Points mechanisms and dynamic contracting with limited commitment Emil Temnyalov∗ September 19, 2017

Abstract I study the role of points programs, such as frequent flyer and other reward programs, as a revenue management tool. I develop a two-period contracting model with limited or no commitment, where a capacity-constrained firm faces consumers who learn their willingness to pay over time. The firm cannot commit to future contracts ex ante, but it can commit to allocate any unsold capacity through a points program. I characterize the optimal mechanism in this setting: the firm sells “points” to consumers ex ante, it posts a price at the interim stage, and finally it allocates any unsold units to members of its points program through a lottery. The points scheme creates an endogenous and type-dependent outside option for consumers, which generates novel incentives in the firm’s pricing problem. It induces the firm to do less screening at the interim stage, and to offer lower equilibrium prices, reversing the intuition of demand cannibalization. The optimal mechanism with limited commitment yields larger revenue than the canonical static screening mechanism and is more efficient, but does worse than the best full-commitment mechanism, and features some distortion. Keywords: Reward programs, loyalty programs, dynamic contracting, revenue management, limited commitment, points mechanisms. JEL: D21, D42, D61, D86, L10, L93

∗

University of Technology Sydney, Economics Discipline Group; [email protected]. I have benefited greatly from comments and suggestions by David Besanko, Jeff Ely, Bill Rogerson and Asher Wolinsky, and from discussions with Kevin Bryan, Keiichi Kawai, Anton Kolotilin, Mara Lederman, Jorge Lemus, and John Wooders, as well as from the expertise of Ravindra Bhagwanani of Global Flight. I have also benefited from comments by seminar audiences and discussants at IIOC 2015, AETW 2016, the Econometric Society Asia Meeting 2016, the UQ IO–theory workshop 2016, Drexel University, Monash University, Northwestern University, University of Melbourne, UNSW, University of Technology Sydney, University of Toronto, and University of Washington Bothell.

1

Introduction

Points programs are schemes where a firm creates some form of currency (e.g. frequent flyer miles or hotel guest points), designed to influence consumers’ choices and incentivize particular behavior. Such points are typically either bundled with the purchase of another good (e.g. a flight or a hotel stay), or sold separately (e.g. through a credit card issuer or a bank). Consumers accumulate this currency over time and eventually can redeem it for goods. Such points programs are thus a type of indirect allocation mechanism through which the firm allocates goods to some types of customers, and one can naturally analyze such “points mechanisms” in the framework of mechanism design. The economics literature on reward programs has generally focused on their role as a tool to build customer loyalty, or as a way for the firm to introduce switching costs into the consumer’s choices. In this paper I focus on a different, complementary function of rewards programs as a way to implement a form of dynamic pricing, where the firm offers consumers the option to participate in a points program and to obtain the good through it with some probability. Consider, for example, frequent flyer programs, one of the most well known types of such points mechanisms. In practice, industry experts often describe such programs as a tool for airlines to sell “distressed inventory,” i.e. residual unsold capacity.1 In recent decades such programs have become one of the most significant drivers of profitability in the airline industry, and also account for a significant share of seats.2 Moreover, there is now a very sizable market for airline miles: a majority of miles are in fact not earned by customers who fly with an airline, but sold to banks, who pre-purchase them from the airline and award them to their own customers as promotional incentives.3 On average consumers collect points over a period of 30 months before they redeem them for an award seat, which suggests that points programs also introduce a novel dynamic aspect into the firm’s pricing problem, in addition to the usual revenue management question of how to set prices over time.4 In the airline market, as well as in other industries that fit my model, such as hotels and car rentals, 1 See for example “Persuasive Perks: The World of Loyalty Programs” (CBC, May 26 2016); “Exploring the leverage in airline loyalty” (Tnooz, Jan 22 2015); “The Information Company” (Oracle report, 2008). 2 See for example “Airlines Make More Money Selling Miles Than Seats” (Bloomberg, March 31 2017). In 2007 financial analysts estimated the value of United’s MileagePlus program to be approximately $7.5 billion, and concluded that the loyalty program, if it operated as a stand-alone business, could be worth more than United itself, which at the time had a market value of $5.45 billion. See for example “Getting mileage out of frequent fliers” (Chicago Tribune, July 24 2007). MileagePlus had annual revenues of $3 billion in 2010, according to IdeaWorks: “Loyalty by the Billions” (IdeaWorks, September 12 2011). At American Airlines, on average 8% of seats are now taken up by consumers who redeem miles, rather than buy a ticket. 3 Industry expert Ravindra Bhagwanani of Global Flight estimates that, for a typical major U.S. airline, approximately 30-45% of miles issued are awarded to the airline’s own flyers and 5-10% to partner airlines customers, 40-55% are sold to credit card issuers, 3-8% to hotel programs, 2-5% to car rental companies, and 2-10% to retail and other partners. 4 “The Price of Loyalty” (IATA special report, August 1 2012).

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the firm determines the rate at which points are issued, it sets the prices that it charges for redemptions in terms of points, and it chooses how much capacity to make available for such redemptions. Points mechanisms can therefore be a valuable revenue management and dynamic price discrimination tool. A natural question in this setting is how the firm can use a points program in conjunction with its usual pricing mechanism, to maximize revenues. Consider the pricing problem of a capacity-constrained firm, as in the examples of the airline and hotel industry. If the firm anticipates that demand in some particular period is low, it may optimally decide to set prices such that it does not sell all of its capacity. The firm could in this case offer a points program and allow some of its customers who participate in this program to redeem their points for some of the residual unsold capacity. However, this introduces the possibility of demand cannibalization: sophisticated consumers should anticipate that they may obtain the good through the points program, which will affect the price they would be willing to pay for the good outside of this program. The terms of the points program thus clearly affect the firm’s pricing decisions. I show that this intuition for demand cannibalization is in fact incomplete: although at any given price the revenue-generating demand for the good indeed decreases when the firm allows consumers to redeem their points for the good, the firm’s optimal price is in fact lower when it offers this additional channel to obtain the good, and moreover in equilibrium the firm sells more units, in addition to whatever residual capacity is allocated through the points program. The firm can use this as a form of endogenous pricing commitment and extract larger revenue overall, even in the absence of competition, i.e. in a setting where loyalty and switching costs cannot play any role. I develop a screening model of points programs, to explain why such programs are profitable and how firms can design them optimally. I consider a model where a seller, the firm (e.g. an airline), has a fixed capacity of a good to sell to a population of buyers, the consumers. The seller and the buyers interact over two periods: period 1 is an ex ante contracting stage; period 2 is a standard spot market where the seller offers a mechanism to allocate its capacity among the buyers. In period 1 neither side of the market has any private information; in period 2 consumers privately learn their valuations for the good, while the seller only knows the distribution of buyers’ valuations (i.e. types), as in the standard mechanism design problem. In this setting, I study the problem of dynamic contracting with limited commitment: suppose that in period 2 the seller has full commitment regarding the contract it offers at that stage, but it cannot commit to it ex ante, in period 1. Instead, suppose the seller can only make a specific kind of commitment in period 1: it can sell “points” to the potential buyers and promise to allocate any unsold goods in period 2 to buyers who participate in this points scheme. How will the ability to offer such a scheme affect the optimal mechanism that the seller offers in the later stage, and how should the seller sell access to this scheme? 2

In my model the points scheme provides the seller with one more tool to allocate its capacity dynamically, in addition to the mechanism that it can offer in period 2. This is a natural way to introduce limited commitment into a sequential screening model, which is motivated by the many examples of such points schemes in practice, such as frequent flyer and hotel programs. Here, the seller has full commitment regarding the contract that it offers in period 2, but cannot commit to it ex ante in period 1, and instead can only offer some restricted types of promises. This setting falls inbetween two alternative canonical assumptions regarding commitment: in the sequential screening literature the seller is generally assumed to have full commitment, and hence the ability to offer a two-period mechanism in period 1; in the literature on dynamic mechanism design without commitment the seller is assumed to have full commitment within each period, but no commitment at all regarding future periods. I characterize the seller’s optimal points mechanism, where it chooses how many points to sell in period 1 and at what price, it offers a posted price contract for the good in period 2, and finally it allocates any unsold units through a lottery among buyers who bought points in period 1 but did not buy the good in period 2. I find that the seller’s revenue is larger than that in the optimal static mechanism, but smaller than that in the optimal dynamic mechanism with full commitment. That is, the points program provides the seller with a valuable type of commitment, which it can use to partially contract with future buyers ex ante and increase its overall revenue. On the other hand, this type of program is of course not as valuable as the dynamic mechanism that the seller would offer if it could somehow fully commit to all future contracts ex ante. To develop some intuition it is useful to first think about the optimal dynamic mechanism with full commitment, and about the optimal static mechanism in the absence of a points scheme. It is easy to see that with full dynamic commitment, a la Courty and Li (2000), in period 2 the seller would allocate all units efficiently, to the set of highest types such that all of its capacity is used up, and in period 1 it would charge all potential buyers a price for the right to participate in the period 2 allocation, where this price extracts all of their expected surplus from period 2, as the consumers are homogeneous in period 1. On the other hand, in a static setting the optimal mechanism is a simple posted price, where the seller’s capacity constraint may or may not be binding, depending on the parameters of the model. My model has a continuum of buyers, and hence there is no uncertainty regarding the aggregate distribution of realized types in period 2, so the seller can implement the optimal allocation with a posted price.5 If this leaves some capacity unsold, it is natural to ask whether the seller could do better by allocating any unsold capacity through some other means, such as a points scheme. 5 I.e. there is no gain from using anything more sophisticated, such as an auction for example, since there is no aggregate uncertainty.

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Now consider the points mechanism with limited commitment. The seller sells points to the buyers in period 1, and promises that any unsold capacity in period 2 will be allocated among those who buy points. The seller cannot commit in period 1 as to what contract it will offer in period 2, so the latter must be sequentially optimal, given the points scheme that the seller offers in period 1 and the buyers’ choices.6 Given the existence of the points program, in period 2 a consumer has two possible ways to obtain the good: through the spot contract that the seller offers at that stage, or through the points program. The latter introduces a typedependent endogenous outside option. Redemptions through the points program generally involve rationing, since any residual capacity is allocated among all points buyers, and each of them can only receive the good with some probability. Moreover, I show that the period 2 allocation is monotonic in type, which helps to characterize the relationship between prices and each type’s equilibrium purchase decision. In equilibrium consumers buy at the posted price if and only if their type exceeds a common threshold: relatively high types buy a unit at the posted price, whereas low types choose the points lottery and only receive a unit with some probability. The lottery creates a positive outside option, so the posted price in the spot market must leave the marginal buyer some rents, to induce her to buy at that price rather than take the lottery. Hence by offering a points program ex ante, the seller seemingly cannibalizes some of its demand in period 2: at any price, fewer types would buy than would if the points program did not exist. Importantly, the optimal price is in fact lower than the optimal price in the standard static monopolistic screening mechanism, and moreover the marginal buyer type is lower than in the static mechanism. In other words, if the seller sells points in period 1 it will subsequently offer a contract whereby it sells more units in the spot market, at a lower price (relative to the static screening mechanism), and in addition it will give away any remaining unsold capacity. The sequentially optimal contract in period 2 is thus more efficient than the static benchmark. Intuitively, in the optimal points mechanism the seller sets a lower price and sells more units, because it has an additional incentive to increase the quantity sold that is not present in the usual monopolistic screening trade-off. When deciding whether to sell an extra unit on the margin, the seller takes into account the typical effects this will have: the additional revenue from the marginal unit, and the necessary decrease in price for infra-marginal units. But it now also faces an additional consideration: if it sells an extra unit, it induces less generous rationing in the points lottery, i.e. it decreases the equilibrium probability that a consumer who takes the lottery will obtain a unit. This implies that the outside option of each type decreases, and so the price that the seller can charge the marginal buyer type increases, thus 6 In contrast, in the dynamic mechanism with full commitment the seller would want to commit to a fully efficient contract in period 2, which is not sequentially optimal.

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providing the seller with an additional incentive to increase the quantity sold on the margin.7 While this suggests at first that the adoption of a points program is costly, because it leads to some demand cannibalization and induces equilibrium prices that are below the optimal static screening prices, the seller does better with the points mechanism overall. In particular, total surplus in the points mechanism is larger than in the optimal static mechanism, and the seller can capture all of this incremental gain in surplus as revenue when it sells points in period 1. Because the period 2 contract depends on whether the consumers buy points or not in period 1, the price of points extracts all of the expected incremental surplus from period 2. Moreover, I show that it is optimal for the seller to sell points to all consumers in period 1, as this maximizes the price it can charge for points. Consumers are homogeneous ex ante, so at the optimal price of points in period 1, each of them is in fact indifferent between buying points or not. As an extension, I also consider the question of what happens if the seller has no dynamic commitment at all: i.e. if it offers a points program in period 1, but cannot commit that it will allocate all unsold units through it in period 2. Instead, suppose the seller decides in period 2 how much of its unsold inventory to allocate through the points program. This model has multiple equilibria, which can however be ranked according to the seller’s revenue. Interestingly, the seller’s revenue-maximal equilibrium without dynamic commitment is equivalent to the optimal points mechanism with limited commitment. On the other hand, the seller’s revenue-minimal equilibrium is equivalent to the optimal static screening mechanism. Thus the assumption of limited commitment is in some sense non-essential—even if the seller could not perfectly commit to allocate all unsold units through its points mechanism, the best-case equilibrium without commitment would achieve the same allocation and revenue. How to model limited commitment in a dynamic contracting setting is largely still an open question. This paper proposes one type of limited commitment, which is motivated by the examples of loyalty or points programs in the airline, hotel, and other industries. I study why such programs are profitable and how they should be designed, so the features of my model are motivated by the features of the industries where we find such programs. First, I study a setting where sellers can interact with potential future buyers in some limited way, by offering a points program ex ante, before buyers have private information. This is largely the case in practice: consumers generally choose which programs to participate in well before they precisely know their future demand for specific flights, hotels and other goods, and before the seller has offered any specific contracts for these goods. Second, at the time the seller 7

Notice that if the firm could commit to a period 2 mechanism ex ante, this consideration would be absent. In the optimal mechanism with commitment the firm fully internalizes the effect of the period 2 allocation, and has no incentive to increase the quantity sold in order to decrease the buyers’ outside options.

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offers such a points program, it cannot commit to what contracts it will offer in the future, in the spot market for the good. It is natural to think of the ex ante stage as one where the seller designs its points program (e.g. the earning and redemption rates of its frequent flyer program), and to think of the spot market as one where it sets the prices of the goods it sells (e.g. the prices of seats in different airline route markets). Finally, this paper shows that points mechanisms are useful even in the absence of competition, and can create value above and beyond that which a seller could create by using points schemes as a loyalty building tool or as a way to create switching costs. Indeed, my model features a single seller, so such competition-softening motivations clearly do not apply, and yet the points mechanism turns out to be a valuable revenue management tool.8

2

Related literature

Commitment is an essential assumption in the mechanism design literature: when a seller offers a mechanism to sell a good to privately informed buyers, it must be able to fully commit to the rules of the mechanism. This paper presents a setting where we can partially relax the assumption of full commitment, in the sense that in my model the seller cannot commit to a mechanism in period 1, but it can only sell the promise to allocate any unsold capacity from period 2 through its points program. The seller only has commitment in period 2 regarding the mechanism that it offers at that stage, and the mechanism that is offered in period 2 endogenously determines the value to consumers of the seller’s points program in period 1. One can think of this setting as a standard static contracting problem augmented with an ex ante stage where the seller can sell to consumers the promise to allocate all of its units through some mechanism. Hence this model provides a novel setting of dynamic contracting with limited commitment. I consider a two-period model where consumption happens in the second stage, while the first stage is only a contracting device. The model is thus most closely related to the literature on sequential screening. Courty and Li (2000), a seminal paper in this area, studies optimal dynamic screening when consumers learn about their valuation for a product gradually. In their setting the firm has full commitment power when it contracts in the first stage, and the optimal contract is a refund contract whereby a consumer whose final realized valuation is low receives a refund for her purchase. Intuitively, the less informative a consumer’s initial private information is about her final valuation, the more surplus the firm can extract. Akan, Ata and Dana (2015) generalize this model and extend the intuition to a setting where consumers 8

In practice many industry commentators discuss frequent flyer and other loyalty programs in a similar way: airlines typically use their points programs to allocate any unsold (or anticipated to be unsold) inventory to consumers who redeem points for it, while also balancing the risks of demand cannibalization.

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learn about their types at different times. In that case, the optimal mechanism is also an initial contract with a refund clause. In contrast to Courty and Li (2000) and Akan et al. (2015), I model the case where the firm cannot commit to a period 2 mechanism and cannot sell full contracts ex ante, and where the firm has fixed capacity. Deb and Said (2014) weaken some of the commitment assumptions of the sequential screening literature. In particular, in their model consumers arrive in two cohorts, and when the firm sells contracts to the first cohort it cannot commit to the prices that it will offer in the second cohort. The anticipated second-period contract thus offers an endogenous outside option to period 1 buyers. I further weaken the firm’s commitment power and contracting ability: I consider a setting where the firm cannot offer a long-term contract in the first stage and cannot commit to future prices. Instead, the firm’s commitment to a points program in period 1 creates a type-dependent outside option that consumers have when contracting in period 2. Hua (2007) studies an auction where the seller has the ability to contract with one uninformed buyer prior to the auction. The optimal contract in this setting has several similarities to the optimal mechanism in my setting: it induces less rent-seeking by the auctioneer in the auction stage, it increases the probability of trade by favoring the contracted buyer, and it may increase total surplus, which is intuitively similar to the effect of ex ante contracting that I consider. Unlike Hua (2007), in my model the firm can interact with all consumers ex ante, and has a more limited form of commitment regarding the actual trade stage. This paper is also related to the large literature on revenue management with strategic buyers, such as Aviv and Pazgal (2008), Jerath, Netessine and Veeraraghavan (2010) and Board and Skrzypacz (2016), among others. Hörner and Samuelson (2011) study the problem of a monopolist who posts prices over time in order to sell her inventory before a deadline, where the seller lacks commitment. Dilme and Li (2016) also study revenue management in a dynamic model with no commitment: a seller has some fixed number of units to sell before a deadline, buyers arrive over time and can strategically time their purchases, and the seller cannot commit to what prices it will offer in the future. Öry (2017) studies a dynamic pricing model where a seller can offer targeted sales over time, which induces lower regular prices due to lack of commitment. Unlike the existing revenue management literature, which has generally focused on models with either full or no commitment, I introduce a new notion of limited commitment and explore the implications for the seller’s dynamic pricing problem. A recent literature in management considers the question of how a firm could exploit unsold inventory to incentivize customer purchases. Kim, Shi and Srinivasan (2001) study the design of reward programs and their effect on competition in a setting where consumers differ in their ability to participate in rewards programs and in their price sensitivity. Kim, Shi and

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Srinivasan (2004) study the role of rewards programs in capacity management with competing firms. By offering some of its capacity through a rewards program, in a setting with demand volatility, the firm can induce higher equilibrium prices. I consider a setting with a single firm and no aggregate demand uncertainty, and show that rewards programs can also be a valuable revenue management tool, even absent capacity management and competitive effects. This paper also relates to the extensive literature on endogenous switching costs in a horizontally differentiated market. Such models typically study a two-period oligopolistic model where competing firms sell a product in both periods and the firm benefits either from the ability to write long term contracts that lock consumers in, or from the ability to learn about a buyer’s unchanging preferences. Fudenberg and Tirole (2000) discuss a setting where consumers’ preferences are fixed over time and firms cannot commit to future prices or write long term contracts. The firm learns about a consumer’s preferences based on her first period purchase, and uses that information to price discriminate in the second period. Caminal and Matutes (1990) study a model where consumer preferences change over time, so a buyer’s first period purchase reveals no information about her second period preferences. However the firm can commit to offer a discount to repeat buyers in period 2, which softens the effect of competition in the latter stage. Fudenberg and Tirole (2000) also consider a setting where the firm offers both long term and short term contracts, which induce consumers who strongly prefer one firm to purchase the former type of contract, while those with weaker preferences buy the latter. Stole (2007) provides a comprehensive overview of the literature on switching costs in horizontal differentiation models. While the above papers study dynamic contracts as a way to implement switching costs, I focus instead on a monopolistic setting where there is no scope for switching costs to play any role, and I study the dynamic price discrimination aspect of points and loyalty programs. More generally, the paper relates to the extensive literature on dynamic mechanism design. Among the earliest papers on this subject, Baron and Besanko (1984) study a multiperiod model where a firm has private information about its costs and repeatedly interacts with a regulator who designs a mechanism to set the price of the firm’s output. In their paper, the regulator can fully commit to a dynamic mechanism. Pavan, Segal and Toikka (2014) and Battaglini and Lamba (2015) study incentive compatibility in dynamic mechanism design settings with commitment. The former provide a dynamic envelope formula for local incentive compatibility and discuss revenue equivalence and implementability, while the latter study global incentive compatibility in settings where local constraints are insufficient. In contrast to the above papers, which feature dynamic information, Said (2012) and Board and Skrzypacz (2016) study models with a dynamic population of agents with fixed private information, where the designer can commit to a mechanism. Skreta (2006) considers optimal dynamic mechanisms in a model where the seller has no commitment, and shows that posted prices 8

maximize revenue in this setting. The results in this paper also share some intuitive similarity with models of monopoly pricing with demand uncertainty, such as in Nocke and Peitz (2007). The latter study clearance sales, whereby a monopolist sells to high consumer types with certainty at a high initial price, and later sells to low types with some equilibrium probability. Hence their results are qualitatively similar to the type separation that occurs in period 2 of the model in this paper. In contrast to Nocke and Peitz (2007), I study a two-period model where the firm deals with all consumers in the first period, and it is the interaction between period 1 actions and the period 2 mechanism that drives firm revenue.

3

Model

Consider a seller (or firm) who serves a continuum of mass 1 of buyers (or consumers), each of whom demands 1 unit of a good. The seller has a capacity of k < 1 units and a marginal cost normalized to 0. The seller and its potential buyers interact in two periods, t ∈ {1, 2}, where period 1 is an ex ante contracting stage and period 2 is a spot market for the good. Consumption happens only in period 2. In period 1 the seller and the buyers have no private information: both sides of the market only know the distribution of future realizations of uncertainty. In period 2 each consumer privately learns her type, i.e. her valuation or consumption utility for the good; a mass m ≤ 1 of consumers draw their valuations, v, from some commonly known and differentiable distribution F with support normalized to [0, 1], while the remaining mass 1 − m of consumers have a valuation of v = 0. This environment has two main features: first, consumers privately learn their types over time, and second, only some subset of potential buyers will be active in the market ex post.9 The seller cannot sell long-term contracts, and in particular cannot commit in period 1 as to what pricing scheme it will offer in period 2. Instead, suppose the seller can create a “points program” in period 1, and promise that it will allocate any residual unsold capacity in period 2 among consumers who participate in this points scheme. At t = 1 the seller sells points to consumers who participate in this scheme—it chooses what subset of consumers, φ, to sell points to, and what fee to charge for points, denoted by q.10 At t = 2 each consumer learns her private valuation, the seller announces a mechanism to sell the good in the spot market, 9

We can think of the entire set of agents as the set of all potential consumers in a market, and the subset m as those who turn out to be active in some period of time. For example, in a particular airline route market, the mass of agents m may be those looking to get a seat for a specific date. 10 In the example of the airline industry, this fee represents the implicit price of frequent flyer miles which are bundled into the tickets that a consumer has bought in the past. Without loss of generality in my model a consumer buys a single point, which represents some number of frequent flyer miles required for a redemption.

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buyers participate simultaneously, and finally any unsold units are allocated through a lottery among those consumers who bought points at t = 1 and did not obtain a unit in the spot market at t = 2. The seller will generally need to ration unsold units among participants in the points scheme, and I naturally model this as a lottery.11 The seller can condition the contracts it offers at t = 2 on whether a consumer participates in the points scheme or not, because it has access to this information, and it is relevant to the consumer’s purchase decision at t = 2 since it determines the consumer’s endogenous outside option.12 Consumers are fully rational, so the value of the points scheme at t = 1 is endogenously determined by the sequentially rational mechanism that they anticipate the seller will offer at t = 2. In this model, period 1 represents the period when the seller sets the terms of its points scheme (e.g. an airline sets the earning and redemption rates for its frequent flyer program), while period 2 represents the market for the good itself (e.g. the market for some airline route on a specific date), where consumers have private information about their willingness to pay. Furthermore, the seller cannot sell long-term contracts in period 1.13 The commitment to allocate any unsold units through the points program is the only commitment or contract that the seller can offer at t = 1. As an extension, I later relax this assumption and study a version of the model where the seller decides ex post how much of its unsold units to allocate through the points scheme, rather than committing at t = 1 to allocate all unsold units.

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Analysis

I begin with a remark which points out that any mechanism that the seller offers in equilibrium at t = 2 must be a set of posted prices, since the model has a continuum of buyers. Remark 1. The sequentially optimal mechanism at t = 2 is a pair of posted prices: p for consumers who have bought points at t = 1, and pe for consumers who have not. At t = 2, for any history of play, the seller faces 2 populations of buyers—some mass φ of consumers who have bought points at t = 1, and the remaining mass 1 − φ who have not. Notice also that consumers are homogeneous at t = 1, so whether a particular consumer 11

In the example of frequent flyer programs, a consumer who tries to redeem her airline miles for a seat generally does face a lottery, in that only some seats are made available for redemption at all, and they are generally made available at random times, so she will only receive a seat with some probability. 12 In the example of frequent flyer programs, airlines indeed offer prices that discriminate between members and non-members of their frequent flyer program, since tickets are typically bundled with miles, and members effectively receive a rebate when they purchase a ticket, in the form of miles. 13 In the airline example such contracts are indeed not sold, as they would be too complicated to implement. In practice, selling long-term contracts ex ante would mean selling to each consumer contracts for every possible flight that she might want to buy, for any route and any future time.

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has bought points or not carries no information regarding their type at t = 2. The optimal mechanism in this setting must be a set of posted prices, one price for each group of buyers, because within each non-empty group of buyers we have a continuum of agents, the seller knows the distribution of realized types, and there is no aggregate uncertainty. Hence we can restrict attention without loss of generality to contracts where the seller posts a price for consumers who have bought points, and a price for those who have not, denoting the former by p and the latter by pe.

4.1

The static and full commitment benchmarks

As a benchmark for comparison I consider the optimal mechanisms in two different settings: in a static model where t = 2 is the only period, and in a dynamic model with full commitment where at t = 1 the seller can commit to a mechanism for t = 1, 2. The usual monotone hazard rate property (MHRP) on the distribution of types, F , is sufficient to guarantee the existence of an optimal mechanism.14 Assumption 1. Assume the hazard rate to v¯ =

F 0 (v) 1−F (v)

is increasing in v, and denote by v¯ the solution

1−F (¯ v) F 0 (¯ v) .

The following remark characterizes the optimal posted price that maximizes the seller’s revenue: the seller sets p¯ = v¯ and sells a unit to all types with non-negative virtual surplus if the capacity constraint is not binding, or sets p¯ such that it sells all k of its capacity to the k highest types with positive virtual surplus. Demand in each case is equal to m(1 − F (¯ p)). This is the usual static monopoly pricing problem with a capacity constraint. Remark 2. The optimal static mechanism under Assumption 1 is a posted price p¯:

p¯ =

 v¯

if

k ≥ m(1 − F (¯ v )),

F −1 (1 − k ) m

if

k < m(1 − F (¯ v )).

The corresponding seller’s revenue is

¯ = Π

 (1 − F (¯ v ))2  m

F 0 (¯ v)

kF −1 (1 −

 

k m)

if

k ≥ m(1 − F (¯ v )),

if

k < m(1 − F (¯ v )).

(v) More generally, we only need F to be regular, i.e. the virtual value v− 1−F to be increasing. The MHRP F 0 (v) is a more intuitive sufficient condition for F to be regular, and is satisfied for a wide variety of distributions of interest, such as the uniform, power, normal, logistic, exponential, etc. 14

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In contrast, now consider a two-period setting with full commitment. Suppose the seller can offer a two-period mechanism at t = 1, so it can commit ex ante to the prices it will set at t = 2. The following remark characterizes the optimal mechanism in this setting, which consists of a participation fee at t = 1 and an ex post efficient price at t = 2. The reason for this result is obvious: recall that consumers are homogeneous at t = 1, so the seller can offer a fully efficient allocation at t = 2, and it can extract all of the expected surplus at t = 1, by charging a fee to participate in the period 2 allocation. This mechanism achieves the maximal feasible total surplus, and the seller captures all of this surplus. The mechanism is essentially the limit case of Courty and Li (2000), in the limit where buyers are homogeneous at t = 1. Remark 3. The optimal dynamic mechanism with full commitment consists of an ex post efficient posted price p∗ at t = 2: ∗

p =F

−1



k 1− , m 

and a participation fee q ∗ at t = 1 ∗

q =m

4.2

Z 1 p∗

(v − p∗ )dF (v).

Points mechanisms with limited commitment

I now turn to the main setting of this paper. Suppose the seller only has limited commitment: at t = 2 it can offer a contract (p, pe) to sell its inventory, and at t = 1 it can offer a points program, whereby it promises to allocate any unsold capacity at t = 2 to consumers who participate in this program. In particular, the seller cannot commit at t = 1 as to what contract it will subsequently offer at t = 2, so the value of its points program to consumers will be determined endogenously by the sequentially optimal contract that the seller offers at t = 2 in equilibrium. After consumers simultaneously decide whether to buy at the posted prices, any residual unsold capacity is allocated among consumers who bought points at t = 1 and did not buy a unit of the good at t = 2. Generally, this residual capacity must be rationed among such consumers, i.e. the seller must offer a points lottery to allocate this unsold capacity. First consider the case where k < m(1 − F (¯ v )). This is the case where the seller’s capacity constraint is binding for the optimal period 2 contract. I show that in this case the seller cannot allocate any units through the points program at t = 2, since the sequentially optimal contract allocates all k units at the posted prices, i.e. there is no residual unsold capacity. In fact, in this case the optimal contract at t = 2 is equivalent to the optimal static contract,

12

and the seller cannot do better by offering a points program at t = 1. The intuition for this is simple: when the capacity constraint is binding, the marginal buyer type has positive virtual surplus, so the seller can increase its revenue by selling all k of its capacity, hence setting all buyers’ outside options to 0, as there is no unsold capacity to be allocated through the points mechanism. If the seller were to set higher prices, so that not all capacity is sold at the posted prices, this would mean the marginal buyer has a positive outside option. Hence by lowering prices, the seller captures the usual (positive) virtual surplus, plus some additional surplus from the fact that the marginal buyer’s outside option decreases, as there is less capacity allocated through the points mechanism. The sequentially optimal prices at t = 2 in this case are such that all k capacity is sold. Proposition 1. Suppose k < m(1 − F (¯ v )) and Assumption 1 holds. The optimal contract at t = 2 consists of (p, pe) such that 

p = pe = F −1 1 −

k , m 

the optimal fee at t = 1 is q = 0, and the seller’s overall revenue is Π = kF

−1



k 1− . m 

Proposition 1 shows that if the seller’s capacity constraint is binding at the optimum, the best it can do is to offer the optimal static price, and a points mechanism cannot do better. The ability to commit to allocate any residual capacity ex post has no value to the seller, since it cannot credibly offer a contract where there will be any residual capacity. Buyers anticipate this, and hence at t = 1 are willing to pay 0 for points. Since in equilibrium no units are allocated through the points program, every consumer’s outside option is indeed 0. For the remainder of this section I therefore focus on the case where k ≥ m(1 − F (¯ v )), i.e. where the capacity constraint is not binding at the optimum, which is the natural case to study points mechanisms. As in the static and full commitment benchmarks, I impose a kind of hazard rate condition on F , which guarantees that an optimal mechanism exists.15 Assumption 2 implies Assumption 1, and moreover it can easily be shown that vˆ ≤ v¯. Assumption 2. Assume that the modified hazard rate F 0 (v) 1 1 − F (v) 1 − m(1 − F (v)) 15

As before, this condition is just an intuitive sufficient condition. More generally, F only needs to be regular, (v) in the sense that there is a unique solution to the expression v = 1−F (1 − m(1 − F (v))), which is analogous F 0 (v) to the standard regularity assumption. Notice that this condition can hold for a variety of distributions, e.g. when F is uniform, or a power distribution. Assumption 2 is a sufficient condition for this to hold.

13

is increasing in v, and denote by vˆ the solution to vˆ =

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] . 0 F (ˆ v)

Now consider the seller’s choice at t = 1 of what fraction of buyers to sell points to, i.e. the optimal choice of φ. I show that it is optimal for the seller to set φ = 1 and sell points to all consumers ex ante. Lemma 1. Under Assumption 2 any points mechanism with φ ≥

k−m(1−F (ˆ v )) 1−m(1−F (ˆ v ))

is feasible, and

φ = 1 is optimal. The proof of this lemma shows that for any feasible φ the seller posts prices at t = 2 such that it sells to the same subsets of consumer types among the set who bought points and among the set who did not buy points.16 That is, in the optimal mechanism a consumer buys a unit at t = 2 if and only if her type v is greater than some threshold type, which does not depend on whether she owns a point or not. Intuitively, when some residual units are allocated through the points lottery, consumers who own points are charged a lower price than those who do not, when the seller sets prices optimally. But while the price that points owners pay is lower, the allocation of units as a function of type is in fact independent of φ, and points owners and non-owners both obtain a unit according to the same type threshold rule. Hence the seller’s optimal prices induce a points owner to buy a unit if and only if her type is above a particular cutoff, and they induce a non-owner to buy a unit if and only if her type is above that same cutoff. This feature of the mechanism is optimal for the seller, because both points owners and non-owners count against its capacity constraint equally, and selling a unit to either type of consumer has the same effect on the total amount of unsold capacity which will be allocated through the points lottery. Thus the total number of units that is optimally allocated through the points lottery is independent of φ, for φ < 1. This further implies that the total expected surplus from all units allocated through the lottery is constant for φ < 1: decreasing φ, for example, merely divides this surplus across a smaller mass of consumers. Notice also that the seller captures all of this surplus as revenue, through the sale of points ex ante, and hence the fact that it is constant in φ means that the seller cannot do better by decreasing φ, so it is indifferent among all feasible φ < 1. On the other hand, setting φ = 1 means that only a measure 0 of consumers do not own points in period 2, and hence any posted price pe is sequentially optimal. This allows the seller to credibly set pe = 1 at t = 2, and to exclude any consumer who does not buy points, which 16 For φ to be feasible it must be that the mass of points owners who do not buy a unit at t = 2 is at least as large as the mass of unsold units. This condition is necessary, or else the seller’s commitment to allocate all unsold units through the points program would be meaningless.

14

in turn implies that the seller charges a higher fee for points at t = 1. The total surplus generated when φ = 1 is the same as for any feasible φ < 1, provided the seller sets prices optimally in period 2, but in the former case the seller extracts more revenue, because it can credibly exclude a consumer who does not participate in the points program, and thus charge a higher price for points. Hence φ = 1 is optimal. As a consequence of Lemma 1 we can restrict attention to mechanisms with φ = 1 and characterize the optimal posted prices (p, pe) at t = 2. Lemma 2. Under Assumption 2, if k ≥ m(1 − F (ˆ v )), the optimal contract at t = 2 is a pair of posted prices, (p, pe): p=

(1 − k)(1 − F (ˆ v )) 0 F (ˆ v)

and

pe = 1.

The seller’s revenue at t = 2 is Π=

m(1 − k)[1 − F (ˆ v )]2 F 0 (ˆ v)

Intuitively, the lemma shows that there exists a threshold type, vˆ, such that types above it buy a unit at the posted price p, while types below it take the points lottery and only obtain the good with some probability. Since a consumer can obtain a unit through the points lottery, her outside option is better than 0, so for the marginal buyer type, vˆ, the maximum that the seller can charge that type is less than vˆ, i.e. the seller must set some p < vˆ and leave the marginal buyer type some additional rents, to induce that type to buy at the posted price, rather than take the points lottery. Figure 1 provides a graphical representation of how this cutoff type is determined, and the rents that type receives due to the endogenous outside option. Figure 1: The cutoff type vˆ as a fixed point.

The solid line represents the payoff to a type v from buying a unit at price p. The dashed line represents the payoff to type v from taking the points lottery, in the symmetric equilibrium 15

of the period 2 game. Notice that if vˆ is the equilibrium cutoff type, the slope of the dashed line, which is the value of the outside option to a type v, must be equal to the equilibrium probability of obtaining a unit through the points lottery, which must be consistent with the cutoff type vˆ. I.e. the slope of the dashed line itself depends on the cutoff vˆ, so the latter is a fixed point. The precise price p that is optimal is determined by the standard monopoly screening tradeoff, plus the additional consideration that when the seller lowers its price on the margin, it induces more consumers to buy units, which lowers the endogenous equilibrium probability of obtaining a unit through the points lottery. Hence the outside option of those types that buy a unit decreases, and the seller can charge a marginally higher price. Thus the seller now has an additional incentive to lower its posted price and to increase the number of units sold, as that allows it to leave smaller rents to types that buy at the posted price, given the cutoff strategy that buyers play in equilibrium. Next, I consider the seller’s points pricing decision at t = 1. By Lemma 1 it is optimal to sell points to all consumers, i.e. to set φ = 1. The following lemma characterizes the price that the seller can charge for a point ex ante. Lemma 3. Under Assumption 2 the optimal fee for points at t = 1 is q

=

m · [1 − F (ˆ v )] ·

Z 1

(v − p)dF (v)+

vˆ

+

m · F (ˆ v)

Z vˆ

vdF (v) ·

0

where vˆ solves vˆ =

k − m(1 − F (ˆ v) 1 − m(1 − F (ˆ v ))

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] (1 − k)(1 − F (ˆ v )) and p = . 0 0 F (ˆ v) F (ˆ v)

The optimal points fee at t = 1 captures the expected difference in surplus for each consumer between two continuation games—the one where she buys a point at t = 1 and the one where she does not. In the former subgame, she faces the posted price p from Lemma 2 and either obtains a unit at that price, or with some probability obtains a unit through the points lottery. In the latter subgame, she faces the posted price pe from Lemma 3, and is excluded from the points lottery. This difference in expected payoffs is due to two parts: the first term in Lemma 3 is the surplus that each buyer type above vˆ anticipates from buying at the posted price at t = 2; the second term in Lemma 3 is the expected surplus from participating in the points lottery, for types below vˆ. As a consequence of Lemmas 1–3 we can now fully characterize the optimal points mechanism. Proposition 2. Under Assumption 2, if k ≥ m(1 − F (ˆ v )), the optimal points mechanism consists of: 16

• At t = 1 the seller sets φ = 1 and charges a fee q to participate in the points program: q = m[1 − F (ˆ v )][E(v − p|v ∈ [ˆ v , 1])] + mF (ˆ v )E(v|v ∈ [0, vˆ])

k − m(1 − F (ˆ v) . 1 − m(1 − F (ˆ v ))

• At t = 2 the seller posts prices, p for points owners and pe for non-owners, with p=

(1 − k)(1 − F (ˆ v )) F 0 (ˆ v)

and pe = 1, where vˆ is the solution to vˆ =

[1−F (ˆ v )][1−m(1−F (ˆ v ))] . F 0 (ˆ v)

The points lottery

allocates k − m(1 − F (ˆ v )) units. • The expected total revenue from the mechanism is TR

=

q+Π

=

q+

m(1 − k)[1 − F (ˆ v )]2 . F 0 (ˆ v)

In the optimal points mechanism the seller sells points to all consumers in period 1, and then allocates any unsold capacity in period 2 through a points lottery among those consumers who have not bought a unit. The sequentially optimal contract in period 2 consists of a pair of posted prices, and the seller trades off the additional consideration that the prices it sets also determine each consumer’s endogenous and type-dependent outside option. In particular, if the seller marginally lowers prices, consumers anticipate that fewer units will be allocated through the points lottery, hence the value of their outside option decreases, and therefore the seller leaves less rents to the types who obtain a unit at the posted prices. Having characterized the optimal points mechanism, I next discuss its efficiency and revenue in comparison to the static and full commitment benchmarks from Remarks 2 and 3, respectively. Proposition 3. The optimal points mechanism is more efficient and yields larger revenue than the optimal static mechanism. Intuitively, there are two reasons why the points mechanism is more efficient than the static mechanism: first, more units are allocated to a larger set of high types who buy at the posted price at t = 2; second, any residual capacity is allocated among the subset of types who do not buy at the posted price. In both mechanisms buyers play a cutoff strategy when deciding whether to buy a unit in period 2, and the seller’s choice of price induces a particular cutoff type. In the static benchmark types between v¯ and 1 buy a ticket, while types below v¯ do not. In the points mechanism types between vˆ and 1 buy a ticket, while types below vˆ take the points lottery instead. As Proposition 3 shows, vˆ < v¯, which further implies p < vˆ < v¯ = p¯. 17

Hence the points mechanism is more efficient, since the optimal price and respectively the optimal type cutoff, are lower than in the static benchmark. Proposition 3 also shows that the monopolist does better with the points mechanism than with the static mechanism. The intuition for this is simple: because points are sold ex ante, the points fee extracts all of the incremental surplus generated by the points mechanism, in comparison to the static mechanism. Proposition 4. The optimal points mechanism is less efficient and yields less revenue than the optimal dynamic mechanism with full commitment. The optimal dynamic mechanism with full commitment is fully efficient ex post, so it attains the maximal possible total surplus, and moreover the seller captures all of this surplus. In contrast, the optimal points mechanism is not fully efficient, since some units of the good are allocated to relatively low types. In this setting the seller still has an incentive to screen buyers at t = 2, and hence prices are distorted in order to maximize the seller’s period 2 revenue. While this distortion is not as large as in the optimal static mechanism, it means that some surplus is destroyed because of the seller’s inability to commit to prices ex ante. In the mechanism where the seller can fully commit to period 2 prices in period 1, it completely internalizes the effect of any distortions on total surplus, and hence the full commitment mechanism features no distortions. Because the latter attains the maximal possible total surplus, and the seller captures all of this surplus, it of course does better than the optimal points mechanism. However, Proposition 2 shows that the seller can still benefit from a limited form of commitment that is less demanding than assuming that it can fully screen buyers ex ante.

4.3

Points mechanisms without commitment

A natural question in this setting is how important is the precise type of commitment that I assume the seller can make in period 1 regarding its points program. Throughout the paper I assume that the seller cannot commit to a period 2 mechanism ex ante, and it cannot offer type-contingent contracts at t = 1. Instead, period 1 is simply meant to represent a period of time when the consumer does not know whether she will be in the market at all, or what her type will be. I assume that the only commitment the seller can offer ex ante, at t = 1, regarding the mechanism offered at t = 2, is that it can sell “points” and commit to allocate its unsold capacity ex post to some consumers who participate in the points program, i.e. who buy a point at t = 1. As an alternative assumption, one could imagine that the seller cannot fully commit to allocate 18

all unsold units through its points program. Suppose the seller chooses how much of its unsold capacity to allocate to points owners ex post, in a sequentially rational manner. What equilibria can be sustained in this case, and what revenue can the seller capture in this case? To answer this question I first have to slightly generalize the timing of the events in the model. In particular, I assume the following: in period 1 the seller sells points; in period 2 consumers privately learn their types; the seller then posts a price for the good; consumers then simultaneously decide whether to buy a unit or not; finally, the seller decides how much of its unsold capacity to give away to consumers who have not bought a unit. This timing of the model essentially takes my main model and adds a stage at the end, where the seller chooses how much of its unsold capacity to give away. I denote by r the fraction of the good that the seller decides to give away ex post, where r ∈ [0, k − d(p)] and d(p) is the number of units sold at a price of p. For simplicity, I will restrict attention to the case where the seller sells points to all consumers in period 1, i.e. it sets φ = 1 at t = 1. More generally, one can also show that setting φ = 1 is optimal if the firm can choose φ, following the same logic as in the limited commitment case discussed in Lemma 1. The following proposition shows that this extension of the model has a continuum of equilibria, which span the full range of allocations between the static benchmark in Remark 2 and the points mechanism in Proposition 2. Specifically, we have a continuum of equilibria with price ranging between the optimal p in the points mechanism and p¯ in the static benchmark. Correspondingly, in all equilibria consumers play a cutoff strategy when deciding whether to buy a unit or not, and the equilibrium cutoff types cover the range between vˆ and v¯ from Proposition 2 and Remark 2. In this sense, any outcome that is between the outcomes of the static mechanism and of the points mechanism is an equilibrium outcome. Moreover, the revenue-maximal equilibrium for the seller is the one that corresponds to a price of p and a cutoff type vˆ. I.e. the optimal points mechanism in Proposition 2 is identical to the revenue-maximal equilibrium in this extension. Therefore the features of the optimal points mechanism do not depend critically on the exact form of commitment that I assume.17 Proposition 5. Under Assumption 2, the extension of the model where the seller chooses r ex post has a continuum of equilibria. For r ∈ [0, k − m(1 − F (ˆ v ))] there exists an equilbirium with price pr given by: pr ∈ [p, p¯] 17

In this model it is natural to focus on the revenue-maximal equilibrium for the seller, as the model features one large player, the seller, and a continuum of small, non-atomic buyers.

19

where p =

(1−k)(1−F (ˆ v )) , F 0 (ˆ v)

vˆ =

p¯ =

1−F (¯ v) F 0 (¯ v) ,

and vˆ and v¯ solve:

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] F 0 (ˆ v)

and

v¯ =

1 − F (¯ v) . F 0 (¯ v)

The revenue-maximal equilibrium, which is weakly Pareto dominant, coincides with the optimal points mechanism and has: pr =

(1 − k)(1 − F (ˆ v )) F 0 (ˆ v)

and

r = k − m(1 − F (ˆ v ))

Intuitively, when the seller chooses how much of its unsold capacity to give away ex post, it is indifferent among all feasible r because there is no renegotiation at that point. This generates a multiplicity of equilibria, depending on which r the seller chooses at the final stage of the game. In each equilibrium consumers must correctly anticipate the seller’s choice of r. For any r consumers must then play a cutoff strategy when deciding whether to buy a unit or not, with the cutoff v r (p) depending on their belief about r and on the posted price p. Given this, the seller chooses p to maximize its revenue, anticipating that consumers will decide according to the cutoff strategy v r (p). Different equilibrium values of r determine different equilibrium prices pr , ranging between the price p from Proposition 2, and the price p¯ from Remark 2. Hence any outcome that is inbetween the outcomes of the points mechanism and of the static mechanism can be an equilibrium outcome. However, one can rank all of these equilibria and show that the revenue-maximal equilibrium is the one where pr = p, and the seller allocates all unsold capacity through its points program, i.e. r is such that m(1 − F (ˆ v )) + r = k. This is because in all equilibria of the game consumers have the same expected surplus, and the airline extracts all of the total surplus through the sale of points at t = 1. Thus the revenue-maximal equilibrium is the one that generates the largest total surplus, subject to prices being sequentially optimal at t = 2. Hence the seller is better off in the equilibrium where pr is as low as possible, given that it is in the set [p, p¯], and r is as large as possible, given pr .

5

Discussion

This paper studies a novel aspect of points programs, such as frequent flyer and other reward programs: points programs offer firms an additional channel to implement dynamic pricing, and can be profitable even in the absence of competition. While the economics literature has generally focused on the role of points programs in creating customer loyalty and implementing

20

switching costs, I show that points programs are also a valuable sequential screening tool. In particular, a firm can use its points program as a way to indirectly allocate its capacity dynamically, before consumers have private information regarding their valuation for the good. This mechanism works even in the absence of commitment, i.e. even if the seller cannot fully commit as to what kind of mechanism or contract it will subsequently offer in the spot market for the good. I study a setting where a firm can offer consumers the ability to participate in a points program ex ante, and can commit to allocate any residual unsold capacity through this program. I characterize the optimal points mechanism, where the firm sells points ex ante, it then offers a contract to sell the good to consumers who have private information about their valuations, and finally any unsold capacity is allocated through a lottery among consumers who participate in the points program. The introduction of the points program creates an endogenous and type-dependent outside option for consumers at the interim stage, in the spot market for the good. This changes the firm’s pricing incentives, in a way that induces less screening at that stage, and hence a more efficient allocation. The model yields new insights regarding the management and design of reward programs: I show that the optimal mechanism in some sense reverses the standard intuition of demand cannibalization. In general one might expect that if the firm offers consumers two different channels to obtain the good, the introduction of the points program may cannibalize some of the demand in the spot market for the good. This is only partially true—at any given contract, demand in the spot market is indeed lower when the firm offers a points program, since some consumers optimally choose to not purchase, and instead obtain the good through the points mechanism. However, the optimal equilibrium contract changes when the firm offers a points program. In fact, I show that the firm will set lower prices in the game where it also offers a points mechanism. Hence overall we see the opposite of demand cannibalization: by offering a points program, the firm indirectly commits to more efficient pricing in the spot market, and ultimately screens less and sells more of its capacity. The predictions of this model are generally consistent with several important observations from industries that feature loyalty and points programs. First, firms tend to use their loyalty programs as a way to allocate unsold capacity, or “distressed inventory.” This is often noted by industry commentators who observe that airlines, for example, tend to release much of their unsold seats to members of their frequent flyer programs, who can book these seats by redeeming frequent flyer miles. Second, in addition to using pricing algorithms to implement revenue management, in practice airlines also use algorithms to decide how much capacity to release to members of their loyalty programs, and at what time to do so. The design of such algorithms is clearly an important economic problem, because the firm faces a trade-

21

off between its pricing mechanism and its points program. Third, firms generally face a commitment problem in terms of the operation of their loyalty program: when deciding what prices to set, the firm would myopically prefer to make this program less generous, although in the long run this would undermine the value of the program and the premium that consumers would be willing to pay to participate in it. This paper shows that while loyalty programs can still be valuable in a setting with limited commitment, they would indeed be even more valuable if the firm could commit to the terms of its loyalty program. My results also provide some surprising implications for firm strategy: it is tempting to think of loyalty programs as a mechanism that induces demand cannibalization, but this is in fact a primary reason why they are profitable overall, and firms which use such programs as a dynamic mechanism in fact benefit from increasing participation and redemptions through their loyalty program. Furthermore, the results and intuition apply more broadly to industries that share the main features of my model, where consumers have heterogeneous willingness to pay that is revealed over time, the firm has a fixed capacity to sell, and the firm can interact with consumers ex ante, or repeatedly over time, through its loyalty program.

22

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25

Appendix Proposition 1 First consider the case where φ = 0, i.e. at t = 2 all consumers have an outside option of 0. At any posted price pe the total demand by buyers is m(1 − F (pe)). Denote by p0 the price such that m(1 − F (p0 )) = k, the highest price at which the seller sells all k of its capacity. Then any price pe < p0 is dominated by p0 . Furthermore, if k < m(1 − F (¯ v )), then p0 > v¯, and hence p0 −

1−F (p0 ) F 0 (p0 )

> 0. Hence the m(1 − F (p0 )) mass of k highest types all have positive

virtual surplus under Assumption 1. Therefore any price pe > p0 is yields less revenue than p0 , since m[(1 − F (pe)) − peF 0 (pe)] < 0 for any pe > p0 . Hence p0 dominates any other possible posted price, so the optimal set of prices is 

p = pe = p0 = F −1 1 −

k . m 

Now analogously consider the case where φ > 1, i.e. a subset of buyers have a non-negative outside option. At any pair of prices (p, pe) the total demand by buyers is D(p, pe) ≡ φm(1 − F (vφ )) + (1 − φ)m(1 − F (pe)), where vφ ≥ p. Hence total demand is weakly lower at any given price, compared to the case where φ = 0. Moreover, if D(p, pe) < k, some buyers have a strictly positive outside option, and vφ > p. Hence D(p, pe) < k implies D(p, pe) ≤ D(p0 , p0 ). Note that at p = pe = p0 , total demand is exactly D(p, pe) = k, and all buyers have an outside option of 0. Thus (p, pe) dominates any other posted prices. 

Finally, total revenue with p = pe = F −1 1 − Π = kF

k m

−1





is given by

k 1− . m 

Lemma 1 Proof. First consider the case that the seller sets some φ < 1, so that a positive measure of consumers do not buy points in period 1. Given prices p and pe, a consumer who does not own a point at t = 2 buys a unit of the good if and only if her type is v ≥ pe, while a consumer who owns a point must play a threshold strategy with some threshold type vˆ such that she buys a ticket if and only if v ≥ vˆ. A consumer of type v ≤ vˆ takes the points lottery, where vˆ is the cutoff type, which must satisfy the indifference condition: vˆ − p = vˆl

26

where l is the equilibrium probability that a consumer who takes the points lottery obtains a unit, i.e.: l=

k − m(1 − φ)(1 − F (pe)) − mφ(1 − F (ˆ v )) φ − mφ(1 − F (ˆ v ))

Thus we obtain the following indifference condition for the marginal type of a consumer who owns a point and buys a unit at the posted price p: vˆ − p = vˆ

k − m(1 − φ)(1 − F (pe)) − mφ(1 − F (ˆ v )) φ − mφ(1 − F (ˆ v ))

Rearranging, we obtain p = vˆ

φ − k + m(1 − φ)(1 − F (pe)) φ − mφ(1 − F (ˆ v ))

We can now express the seller’s maximization problem in terms of finding the optimal allocation, i.e. the optimal marginal buyer types, pe and vˆ: max m(1 − φ)(1 − F (pe))pe + mφ(1 − F (ˆ v ))ˆ v p e,ˆv

φ − k + m(1 − φ)(1 − F (pe)) φ − mφ(1 − F (ˆ v ))

The first order conditions for this problem give us the following system of equations: 1 − F (ˆ v ) − m + 2mF (ˆ v ) − mF (ˆ v )2 F 0 (ˆ v) (1 − m + mF (ˆ v ))(1 − F (pe)) − mˆ v (1 − F (ˆ v ))F 0 (pe) pe = F 0 (pe)((1 − m + mF (ˆ v )))

vˆ =

We can then verify that the unique solution to the system is pe = vˆ =

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] F 0 (ˆ v)

Notice that under Assumption 2 there exists a unique vˆ that solves this condition. Because in the optimal contract vˆ = pe at t = 2, we see that the set of consumer types who own points and buy a unit is the same as the set of consumer types who do not own points (p e)) , while it charges and buy a unit. The seller charges each point owner p = vˆ φ−k+m(1−φ)(1−F φ−mφ(1−F (ˆ v ))

each non-owner pe = vˆ. Thus the total number of units sold at t = 2 is m(1 − F (ˆ v )), which is independent of φ. The remaining unsold capacity, k − m(1 − F (ˆ v )) is allocated to points owners through the points lottery. It follows that each consumer who buys a point at t = 1 expects the following surplus from the points lottery: mF (ˆ v )E[v|v ∈ [0, vˆ]]

k − m(1 − F (ˆ v )) . φ − mφ(1 − F (ˆ v ))

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Notice that mF (ˆ v ) is the probability that the consumer’s realized type is v ∈ [0, vˆ] in period 2, E[v|v ∈ [0, vˆ]] is her expected type in that event, and

k−m(1−F (ˆ v )) φ−mφ(1−F (ˆ v ))

is her equilibrium

probability of obtaining a unit through the points lottery, given that her type is v ∈ [0, vˆ] and she participates in the lottery. Hence this surplus is included in what the consumer is willing to pay for a point at t = 1. k−m(1−F (ˆ v )) For any feasible φ, the seller charges mF (ˆ v )E[v|v ∈ [0, vˆ]] φ−mφ(1−F (ˆ v )) for each point, and

therefore it captures the following surplus through points sales: mF (ˆ v )E[v|v ∈ [0, vˆ]]

k − m(1 − F (ˆ v )) 1 − m(1 − F (ˆ v ))

Notice that this expression does not depend on φ, so the seller is indifferent among all feasible φ < 1. Next, consider the case where the seller sets φ = 1, so that the set of consumers who do not buy points at t = 1 is of measure 0. Following the analogous derivation from the previous case, at t = 2 the seller’s problem is: max m(1 − F (ˆ v ))ˆ v vˆ

1−k 1 − m(1 − F (ˆ v ))

The first order condition for this problem gives us the same cutoff as before, i.e.: vˆ =

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] 0 F (ˆ v)

Therefore the period 2 allocation is the same for φ = 1 and for any feasible φ < 1. However, setting φ = 1 imposes no restriction on pe, since the set of consumers who do not buy points in this case is of measure 0, and so in period 2 any pe is sequentially optimal. In particular, if the seller sets φ = 1 in period 1, it can credibly set pe = 1 in period 2, hence excluding any consumer who does not buy a point. This threat is larger than the threat that the seller could make if it set any φ < 1, since in that case the optimal period 2 price is pe = vˆ < 1. Hence setting φ = 1 allows the seller to charge a higher price for points in period 1, as it lowers the continuation value to a consumer of not buying a point, given that all other consumers buy points. Thus φ = 1 and any feasible φ < 1 induce the same period 2 allocation and generate the same total surplus, but setting φ = 1 allows the seller to extract more of that surplus as revenue, by allowing it to set a higher fee for points at t = 1. Thus setting φ = 1 is optimal. Finally, we can explicitly characterize which φ are feasible, i.e. φ such that the seller can carry out its commitment to allocate all unsold capacity among points owners. Since vˆ = pe,

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φ is feasible if and only if φ + (1 − φ)m(1 − F (ˆ v )) ≥ k. Equivalently, φ is feasible if and only if φ≥

k − m(1 − F (ˆ v )) 1 − m(1 − F (ˆ v ))

Lemma 2 Proof. By Lemma 1 we can restrict attention without loss of generality to the case where φ = 1. At any posted price p consumers simultaneously decide whether to buy a unit or to enter the points lottery. Let l be the equilibrium probability that a consumer who enters the points lottery obtains a seat. If in equilibrium a type v buys a unit, it must be that v − p ≥ vl, and hence any type v 0 > v must also buy a unit, since v 0 − p − v 0 l > v − p − vl ≥ 0, i.e. v 0 − p > v 0 l. Analogously, if in equilibrium a type v does not buy a ticket, it must be that vl ≥ v − p, and hence any type v 00 < v must also not buy a ticket, since v 00 − p − v 00 l < v − p − vl ≤ 0, i.e. v 00 − p < v 00 l. Hence at any price p there exists a unique cutoff type such that all types above it buy a unit, and all types below it take the points lottery instead. Denote this cutoff type by vˆ. Clearly vˆ must satisfy the indifference condition vˆ − p = vˆ ·

k − m(1 − F (ˆ v )) 1 − m(1 − F (ˆ v ))

where the last term is the equilibrium probability that a consumer who enters the lottery gets a seat, which must be consistent with vˆ. Let H(ˆ v) ≡

k − m(1 − F (ˆ v )) 1 − m(1 − F (ˆ v ))

be the probability that a consumer gets a seat through the FFP lottery, given that all types above vˆ buy a ticket and all types below vˆ take the FFP lottery. Notice that p = vˆ · (1 − H(ˆ v )) and also H 0 (ˆ v) =

(1 − k)mF 0 (ˆ v) mF 0 (ˆ v) = [1 − H(ˆ v )] 2 [1 − m(1 − F (ˆ v ))] 1 − m(1 − F (ˆ v ))

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We can now re-write the seller’s revenue maximization problem at t = 2 in terms of the cutoff type vˆ that it wants to induce in equilibrium to buy a unit: max vˆ(1 − H(ˆ v )) · m(1 − F (ˆ v )) vˆ

The FOC for this problem implies that mˆ v (1 − F (ˆ v ))F 0 (ˆ v) m[1 − H(ˆ v )] (1 − F (ˆ v )) − vˆF (ˆ v) − =0 1 − m(1 − F (ˆ v )) 

0



and hence the revenue maximizing vˆ solves the following equation: vˆ =

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] F 0 (ˆ v)

Notice that under Assumption 2 this vˆ exists and is unique. Next, since p = vˆ[1 − H(ˆ v )], we have that p =

[1−F (ˆ v )][1−m(1−F (ˆ v ))] 1−k F 0 (ˆ v) 1−m(1−F (ˆ v )) ,

so the optimal

posted price is p=

(1 − k)(1 − F (ˆ v )) 0 F (ˆ v)

Demand at p is equal to m(1 − F (ˆ v )), so the seller’s revenue is Π=

m(1 − k)[1 − F (ˆ v )]2 F 0 (ˆ v)

Lemma 3 Proof. Consider wlog the equilibrium where φ = 1, i.e. all consumers buy a point at t = 1. A consumer’s ex ante expected payoff from the continuation game where she buys a point is that of a continuation game where, by Lemma 2, she will face p= where vˆ solves vˆ =

(1 − k)(1 − F (ˆ v )) 0 F (ˆ v)

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] . 0 F (ˆ v)

On the other hand, if a consumer does not buy a point at t = 1, in the equilibrium where φ = 1 it is credible for the firm to charge her pe = 1 in period 2, i.e. the firm will exclude 30

the consumer in period 2, since this minimizes her continuation payoff in the game where she does not buy a point, and therefore maximizes the fee that the seller can charge for points in period 1. Thus the revenue-maximizing fee for points at t = 1 is equal to the difference between the consumer’s continuation payoff if she buys a point and her continuation if she does not buy a point, i.e. her expected surplus in period 2 on the equilibrium path. This expected surplus is composed of the surplus of types who buy a unit in equilibrium, and of types who do not buy a unit but participate in the points lottery. Hence the optimal fee is q

= m · [1 − F (ˆ v )] · [E(v − p|v ∈ [ˆ v , 1])]+ +

m · F (ˆ v )E(v|v ∈ [0, vˆ]) ·

k − m(1 − F (ˆ v) 1 − m(1 − F (ˆ v ))

Proposition 2 Proof. The parts of the proposition follow as a consequence of Lemmas 1–3. In particular, by Lemma 1 it is optimal in period 1 to set φ = 1. By Lemma 3 it is optimal to set q

= m · [1 − F (ˆ v )] · [E(v − p|v ∈ [ˆ v , 1])]+ +

m · F (ˆ v )E(v|v ∈ [0, vˆ]) ·

k − m(1 − F (ˆ v) 1 − m(1 − F (ˆ v ))

By Lemma 2 it is optimal to charge p= where vˆ is the solution to vˆ =

(1 − k)(1 − F (ˆ v )) , F 0 (ˆ v)

[1−F (ˆ v )][1−m(1−F (ˆ v ))] . F 0 (ˆ v)

Notice that under Assumption 2, vˆ exists

and is unique. The corresponding revenue from period 2 is Π=

m(1 − k)(1 − F (ˆ v ))2 . F 0 (ˆ v)

Finally, total expected revenue from the points mechanism is the sum of the revenue from points sales in period 1 and the expected revenue from period 2: TR

=

q+Π

=

q+

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m(1 − k)[1 − F (ˆ v )]2 . F 0 (ˆ v)

Proposition 3 Proof. By Remark 2, the seller sells m(1 − F (¯ v )) units in the static benchmark, whereas from Proposition 2 it sells m(1 − F (ˆ v )) units in the optimal points mechanism. A consumer in the static benchmark buys a unit if and only if her type is v ≥ v¯, while in the points mechanism she buys a ticket if and only if her type is v ≥ vˆ. Recall that v¯ is the solution to v¯ = while vˆ is the solution to vˆ =

Since the function

1 − F (¯ v) , 0 F (¯ v)

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] . 0 F (ˆ v)

[1−F (v)][1−m(1−F (v))] F 0 (v)

is everywhere below

1−F (v) F 0 (v) ,

we have vˆ < v¯. Hence

more units are allocated at the posted price in the points mechanism than in the static mechanism. Moreover, the remaining k − m(1 − F (ˆ v )) capacity is allocated to consumers who do not buy at the posted price. Therefore the points mechanism is more efficient than the static mechanism, as the set of consumer types who are allocated a unit strictly contains the set of consumer types who are allocated a unit in the static mechanism. Total surplus is larger in the points mechanism, and moreover the seller extracts all of the expected surplus as revenue in the points mechanism, whereas in the static mechanism it does not. It follows immediately that the seller’s revenue is higher under the points mechanism.

Proposition 4 Proof. By Remark 3, the optimal mechanism with full commitment is ex post fully efficient, since all k units are allocated the the k highest types at t = 2. By Proposition 2, the points mechanism is not fully efficient, since less than k units are allocated at the posted price p, and some positive mass of units is allocated randomly through the points lottery. Hence the points mechanism is less efficient than the optimal mechanism with full commitment. More over, the latter extract all of the total surplus, and therefore yields larger revenue than the optimal points mechanism.

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Proposition 5 Proof. We proceed by backward induction. Consider the seller’s choice of r ex post. Suppose the seller posts some price p and a measure d(p) of consumers buy the good. The seller thus chooses r ∈ [0, k − d(p)] ex post. Note that the seller is indifferent among all feasible r at this stage. Therefore consider any equilibrium with some feasible r ∈ [0, k − d(p)]. When consumers decide whether to buy a unit or not, in equilibrium they accurately anticipate r. For any given equilibrium r, a consumer expects some surplus CS r from period 2, taking as sunk the cost of buying a point at t = 1. When the seller sells points at t = 1, the consumer chooses between paying some price q r and obtaining an expected surplus of CS r in period 2, or not buying a point and obtaining a surplus of 0, which is the surplus of a consumer who is excluded at t = 2, as discussed in Lemma 3. The optimal price of a point, q r , at t = 1 is thus q r = CS r , for a given equilibrium r. Hence in any equilibrium the consumer’s ex ante surplus is 0, while the seller captures all of the surplus generated by the mechanism. Thus in the revenue-maximal equilibrium, p and r must maximize total surplus in period 2, subject to the constraint that the price offered is sequentially optimal. Following the proof of Lemma 2, the sequentially optimal price for any r at t = 2 must be in the set [p, p¯], where p=

(1 − k)(1 − F (ˆ v )) 0 F (ˆ v)

and

p¯ =

1 − F (¯ v) , 0 F (¯ v)

and vˆ and v¯ solve: vˆ =

[1 − F (ˆ v )][1 − m(1 − F (ˆ v ))] F 0 (ˆ v)

and

v¯ =

1 − F (¯ v) . F 0 (¯ v)

Notice that total surplus is maximized, subject to the constraint that pr be sequentially optimal, when pr is as small as possible, i.e. equal to p, and when r = k − m(1 − F (ˆ v )). Therefore the revenue-maximal equilibrium is the one where: pr =

(1 − k)(1 − F (ˆ v )) 0 F (ˆ v)

and

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r = k − m(1 − F (ˆ v )).