðd f ;r f Þ¼ð1;0Þ > 3 d 1 ¼ a0 þðk 0 þ 1Þr ;r ¼ a0 k 0 a0 k 0 ; 1 h0 c 1 þ D ; 1 h 0 c 1 þ D ð18þ By using those quantities defined in (18), we obtain the following results: Theorem (Reward Inductor (r > 0)). Suose the firm chooses to reward the inductor under RRP so that r > 0. Then the firm s otimal reward structure ðd 1 ; r Þ can be described as follows:

8 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) Fig.. Numerical analysis of the imacts of imression management factors on the otimal RRP. n o 1. When h < 1þD ; ð1 s 1 Þ ðd 1 ; r Þ¼ min aþc ; a0 ; 1 ; 0 : 1þD. When 6 h < 1þD ð1 s 1 Þ 1 s 1 ; ðd 1 ; r Þ¼argmax 3 ðd a ; r a Þ; 3 d b ; r b ; 3 ðd c ; r c Þg. 3. When h P 1þD 1 s 1 ; ðd 1 ; r Þ¼argmaxf 3 ðd d ; rþ; d 3ðd e ; rþ; e 3ðd f ; rþg. f When the inductor s referral effectiveness is low (i.e., when h < 1þD ð1 s 1 ), Theorem reveals that the firm should never reward Þ the inductor, i.e., r = 0. This result is consistent with the result obtained in Theorem 1 in the base model. It is easy to check from Theorem that the case associated with h < 1þD ð1 s 1 reduces to the Þ base model for the case associated with h < 1= when D = 0 and s 1 = 0. Also, as indicated in Statement 3, it is ossible for the firm to reward none when the inductor s effectiveness is sufficiently high; say, when h P 1þD ð1 s 1 ). We shall investigate this further Þ numerically in the next section. By using the results obtained in Theorems 3 and for the case when the inductor s referral reward r = 0 and r > 0, resectively, we can determine the firm s (global) otimal rofit by selecting the case that yield the higher rofit; i.e., the firm s otimal rofit is equal to P ¼ maxf ; 3g. In addition, one can retrieve the otimal reward structure ðd 1 ; r Þ that yields the maximum rofit for the firm in the resence of various imression management factors..3. Numerical analysis We now examine the otimal reward structure ðd 1 ; r Þ in the resence of imression management factors. Each of the imression management factors catured by the function s = s(r,d 1 )=s 0 s 1 r + s (1 d 1 )I {r=0} given in (8) will have imact on the reward structure. Recall from Section that the inductor s intrinsic reward is catured by the arameter s 0 P 0; the inductor s concerns about being seen as taking advantage of his friend are catured by the arameter 0 6 s and the inductee s susicion is catured by the additional hurdle rate factor D P 0; and the inductor s warm glow, or the ositive imression created for enabling the inductee to obtain a discount (0 < d 1 < 1) while referring without any reward (r = 0), is catured by the arameter s >0. To examine how these imression management factors affect the otimal reward structure, we use the same arameter values as in the base case; namely, we set = 1., c = 0., and h 0 = 0.. To isolate the effect of each of the imression management factors, we conduct our numerical examles as follows: Case 1: The Imact of Intrinsic Reward (s 0 > 0). Using the base model as the benchmark, we examine how the reward structure changes when we vary the arameter s 0 from 0 to. To isolate the effects of changing s 0, we set s 1 =0,s = 0, and D = 0. Therefore, the otimal reward structure in the resence of intrinsic reward will reduce to the base case when s 0 = 0. Recall from Theorem that the otimal reward structure for the base case when the regular rice is sufficiently high can be described as follows: reward the inductee only when the inductor s referral effectiveness is low; reward both when his effectiveness is at a medium level; and reward the inductor only when his referral effectiveness is high. Observe from the to left chart in Fig. that as the intrinsic reward s 0 increases, the otimal reward structure shifts away from rewarding the inductor and shifts towards rewarding none when s 0 exceeds a certain threshold. Therefore, unincentivized RRPs (using word of mouth) can be an effective mechanism for acquiring new customers when the inductors have high intrinsic rewards for referring the firm s services to their friends. Case : The Imact of Negative Imression Effect (s 1 > 0 and D = ks 1 ). Similarly, using the base model as the benchmark, we examine how the otimal reward structure changes as the negative imression effect s 1 increases from 0 to 1. To isolate the effects of changing s 1, we set s 0 =0,s = 0, and D = ks 1 where k = 0.8. The to right chart in Fig. deicts the otimal reward structure. It is easy to check that, as the negative imression effect s 1 increases, the

9 738 P. Xiao et al. / Euroean Journal of Oerational Research 15 (011) otimal reward structure shifts towards rewarding the inductee only. For instance, as we increase s 1 from 0 to 0.8, the otimal reward structure shifts toward rewarding the inductee and rewarding both. Then, as s 1 exceeds a certain threshold, say, (s 1 > 0.8), it is otimal for the firm to reward the inductee only. Case 3: The Imact of Positive Imression Effect (s > 0). Again, using the base model as the benchmark, we examine how the reward structure changes when we vary the arameter s from 0 to. To isolate the effects of changing s, we set s 0 =0,s 1 = 0 and D = 0. Observe from the bottom chart in Fig. that, as the ositive imression effect s increases, the otimal reward structure shifted towards rewarding the inductee only. Secifically, when the ositive imression effect s exceeds a certain threshold, it is otimal for the firm to reward the inductee only. As these cases show, all three tyes of imression management factors affect the otimal design of RRPs and shift otimal reward structures away from the inductor. It becomes otimal to reward none and rely on WOM if strong intrinsic benefits exist and a threshold level of inductor effectiveness is exceeded, and to reward the inductee only if either high negative imression management factors or high ositive imression management factors are resent. Rewarding the inductor is never otimal in these cases. 5. Concluding remarks Although referral reward rograms (RRPs) have become ubiquitous, their effectiveness has been examined only by a handful of exerimental (Ryu and Feick, 007; Tuk et al., 009; Wirtz and Chew, 00) and modeling studies (Biyalogorski et al., 001; Kornish and Li, 010). In this aer, we develo a benchmark model that exlores the otimal reward structure a firm should offer in equilibrium. We found that the otimal design of a RRP deends on the regular selling rice and the referral effectiveness of the inductor. In articular, when the regular selling rice is low, the firm cannot afford to offer rewards to the inductor and the inductee without sreading its rewards too thin and rendering them ineffective. Hence, the firm should not reward both. To reward either the inductor or the inductee but not both, the firm has to choose the layer who offers the highest return for its reward. Consequently, it is intuitive to reward the inductor only when his referral effectiveness is sufficiently high. Otherwise, the firm should offer the incentive to the inductee only. Comaring the firm s rofits under the otimal RRP and under direct marketing, we found that RRP dominates direct marketing when the firm s number of current customers (i.e., its market enetration) or their referral effectiveness is sufficiently high. We extended our model to include imression management factors that were hitherto unexlored in the modeling literature. Our analysis revealed that the otimal reward structure shifts from rewarding the inductor only toward rewarding both, rewarding the inductee only, and there even exist otimal RRP designs that call for reward none. When the latter haens, the firm should rely on unincentivised WOM via inductors social network to acquire new customers. Our model heled us to formalize our understanding of RRPs, but it can also serve as a building block for determining otimal reward structure in more general settings. Secifically, our numerical analysis exlored all three imression management factors in isolation for arsimony, but the model can be used to exlore otimal RRP designs of any combination of the imression management factors. Our study offers imortant managerial imlications. The findings of our base model can hel firms understand (1) when to use RRPs rather than direct marketing (e.g., when their market enetration is high, or when their customers are highly satisfied or delighted), and () how to design an otimal RRP (e.g., reward the inductor, the inductee, both or none). Key variables that determine the otimal design are the inductor s referral effectiveness (increasing effectiveness favors shifting rewards from the inductee to the inductor), and the regular selling rice (increasing the regular selling rice favors rewarding both the inductor and inductee). Our extended model rovides guidelines for firms whose customers consider imression management factors as imortant. Conditions with high intrinsic rewards in articular are interesting as they call for reward none once a certain level of inductor referral effectiveness is exceeded. This means that situations, where inductors objective is to be seen as knowledgeable and/or as helful, and their referral effectiveness is sufficiently high, do not call for rewarding any of the arties. Such situations may include high involvement services which inductors enjoy talking about and make them aear in a ositive light (Arndt, 1967; Dichter, 1966). In contrast, if inductors are concerned about making a otentially negative imression as being motivated by a reward and as benefiting from the inductee s urchase decision, the otimal RRP design shifts with increasing concern from rewarding the inductor, to rewarding both, and finally, to rewarding the inductee only. Imression management concerns may be most imortant in weak to medium-tie relationshis, such as acquaintances and work colleagues, where single actions like a referral can shae imressions more strongly (c.f., Ryu and Feick, 007; Tuk et al., 009). This is in contrast to strong tie relationshis (e.g., such as immediate family members who may be unlikely to shift their imression because of a referral reward as imressions are quite stable) and very weak ties, where inductors may be much less concerned about the imression they may create (e.g., in an anonymous online context). Future research could be extended to consider the otimal designing of RRP under cometitive markets. Also, it can be interesting to study how a firm can design a menu of RRPs considering consumers heterogeneity in imression management concerns. Moreover, as mentioned in Section 3, it would be of interest to develo a -eriod dynamic model for a start-u firm that offers RRP so that each consumer can decide to be an inductor (i.e., an early adoter who signs u the firm s service without referral and then refers to his friends) or an inductee (i.e., a late adoter who signs u later and obtains some rewards). 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