Strategic Consumers, Revenue Management, and the Design of Loyalty Programs

So Yeon Chun McDonough School of Business, Georgetown University, [email protected] Anton Ovchinnikov Smith School of Business, Queen’s University, [email protected]

In this paper, we study the interaction between revenue management and a premium-status loyalty program, as well as the role of strategic consumers in this interaction. Specifically, we consider a contemporaneous change where firms across several industries switch their loyalty programs from quantity/mileage-based (under which consumers obtain status by flying a certain number of miles) to spending-based (under which consumers obtain status by spending a certain number of dollars). This change has been met with fierce opposition from the media and consumers. We present a model for strategic, forward-looking, and status- seeking consumers’ decisions on how much to purchase/fly over a certain time period, and endogenously derive the strategic consumer demand as a function of the firm’s prices and loyalty program parameters. We then incorporate such demand into the firm’s pricing and loyalty program design problem and compare the solutions under different loyalty program designs: the mileage- and spending-based programs, as well as the enhanced mileage programs with the selling of miles and bonus miles features. We show that the spending- based program is more profitable than the mileage-based program, and that a switch to a spending-based program can be Pareto-improving: the firm’s profit can increase, yet no consumers will be worse off. We also show that the enhanced mileage programs can be even more profitable than the spending-based program. Finally, we demonstrate that when the firm coordinates its pricing and loyalty decisions, it can benefit from strategic consumer behavior, and provide a detailed examination for how firms’ loyalty program designs could alter strategic consumer behavior to increase profitability.

Key words : marketing – operations interface, loyalty (reward) programs, premium status, revenue management, pricing, strategic consumer behavior

1. Introduction Loyalty programs have been widely used in many industries. In the airline industry, such programs are commonly known as “frequent-flyer programs” or “mileage programs” – for example, Delta Airlines’ SkyMiles, or United Airlines’ MileagePlus. As these programs’ titles suggest, they reward customers for the miles flown with the airline. One particular kind of reward that we will focus on in this paper is that of premium status, which is usually awarded to consumers who fly a required number of qualified miles or segments within a qualification period. For example, a Gold status was typically awarded to those who fly 50,000 miles or 60 segments within a calendar year. The benefits

1 2 of a premium status in the airline industry vary, but they often include dedicated customer service, priority check-in, free checked bags, upgrades, access to lounges, and so on. Similar programs are popular in other industries as well: Starbucks Rewards is one notable example. This mileage/quantity-based design of premium-status loyalty programs, however, is undergoing a fundamental change. In the airline industry, several firms started switching their frequent-flyer programs from mileage-based (under which consumers obtain premium status by flying a certain number of miles/segments) toward spending-based (under which consumers obtain premium status by spending a certain number of dollars). For instance, in Virgin America’s frequent-flyer program, “Elevate”, members earn five points for every $1 spent and need to accumulate 50,000 points within a year (i.e., spend $10,000) to qualify as Gold. Delta and United also recently introduced changes that make their loyalty programs more spending-based. Similar changes occur in other industries: for instance, Starbucks used to award 1 star per transaction and members required 30 stars for Gold status (a quantity-based design). However, in April 2016, the program changed to awarding 2 stars per dollar spent and Gold status now requires 300 Stars, i.e., a consumer needs to spend $150 to qualify for Gold status (a spending-based design). The media and public reaction to these changes has been rather negative. Reactions range from simply expressing the irony, e.g., “Delta SkyDollars? United DollarsPlus?”(Airline Weekly 2015), to claiming that “basically you’re hosed”(Benderm 2014), and that the “changes are bad for everyone”(Mackenzie 2014). CNN reports a “customer outcry,” E!online states that “people are so pissed,” Fortune writes that “customers are furious" and social media is full of posts from angry customers who call the changes to spending-based programs “awful” and announce a “boycott”(Doublewides fly(2014), Vasel(2016), Mullins(2016), Snyder(2016)). In parallel to the recent trend toward spending-based designs, firms have also been making more subtle changes to their loyalty programs. For instance, airlines have been adding enhancing features to their loyalty program designs while keeping them essentially mileage-based; Air Canada offers a 25% mileage premium (a “bonus miles” design) on tickets in the Latitude fare class, while status miles are still based on distance flown1. Turkish Airlines also awards status based on distance flown, but allows its customer to purchase status-qualifying miles (a “selling of miles” design.)2 Many other firms over the years also introduced similar enhanced features. The goal of this paper is to analyze these different loyalty program design choices, and to assess their impact on firms and consumers. First, with regards to the recent changes, we seek to answer if, why, and by how much firms benefit from switching from mileage- to spending-based designs and,

1 https://www.aircanada.com/en/aeroplan/earn.html 2 http://www.turkishairlines.com/en-int/corporate/news/news/complete-your-miles-keep-your-status-301 Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 3 consequently, from re-optimizing pricing and revenue management (RM) decisions. We further seek to understand the impact of the switch on consumers – specifically, whether the aforementioned negative reactions are warranted: Will consumers indeed be worse off? Second, with respect to the enhanced mileage-based designs (with bonus miles or selling of miles features), we investigate whether firms could benefit more from enhancing their mileage-based programs rather than from switching to a spending-based program, and we further examine how strategic consumer behavior could be affected under these enhanced designs. A central phenomenon captured in our analysis is that consumers are forward-looking and status- seeking (i.e., strategic); rather than viewing each transaction/flight in isolation, they look forward to possibly obtaining premium status and optimize their purchasing so as to maximize their total surplus over a period of time, taking into account both the prices they pay and the loyalty status they may obtain as a result. Under the airline mileage-based programs, such strategic behavior manifests itself in so-called “mileage-runs” – situations where consumers take longer/incovenient/unnecessary flights in order to accumulate the needed miles to maintain or enhance their status levels (in the hotel industry, a similar practice is called “mattress runs”). The evidence for mileage runs is abundant. A survey reported in Del Nero(2014) shows that 60% of frequent travelers engaged in a mileage run in December 2013, and 25% did a mattress run. A New Yorker article by Sernovitz(2016) mentions a customer who took three flights during the same day between different airports in the same city to accumulate the requisite number of flight segments, while noting that “[t]he planes were filled with others doing the same.” Beyond the airline industry, a Fortune magazine article by Wahba(2016) states that prior to the recent change, “Starbucks was finding that many customers were asking baristas to ring up one item at a time to collect more points” – clearly an unnecessary and inconve- nient activity in spirit quite similar to a mileage run, performed solely to earn loyalty points. Mileage running increases demand, but its impact on the firm’s profitability is not obvious because the firms incur status-related costs to serve premium customers (handling checked bags, lounge access, etc.), in addition to the cost of operating flights. For example, imagine a situation where the premium qualification threshold is 60 flights, and consider a consumer who needs to take 59 flights per year. Without being strategic, this consumer would take 59 flights per year and would therefore fly without premium status; the firm would incur no status-related costs. If, however, this customer becomes strategic and “runs” for one additional flight every year, then (s)he will maintain premium status. Consequently, every year, the firm will obtain additional revenue from one flight, but will incur the status-related costs for all 60 flights that this customer will take. Therefore, even though demand increases, it is unclear whether profit increases as well. 4

In this paper, we consider a monopolistic3 firm that sets prices and determines the loyalty program design and qualification thresholds. In response to the firm’s prices and loyalty decisions, consumers strategically decide on how much to fly and at what prices to maximize their total surplus over the status qualification time period. That is, consumers are forward-looking and status- seeking, and they optimize their flying activities by taking into account both the prices they pay and the loyalty status they obtain as a result. To describe this complex phenomenon, we present a novel model of consumer behavior, which captures consumer response to prices and loyalty program design. This allows us to characterize rich patterns of consumer behaviour, analytically solve the customers’ problems under different loyalty program designs, and endogenously derive the demand faced by the firm as a function of its RM and loyalty decisions. We then formulate models for how the firm decides on prices and loyalty program design decisions, factoring in the consumer response. We solve the firm’s profit maximization problems under different loyalty program designs, and discuss managerial implications. We first compare mileage-based and spending-based programs and then consider enhanced mileage-based program designs with selling-of-miles and bonus-miles. Our paper is the first to consider the impact of strategic consumer behavior on joint pricing and loyalty program decisions. For this reason, we focus on a single aspect of the loyalty program design where we believe strategic behavior is most critical: premium status. Other aspects of the loyalty program design, such as the accumulation and redemption of points/miles, or an issue of externality (where the status becomes less valuable if many consumers have it) offer interesting research questions; however, they are outside the scope of this paper.

1.1. Summary of findings First, we characterize the optimal strategic response of consumers to the firm’s pricing and loyalty program design. Specifically, we show that under the mileage-based design “flying-up”, i.e., flying more to obtain or maintain premium status (mileage running), is indeed optimal for a segment of consumers. Under the spending-based design, “spending-up” (paying more for the same

3 While the airline industry is clearly competitive, it is also highly concentrated; many large airports are dominated by a single airline. For example, according to www.transtats.bts.gov, in January 2016, Delta’s share at its hub airport in Atlanta was 73.44%. The second-largest major carrier in Atlanta, American, had a share of only 2.54%. Similarly, American’s share at its hub in Dallas/Fort Worth was 67.39% while Delta’s was under 3%. As a result, a frequent traveller residing in the Atlanta metro area would experience major inconvenience by being loyal to any airline other than Delta, while one residing in Dallas would have a hard time flying anything but American. This pattern holds for many major cities around the world: residents of Paris (hub of AirFrance) have much fewer choices on other airlines, such as Lufthansa, while those who live in Frankfurt (hub of Lufthansa) have much fewer choices with AirFrance. Supporting this logic Sernovitz(2016) writes that customers are extremely loyal to United, either because they like it “... or, more probably, because the airline has a hub near your home.” In other words, for many frequent travellers – who are the primary object of our study – the choice of airline is mainly determined by where they live and not by the airlines’ competitive behaviors. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 5 product or service in order to qualify for premium status) is also an optimal strategic choice for some consumers. Spending-up options include: paying-up (strategically purchasing the same products/services at higher prices, even when lower prices are available), flying-up (flying more than needed at low prices – i.e., “mileage-running”), or mixing the two. That is, a switch to the spending-based program will reduce, but not eliminate, mileage-running completely since one way to spend more is simply to fly more. For the enhanced mileage-based designs, we similarly characterize which consumers will pay more to get “bonus-miles” and which consumers will “‘buy miles” without flying to qualify for the status. For both enhanced designs, flying-up could still be an optimal choice for some consumers. We emphasize that in all these cases, the consumers who decide to fly or spend more do so strategically because it increases their surplus. Second, regarding the major change occurring in the industry, we compare the mileage- and spending- based designs. We show that the firm’s profit is higher under the spending-based design. One might intuit that additional profit comes from an increase in revenue from consumers who spend more. We show, however, that an equally important driver is the decrease in costs. The spending-based program expands the segment that qualifies for the premium status strategically and simultaneously reduces the segment of consumers who mileage-run. Replacing mileage-running (flying-up) with spending-up could result in the same revenue, but from fewer flights, which decreases costs, rendering the change to the spending-based program profitable. This result may thus provide the rationale for the switch to spending-based programs that some firms implement in practice. Third, and speaking directly to the customer outcry that was generated by the change, we show that a spending-based program could be Pareto-improving: the firm can increase its profit without any consumers being worse off. To achieve a Pareto-improving solution, however, the firm might need to sacrifice a portion of its additional profit – i.e., settle with the profit that is above the optimal mileage-based solution but below the optimal spending-based one. While it is not clear if all firms would do that, some firms could consider such an approach; e.g., as Wahba (2016) writes, “Starbucks... is being careful not to irritate customers or make them feel the new program is less generous." With the optimal spending-based solution, some consumers may suffer because it discourages mileage-running by possibly making the effective qualification threshold for such consumers higher (thus making mileage-running harder). Therefore, the negative public reactions mentioned earlier are justified: some consumers may be worse off if the firm switches to the spending-based design. Fourth, and perhaps most importantly, we show that the firm does not need to make a drastic switch to the spending-based program to improve its profit: enhancing the mileage-based program with the selling-of-miles or bonus-miles features can achieve profits that are even higher than the 6 spending-based optimum. This happens because the two enhanced designs further decrease the segment of consumers who mileage-run, and simultaneously expand the segment that strategically qualifies for the status, both of which lead to further increases in profits. This may explain why some firms have enhanced their mileage-based programs instead of changing their programs to spending-based. Finally, we make an interesting observation that transcends all the program designs we considered. When the firm coordinates its RM and loyalty program, it can, in fact, benefit from strategic consumer behavior: its profit is higher when consumers are strategic than when they are not. Furthermore, the spending-based program benefits more from strategic behavior than the mileage-based program, and the enhanced mileage-based programs benefit even more than the spending-based program. This finding is particularly interesting because in much of the literature, strategic consumers are known to negatively impact the firm (e.g., Cachon and Swinney 2009, Levin et al. 2009, Ovchinnikov and Milner 2012). We thus identify an important element, the interaction of RM and the loyalty program, that can reverse conventional wisdom about the impact of strategic consumers on the firm’s profitability. In a complementary manner, our work also provides a new insight into the sources of a loyalty program’s value: while traditionally viewed as means of softening competition (e.g., Kim and Srinivasan 2001, Kim et al. 2004), we show that even without competition, loyalty programs can be valuable because of how different designs alter strategic consumer behavior in various ways. To conclude the summary of findings, our paper is the first to rigorously examine loyalty program design in the presence of strategic consumers and provide insights that are relevant for researchers and industry professionals as they rethink and redesign loyalty programs to improve their firms’ profits and customer satisfaction.

2. Related Literature Our paper is on the intersection between marketing and operations, and builds on prior findings from both fields. Marketing researchers examined consumer purchase behavior induced by loyalty programs. For example, Lewis(2004) empirically found that loyalty programs encourage consumers to shift from myopic (single-period) decision making to forward-looking (multiple-period) decision making so as to maximize the total surplus over a certain time period. Liu(2007) further observed that different consumers react differently to the loyalty program, suggesting a need to consider consumer heterogeneity when studying loyalty programs. Kivetz et al.(2006) demonstrated the “points pressure” effect – i.e., that consumers purchase more as they get closer to the reward threshold in a frequency reward program, and Kopalle et al.(2012) confirmed that point pressure Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 7 exists in the premium-status programs. Building on these empirical findings, we model forward- looking, strategic, and heterogeneous consumer behavior, and endogenize the point-pressure effect. Strategic consumer behavior induced by revenue management has also been studied in marketing (e.g., Shugan and Xie 2005, Koenigsberg et al. 2008), but it received more attention in operations, where many researchers studied how a firm should anticipate strategic consumer behavior and integrate it into its RM decisions (e.g., Su 2007, Zhang and Cooper 2008, Liu and van Ryzin 2008, Sahin et al. 2008, Cachon and Swinney 2009, Levin et al. 2009, Osadchiy and Vulcano 2010). These studies comprehensively address the RM problem, but they do not consider loyalty programs. Our model is less detailed than those used in these studies when dealing with the RM side of the problem, as we consider a menu of two prices that are static throughout the time period. However, in our model, the firm also decides on the premium status qualification requirement, and consumers strategically take it into account, alongside prices, when making their purchasing decisions. Combining consumer response to loyalty programs and RM, our work is related to many existing studies in marketing and economics, which have studied the impact of loyalty programs. These include the implications of loyalty programs on customer retention and switching costs (Carlsson and Löfgren 2006, Meyer-Waarden 2007), brand loyalty and attitudinal loyalty (Uncles et al. 2003, Shugan 2005), demand and market share (Meyer-Waarden 2007, Lederman 2007), and competition (Kim and Srinivasan 2001, Kim et al. 2004). Our paper contributes to this literature by identifying another important element that makes loyalty programs beneficial even in the absence of competition: strategic consumer behavior with respect to the interaction of loyalty programs and RM. Our paper is also related to the stream of research on the design of loyalty programs, see Chapters 22 and 23 in Blattberg et al. 2008 and Dorotic et al. 2012 for reviews. While different design components of loyalty programs have been considered, e.g., reward types (Kivetz 2005, Keh and Lee 2006), reward timing (Zhang et al. 2000, Roehm and Roehm 2010), and frequency and magnitude (Kivetz 2006, Nunes and Drèze 2006), these papers do not consider a change from mileage- to spending-based loyalty programs, which is the main focus of our paper. Finally, our paper also contributes to the growing body of literature on the interface between marketing and operations (for a general review, see Coughlan and Shulman 2010 and Tang 2010), which integrates broadly-defined concepts from RM and CRM (e.g., Aflaki and Popescu 2014, Ovchinnikov et al. 2014). Specific to RM and loyalty programs, Chung et al.(2014) examine how the terms of reward point redemption affect the firm’s pricing and inventory control, Sun and Zhang (2014) study the tradeoff between the long and the short expiration terms for loyalty program rewards, and Chun et al.(2014) study the long-term management of loyalty programs from a unique perspective that touches upon operations, marketing, and finance. These papers, however, do not investigate different loyalty program designs. 8

3. The Model We consider a monopolistic firm with an infinite capacity that sells a single product and decides on a menu of two prices, p and p+ ≥ p. The firm also decides on the structure of its loyalty program, either quantity/mileage/segment-based4 or spending-based, and sets the premium status qualification thresholds, G, and S, respectively. Under a mileage-based program, and its enhanced variants considered in Section5, a consumer who takes a minimum of G flights qualifies for a premium status, which we refer to as “Gold” from now on. Under the spending-based program, a consumer who spends S or more dollars qualifies for Gold. Under all programs, the firm incurs cost c for flying a Gold5 customer; other costs are normalized to zero. Motivated by the empirical observations introduced earlier (Lewis 2004, Liu 2007, Kopalle et al. 2012), we model the behavior of consumers who are forward-looking (strategic) and therefore optimize their flying activity over a period of time so as to maximize the total utility from flying over that time period. In consumers’ decision-making processes, both the prices they pay and the loyalty program status they obtain play an important role. Next, we discuss two critical components of our model: heterogeneity and extra utility of the premium status.

3.1. Model of Consumer Heterogeneity We first restrict our attention to the case of flying without a premium status; the impact of the Gold status on utility is considered in the next subsection. Forward-looking behavior in our setting implies that a consumer has a list of trips/flights (s)he would like to take over some time period6. (S)he then sorts this list from the highest-utility flight to the lowest – most important to the least important – and going down the list decides on the number of flights to take (equivalently, which flight to take) in order to maximize the total utility. Intuitively, the marginal utility of a flight decreases in the number of flights already taken, leading to satiation. The satiation point, however, is naturally different among consumers due to varying travel needs and the importance of travel. To capture this logic, we assume that consumers are heterogeneous with respect to the satiation th parameter ai so that consumer i’s marginal utility of the (n + 1) flight is ai − n: the marginal

4 Since the firm only sells a single product, all flights are of the same distance, which implies that miles and segments are interchangeable. To avoid duplication, we will simply call such a program “mileage-based.” 5 This reflects the cost to provide status benefits, including handling free-checked baggages, food/drinks/services in lounges, dedicated customer service agents, etc. 6 For simplicity, we assume this period equals the qualification period for Gold status. A typical qualification period is one year, and one might argue that consumers are not looking that far ahead. In that case, our model can be modified and corresponds to the flights in the shorter period (for example, in the last 3 months of the year with the qualification threshold adjusted for the remaining qualification balance). Also, in practice, crossing the qualification threshold in one period leads to Gold status in the next, but, for simplicity, we assume that the system is in a steady state where the consumers’ recently observed behaviors will continue into the future. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 9

utility is decreasing in the number of flights n, and at n = ai flights, the marginal utility becomes zero. Note that parameter ai can be also interpreted as consumer i’s willingness-to-pay (WTP) for the most important flight. Then the total utility from taking the first n flights from consumer-i’s list equals: Z n n2 Ui(n) = (ai − ω) dω = ain − . (1) 0 2 We note that this quadratic utility structure is widely used when modeling multiple product uses, and is has been empirically calibrated/verified in various applications (see, e.g., Lambrecht et al. 2007, Economides et al. 2008, Kim et al. 2010, Gilbert and Jonnalagedda 2011.) The firm charges price p per flight and the consumer-i surplus from taking n flights equals:

Surplusi,R(n) = Ui(n) − pn, (2) where the subscript R refers to ﬂying “as needed” with a regular status. Consumer-i determines

∗ 7 the optimal number of ﬂights to take, ni,R, by solving maxn{Surplusi,R(n)}, which leads to :

∗ ni,R = max[0, (ai − p)]. (3)

3.2. Model of Premium Status (Gold) Utility The firm introduces a loyalty program and rewards some consumers with a premium (Gold) status. The Gold status comes with many benefits, such as dedicated customer service, priority check-in and boarding, the ability to check multiple and heavy bags with priority tags, complementary or nominally priced upgrades, and access to lounges8 – all of which undoubtedly provide extra utility. To capture this utility, we introduce parameter g ≥ 0, which represents the benefit of flying with the Gold status. In practice, many of the status benefits have a fixed (purchase) price9, and we therefore assume that the value of the status benefit is added to ai, which increases Gold consumers’

WTP to ai + g. The surplus of consumer i who ﬂies with the Gold status (denoted by a subscript ‘G’) then becomes: n2 Surplus (n) = (a + g − p)n − (4) i,G i 2

7 We treat the “number of flights” as a continuous variable. In practice, the number of flights needed to qualify for Gold is quite large, typically 60; hence, a consumer who may strategically decide to fly-up is, arguably, taking many flights. Therefore, approximating such a discrete, but rather large number, with a continuous variable is reasonable. 8 For example, see http://www.staralliance.com/en/benefits/status-benefits/ for the benefits of Star Alliance Gold (United, Air Canada, Lufthansa, Air China) A nearly identical set benefits exist for other programs, such as Sky Team Elite (Delta, Air France, Aeroflot) and One World Emerald (American, British Airways, Cathay Pacific). 9 For example, United Airlines offers expedited screening and priority boarding through Premier Access, which costs $15 on most domestic flights https://www.united.com/CMS/en-US/products/travelproducts/Pages/ PremierAccess.aspx, and United Club access can be purchased for $50 https://www.united.com/web/en-US/ content/travel/airport/lounge/rates.aspx. We can find many similar examples. 10

This is consistent with abundant empirical evidence that consumers with status are willing to pay more for travel. Mathies and Gudergan(2012) (using lab experiments) and McCaughey and Behrens(2011) and Brunger(2013) (using secondary data) demonstrated that premium-status consumers have a higher WTP even after controlling for the eﬀects of trip purpose and time of booking.

∗ Under this model, a Gold consumer determines the number of ﬂights to take, ni,G, by solving the optimization problem maxn{Surplusi,G(n)}, which leads to:

∗ ni,G = max[0, ai − p + g]. (5)

Comparing this equation with (3), notice that the optimal number of ﬂights increases if the consumer obtains Gold status – an important observation driven by the increased WTP10.

4. Mileage-based and Spending-based Programs We first formulate and solve the consumer and the firms’ problems for the two “basic” program designs and then compare the resultant solutions and discuss managerial intuition. Specifically, we seek to understand the impact of the switch from a mileage-based to a spending-based program in light of strategic consumer behavior. We then extend our analysis and consider “enhanced” mileage-based designs (with the selling of miles and with bonus miles features) in Section5.

4.1. Mileage-based Program Under a mileage-based program design, there is no benefit for a consumer to pay a higher price p+ because the high price lowers the surplus without contributing to the Gold utility. Therefore, a customer who takes G or more flights with the regular price p qualifies for Gold status. That is,

∗ consumers with ni,G ≥ G qualify for the Gold status automatically, simply by ﬂying “as needed.” However, those are not the only consumers who will qualify for Gold: a consumer may strategically

∗ fly-up to Gold by taking G > ni,R flights. The additional flights will have a negative surplus, but this may be beneficial if the total surplus with the Gold status exceeds the total surplus of taking

∗ ni,R ﬂights with the regular status. This strategic behavior is characterized by Lemma1 and is illustrated in Figure1 (all proofs are in the Appendix):

10 We note that this additive formulation is certainly not the only possible way to model the benefit of the Gold status; in fact, while working on the paper, we also considered a multiplicative surplus model with Surplusi,G(n) = n2 (ai − p)n − 2 × g for g ≥ 1 and obtained qualitatively similar results. The main difference between the two models is that with the multiplicative model, obtaining a status does not increase the optimal number of flights – i.e., status does not increase WTP (mathematically, the surplus at the satiation point is still zero if multiplied by g). Calibration of different Gold utility formulations is an interesting future research question. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 11

1 2 3 Lemma 1. Under a mileage-based program, there exist up to three thresholds a¯M ≤ a¯M ≤ a¯M , j which lead to up to four segments (AM , j = I,...,IV ) of consumer behaviors: I 1 AM : If 0 ≤ ai ≤ a¯M , then it is optimal not to fly. II 1 2 AM : If a¯M ≤ ai ≤ a¯M , then it is optimal to fly as needed, without Gold status. III 2 3 AM : If a¯M ≤ ai ≤ a¯M , then it is optimal to fly-up to Gold. IV 3 AM : If a¯M ≤ ai, then it is optimal to fly as needed (and automatically qualify for Gold.)

# of flights:

n=0 n=ai -p n=G n=ai -p+g

I II III IV Demand segment: AM AM AM AM Fly as needed Fly regular Fly-up Action: Do not fly and qualify as needed to Gold for Gold

ai 1 2 Ö2 3 aM =p aM =p+G- gG aM =p+G-g Figure 1 Strategic consumer behavior under the mileage-based loyalty program.

This Lemma implies that depending on ai there could be up to three different optimal customer behaviors: do not fly, fly as needed (some will qualify for Gold) and, most importantly, fly-up to Gold. The latter corresponds to the mileage-runners – consumers who strategically fly more than is needed in order to qualify for the Gold status. We emphasize that mileage-running is optimal for these consumers and is a result of utility-maximizing behavior. Their total surplus is higher with mileage runs (because they obtain Gold status), even though some flights they take have a negative surplus. Next, we examine consumers’ response to changes in Gold qualification threshold and benefit.

I,...IV Corollary 1. If all demand segments AM are non-empty (as depicted on Figure1) then: 2 3 3 2 II III (i) a¯M and a¯M increase in G, and a¯M increases faster than a¯M . That is, segments AM and AM IV expand and segment AM shrinks in G. 2 2 2 II (ii) a¯M , a¯M decrease in g; more so, a¯M decreases faster. That is, segment AM shrinks and seg- III IV ments AM ,AM expand in g.

This result shows that the total number of consumers who qualify for Gold decreases when the status qualiﬁcation requirement becomes more stringent, and increases when the Gold status becomes

III more valuable – both agree with business intuition. Because AM expands while AM IV shrinks, the proportion of strategic Golds increases in qualification requirement. Under the mileage-based 12 program, strategic consumers fly-up (mileage-run), so if the firm makes it harder to qualify for Gold, then the proportion of mileage runners increases. This points to a distinctive feature of the mileage-based design; later, we will show that this is not the case under other designs. The firm’s problem under the mileage-based design is to select the qualification requirement G

+ and the regular price pM (as argued above, there is no demand at high price p ) that maximize its profit: " # X j III IV max πM (pM ,GM ) = max pM × DM − c × DM + DM , (6) pM ,GM pM ,GM j=II,...IV II R ∗ III R where DM = II ni,RdF (ai) is the demand from the fly-regular segment, DM = III GdF (ai) is AM AM IV R ∗ the demand from the strategic Gold (fly-up) segment, DM = IV ni,GdF (ai) is the demand from AM auto-Golds as per Lemma1, and F (ai) denotes the CDF of the distribution of ais. Note that the II second term excludes DM since the consumers in that segment are flying regular and thus the firm does not incur the status-related cost for servicing them.

4.2. Spending-based Program Under the spending-based loyalty program, a consumer who spends S or more dollars with the

firm qualifies for Gold status. That is, consumers with n∗i,G ×p ≥ S qualify automatically by flying as needed, but there may be other consumers who may strategically spend-up to Gold, i.e., they may spend more than their flying need, to obtain or maintain the Gold status. Different from the mileage-based program case, under the spending-based program, there are three different ways to spend-up. The first way is simply to fly more at the regular price p; this is analogous to flying-up (mileage-run) under the mileage-based program. The second way, – and this might be the main premise of the spending-based loyalty program design – is to choose to pay p+ for (fewer) flights instead of taking (more) flights at a price p. We emphasize that the product/service a consumer receives is identical: that is, paying fare p+ is seemingly “stupid” – why would someone pay more for exactly the same product/service? Below, we discuss how such pay-up behavior could indeed be optimal for some consumers. The third way is to mix the two. Let n, n+ denote the number of flights taken at prices p, p+, respectively. From (4), these solve:

+ 2 + (n + n ) + + + + max (ai + g)(n + n ) − − (pn + p n ), s.t. pn + p n ≥ S, (7) n,n+∈R+ 2 which is maximized at:

−S + (a + g)p+ S − (a + g)p n∗ = max[0, i ], and n+∗ = max[0, i ], (8) i,S p+ − p i,S p+ − p

∗ + +∗ where ni,S ≥ 0 if ai ≥ S/p − g and ni,S ≥ 0 if ai ≤ S/p − g; if only one of these two conditions is satisﬁed, the optimal solution is (0, S/p+) and (S/p, 0), respectively. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 13

+∗ ∗ Observe that the number of flights taken at a high (regular) price, ni,S (ni,S), increases (decreases) in S and decreases (increases) in ai and g. For those consumers who attempt to strategically qualify for Gold, a higher qualification spending requirement makes purchasing higher-fare flights more attractive, and makes purchasing low-fare flights less attractive. This is because in order to spend- up to Gold, consumers can fly less if they purchase higher-fare flights; however, they need to fly more if they purchase regular-fare flights. Since these additional flights have increasingly negative surplus, the intuition for this result follows. As in the case with the mileage-based program, consumers solve (7) and compare the optimal

2 surplus with that of ﬂying regular, (ai − p) /2, and that determines whether it is optimal to strategically spend-up to Gold. The following Lemma summarizes strategic consumer behavior under the spending-based program:

1 2 Lemma 2. Under a spending-based loyalty program there exist up to five thresholds, a¯S ≤ a¯S ≤ 3 4 5 j a¯S ≤ a¯S ≤ a¯S, which lead to up to six segments (AS, j = I,II...,VI) of consumer behavior: I 1 AS: For 0 ≤ ai ≤ a¯S, it is optimal not to fly. II 1 2 AS : For a¯S ≤ ai ≤ a¯S, it is optimal to fly as needed at price p without Gold status. III 2 3 ∗ +∗ AS : For a¯S ≤ ai ≤ a¯S, it is optimal to spend-up to Gold with ni,S = 0, ni,S > 0 (pay-up). IV 3 4 ∗ +∗ AS : For a¯S ≤ ai ≤ a¯S, it is optimal to spend-up to Gold with ni,S > 0, ni,S > 0 (mix). V 4 5 ∗ +∗ AS : For a¯S ≤ ai ≤ a¯S, it is optimal to spend-up to Gold with ni,S > 0, ni,S = 0 (fly-up). VI 5 AS : For a¯S ≤ ai, it is optimal to fly as needed at price p and automatically qualify for Gold.

n=0 n>0 n>0 n >0+ n+ >0 n+ =0

# of flights:

+ + n=0 n=ai -p np+n p =S n=ai -p+g

I II III IV V VI Demand segment: AS AS AS AS AS AS Fly as needed Fly regular Spend-up Action: Do not fly and qualify as needed to Gold for Gold

1 2 3 4 5 aS =p aS aS aS aS =p+S/p-g Figure 2 Strategic consumer behavior under the spending-based loyalty program.

I II VI Figure2 illustrates this Lemma. Segments AS,AS , and AS are identical to the equivalent I II IV III V segments AM ,AM , and AM for a mileage-based program. However, segments AS − AS are new and reflect the three different ways to spend-up: “pay-up” – take n+∗ flights at a high price p+; “fly-up” – take n∗ flights a regular price p; or “mix” – take a mixture of flights, n∗ at price p and 14

+∗ + n at p . Observe from Lemma2 and Figure2 that consumers who pay-up have the lowest ai among those who spend-up. This is because should they decide to fly-up, they need to take the largest number of “unnecessary” flights; hence, the option to pay-up is most valuable for them. Next, we examine consumers’ response to changes in Gold qualification threshold and benefit.

II,...V I Corollary 2. If all demand segments AS are non-empty (as depicted in Figure2) then: 4 5 3 2 (i) a¯S and a¯S increase in S at the same rate; a¯S increases slower, and a¯S even slower. That is, VI V II IV segment AS shrinks, segment AS is unaﬀected, while segments AS ...AS expand in S. 3 4 5 2 II (ii) a¯S, a¯S, and a¯S decrease in g at the same rate, and a¯S decreases faster. That is, segment AS III VI IV V shrinks while AS and AS expand in g. AS and AS are unaﬀected.

Similar to the mileage-based program, the total number of consumers who qualify for Gold decreases when the status qualification requirement becomes more stringent and increases when the Gold status becomes more valuable. Further, the proportion of strategic Golds increases. Different from the mileage-based program case, however, under the spending-based program, an increase in qualification primarily translates into high-price purchases and not in mileage-running: the proportion of those who purchase at least some flights at high price p+ increases and the proportion of those who purchase all flights at p+ increases the fastest, while the number of mileage-runners is not affected. This relates to the important difference between the mileage- and the spending-based programs that we explore in more detail in Section 4.3.

The ﬁrm’s problem under the spending-based design is to select the regular price pS, high price + pS , and the spending requirement S to maximize its proﬁt: " Z # + II VI X j max πS(pS, pS ,S) = max pS × DS + DS + S × dF (ai) − c × DS , (9) p ,p+,S p ,p+,S III...V S S S S AS j=III,...V I

II R ∗ III S R where DS = AII ni,RdF (ai) is the demand from the fly-regular segment, DS = + AIII dF (ai), S pS S IV R ∗ +∗ V S R DS = IV ni,S + ni,S dF (ai) and DS = V dF (ai) are the demands from the three strategic AS pS AS VI R ∗ Gold segments, and DS = VI ni,GdF (ai) is the demand from auto-Golds as per Lemma2. Note AS III V that the second term in (9) reflects the fact that each customer in segments AS ...AS spends S II with the firm, and the third term reflects that consumers in segment AS are flying regular and thus the firm does not incur status-related costs for servicing them.

4.3. Comparison of Mileage-based and Spending-based Programs Proposition 1. The optimal proﬁt under the spending-based program is higher than under the mileage-based program.

This proposition implies that by switching from a mileage-based program to a spending-based program, the firm can increase its profit, which might explain why some firms are making changes Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 15 to their program designs, as we discussed in the Introduction. While one may anticipate this result from a structural point of view (the spending-based program has one more effective decision + variable, pS ), the driving force behind this profit increase is related to strategic consumer behavior. To elaborate on this further, we compare consumer segments under the two programs with pM = pS and S = GM × pM (i.e., so that the regular prices and effective Gold qualifications are the same). In this case, it is not difficult to verify from the proof of Corollaries1 and2 that

3 5 2 4 2 2 a¯M =a ¯S, a¯M ≤ a¯S anda ¯M ≥ a¯S, as depicted in Figure3.

1 2 3 aM =p aM aM =p+G-g

Mileage: ai Fly-up

Pay-up

Spending: ai Mix

1 2 3 4 5 aS =p aS aS aS aS =p+S/p-g Figure 3 Schematic comparison of strategic consumer behavior under mileage- versus spending-based designs.

As shown, the spending-based program affects consumers in two different ways. First, there are fewer mileage-runners under the spending-based program. Some consumers substitute flying-up with paying-up and this is crucial: flying-up under the mileage-based program increases the firm’s revenue, but also costs, while paying-up under the spending-based program results in the same revenue (because S = G × p in this comparison), but from fewer flights, which decreases costs. This is naturally more profitable. Second, the total strategic segment expands under the spending-based program; that is, some consumers who opted to fly regular under the mileage-based program find it beneficial to strategically qualify for Gold under the spending-based design. This increases both + the revenue and the cost, and the firm sets the optimal pS > pS so as to benefit from the increased number of strategic consumers. The bottom-line of this discussion is that the firm would prefer a spending-based rather than a mileage-based program. In this sense, the changes in the loyalty program design discussed in the Introduction are natural: firms switch to spending-based programs because they are more profitable. On the other hand, as is also mentioned in the Introduction, this switch raised significant attention in the media, and invited generally negative public reactions. This calls for another question: do consumers also benefit from the spending-based program? The following analyses provide further insight into the potential benefits of the spending-based program design.

Proposition 2. Spending-based loyalty programs can either maintain the ﬁrm’s proﬁt and surplus of each and every consumer at the same level as under the optimal mileage-based program, or can improve them, thus leading to a Pareto improvement over the optimal mileage-based program. 16

This proposition implies that a spending-based loyalty program could lead to a win–win situation for the firm and its consumers: by switching from a mileage-based to a spending-based program, the firm can increase its profit and increase surplus for each and every consumer (both regular and Gold customers). We emphasize “can” because Proposition2 refers to a feasible spending-based program solution, not necessarily one in which the firm maximizes its profit. To achieve such a Pareto-improving solution the firm might need to sacrifice some of the profit improvement from the spending-based program, so that its profit is above the mileage-based optimum, but below the spending-based optimum. Under the optimal solution, the firm can clearly benefit even more, but some consumers might suffer. To see why, note that if pM = pS and S = GM × pM , as assumed above, the “effective” qualifications are identical. But if pS < pM , then the number of flights that must be taken at regular price to qualify for Gold S/pS will exceed G. It would therefore result in a further decrease in mileage-running even if the spending requirement, S, is intact. Because some of the flights taken by mileage-runners, by definition, have a negative surplus, a lower pS implies that more such flights must be taken, which has an even larger total negative impact on the surplus of those who were flying-up to Gold; some of these consumers might even lose their status. In other words, by making mileage-running harder, the optimal spending-based program improves the firm’s profitability, but it may decrease the surplus for some consumers. Thus, the negative public reactions reactions mentioned in the Introduction are justified: some consumers may be worse off if the firm changes its loyalty program design. This discussion naturally leads to the following questions. If the firm wants to avoid negative media attention and consumer backlash against the drastic switch to the spending-based program, what are the alternatives? Specifically, could the firm improve its profit without a fundamental change in its loyalty program structure? The answer is “yes,” as we discuss next.

5. Enhanced Mileage-based Loyalty Program Designs As discussed in the Introduction, some ﬁrms have enhanced their otherwise mileage-based programs. Next we consider two such enhanced designs: mileage-based programs with selling of miles (SOM) and with bonus miles (BM). The goal of this analysis is to understand if there are ways to improve proﬁtability to the spending-based optimum, or even beyond, without drastic changes to the loyalty program design.

5.1. Enhanced Mileage-based Program With Selling of Miles (SOM) One way to enrich a mileage-based program is to allow consumers to purchase Gold “status qualifying” credits/miles/segments at price q in addition to awarding them for flying at price p. As Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 17 under the regular mileage-based program, consumers who take G or more flights by flying “as needed” qualify for Gold automatically. However, there may be other consumers who qualify for Gold strategically either by flying more (fly-up) at price p or by purchasing qualifying credits/miles at q (buy-up), or by mixing the two. Let n and nq denote the number of flights taken at price p and the miles purchased at price q, respectively. Then, consumers solve the following optimization problem: 2 n q q max (ai + g)n − − (pn + qn ), s.t. n + n ≥ G, (10) n,nq∈R+ 2 and obtain the optimal solutions:

∗ q∗ ∗ ni,SOM = ai + g − p + q, and ni,SOM = G − ni,SOM = G − ai − g + p − q. (11)

Observe that the number of Gold qualification miles/credits purchased by consumer i decreases in g and increases in G. When g is high, flying with Gold status brings more utility; hence, consumers who choose to qualify for Gold strategically naturally prefer flying (and earning) more and buying less. Similarly, when G is high, consumers need more miles to qualify for Gold, and so they purchase more. Both effects agree with business intuition. Consumers compare the optimal surplus from substituting (11) into (10) with that of flying

2 regular, (ai − p) /2, and that determines if it is optimal to strategically qualify for Gold under the SOM design as summarized in the following Lemma:

Lemma 3. Under the enhanced mileage-based loyalty program with selling of miles (SOM), there 1 2 3 4 j exist up to four thresholds: a¯SOM ≤ a¯SOM ≤ a¯SOM ≤ a¯SOM , which lead to up to ﬁve segments (ASOM , j = I,II...,V ) of consumer behavior:

I 1 ASOM : If 0 ≤ ai ≤ a¯SOM , then it is optimal not to fly. II 1 2 ASOM : If a¯SOM ≤ ai ≤ a¯SOM , then it is optimal to fly as needed, without Gold status. III 2 3 q∗ ASOM : If a¯SOM ≤ ai ≤ a¯SOM , then it is optimal to mix fly-up and buy-up to Gold, ni,SOM > 0. IV 3 4 q∗ ASOM : If a¯SOM ≤ ai ≤ a¯SOM , then it is optimal to fly-up to Gold, ni,SOM = 0. V 4 ASOM : If a¯SOM ≤ ai, then it is optimal to fly as needed and automatically qualify for Gold.

I II IV V Figure4 illustrates the Lemma. Segments ASOM ,ASOM , ASOM , and ASOM correspond to the I IV equivalent segments AM − AM for a mileage-based program presented in Section4. However, III q∗ segment ASOM is new and reﬂects the option of buying ni,SOM qualifying miles.

It is interesting to note that from (11), ni,SOM > ni,R. That is, no consumer would ﬂy as needed and buy some miles on top: those who decide to buy miles also adjust their ﬂying pattern and

2 ﬂy more; see a discontinuity ata ¯SOM in Figure4. Observe also that the possibility to buy miles is most valuable for consumers with low ai; for them, ﬂying-up requires taking a large number 18

n>0 n>0 n>0q n=0q

# of flights:

q n=0 n=ai -p n+n=G n=ai -p+g

I II III IV V Demand segment: ASOM ASOM ASOM ASOM ASOM Fly+buy Fly as needed Fly regular Fly-up Action: Do not fly miles and qualify as needed to Gold to Gold for Gold

1 2 3 4 aSOM =p aSOM aSOM aSOM =p+G-g Figure 4 Strategic consumer behavior under the mileage-based loyalty program with SOM. of “unnecessary” ﬂights. This is similar to the pay-up option under the spending-based program, which was most beneﬁcial for the consumers with low ai. The following corollary describes the comparative statics for consumer behavior thresholds under the SOM design:

I,...V Corollary 3. If all demand segments ASOM are non-empty (as depicted in Figure4), then: 3 4 2 (i) a¯SOM and a¯SOM increase in G at the same rate, and a¯SOM increases slower. That is, segment V IV II III ASOM shrinks, segment ASOM is unaﬀected, while segments ASOM and ASOM expand in G. 3 4 2 (ii) a¯SOM and a¯SOM decrease in g at the same rate, and a¯SOM decreases faster. That is, segments III V IV II ASOM and ASOM expand, segment ASOM is unaﬀected, and segment ASOM shrinks in g.

From part (i), when the Gold qualification is more stringent (higher G), the total number of Gold customers decreases, which is intuitive. However, the proportion of strategic Gold customers increases, among which, the proportion of those who fly and buy miles increases faster. An increase in qualification therefore translates primarily in the increase in miles purchased, while the number of mileage-runners does not change. From part (ii), the total number of consumers who qualify for Gold increases when the status becomes more valuable; furthermore, the increase mainly comes from consumers who fly and purchase miles. It is interesting to note that although SOM is a mileage- based design, these effects are similar to those observed under the spending-based program, where an increase in the qualification threshold motivates consumers to pay more and not mileage-run. The firm’s profit maximization problem under the SOM design is given by: " X j max πSOM (pSOM , q, GSOM ) = max pSOM × DSOM p ,q,G p ,q,G SOM SOM SOM SOM j=II,...V Z # q∗ X j +q × ni,SOM dF (ai) − c × DSOM , (12) III ASOM j=III,...V Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 19

II R ∗ III where DSOM = II ni,RdF (ai) is the demand from the fly-regular segment, DSOM = ASOM R ∗ IV R III ni,SOM dF (ai) is the demand from the buy-up segment, DSOM = IV GdF (ai) is the ASOM ASOM R ∗ demand from the fly-up segment, and DV = V ni,GdF (ai) is the demand from auto-Golds. Note ASOM III that the second term reflects the additional revenue from the ASOM through the selling of miles II and the third terms omits ASOM because these consumers fly without the Gold status.

5.2. Enhanced Mileage-based Program With Bonus Miles (BM) Another way to enrich a mileage-based program is to award 1 Gold mile/credit for a ticket purchased at price p, but award B > 1 miles for a ticket purchased at pB > p. As before, consumers who accumulate G or more miles qualify for Gold; some may do so by flying “as needed” and qualify automatically, while others do so strategically. With the Bonus design, miles can be accumulated by flying more at price p (fly-up) or by purchasing more expensive tickets at price pB (buy-up), or by mixing the two. Let n and nB denote the number of flights taken at prices p and pB, respectively. Then, consumers solve the following optimization problem:

B 2 B (n + n ) B B B max (ai + g)(n + n ) − − (pn + p n ), s.t. n + B × n ≥ G, (13) n,nB ∈R+ 2 and obtain the following solutions:

(pB − p) − (B − 1)(a − p + g − G) (pB − p) − (B − 1)(a − p + g − G) n∗ = G − B i and nB∗ = i . i,BM (B − 1)2 i,BM (B − 1)2 (14) Observe again that the number of Bonus miles purchased decreases in g and increases in G. When g is high, flying brings more utility; hence, consumers prefer to fly more. But when G is high, buying expensive tickets so as to earn Bonus miles becomes more attractive to strategically qualify for Gold, which agrees with business intuition. Consumers compare the optimal surplus from substituting (14) into (13) with that of flying

2 regular, (ai − p) /2, and that determines if it is optimal to strategically qualify for Gold under the Bonus miles design, as summarized in the following Lemma:

Lemma 4. Under the enhanced mileage-based loyalty program with bonus miles (BM), there exist 1 2 3 4 5 j up to ﬁve thresholds, a¯BM ≤ a¯BM ≤ a¯BM ≤ a¯BM ≤ a¯BM , which lead to up to six segments (ABM , j = I,II...,VI) of consumer behavior:

I 1 ABM : If 0 ≤ ai ≤ a¯BM , then it is optimal not to fly. II 1 2 ABM : If a¯BM ≤ ai ≤ a¯BM , then it is optimal to fly as needed at price p without Gold status. III 2 3 ∗ +∗ ABM : If a¯BM ≤ ai ≤ a¯BM , then it is optimal to buy-up to Gold with ni,BM = 0, ni,BM > 0 (pay-up). IV 3 4 ∗ +∗ ABM : If a¯BM ≤ ai ≤ a¯BM , then it is optimal to buy-up to Gold with ni,BM > 0, ni,BM > 0 (mix). V 4 5 ∗ B∗ ABM : If a¯BM ≤ ai ≤ a¯BM , then it is optimal to buy-up to Gold with ni,BM > 0, ni,BM = 0 (fly-up). 20

n=0 n>0 n>0 n >0B nB >0 nB =0

# of flights:

B n=0 n=ai -p n+Bn =G n=ai -p+g

I II III IV V VI Demand segment: ABM ABM ABM ABM ABM ABM Fly as needed Fly regular Buy-up Action: Do not fly and qualify as needed to Gold for Gold

1 2 3 4 5 aBM =p aBM aBM aBM aBM = p+G-g Figure 5 Strategic consumer behavior under the mileage-based loyalty program with Bonus miles.

VI 5 ABM : If a¯BM ≤ ai, then it is optimal to ﬂy as needed at price p and automatically qualify for Gold.

I II V VI Figure5 illustrates the Lemma. Segments ABM ,ABM , ABM and ABM correspond to the equiv- I IV alent segments AM − AM for a mileage-based program presented in Section4. However, segments III IV ABM and ABM are new and reflect consumers who benefit from the BM offer. Similar to the B∗ B spending-based program, these consumers either take all their ni,B flights at the high price p and get bonus miles on every flight, or take a mixture of regular- and high-priced flights. In either case, these consumers take fewer than G flights and qualify for Gold due to the bonus miles they collect.

Observe from Lemma4 and Figure5 that consumers who buy-up have the lowest ai among those who strategically obtain status. This is because, should they decide to ﬂy-up, they need to take the largest number of “unnecessary” ﬂights; hence, the option to buy-up is most valuable for those customers (this is, again, similar to the spending and SOM designs analyzed earlier). The following corollary describes the comparative statics for consumer behavior thresholds under the BM design:

I,...V I Corollary 4. If all demand segments AB are non-empty (as depicted on Figure5), then: 4 5 3 2 (i) a¯B and a¯B increase in G at the same rate, a¯B increases slower, and a¯B increases even slower. VI V II...IV That is, segment AB shrinks, segments AB is unaﬀected, while segments AB expand in G. 3 5 2 III (ii) a¯B...a¯B decrease in g at the same rate, and a¯B decreases faster. That is, segments AB and VI IV V II AB expands in g, segments AB and AB are unaﬀected, and segment AB shrinks.

From part (i), when the Gold qualification is more stringent (higher G), the total number of Gold customers decreases, which is intuitive. However, the proportion of strategic Gold customers increases and the proportion of those who only fly with the bonus-miles flights (i.e., those who only purchase high-priced tickets) increases fastest; the number of mileage-runners is not affected. That is, an increase in qualification thus translates primarily in the increase in the high-price flights purchased, which is good news for the firm. From part (ii), the total number of consumers who Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 21 qualify for Gold increases when status becomes more valuable. Furthermore, the increase in status benefit mainly affects customers who only purchase high-priced tickets. These effects are similar to the spending-based designs which is, again, interesting because the BM design is a mileage-based program, yet it affects consumer behavior effectively as a spending-based program. The profit maximization problem of the firm under the bonus design is given by: " Z ! B II V ∗ VI max πBM (pBM , p ,B,GBM ) = max pBM × DBM + DBM + ni,BM dF (ai) + DBM B B p ,p ,B,G pBM ,p ,B,GBM IV BM BM ABM Z ! B III B∗ + p × DBM + ni,BM dF (ai) IV ABM # X j − c × DBM , (15) j=III,...V I

II R ∗ III where DBM = II ni,RdF (ai) is the demand from the regular segment, DBM = ABM R GBM IV R ∗ B∗ V R III dF (ai), DBM = IV ni,BM + ni,BM dF (ai) and DBM = V GdF (ai) are the demands ABM B ABM ABM VI R ∗ from the three strategic Gold segments, and DBM = VI ni,GdF (ai) is the demand from auto- ABM II Golds. Note that the second term reﬂects the bonus miles option and the third excludes ABM as these customers ﬂy without the Gold status.

5.3. Comparison of Enhanced Mileage-based and Spending-based Programs

For the following proposition, we assume a two-point distribution F (ai) with aH (H-type consumer) and aL (L-type consumer) for analytical tractability. Numerical simulations in Section7 conﬁrm that the same result holds for more general distributions.

Proposition 3. The optimal proﬁt under the enchanted mileage-based designs with selling of miles or bonus miles is higher than under the spending-based program.

The implication of this proposition is profound: by enhancing its mileage-based program, the firm can obtain an even higher profit than from changing it to a spending-based design. This perhaps explains why some firms mentioned in the Introduction enhanced their otherwise mileage-based programs with features like the ones we considered. To understand the drivers for higher profit, we next consider the underlying consumer behavior in more detail.

5.3.1. SOM versus Spending-based programs. To build intuition for the profit increase under the SOM design, consider consumer behavior in the case where pS = pSOM and S = GSOM × pSOM , i.e., when the regular prices and effective Gold qualifications are the same.

If we further assume that pS = pSOM = q, by comparing the two sets of thresholds from the 2 2 4 3 proofs of Corollaries2 and3, it is not diﬃcult to establish that ¯ aS ≥ a¯SOM and aS = aSOM . That is, the SOM design keeps the same number of mileage runners as the spending-based program, 22

but it converts some of the regular customers into strategic mile-purchasers. However, if q < pS = pSOM = q (as is supported analytically in Proposition3 and numerically in Section7), then the SOM design further converts some of the mileage-runners under the spending-based program into miles purchasers. See Figure6 for an illustration.

1 2 3 4 5 aS =p aS aS aS aS =p+S/p-g

Spending: ai Fly-up Pay-up Mix

Selling of miles: ai Buy-up (SOM)

1 2 3 4 aSOM =p aSOM aSOM aSOM =p+G-g Figure 6 Schematic comparison of strategic consumer behavior in the SOM vs. spending-based designs.

Theretofore, SOM impacts consumer behavior in three ways. First, it even further decreases the number of mileage-runners. Second, its converts consumers who pay-up (purchase high-priced tickets and fly) under the spending-based program to buying miles. Both of these effects reduce costs. Finally, SOM further expands the region of strategic Gold behavior: some consumers who did not find it worthwhile to spend-up to Gold under the spending-based program choose to fly/buy-up + to Gold under SOM. Similar to pS under the spending-based program, q controls the revenue–cost trade-off from the strategic consumer behavior; thus, the firm can benefit from a larger number of consumers who strategically qualify for Gold.

5.3.2. Bonus miles versus Spending-based programs. To compare the BM and spending- + + based designs, similarly to SOM, consider a case where pS = pB, pS = pB, and S = GB × pB, i.e., + the prices and qualiﬁcations are the same. Structurally, if we further assume B = pS /pS, the two + designs become identical. However, if B > pS /pS (as is supported analytically in Proposition3 and 2 numerically in Section7), then from the proof of Corollary4, ¯ aB decreases (shifts to the left) and 4 a¯B increases (shifts to the right), as shown in Figure7.

1 2 3 4 5 aS =p aS aS aS aS =p+S/p-g

Spending: ai Fly-up

Pay-up (S, BM)

Bonus miles: ai Mix (S, BM)

1 2 3 4 5 aB =p aB aB aB aB = p+G-g Figure 7 Schematic comparison of strategic consumer behavior in the Bonus vs. spending-based designs.

The comparison between the bonus miles and spending-based designs is therefore similar to the SOM case. The notable difference, however, is that while cost reduction is more pronounced under SOM, both revenue increase and cost reduction effects are prominent under bonus miles. The BM Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 23 design further reduces the number of mileage-runners by converting them to high-priced ticket purchasers. Those consumers contribute to revenue increases while decreasing the cost since they possibly fly less. That is, the BM design encourages buy-up even more than the spending-based design; in this sense, it could be also considered as an enhanced spending-based design even though it is based on the mileage-based program. As before, the BM design enjoys the expanded region of strategic Gold consumers by controlling the bonus parameter B. The bottom line of this discussion is that the enhanced mileage-based program with the selling of miles or bonus miles features could be more profitable than the optimal spending-based program. Comparing the two enhanced designs to each other did not reveal a clear pattern or profit ordering; this is also confirmed by the numerical simulations in Section7. The implication, though, is clear: rather than making a drastic switch to spending-based programs, firms could instead enhance their mileage-based programs and obtain an even higher profit. We lastly note that by following the same logic as in the proof of Proposition2 both enhanced designs could be Parteo-improving over the optimal mileage-based program. However, comparing the optimal enhanced designs and the optimal spending-based design did not reveal a clear pattern. While in some cases, Pareto improvement is observed at optimality, it is not always guaranteed, as shown in Section7.

6. Implications of Strategic Consumer Behavior To conclude the analytical part of our paper, we present a result that transcends all the designs we considered. Specifically, while we considered how strategic consumer behavior changes under different loyalty programs, we have not yet examined whether such a behavior is beneficial in the first place. To do so, consider a case where consumers are not strategic; that is, they optimize instantaneous surplus and do not look forward to the benefits of Gold status nor do they alter their behavior to obtain status – i.e., they act myopically. Myopic behavior implies two things. First, no consumer will purchase anything other than flights at price p because doing so decreases instantaneous surplus. Thus, all program designs we considered become equivalent, and it is sufficient to only consider a mileage-based program. Second, only those consumers with ai − p + g ≥ G will qualify for Gold. As in the case with Proposition3, for the following result we assume a two-point distribution

F (ai) with ai ∈ {aL, aH } for analytical tractability. Numerical simulations in Section7 conﬁrm that the same result holds for more general distributions.

Proposition 4. The ﬁrm’s proﬁt when consumers are strategic is higher than when consumers are myopic. 24

This result offers two insights. First, it shows that when consumers are strategic, by coordinating its revenue management and loyalty programs, the firm can increase its profit – i.e., it can benefit from strategic consumer behavior. This result is particularly interesting because in many studies, strategic consumer behavior hurts the firm’s revenue, e.g., see Aviv and Pazgal(2008), Levin et al. (2009), Ovchinnikov and Milner(2012). To the best of our knowledge, there are very few papers that identified the mechanisms for benefiting from strategic consumers, e.g., fixed costs in Marinesi and Netessine(2014), and flight passes in Wang et al.(2015). We show that the loyalty program is another such mechanism. As argued in the Introduction, it is generally not obvious that an increase in demand from strategic consumers will result in increased profit because strategic behavior also increases costs. Our results, however, show that by optimally coordinating the RM and loyalty program, the additional revenue is necessarily larger than the additional costs, which leads to higher profit. In this sense, our work also provides a new insight into the sources of loyalty programs’ value in a complementary manner: while traditionally viewed as means of softening competition (e.g., Kim and Srinivasan 2001, Kim et al. 2004), we show that even without competition loyalty programs can be valuable because of how they alter strategic consumer behavior with respect to the optimally coordinated loyalty program and revenue management. Proposition4 also expands upon the results of Acquisiti and Varian(2005), who argued that if revealing their “type” leads to future benefits that consumers value, then dynamic discrimination is profitable even with strategic consumers. In our case, consumers who purchase high-price tickets or miles reveal their types, while those who mileage-run conceal their types: all consumers in

III V IV V segments AM ,AS ,ASOM , and AB behave identically, and from the firm’s perspective, they are indistinguishable despite having different ais. However, even under the mileage-based program (i.e., when all strategic consumers conceal their types), we show that the firm benefits from strategic behavior, which is different from their results. Therefore, our results reveal a new mechanism for how a firm can benefit from strategic consumer behavior. The second implication is that by switching from a mileage-based program to a spending-based program, the firm can benefit more from strategic consumer behavior: the profit under the spending- based program is higher than under the mileage-based program, and with the enhanced mileage- based programs (SOM and BM), the firm can benefit even more. As discussed in Sections 4.3 and 5.3 one of the drivers of the profit increases is that these programs increase the number of consumers who qualify for Gold strategically, particularly those who buy high-price tickets or miles. This is consistent with the results of Acquisiti and Varian(2005) because by converting mileage-runners to spend-up/buy-up consumers, these designs effectively incentivize consumers to reveal their types. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 25

7. Numerical Illustrations To further illustrate our results and provide additional intuition, we proceed with numerical simulations. We considered a setting with four 5-point distributions of ai ∈ [100, 300]: uniform, centered, left- and right-skewed; c ∈ {5, 10, 20} and g ∈ {25, 50, 75} – a total of 36 scenarios. For each scenario, we numerically solve11 the firm’s optimization problems under four program designs (mileage, spending, selling-or-miles, bonus miles), as well as for the case with myopic consumers, and examine the firm’s profit and consumers’ surplus. We make the following key observations: Observation 1. The benefit from strategic consumer behavior is in all cases positive and under the mileage-based program is, on average, 16.46%; it is clearly even larger under other designs. Observation 2. Comparing mileage-based and spending-based designs, in 26 out of 36 cases the effective solutions are identical. In the remaining 10 cases, the spending-based profit is higher, on average higher by 0.87% . The regular price is, on average, 2.8% lower, and the effective qualification

∗ ∗ ∗ requirement (GM × pM vs SS) is, on average, 1.46% higher. This confirms the intuition derived earlier: the firm benefits from strategic consumer behavior and with the spending-based program, it discourages mileage-running in favor of spending-up. Therefore, the effective number of flights to qualify for Gold by mileage-running could be larger:

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ that pS < pM and SS > GM × pM implies SS/pS > GM which means that some customers may not be to maintain the same status if the firm switches to a spending-based program. Observation 3. Comparing the SOM and spending-based designs, in 16 out of 36 cases the solutions are identical, and in the remaining 20 cases the profit is higher, on average, by 1.86%. Under SOM, the qualification miles are priced at approximately 43% of regular ticket prices, and the effective qualification requirement is approximately 5.7% higher than under the spending-based solution. Observation 4. Comparing the bonus miles design with selling of miles and spending-based designs, in 12 out of 36 cases the BM solution is identical to SOM, in which cases it is also identical to the spending-based program. In the remaining 20 cases, BM is always more profitable than the spending-based program, but it is less profitable than SOM by an average of 1.66%. In 4 cases, BM is more profitable than SOM by a maximum of 0.04%. These two observations have several implications. First, in the majority of the cases, and on average, the enhanced mileage designs are more profitable than the spending-based program, which confirms our analytical results. Second, the benefit of enhancing a mileage-based design seems much larger than the benefit of switching to a spending-based program. Combined, these suggest that firms do not need to make a drastic switch to spending-based designs; rather, they should

11 A grid search with integer steps was used. 26 consider enhancing their mileage-based designs. Finally, and interestingly, the design with bonus miles, while having the largest number of decision variables, is infrequently more profitable; on average, it is actually less profitable than the selling of miles design. Examining consumer surpluses, our simulation suggests that, when compared with the optimal mileage-based solutions, the optimal spending-based and SOM solutions generally increase consumer surplus, while the BM design tends to slightly decrease the surplus; the SOM design also typically increases the surplus as compared to the optimal spending-based solutions. That being said, in all designs, there are instances when the surplus for at least some customers decreases at the optimal solution. This reinforces the point that in order to keep all customers happy, the firm may need to sacrifice a portion of the incremental profit it generates from a new program design.

8. Conclusions This paper investigates the interaction between the revenue management and the premium-status loyalty program functions of the firm and, specifically, the role that strategic, forward-looking and status-seeking consumers play in this interaction. We consider a contemporaneous change happen- ing in practice, where firms across several industries are switching their loyalty programs from the quantity/mileage-based designs toward spending-based designs. This switch has been receiving significant attention in the media and it has been met with negative consumer reactions. At the same time, other firms, rather than switching, enhance their otherwise mileage-based programs with various features. These include offering the ability to buy points/miles (the “selling of miles” design) and offering extra points/miles for high-priced purchases (the “bonus-miles” design). The goal of our work is to rigorously study the impact of loyalty program design choices on the firm and its consumers. A critical aspect of this problem is that consumers exhibit sophisticated strategic behavior: rather than viewing each purchase in isolation, they consider multiple purchase decisions to maximize the total surplus over a period of time. The prices they pay and the possibility of obtaining or maintaining premium status in the loyalty program both play an important role in customers’ decision making. Our paper contributes to the marketing literature in two ways. First, we present a novel consumer behavior model that captures this strategic behavior and incorporates consumer response to the firm’s prices and loyalty program design decisions. Using this model, we endogenously derive the consumer demand, characterize rich patterns of consumer behavior, and factor them into the firm’s pricing and loyalty program optimization problems. Second, we present a number of managerial insights on how the firm’s pricing and loyalty decisions impact and are impacted by strategic consumer behavior under four loyalty program designs: mileage-based, spending-based, and enhanced mileage-based designs with the selling of miles and bonus miles. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 27

Analyzing the consumers’ problem, we show that the so-called mileage-runs (flying more to obtain or maintain premium status), which are often observed in practice, are indeed utility-maximizing strategic behaviors; that is, some consumers should engage in mileage-running. Similarly, under the spending-based and bonus miles designs, it is optimal for some consumers to purchase at high prices even when the product is available at a lower price, and under the selling of miles design it is optimal for some consumers to purchase miles and fly less. Our model describes how different consumers react to prices and loyalty program parameters, and this then allows us to derive intuition about the drivers of profitability under different loyalty program designs. Comparing the different program design, we shows that the firm’s profit under the optimal spending-based program is higher than under the mileage-based program, which explains why some firms switch. However, we also show that by enhancing its mileage-based programs, the firm can achieve an even higher profit than the optimal profit under the spending-based design. Therefore, instead of making a drastic switch to the spending-based program, firms may be better off by keeping mileage-based programs with proper enhancements. The drivers for these profit increases are the following. The spending-based program enlarges the segment of consumers who strategically qualify for premium status while simultaneously reducing mileage-running, which increases revenue and decreases costs. The enhanced designs with the selling of miles and bonus miles further expand the strategic behavior and further reduce mileage-running. Across all designs, we show that by coordinating revenue management and loyalty decisions, the firm benefits from strategic consumer behavior – i.e., its profit is higher when consumers are strategic than when they are myopic, which is in contrast to most of the existing research. Beyond simply comparing the designs, our work therefore also provides a new insight into the sources of a loyalty program’s value: while traditionally viewed means of softening competition, we show that because of how different program designs alter strategic behavior, loyalty programs can be valuable even in the absence of competition. To summarize, we study the timely issue of changing loyalty program designs, and provide valuable insights for researchers and industry professionals as they rethink and redesign loyalty programs to improve firms’ profits and customer satisfaction. To illustrate the practical implications of our results, consider two kinds of firms discussed in the Introduction. Some firms that may be thinking of changing their premium status programs to a spending-based design might be better off by maintaining their mileage-based programs, but they may wish to add enhancing features such as the selling of miles or bonus miles. In contrast, other firms, that already switched to spending-based programs, could perhaps have sought for opportunities to modify their quantity-based programs instead (e.g., if Starbucks awards “double stars” for large purchases, say, $20 or more, it would be similar to the bonus miles design we discussed). Our results show that this would have incentivized the “right” behaviors and ultimately increased profitability beyond the spending-based design. 28

Future research can expand upon our study in several directions. For example, additional models can incorporate a richer pricing structure or address the problem of allocating capacity among differently priced products. One can also extend our setting to a dynamic model and explicitly consider customer arrivals, as well as address other questions (beyond the premium status) regarding the design and control problem of the loyalty programs. There could include studying the accumulation and redemption of loyalty points/miles. Competition could add another layer to our analysis: for instance, could there be an industry equilibrium where some firms operate mileage-based programs while others use spending-based designs? An empirical investigation of strategic consumer behavior in response to pricing and loyalty program design is clearly of interest as well. In the context of our model, this could mean validating the assumptions of strategic consumer behavior, as well as calibrating utility formulations and distributions of consumers. We are currently in the process of discussing the possibility of obtaining the data needed to perform this type of empirical analysis with several firms, in the hopes that such data would form the basis for a follow-up paper. Finally, designs that disproportionably reward high-price purchases (such as spending-based and bonus-miles design) create the possibility of a moral hazard (a business traveller is incentivized to purchase a higher-priced product when a lower-priced one is available), which poses interesting research questions as well.

References Acquisiti, A., H. R. Varian. 2005. Conditioning prices on purchase history. Marketing Science 24(3) 367–381.

Aﬂaki, S., I. Popescu. 2014. Managing retention in service relationships. Management Science 60 415–433.

Airline Weekly. 2015. Delta SkyDollars? United DollarsPlus? Stakeholders and competitors watch closely as two U.S. giants ditch distance-based awards. January 23.

Aviv, Y., A. Pazgal. 2008. Optimal pricing of seasonal products in the presence of forward-looking consumers. Manufacturing & Service Operations Management 10(3) 339–359.

Benderm, A. 2014. United Airlines Changes Its MileagePlus Frequent Flier Program, And Basically You’re Hosed. Forbes June 10, Retrieved from http://www.forbes.com/sites/andrewbender/2014/06/10/ united-airlines-changes-its-mileageplus-frequent-flier-program-and-basically-youre-hosed/.

Blattberg, R.C., B.D. Kim, S.A. Neslin. 2008. Database Marketing: Analyzing and Managing Customers. Springer, New York, NY.

Brunger, W. G. 2013. How should revenue management feel about frequent ﬂyer programs? Journal of Revenue and Pricing Management 12(1) 1–7.

Cachon, G. P., R. Swinney. 2009. Purchasing, pricing, and quick response in the presence of strategic consumers. Management Science 55(3) 497–511. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 29

Carlsson, F., A.¨ L¨ofgren.2006. Airline choice, switching costs and frequent ﬂyer programmes. Applied Economics 38(13) 1469–1475.

Chun, S. Y., D. Iancu, N. Trichakis. 2014. Dynamic managament of loyalty programs. Tech. rep.

Chung, H., S. Y. Chun, H. S. Ahn. 2014. Pricing when consumers buy with loyalty-points. Tech. rep.

Coughlan, A., J. Shulman. 2010. Creating superior value by managing the marketingoperations management interface. A. M. Tybout, B. J. Calder, eds., Kellogg on Marketing, 2nd ed., chap. 20. John Wiley & Sons, Hoboken, NJ, 393–408.

Del Nero, J. 2014. Poll Finds Majority of Frequent Flyers Do Mileage Runs to Maintain Status. January 6, FlyerTalk and Frequent Business Trav- eler, Retrieved from http://www.frequentbusinesstraveler.com/2014/01/ poll-finds-majority-of-frequent-flyers-do-mileage-runs-to-maintain-status/.

Dorotic, M., T. H. Bijmolt, P. C. Verhoef. 2012. Loyalty programmes: Current knowledge and research directions. International Journal of Management Reviews 14(3) 217 – 237.

Doublewides ﬂy. 2014. Delta Skymiles Program to be 100% Revenue Based Starting Jan 1, 2015. Rocky on Doublewides ﬂy, Febrary 26, Retrieved from http://upgrd.com/blogs/doublewidesfly/ delta-skymiles-program-to-be-100-revenue-based-starting-jan-1-2015.html.

Economides, N., K. Seim, V. B. Viard. 2008. Quantifying the beneﬁts of entry into local phone service. The RAND Journal of Economics 39(3) 699–730.

Gilbert, S. M., S. Jonnalagedda. 2011. Durable products, time inconsistency, and lock-in. Management Science 57(9) 1655–1670.

Keh, H.T., Y.H. Lee. 2006. Do reward programs build loyalty for services? the moderating eﬀect of satisfaction on type and timing of rewards. Journal of Retailing 82(2) 127–136.

Kim, B. D., M. Shi, K. Srinivasan. 2004. Managing capacity through reward programs. Management Science 50(4) 503–520.

Kim, B. D., K. Srinivasan. 2001. Reward programs and tacit collusion. Marketing Science 20(2) 99–120.

Kim, Y., R. Telang, W. B. Vogt, R. Krishnan. 2010. An empirical analysis of mobile voice service and sms: a structural model. Management Science 56(2) 2340252.

Kivetz, R. 2005. Promotion reactance: the role of eﬀort-reward congruity. Journal of Consumer Research (31) 725–736.

Kivetz, R. 2006. The eﬀects of eﬀort and intrinsic motivation on risky choice. Marketing Science (22) 477–502.

Kivetz, R., O. Urminsky, Y. Zheng. 2006. The goal-gradient hypothesis resurrected: Purchase acceleration, illusionary goal progress, and customer retention. Journal of Marketing Research (43) 39–58. 30

Koenigsberg, O., E. Muller, N. J. Vilcassim. 2008. easyjet pricing strategy: Should low-fare airlines oﬀer last-minute deals? QME 6(3) 279–297.

Kopalle, P. K., Y. Sun, S. A. Neslin, B. Sun, V. Swaminathan. 2012. The joint sales impact of frequency reward and customer tier components of loyalty programs. Marketing Science 31(2) 216–235.

Lambrecht, A., K. Seim, B. Skiera. 2007. Does uncertainty matter? consumer behavior under three-part tariﬀs. Marketing Science 26(5) 698–710.

Lederman, M. 2007. Do enhancements to loyalty programs aﬀect demand? the impact of international frequent ﬂyer partnerships on domestic airline demand. RAND Journal of Economics 1134–1158.

Levin, Y., J. McGill, M. Nediak. 2009. Dynamic pricing in the presence of strategic consumers and oligopolis- tic competition. Management Science 55 32–46.

Lewis, M. 2004. The inﬂuence of loyalty programs and short-term promotions on customer retention. Journal of Marketing Research 41(3) 281–293.

Liu, Q., G. van Ryzin. 2008. Strategic capacity rationing to induce early purchases. Management Science 54(6).

Liu, Y. 2007. The long-term impact of loyalty programs on consumer purchase behavior and loyalty. Journal of Marketing 71(4) 19 – 35.

Mackenzie, S. 2014. Uniteds Changes Are Bad for Everyone, and Thats Not News. June 12, Travel Codex, Retrived from http://travelcodex.com/2014/06/uniteds-changes-bad-everyone-thats-news/.

Marinesi, K., S. Girotra, S. Netessine. 2014. Operational advantages and optimal design of threshold dis- counting oﬀers. Insead working paper no. 2013/90/tom, INSEAD.

Mathies, C., S. P. Gudergan. 2012. Do status levels in loyalty programmes change customers’ willingness to pay? Journal of Revenue and Pricing Management 11(3) 274–288.

McCaughey, N. C., C. Behrens. 2011. Paying for Status? - The eﬀect of frequent ﬂier program member status on air fare choice. Monash economics working papers, Monash University, Department of Economics.

Meyer-Waarden, L. 2007. The eﬀects of loyalty programs on customer lifetime duration and share of wallet. Journal of Retailing 83(2) 223.

Mullins, J. 2016. Starbucks Just Launched Its New Rewards Program and People Are Already Pissed. April 12, Retrieved from http://www.eonline.com/ca/news/756107/ starbucks-just-launched-its-new-rewards-program-and-people-are-already-pissed.

Nunes, J.C., X. Drèze.2006. The endowed progress effect: how artificial advancement increases effort. Journal of Consumer Research 32 504–512.

Osadchiy, N., G. Vulcano. 2010. Selling with binding reservations in the presence of strategic consumers. Management Science 56(12) 2173–2190. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 31

Ovchinnikov, A., B. Boulu-Reshef, P. E. Pfeifer. 2014. Balancing customer acquisition and retention spending for ﬁrms with limited capacity. Management Science 60(8) 2002–2019.

Ovchinnikov, A., J. Milner. 2012. Revenue management with end-of-period discounts in the presence of customer learning. Production and Operations Management 21 69–84.

Roehm, M.L., H.A. Jr Roehm. 2010. The inﬂuence of redemption time frame on responses to incentives. Journal of the Academy of Marketing Science 16 1–13.

Sahin, O., G. Gallego, R. Phillips. 2008. Strategic management of distressed inventories. Production and Operations Management 17(4) 402–415.

Sernovitz, G. 2016. The Madness of Airline ‘Elite Status. February 23, Retrieved from http://www.newyorker.com/business/currency/ the-madness-of-airline-elite-status.

Shugan, S. M. 2005. Brand loyalty programs: Are they shams?. Marketing Science 2 185 – 193.

Shugan, S. M., J. Xie. 2005. Advance-selling as a competitive marketing tool. International Journal of Research in Marketing 22(3) 251–273.

Snyder, B. 2016. Customers Are Furious With Starbucks New Rewards Program. February 23, Retrived from http://fortune.com/2016/02/23/starbucks-rewards-program-changes/.

Su, X. 2007. Intertemporal pricing with strategic consumer behavior. Management Science 53(5) 726 – 741.

Sun, Y., D. Zhang. 2014. A model of customer reward program with ﬁnite expiration term. Tech. rep., University of Colorado Boulder.

Tang, C. S. 2010. A review of marketing-operations interface models: From co-existence to coordination and collaboration. International Journal of Production Economics 125(1) 22–40.

Uncles, M. D., G. R. Dowling, K. Hammond. 2003. Customer loyalty and customer loyalty programs. The Journal of Consumer Marketing 20(4) 294–314.

Vasel, L. 2016. New Starbucks Rewards Program Starts Tuesday. April 11, Retrieved from http://money. cnn.com/2016/04/11/pf/starbucks-rewards-program/.

Wahba, P. 2016. Why Starbucks Is Overhauling its Loyalty Rewards Program. February 22, Retrieved from http://fortune.com/2016/02/22/starbucks-loyalty/.

Wang, J., Y. Levin, M. Nediak. 2015. Selling passes to strategic ccustomers. Working paper, Queen’s University.

Zhang, D., W. Cooper. 2008. Managing clearance sales in the presence of strategic consumers. Production and Operations Management 17(4) 416–431.

Zhang, Z. J., A. Krishna, S. K. Dhar. 2000. The optimal choice of promotional vehicles: front-loaded or rear-loaded incentives? Management Science 46(3) 348–3621. 32

Appendix. Proofs Proof of Lemma1. A consumer of type i will “fly-up” to Gold if G2 (n∗ )2 (a − p)2 (a + g)G − − pG ≥ a n∗ − i,R − pn∗ = , i 2 i i,R 2 i,R 2 √ √ which is equivalent to saying that consumers with ai ∈ [G − 2Gg + p, G + 2Gg + p]; note that by design ∗ g ≥ 0. However, consumers with ni,G = ai + g − p ≥ G, i.e., those with ai ≥ G + p − g, are already qualifying √ for Gold without the need to fly-up. That is, consumers with ai ∈ [G − 2Gg + p, G + p − g] are flying up to 1 2 √ 3 Gold status. Therefore,a ¯M = p ≤ a¯M = G − 2Gg + p ≤ a¯M = G + p − g if 2g < G. Q.E.D. 1 2 √ 3 Proof of Corollary1. From the proof of Lemma1, we have ¯aM = p ≤ a¯M = G− 2Gg+p ≤ a¯M = G+p−g 1 2 2,3 and 2g < G (from the natural ordering aM ≤ aM ). That aM decrease in g and increase in G is obvious. Since ∂ 2 g ∂ 3 ∂ 2 G ∂ 3 a = 1− √ √ < a = 1, it proves part (i). For part (ii), a = − √ √ < a = −1 given 2g < G. ∂G M 2 gG ∂G M ∂g M 2 gG ∂G M Q.E.D. Proof of Lemma2. A consumer who flies as needed at fare p will qualify for Gold if

p × ai + g − p ≥ S,

i.e., if S a ≥ + p − g. i p Otherwise, a consumer will consider spending-up to Gold. As discussed above, this can be done by either simply ﬂying more at price p, or spending more at price p+, or, conceivably, with a combination of both. In other words, to determine the best way to spend-up to Gold, a consumer decides on the number of “regular” segments, n, and the number of “plus” segments, n+ that would maximize:

(n + n+)2 (a + g)(n + n+) − − (pn + p+n+), s.t. pn + p+n+ ≥ S. i 2 Let

−S + (a + g)p+ S − (a + g)p n∗ = i and n+∗ = i . (16) i,S (p+ − p) i,S (p+ − p) ∗ + +∗ It is not diﬃcult to verify that n is non-negative when ai ≥ S/p − g and similarly n is non-negative when ai ≤ S/p − g. Therefore: + + • If ai ≤ S/p − g, then the solution to this consumer problem is (0, S/p ) and the resultant surplus of spending up to Gold for a customer of type i equals:

(2(a + g)p+ − S Surplus1 (0, S/p+) = S i − 1 , (17) i 2(p+)2 + + which is non-negative when ai ≥ S/2p − g + p . + ∗ +∗ • If ai ∈ (S/p − g, S/p − g), then the solution to this consumer problem is (ni,S , ni,S ), as given in (16); the resultant surplus of spending up to Gold, equals:

(a + g)2 − 2S Surplus2 (n∗ , n+∗) = i , (18) i i,S i,S 2 √ which is non-negative when ai ≥ 2S − g. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 33

• If ai ≥ S/p − g then the solution to this consumer problem is (S/p, 0) and the resultant surplus of spending up to Gold equals:

(2(a + g)p − S Surplus3 (S/p, 0) = S i − 1 , (19) i 2p2

which is non-negative when ai ≥ S/2p − g + p. The above solution describes how a consumer would spend up to Gold if she decides to do so. The latter decision, however, will depend on whether the surplus from spending up to Gold is (a) non-negative as per the conditions above, and (b) larger than the surplus of ﬂying as needed, but without the Gold status. The 2 latter surplus as per Section3 equals ( ai − p) /2. Comparing this surplus with the surpluses of spending up to Gold from (17)-(19), we obtain that: h i 2 S q 2S + S q 2S + + • Surplus1i ≥ (ai − p) /2 if ai ∈ p + p+ − p+ (g + p − p ), p + p+ + p+ (g + p − p ) and g + p − p > 0, 2 p−g S • Surplus2i ≥ (ai − p) /2 if ai ≥ 2 + p+g , and 2 h S q 2S S q 2S i • Surplus3i ≥ (ai − p) /2 if ai ∈ p + p − p g, p + p + p g .

To summarize, there are potentially up to six segments, depending on ai, with different consumer behaviors: I Segment AS : If ai ≤ p, then the consumer does not fly at all. The number of flights and total spend are both zeros.

II Segment AS : If ai does not satisfy any conditions for other regions (the expressions that deﬁne this region are messy and un-insightful), then the consumer ﬂies as needed at price p with a regular status. The number

∗ ∗ of flights is given by ni,R and the total spend equals ni,R p. Basically, for these consumers, spending-up to Gold is too expensive: the utility of flying as needed without Gold is higher than the utility of (optimally) spending up to Gold. h i III S q 2S + S q 2S + + Segment AS : If ai ∈ p + p+ − p+ (g + p − p ), p + p+ + p+ (g + p − p ) , ai ≤ S/p − g and ai ≥ S/2p+ − g + p+, then the consumer spends up to Gold by flying S/p+ segments at price p+, spending exactly S in total. √ IV p−g S + Segment AS : If ai ≥ 2 + p+g , ai ∈ (S/p − g, S/p − g) and ai ≥ 2S − g then the consumer spends up to ∗ +∗ + Gold by flying ni,S segments at price p and ni,S segments at price p , spending exactly S in total.

V h S q 2S S q 2S i Region AS : If ai ∈ p + p − p g, p + p + p g , ai ≥ S/p − g and ai ≥ S/2p − g + p, then the consumers spends up to Gold by ﬂying S/p segments at price p spending exactly S in total. Note that this case is identical to ﬂying-up to Gold, analyzed in Section 4.1 with G = S/p.

IV Segment AS : If ai ≥ S/p + p − g, then the consumer ﬂies as needed with the Gold status since her total spend exceeds S. Q.E.D. Proof of Corollary2. By carefully comparing the boundaries of the segments from the proof of Lemma 2, one can verify that the condition where all segments are non-empty implies that: s S 2S a¯2 = a2 = p + − (g + p − p+), (20) S S p+ p+ 34

S a¯3 = = − g, (21) S p+ S a¯4 = − g, (22) S p S a¯5 = p + − g. (23) S p 3,4,5 + ∂ 5 ∂ 4 That aS decrease in g and increase in S is obvious. For part 1, because p > p, ∂S aS = 1/p = ∂S aS > + ∂ 3 ∂ 2 + q g+p−p+ + ∂ 3 ∂ 2 q S 1/p = ∂S aS . Finally, ∂S aS = 1/p − 2Sp+ < 1/p = ∂S aS . For part 2, ∂g aS = − 2p+(g+p−p+) , which ∂ 3,4,5 + + is less than −1 = ∂g aS under the condition S ≥ 2p (g + p − p ), which comes from ensuring the natural 1 2 ordering aS < aS . Q.E.D. ∗ ∗ Proof of Proposition1. Let πM and πS denote the optimal profits under the mileage- and spending- ∗ ∗ ∗ ∗ ∗ +∗ based programs, respectively. The proposition states that πS ≥ πM . Let pM ,GM and pS , pS ,S be the + ∗ ˆ ∗ ∗ corresponding optimal decisions. Note that p = pS = pM and S = pM × GM are feasible solutions of the spending-based loyalty program problem. Thus, consider a constrained version of the spending-based loyalty ∗ ∗ ˆ ∗ ∗ + problem with pS = pM and S = pM × GM , but allow p to change as to maximize the firm’s profit. Then, ∗ + ∗ + ˆ ∗ πS = maxp,p+,S πS (p, p ,S) ≥ maxp+ πS (pM , p , S) ≥ maxp,G πM (p, G) = πM . Q.E.D. Proof of Proposition2. By following the same logic as in the proof of Proposition1, consider a con- ∗ ∗ ˆ ∗ ∗ + strained version of the spending-based loyalty problem with pS = pM and S = pM × GM , but allow p to ∗ + ˆ ∗ change to maximize the firm’s profit. Then, maxp+ πS (pM , p , S) ≥ maxp,G πM (p, G) = πM – i.e., the profit of the firm in this constrained spending-based problem is no less than under the optimal mileage-based problem. By design, the regular price did not change; as such, regular consumers are unaffected. Since the Gold thresholds and regular prices are equal to the optimal mileage-based program values, those who fly Gold are either flying as needed or flying-up to qualify for Gold identically under both programs; however, some may decide to fly some segments at price p+. This implies, from Lemma2, that their surplus (weakly) increases as well. Q.E.D. Proof of Lemma3. If consumers do not qualify for Gold automatically, they consider flying-up or buying-up and solve the following optimization problem:

n2 q q max (ai + g)n − 2 − (pn + qn ) s.t. n + n ≥ G. n,nq ∈R+ Let ∗ q∗ ∗ ni,SOM = ai + g − p + q, and ni,SOM = G − n = G − ai − g + p − q. (24)

Therefore,

∗ q∗ • If ai + g − p + q < G, then the solution to this consumer problem is (ni,SOM , ni,SOM ), as given in (24), and the resultant surplus equals:

1 Surplus1 (n∗ , nq∗ ) = 2q(a + g − G − p) + (a + g − p)2 + q2 . (25) i i,SOM i,SOM 2

• If ai + g − p + q > G and ai + g − p < G, then the solution to this consumer problem is (G, 0) and the resultant surplus of ﬂying up to Gold equals: G2 Surplus2 (G, 0) = (a + g)G − − pG. (26) i i 2 Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 35

Note that no consumers will do buying up alone (without flying up). The above solution describes how a consumer would fly/buy up to Gold if she decides to do so. The latter decision, however, will depend on whether the surplus from flying/buying up to Gold is (a) non-negative as per the conditions above, and (b) larger than the surplus of flying as needed, but without the Gold 2 status. The latter surplus as per Section3 equals ( ai − p) /2. Comparing this surplus with the surpluses of flying/buying up to Gold from (25)-(26), we obtain that: 2 −g2+2gp−2gq+2Gq+2pq−q2 • Surplus1i ≥ (ai − p) /2 if ai ≥ 2(g+q) , 2 √ • Surplus2i ≥ (ai − p) /2 if ai ≥ G − 2Gg + p.

To summarize, there are potentially up to ﬁve segments, depending on ai, with diﬀerent consumer behaviors:

I Segment ASOM : If ai ≤ p then the consumer does not ﬂy at all. The number of ﬂights and total spend are both zeros.

II Segment ASOM : If ai does not satisfy any conditions for other regions (the expressions that deﬁne this region are messy and uninsightful), then the consumer ﬂies as needed at price p with a regular status. The number

∗ of flights is given by ni,R. Basically, for these consumers, flying/buying-up to Gold is too expensive; the utility of flying as needed without Gold is higher than the utility of (optimally) buying up to Gold.

III −g2+2gp−2gq+2Gq+2pq−q2 Segment ASOM : If ai < G − g + p − q, and ai ≥ 2(g+q) , then the consumer does ﬂy/buy-up

to Gold by flying ai + g − p + q segments at price p and purchasing G − ai − g + p − q segments at price q, satisfying exactly G segments in total. √ IV Segment ASOM : If ai > G − g + p − q, ai < G − g + p, and ai ≥ G − 2Gg + p, then the consumer is flying up to Gold by flying G segments at price p.

V Segment ASOM : If ai ≥ G + p − g, then the consumer ﬂies as needed with the Gold status. Q.E.D. Proof of Corollary3. In the case where all consumer segments exist, from the proof of Lemma3, we obtain that: (g + q)2 − 2Gg a¯2 = = G + p − g − q + , (27) SOM 2(g + q) 3 a¯SOM = G + p − g − q, (28)

4 a¯SOM = G + p − g. (29)

3,4 ∂ 2 g ∂ 3 That the derivatives of aSOM in G, g are equal is obvious. ∂G aSOM = 1 − 2(g+q) < 1 = ∂G aSOM , which ∂ 2 1 Gq ∂ 3 2 proves part (i). ∂g aSOM = − 2 − (g+q)2 < −1 = ∂g aSOM when 2Gq > (g + q) , which must hold to preserve 1 2 the natural orderinga ¯SOM < a¯SOM . This proves part (ii). Q.E.D. Proof of Lemma4. If consumers do not qualify for Gold automatically, they consider buying-up to Gold. Similar to the spending-based program case, this can be done by either simply ﬂying more at price p, spending more at price p+ or, conceivably, with a combination of both. In other words, to determine the best way to buy-up to Gold, a consumer decides on the number of “regular” segments, n, and the number of “bonus” segments, nb, with a bonus parameter B that would maximize: 36

(n + nB )2 (a + g)(n + nB ) − − (pn + pB nB ), s.t. n + BnB ≥ G i 2 Let (pB − p) − (B − 1)(a − p + g − G) (pB − p) − (B − 1)(a − p + g − G) n∗ = G − B i and nB∗ = i . (30) i,BM (B − 1)2 i,BM (B − 1)2 It can be shown that B B ∗ ai(−B)+ai+(B−1)G+Bp−p −Bg+(B−1)G+Bp+g−p • Only ni,BM is non-negative when g ≥ B−1 , i.e., ai ≥ B−1 . 2 B B ∗ B∗ B (p−g)+B(g+G−p )−G −Bg+(B−1)G+Bp+g−p • Both ni,BM and ni,B are non-negative when (B−1)B < ai < B−1 . 2 2 B 2 B B∗ −aiB +aiB+B p+BG−Bp −G B (p−g)+B(g+G−p )−G • Only ni,BM is non-negative when g ≤ B2−B , i.e., ai ≤ (B−1)B . Therefore: B2(p−g)+B(g+G−pB )−G • If ai ≤ (B−1)B , then the solution to this consumer problem is (0, G/B) and the resultant surplus of ﬂying up to Gold for a customer of type i equals: G(G − 2B(a + g − pB )) Surplus1 (0, G/B) = − i , (31) i 2B2

B G+2B(p −g) B B which is non-negative when ai ≥ 2B (given that g < p ) and ai > p − g. B2(p−g)+B(g+G−pB )−G −Bg+(B−1)G+Bp+g−pB • If ai ∈ ( (B−1)B , B−1 ), then the solution to this consumer problem is ∗ B∗ (ni,BM , ni,BM ), as given in (30), and the resultant surplus of ﬂying up to Gold equals:

2 2 B 2 2 ∗ B∗ a (B−1) +2a(B−1)((B−1)g−Bp+p )+B p Surplus2i(ni,BM , ni,BM ) = 2(B−1)2 (B−1)2g2−2(B−1)g(Bp−pB )+2BGp−2BGpB −2BppB −2Gp+2GpB +pB (32) + 2(B−1)2 ,

B 2 (ai(B−1)+(B−1)g−Bp+(p ) which is non-negative when G < − 2(B−1)(p−pB ) . −Bg+(B−1)G+Bp+g−pB • If ai ≥ B−1 , then the solution to this consumer problem is (G, 0) and the resultant surplus of ﬂying up to Gold equals: 1 Surplus3 (G, 0) = a G − G(−2g + G + 2p), (33) i i 2

1 which is non-negative when ai > 2 (−2g + G + 2p) given that g < p. The above solution describes how a consumer would fly up to Gold if she decides to do so. The latter decision, however, will depend on whether the surplus from flying-up to Gold is (a) non-negative as per the conditions above, and (b) larger than the surplus of flying as needed, but without the Gold status. The latter

2 surplus as per Section3 equals ( ai − p) /2. Comparing this surplus with the surpluses of spending up to Gold from (31)-(33), we obtain that: √ √ √ √ 2 G(g+p−pB ) 2 G(g+p−pB ) • Surplus1 ≥ (a − p)2/2 if a ∈ G + p − √ , G + p + √ and g + p − pB > 0, i i i B B B B 2 (B−1)2g2−2(B−1)g(Bp−pB )+(p−pB )(2(B−1)G+2Bp−p−pB ) • Surplus2i ≥ (ai − p) /2 if ai ≥ − 2(B−1)((B−1)g−p+pB ) , √ √ 2 • Surplus3i ≥ (ai − p) /2 if ai ∈ p + G − 2gG.p + G + 2gG ,

To summarize, there are potentially up to six segments, depending on ai, with diﬀerent consumer behaviors:

I Segment ABM : If ai ≤ p, then the consumer does not ﬂy at all. The number of ﬂights and total spend are both zeros. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 37

II Segment ABM : If ai does not satisfy any conditions for other regions (the expressions that deﬁne this region are messy and uninsightful) then the consumer ﬂies as needed at price p with a regular status. The number

∗ ∗ of flights is given by ni,R and the total spend equals ni,Rp. Basically, for these consumers, buying-up to Gold is too expensive: the utility of flying as needed without Gold is higher than the utility of (optimally) buying up to Gold. √ √ √ √ 2 G(g+p−pB ) 2 G(g+p−pB ) B2(p−g)+B(g+G−pB )−G Segment AIII : If a ∈ G + p − √ , G + p + √ , a ≤ , and a ≥ BM i B B B B i (B−1)B i + G+2B(p −g) B 2B , then the consumer flies up to Gold by flying G/B segments at price p , flying exactly G segments in total.

2 2 B B B IV (B−1) g −2(B−1)g(Bp−p )+(p−p )(2(B−1)G+2Bp−p−p ) Segment ABM : If ai ≥ − 2(B−1)((B−1)g−p+pB ) and ai ∈ 2 B B B 2 B (p−g)+B(g+G−p )−G −Bg+(B−1)G+Bp+g−p (ai(B−1)+(B−1)g−Bp+(p ) ( (B−1)B , B−1 ) , and G < − 2(B−1)(p−pB ) , then the consumer is ∗ B∗ + flying up to Gold by flying ni,B segments at price p and ni,B segments at price p , flying exactly G segments in total.

√ √ B V −Bg+(B−1)G+Bp+g−p 1 Segment ABM : If ai ∈ p + G − 2gG, p + G + 2gG , ai ≥ B−1 , and ai > 2 (−2g +G+2p), then the consumers spends up to Gold by ﬂying G segments at price p ﬂying exactly G segments in total.

IV Segment ABM : If ai ≥ G + p − g, then the consumer ﬂies as needed with the Gold status. Q.E.D. Proof of Corollary4. In the case where all consumer segments exist, from the proof of Lemma4, we obtain that: r G 2G(g + p − p+) a¯2 = p + − , (34) BM B B B2(p − g) + B(g + G − p+) − G a¯3 = , (35) BM B(B − 1) −Bg + (B − 1)G + Bp + g − p+ a¯4 = , (36) BM B − 1 5 a¯BM = G + p − g. (37)

∂ 2 1 q g+p−pB 1 ∂ 3 ∂ 4,5 ∂ 2 q G For part (i): ∂G aBM = B − 2BG) < B = ∂G aBM < 1 = ∂G aBM . For part (ii): ∂g aBM = − 2B(g+p−pB ) < ∂ 3 B B −1 = ∂g aBM when G > 2B(g + p − p ). Note that this implicitly assumes p ≤ g + p, which is a necessary condition for the non-emptiness of all segments. Q.E.D. Proof of Proposition3. Since the proposition compares the enhanced designs with SOM and BM to the spending-based program design, we first establish the optimal solution for the latter for the case with ai ∈ {aL, aH }. Spending-based design. Assume the scenario where the L-type is flying regular and H-type is spending- up to Gold by mixing n∗ flights at p and n+∗ flights p+; we will soon show that this is optimal. With this,

∗ +∗ L-type takes aL − p flights and H-type takes n + n flights for a total revenue of S. The firm’s profit then equals: + ∗ +∗ πS (p, p ,G) = p(aL − p) + S − c(n + n ). (38)

Note that p+ factors the proﬁt function only through the numbers of ﬂights; further, from equation (8) it

∗ +∗ is not diﬃcult to verify that n + n = aH + g. In other words, πS is only a function of p, S. 38

Because πS increases in S while the consumer surplus from mixing (equation7), is decreasing in S, the optimal function S∗(p) is selected such that the H-type is indiﬀerent between mixing and ﬂying regular.

2 2 ∗ 1 2 Equating the two surpluses and solving (aH − p) /2 = (aH + g) /2 − S, we obtain S (p) = 2 (2aH g + g + 2 2aH p − p ). Substituting this expression for S∗(p) into (38) and diﬀerentiating we obtain that the function is concave

FOC in p and the value that solves the first-order condition is pS = (aH + aL)/3. However, to ensure that FOC ∗ the L-type flies regular, if p > aL then pS = aL; that this happens when aH > 2aL, which is a familiar ∗ ∗ 1 condition we assumed earlier. With this condition, pS = aL, SS = 2 (2aH − aL + g)(aL + g). To determine the optimal value of p+, note that while it does not impact the profit function, we must ensure that the H-type

∗ + ∗+ S is indeed mixing, i.e., that n ≥ 0. Solving (aH + g)p − S ≥ 0 yields to p ≥ . To avoid ambiguity, we aH +g ∗ set p∗+ = SS . S aH +g As is the case with the mileage-based program solution, we again check if the assumed scenario indeed occurs at this optimal solution. 2 • For the H-type, the surpluses from spending-up, mixing, and flying regular are equal (aH − aL) /2; hence, we only need to check if the surpluses from flying-up or auto-Gold are below that. For flying up, the 2 2 2 2 2 2 surplus at the optimal solution equals ((aL − g )(4aH − 8aH aL + 3aL + 4aH g − 4aLg + g ))/(8aL ), which 2 2 2 aL+2aLg−g is below (aH − aL) /2 when aL > g > c and aH > 2g . For auto-Gold, it is not difficult to verify that ∗ H-type’s spending p(aH + g − p) is always below SS . ∗ • For the L-type, flying regular leads a zero surplus by the selection of pS . The other L-type surpluses 1 are: for spending-up − 2 (aH − aL)(aH + aL + 2g) < 0, for mixing −(aH − aL)(aL + g) < 0, and for flying- 3 2 2 3 2 2 2 2 up −(((aL + g)(aL + 3aLg − 5aLg + g + 4aH (aL + g) − 4aH (aL + 2aLg − g )))/(8aL)) < 0 after tedious verification. That is, at the optimal solution, the H-type is mixing and the L-type is flying regular, and the assumed scenario occurs. The optimal solution to the spending-based program is summarized below:

2 2 aL+2aLg−g Lemma 5. If aH > max[2aL, 2g ], then the optimal solution under the spending-based program ∗ ∗ 1 +∗ (2ah−aL+g)(aL+g) p = aL, S = (2aH − aL + g)(aL + g) and p = , and the optimal profit equals: S S 2 2(aH +g) 1 π∗ ≡ π(p∗ , p+∗,S∗ ) = (2a − a + g)(a + g) − c(a + g). (39) S S S S 2 H L L H Interpreting this solution, observe that the L-type consumer flies regular and the H-type spends-up to Gold – i.e., takes n+∗ flights at price p+∗. Further, an infinite number of alternative optima exist with p+ > p+∗, where the H-type is mixing flights at p∗ and p+. Both customer types are, on the margin, indifferent between no-flying and flying regular, respectively, but we assume that decisions that benefit the firm occur. SOM design. As with the preceding analysis, assume the scenario where the L-type is flies regular and the H-type buys status-qualifying miles/segments/credits; we will soon show that this is optimal. With this,

∗ q∗ ∗ the L-type takes aL − p flights and the H-type takes n = aH + g − p + q flights and purchases n = G − n status-qualifying miles. The firm’s profit then equals:

πSOM (p, q, G) = p(aL − p) + (p − c)(aH + g − p + q) + q(G − aH − g + p − q). (40) Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 39

The profit function is increasing in G; hence, the optimal G will be selected so that H- type is indifferent between buying miles and flying regular. The surplus from the former equals

1 2 2 2 2 (2q(a + g − G − p) + (a + g − p) + q ) and the surplus from the latter is the familiar (aH − p) /2. Equating 2 2 ∗ 2aH g+g −2gp+2aH q+2gq−2pq+q the two and solving for G yields G (p, q)SOM = 2q .

Assume p = aL; this clearly implies that the resultant solution will be a lower bound on the optimal

∗ 1 2 proﬁt. Then, πSOM (aL, q, G (aL, q)) = aH (aL − c + g) + 2 (−2aL + 2aL(c + q) − (2c − g + q)(g + q)), which ∗ q is concave in q with a unique maximum at q = aL − c. At this solution, n = aH − c + g > 0 and n = 2 2 2 2 2 aL +c −2aL(c−g)−g(2aH+g) aL−2aLc+c +2aLg−g ≥ 0 if aH ≥ , which we will assume to hold to ensure that H-types 2(aL−c) 2g are indeed buying up miles and the proposed solution holds.

∗ As with other designs, we need to verify if the assumed scenario will occur. By the solution for G (p, q)SOM ,

H-types are no better oﬀ by ﬂying regular than by buying miles. By the solution for q, GSOM > aH + g − aL, i.e., H-types do not automatically qualify for Gold. Finally, for the L-type, the surplus from buying miles

equals −(aH − aL)(aL − c + g) < 0; the surplus from ﬂying regular is also negative (the expression is messy and uninsightful and is not presented here). That is, under the optimal solution, the H-type is indeed buying miles to qualify for Gold and the L-type ﬂies with a regular status. Summarizing, there exists a feasible solution to the SOM-enhanced design as follows:

2 2 2 aL−2aLc+c +2aLg−g Lemma 6. If aH > 2g , then under the mileage-based program with selling of miles there exists a feasible solution with pSOM = aL, qSOM = aL − c and GSOM = 2 2 2aH (aL−c)−2aL(aL−c)+(aL−c) +2aH g−2aLg+2(aL−c)g+g , and the profit of: 2(aL−c) 1 πˆ = (2a − a + g − c)(a − c + g). (41) SOM 2 H L L Interpreting this solution, first note that the ‘hat’ symbol in (41) signifies that this is a feasible, not necessarily optimal solution. At that solution, however, the regular price and the L-type behavior are identical to those under the spending-based program discussed earlier, but the price of qualifying miles is strictly below the regular ‘ticket’ price. The H-type therefore finds it beneficial to substitute (some of) her flying-up with purchasing miles, which consequently saves some cost to the firm.

BM design. As with the preceding analysis, assume the scenario where the L-type is flying regular and H-type mixes flying at p and pB ; we will soon show that this is optimal. The firm’s profit equals:

B + B πB = max (aL − p)p + (pn + p n ) − c(n + n ). (42) p,pB ,B,G

B B B B B (aH +g)(B−1)−Bp+p ∂(np+n p ) p −p Because n + n = B−1 is independent of G, while ∂G = B−1 ≥ 0, the ﬁrm’s proﬁt ∂n 1−B ∂nB 1 is increasing in G. Further, since ∂G = (B−1)2 < 0 and ∂G = B−1 > 0, the optimal G is such that n = 0. Solving and rearranging we obtain: B G∗ = B(a + g − p) + (pB − p). (43) B H B − 1

B 2 ∂SurplusB B(p −p) With this, | ∗ = > 0 when B ≥ 1 – i.e., consumer surplus is increasing in B. At the ∂B G=GB (B−1)3 B B ∂πB (p −p)(p −c) same time, | ∗ = − < 0 – i.e., the ﬁrm’s proﬁt is decreasing in B. Hence, the optimal ∂B G=GB (B−1)2 40

1 2 B will be selected such that the Gold surplus equals the surplus of ﬂying regular, 2 (aH − p) . Solving and rearranging we obtain:

1 B∗ = . (44) 1 − √ pB −p B B (p+g−p )(2aH +g−p−p )

B The square root in the denominator exists when p is either quite small (below min[p + g, 2aH + g − p]) or

B quite large (above max[p + g, 2aH + g − p]), but not for moderate values of p . However, since by deﬁnition 2 2 B ≥ 1, an implicit condition is that the denominator is positive, which is true for pB ∈ [p, g −2p +2aH (g+p) ] ≤ 2(aH +g−p) B min[p + g, 2aH + g − p]. In other words, the above solution holds only for relatively small p that fall within the aforementioned interval.

∂π Over that interval, it is possible to establish that | ∗ ∗ ≥ 0 (the expressions are tedious and are ∂pB G=GB ,B=B thus not presented here). Therefore:

g2 − 2p2 + 2a (g + p) pB∗ = H . (45) 2(aH + g − p)

∗ ∗ +∗ To summarize, we obtained the optimal GB ,B , pB as a function of p. Unfortunately, substituting the resultant expressions into the ﬁrm’s proﬁt function leads to a fourth-degree polynomial, which does not have an easily interpretable solution for the optimal p. However, an interesting property of the optimal solution for the Bonus miles design can be estab-

∗ p+∗S ∗ lished without having the complete solution. Specifically, it is easy to see that B > ∗ In fact, B → pS ∗ B∗ + ∞ and GB → ∞. The resultant number of flights, however is finite: the optimal n = aH + g − p +

p + + (2aH + g − p − p )(g + p − p )..

Combining the results and proving the proposition. Having established the solutions for the three designs, we note that comparing the expressions for proﬁts in Lemmas5 and6, it is not diﬃcult to verify

∗ 2 thatπ ˆSOM −πS = c /2 > 0 – i.e., the heuristic solution for the SOM problem leads to a higher proﬁt than the

∗ optimal spending-based solution. Similarly, considering a heuristic solution to the BM problem with pB = aL, and other variables taking their respective conditionally optimal values as per equations (43)-(45), it can

∗ be veriﬁed that the set of parameters for which such a heuristic solution’s proﬁt exceeds πS is non-empty

∗ (interestingly, such a heuristic BM solution cannot exceed the SOM solution with pSOM = aL).

Summarizing this long proof, we established that under the case with ai ∈ {aL; aH }, and subject to technical

conditions which intuitively mean that the difference between aH and aL cannot be too small, that the optimal profit under the SOM and BM designs exceeds the optimal profit under the spending-based design. Q.E.D. Proof of Proposition4. Since this proposition compares the loyalty program with myopic consumers to the optimal mileage-based program, we first have to derive the mileage-based program solution for the case

with ai ∈ {aL, aH }. Chun, Ovchinnikov: Strategic Consumers, RM and the Design of Loyalty Programs 41

Mileage-based design The firm sets the qualification requirement G and charges price p per flight (as argued above, there is no demand at price p+ under the mileage-based program). Assume a scenario where the L-type is flying regular and the H-type is flying-up to Gold; we will soon show that this is indeed optimal. From Lemma1, the L-type takes aL − p flights and the H-type takes G flights for the total firm’s profit of:

πM (p, G) = p(aL − p) + G(p − c). (46)

2 From the proof of Lemma1, the L-type surplus from flying regular is ( aL −p) /2, and the H-type surpluses 2 2 (aH −p) G from flying regular and flying-up to Gold are 2 and (aH + g − p)G − 2 , respectively. Since the profit is increasing in G, the H-type’s surplus from flying regular is independent of G, while the surplus of flying-up is decreasing12 in G; the optimal function G∗(p) will be such that the H-type is indifferent between flying regular and flying-up to Gold. Equating the H-type surpluses and solving for G leads to two roots, the larger √ ∗ 2 of which equals: GM (p) = aH + g − p + 2aH g + g − 2gp. ∗ 2 ∗ ∂πM (p,GM (p)) Substituting GM (p) into (46) and differentiating, we obtain that ∂p2 = −2 – i.e., the optimal ∗ ∂πM (p,G (p)) profit is concave in p. Solving the first-order condition ∂p = 0 we obtain the unique maximizer FOC p FOC ∗ ∗ FOC pM = 1/2(aH + g + g(2aH − 2aL + g)). Therefore, if pM > aL then pM = aL and otherwise pM = pM . ∗ The former is true, for example, when aH > 2aL, which we assumed above. That is, if aH > 2aL then pM = aL, ∗ p 2 GM = aH − aL + g + 2g(aH − aL) + g . We next check if the assumed scenario indeed occurs under this optimal solution. For that we need to

∗ ∗ show that at (pM ,GM ), the H-type will not automatically qualify for Gold or benefit from flying regular, and the L-type will not benefit from flying-up (as noted earlier, the L-type cannot qualify automatically if the H-type does not). • For the H-type:

∗ — When flying as needed with the Gold status, the H-type will take aH + g − p flights. Because G (p) = √ 2 aH + g − p + 2aH g + g − 2gp > aH + g − p, the H-type will not automatically qualify for Gold. — By the choice of G∗(p) the surplus from flying regular is no greater than from flying-up. 2 • For the L-type, the surplus of flying-up to Gold is (aL + g − p)G − G /2 which, at the optimal solution, p p equals 1/2(−aH + aL + g − g(2aH − 2aL + g))(aH − aL + g + g(2aH − 2aL + g)). It is easy to verify that this expression is non-positive – i.e., it does not exceed a surplus from flying regular, which equals zero since

∗ p = aL. That is, at the optimal solution the H-type is ﬂying-up and the L-type is ﬂying regular, and the assumed scenario occurs. The optimal solutions to the mileage-based program is summarized below:

∗ ∗ Lemma 7. If aH > 2aL, then the optimal solution under the mileage-based program pM = aL, GM = aH − p 2 aL + g + 2g(aH − aL) + g and the optimal proﬁt equals:

∗ ∗ ∗ p πM ≡ π(pM ,GM ) = (aL − c) aH − aL + g + g(2(aH − aL) + g) . (47)

12 That the H-type is ﬂying-up and not qualifying for Gold automatically implies G > aH − p + g, so that ∂ G2 ∂G (aH + g − p)G − 2 = (aH + g − p) − G < 0. 42

Loyalty program with myopic consumers In this case the ﬁrm has two choices:

1) Set a “low” G so that the H-type ﬂies Gold – i.e., G ≤ aH + g − p. Then, πMyopic,1 = (aL − p)p + (aH +

∗ 1 ∗ g − p)(p − c), which is maximized at pMyopic,1 = 4 (aH + aL + c + g) with the optimal value of πMyopic,1 = 1 2 2 2 2 8 (aH + aL + c − 6cg + g + 2aH (aL − 3c + g) + 2aL(c + g)) .

2) Set a “high” G so that the H-type ﬂies regular. Then, πMyopic,2 = (aL − p)p + (aH − p)p, which is maxi- ∗ 1 ∗ 1 2 mized at pMyopic,2 = 4 (aH + aL) with the optimal value of πMyopic,2 = 8 (aH + aL) . Note that the L-type will ∗ ﬂy regular if p < aL, which implies aH < 3aL − c − g.

∗ ∗ It is not diﬃcult to verify that πMyopic,2 ≥ πMyopic,1 if c ≥ g/3. Under this condition, comparing the optimal ∗ ∗ solution with myopic consumers, πMyopic,2, with the optimal solution under the mileage program, πM , it ∗ ∗ p 2 2 is possible to show that πMyopic,2 > πM only if aH > 3aL − c + 3g + 2 (aL − 2aLc + 4aLg − 2cg + 3g ).

However, the myopic solution holds when aH < 3aL − c − g. Since the intersection of these intervals in null, the statement of the proposition follows. Q.E.D.