sequential-screening and airline example

Review of Economic Studies (2000) 67, 697–717 0034-6527�00�00320697$02.00 2000 The Review of Economic Studies Limited

Sequential ScreeningPASCAL COURTY

London Business School

and

LI, HAOUniûersity of Toronto

First ûersion receiûed June 1997; final ûersion accepted March 2000 (Eds.)

We study price discrimination where consumers know at the time of contracting only thedistribution of their valuations but subsequently learn their actual valuations. Consumers aresequentially screened, as in a menu of refund contracts. Initial valuation uncertainty can differin terms of first-order stochastic dominance or mean-preserving-spread. In both cases, optimalmechanisms depend on informativeness of consumers’ initial knowledge about their valuations,not on uncertainty that affects all consumers. It can be optimal to ‘‘subsidize’’ consumers withsmaller valuation uncertainty through low refund to reduce the rent to those who face greateruncertainty and purchase more ‘‘flexible’’ contracts.

1. INTRODUCTION

The mechanism design literature has shed light on many commonly used price discrimi-nation schemes.1 However, most models developed in this literature are static in thatconsumers are assumed to make consumption decisions at the same time as they select acontract. This assumption is not innocuous when consumers learn new information abouttheir demand over time. Consider the demand for plane tickets. Travellers typically donot know their valuations for tickets until just before departure, but they know in advancetheir likelihood to have high and low valuations. A monopolist can wait until the travellerslearn their valuations and charge the monopoly price, but more consumer surplus can beextracted by requiring them to reveal their private information sequentially. An illus-tration of such monopoly practice is a menu of refund contracts, each consisting of anadvance payment and a refund amount in case the traveller decides not to use the ticket.By selecting a refund contract from the menu, travellers reveal their private informationabout the distribution of their valuations, and by deciding later whether they want theticket or the specified refund, they reveal what they have learned about their actualvaluation.

The following example of airplane ticket pricing illustrates sequential pricediscrimination.2 Suppose that one-third of all potential buyers are leisure travellers whosevaluation is uniformly distributed on [1, 2], and two-thirds are business travellers whosevaluation is uniformly distributed on [0, 1] ∪ [2, 3]. Intuitively, business travellers face

1. An early contribution to this literature is Mussa and Rosen (1978). More recently, Wilson (1993) pro-vides applications to real-life pricing problems.

2. The numbers of this example are chosen to expedite exposition. In Section 2, we formally derive theoptimal menu of refund contracts and revisit this example to illustrate some important results of the paper.Similar examples are studied in Courty (1996).

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698 REVIEW OF ECONOMIC STUDIES

greater valuation uncertainty than leisure travellers. Suppose that cost of flying anadditional traveller is 1. If the seller waits until travellers have privately learned theirvaluations, she faces a valuation distribution that is uniform on [0, 3]. The optimal mono-poly price is 2 with expected profit of 1

3 , thus excluding all leisure travellers as well as halfof business travellers who turned out to have low valuations. Suppose instead that theseller offers two contracts before the travellers learn their valuation, one with an advancepayment of 1·5 and no refund and the other with an advance payment of 1·75 and apartial refund of 1. Leisure travellers strictly prefer the contract with no refund. Businesstravellers are indifferent between the two contracts so we assume that they choose thecontract with refund. The monopolist separates the two types and earns an expected profitof 2

3 , twice as much as the profits of charging the monopoly price after travellers havelearned their valuations.

This paper considers a class of monopolist screening problems where consumerssequentially learn their demand and contracts are signed when consumers only have par-tial private information. Such pricing problems are not unique to airplane ticket pricingand refund contracts. Sequential mechanisms take different forms in hotel reservations(cancellation fees), car rentals (free mileage ûs. fixed allowance), telephone pricing (callingplans), public transportation (day pass), and utility pricing (optional tariffs). Sequentialprice discrimination can also play a role in contracting problems such as taxation andprocurement where the agent’s private information is revealed sequentially.

Surprisingly, sequential screening has not received much attention in the screeningliterature. There are research works on two-part tariffs, which share with sequential mech-anisms the feature that consumption decisions are made sequentially, but learning playsno role in this literature. Indeed, a typical textbook discussion of price discriminationintroduces two-part tariffs only as a simple way of implementing concave nonlinear tariffs.The literature on two-part tariffs does not capture the learning feature that consumers donot know yet the quantity they wish to consume or the valuation they place on the goodwhen they choose a contract. Also absent in the literature, is the implication of learningthat consumers often suffer from ‘‘regret’’ when they make sequential decisions: abusinessman could have bought the same ticket at a lower advance price had he knownthat he would fly for sure, and a tourist could have avoided the cancellation fee had heknown his itinerary when he reserved the hotel room.

The primary goal of this paper is to show that sequential mechanisms help producersto price discriminate when consumers learn private information about their demand overtime. Although sequential mechanisms can take different forms, we restrict our attentionto situations where consumers have unit demands as in the airplane ticket pricing problem.In these situations, optimal ex post pricing scheme (after consumers have complete privateinformation about their demand) degenerates to standard monopolist pricing, and themonopolist sequentially discriminates only by choosing the probability of delivering thegood. Refund contracts constrain the delivery probabilities to zero or one, but a generalsequential mechanism is a menu of contracts consisting of pairs of delivery probabilityand payment to the monopolist. Efficiency is achieved by delivering the good if and onlyif the consumer’s valuation exceeds the production cost, but the optimal sequential mech-anism can generate either downward or upward distortions. Downward distortions insequential mechanisms (i.e. not delivering the good when the consumer’s valuation isgreater than the cost) are similar to the standard ‘‘rationing’’ result in nonlinear pricingmodels: under-provision of quality or quantity is used to extract more surplus from moreeager consumers (Mussa and Rosen (1978), Maskin and Riley (1984)). More surprisingly,it can be optimal to ‘‘subsidize’’ consumption by some consumers (i.e. providing the good

COURTY & LI SCREENING 699

when the consumer’s valuation is below the cost). Over-production typically arises onlyin multi-product price discrimination problems when better separation is achieved by sub-sidizing consumption of some goods.3 Although there is only one product in our problem,inefficient over-production can be used effectively as a price discrimination instrumentwhen the production cost is relatively low and consumers differ sufficiently in the degreeof valuation uncertainty they face.

In Section 2, we consider the problem of designing the optimal menu of refund con-tracts for two ex ante types of potential buyers. To continue with the airplane ticketpricing example, the business traveller is either an eager consumer who is more likely todraw higher valuations than the leisure traveller, in the sense of first-order stochasticdominance, or a consumer who faces greater valuation uncertainty than the leisure travel-ler, in the sense of mean-preserving-spread (Rothschild and Stiglitz (1970)). In either case,we show that there is no consumption distortion for the business type in the optimal menuof refund contracts. In the case of first-order stochastic dominance, rationing the leisuretype is the optimal way of reducing the rent to the business type. In the case of mean-preserving-spread, subsidy as well as rationing can be optimal. Sufficient conditions areprovided such that when the production cost is low, subsidizing the leisure type with arefund lower than the cost of the ticket is cost-effective in reducing the rent to the businesstype. For airplane ticket pricing, the marginal cost is low when capacity constraint is notbinding, so our result that the business type purchases a contract with a higher refundexplains why it can be optimal for airlines to offer business travellers more ‘‘flexible’’contracts.

Section 3 examines the general problem of sequential price discrimination with con-tinuous types. This generalization enables us to discuss how type distribution and con-sumer learning affect the design of sequential mechanisms. We characterize the optimalsequential mechanism for a case where consumers face the same valuation uncertainty butdiffer in expected valuation, and a case where consumers have the same expected valuationbut differ in valuation uncertainty. In both cases, the delivery rule is shown to be deter-ministic, and can therefore be interpreted as a menu of refund contracts, or a two-parttariff. The size of distortion depends on the informativeness of consumers’ initial privateknowledge about their valuations, as well as on the type distribution. Distortions are smallif consumers’ initial private knowledge is not informative in that valuation distributionsconditional on types do not vary much across different types of consumers. The size ofdistortion does not depend on any valuation uncertainty that affects all consumers. In thefirst case, where consumers differ in their expected valuation, consumers with greaterexpected valuations are less likely to be rationed and choose the refund contract withlower advance payment and lower refund. In the second case, where consumers differ invaluation uncertainty they face, types facing smaller valuation uncertainty have largerconsumption distortions. As in Section 2, distortions can be either rationing or subsidy,and the latter is optimal when production cost is low.

Section 4 offers further comments on sequential screening when neither first-orderdominance nor mean-preserving-spread is sufficient for reducing the dimension of thedesign problem. This insufficiency results from the multi-dimensional nature of thesequential mechanism pricing problem, which also makes it possible that random deliveryrules are optimal. We study a two-type example of airplane ticket pricing where businesstravellers face greater valuation uncertainty but have a smaller mean valuation than leisure

3. See e.g. Adam and Yellen (1976), or Rochet and Chone (1998). Over-production can also occur insingle-product monopoly models when reservation utility in participation constraints varies with the type ofagents. See Lewis and Sappington (1989).


travellers. We show that a tension arises under refund contracts between exploiting thehigh mean valuation of the leisure type and exploiting the fat tail of the business type,and that randomization helps fine tune sequential discrimination. Section 5 concludes withsome discussion on related works and future direction of this line of research.

2. OPTIMAL MENU OF REFUND CONTRACTS: TWO-TYPE CASE

Consider a monopoly seller of airplane tickets facing two types of travellers, B and L,with proportions fB and fL respectively. Throughout this section, we will think of type Bas the ‘‘business traveller,’’ and type L as the ‘‘leisure traveller.’’ There are two periods.In the beginning of period one, the traveller privately learns his type which determinesthe probability distribution of his valuation for the ticket. The seller and the travellercontract at the end of period one. In the beginning of period two, the traveller privatelylearns his actual valuation û for the ticket, and then decides whether to travel. Each ticketcosts the seller c. The seller and the traveller are risk-neutral, and do not discount. Thereservation utility of each type of traveller is normalized to zero. The business type mayvalue the ticket more in the sense of first-order stochastic dominance (FSD): type B’sdistribution of valuation GB first-order stochastically dominates the leisure type’s distri-bution GL if GB (û)‰GL(û) for all û in the range of valuations [ûq , û̄ ]. Alternatively, thebusiness type may face greater valuation uncertainty in the sense of mean-preserving-spread (MPS, Rothschild and Stiglitz, (1970)): GB dominates GL by MPS if they have thesame mean and �û

ûq(GB (u)AGL(u))duÂ0 for all û ∈ [ûq , û̄]. Let gB and gL be the density func-

tions of two types.A refund contract consists of an advance payment a at the end of period one and a

refund k that can be claimed at the end of period two after the traveller learns his valu-ation. Clearly, regardless of the payment a, the consumer travels only if he values theticket more than k. The seller offers two refund contracts {aB, kB, aL, kL}. The profit maxi-mization problem can be written as

maxaB ,kB ,aL ,kL

∑tGL,B ft (atAktGt (kt )Ac(1AGt (kt ))),

subject to

(IRt ) ∀ tGL, B,AatCktGt (kt )C�û̄

kt

ûdGt (û)Â0,

(ICt,t ′) ∀ t ≠ t ′, AatCktGt (kt )C�û̄

kt

ûdGt (û)ÂAat ′Ckt ′Gt (kt ′)C�û̄

kt′

ûdGt (û).

The first set of constraint (IR) are the individual rationality constraints in period one. Thesecond set (IC ) are the incentive compatibility constraints in period one.

Lemma 2.1. Under either FSD or MPS, IRL and ICB,L imply IRB.

Proof. The two individual rationality constraints can be rewritten as

∀ tGL, B, AatC�û̄

ûq

max {kt , û}dGt (û)Â0.


Then, ICB,L implies

AaBC�û̄

ûq

max {kB , û}dGB (û)ÂAaLC�û̄

ûq

max {kL , û}dGB (û).

Since max {kL , û} is an increasing function of û, if GB dominates GL by FSD,

�û̄

ûq

max {kL , û}dGB (û)Â �û̄

ûq

max {kL , û}dGL (û).

Since max {kL , û} is a convex function of û, the above condition holds also if GB (û) domi-nates GL (û) by MPS. The lemma then follows from IRL . � �

Thus, the business type gets more utility than the leisure type from any refund con-tract, whether it is defined by greater mean valuation or by greater uncertainty. Indeed,we can define the business type by combining FSD and MPS. For example, take a distri-bution GB that dominates GL by MPS. Shifting the whole distribution GB to the rightgives a new distribution that has both greater average valuation and greater valuationuncertainty. It is eady to see that the above lemma continues to hold for this combinationof FSD and MPS. Note that Lemma 2.1 does not hold under general second-order stoch-astic dominance, i.e. greater dispersion without the restriction of the same mean. This canbe seen from the proof of the lemma. Under general second-order stochastic dominancethe integration of the function max {k, û} over [ûq, û̄] can be either greater or smaller for adominant distribution. An example where the business type faces greater valuation uncer-tainty but has a smaller mean than the leisure type will be analysed in Section 4.

Lemma 2.1 implies that IRL binds (holds with equality) in the optimal menu of refundcontracts, otherwise increasing both aL and aB by the same amount would increase profits.Also, ICB,L binds in the optimal menu of refund contracts, otherwise profits could beincreased by increasing aB. Substituting IRL and ICB,L into the objective function andignoring ICL,B, we obtain the following ‘‘relaxed’’ problem

maxkL ,kB

�û̄

kL

( fL (ûAc)gL (û)AfB (GL (û)AGB (û)))dûC�û̄

kB

fB (ûAc)gB (û)dû.

Let S(kL)G�û̄

kL(ûAc)gL (û)dû be the surplus from the leisure type, and

R(kL)G�û̄

kL(GL (û)AGB (û))dû be the rent to the business type, both as function of the

refund to the leisure type. Since the choice of refund for the business type is unconstrained,it should be equal to c to maximize the surplus from the business type. Refund for theleisure type should be chosen to maximize the surplus from the leisure type less the rentto the business type, so the solution to the relaxed problem is given by argmaxk

fLS(k)AfBR(k). Now we are ready to characterize the solution to the original profit maxi-mization problem.

Proposition 2.2. Under either FSD or MPS, in the optimal menu of refund contracts,kBGc and kLGargmaxk fLS(k)AfBR(k).

Proof. It suffices to show that the ‘‘upward’’ constraint ICL,B is satisfied by the given{kB , kL}. Since ICB,L binds, we have

aLAaBG�kL

kB

GB (û)dû,


which implies

AaLCkLGL (kL)C�û̄

kL

ûdGL (û)

G−aBCkBGL (kB )C�û̄

kB

ûdGL (û)A�kL

kB

(GB (û)AGL (û))dû.

Thus, ICL,B is satisfied if and only if �kLkB(GB (û)AGL (û))dû‰0. If the solution to the relaxed

problem has kLGc, the proposition follows immediately. Suppose that the solution haskL ≠ c, and �kL

c (GB (û)AGL (û))dûH0. Consider an alternative menu where the two typeshave the same refund c. The surplus S(c) from type L is greater in the alternative menu.The rent to type B is smaller in the alternative menu, because

R(c)G�û̄

kL

(GL (û)AGB (û))dûC�kL

c

(GL (û)AGB (û))dûF�û̄

kL

(GL (û)AGB (û))dû.

This contradicts the assumption that {kL , kB} solves the relaxed problem. Thus, the solu-tion to the relaxed problem satisfies ICL,B . � �

The derivation of the optimal menu of refund contracts is completed by finding theadvance payments aL and aB from IRL and ICB,L . Proposition 2.2 shows there is no con-sumption distortion for the business type, either when it’s defined by FSD or MPS, or acombination of the two as described previously. Under FSD, the rent function R(kL ) isdecreasing for any kL . Since the surplus S(kL ) from type L is increasing for any kLFc,the solution to the relaxed problem has kLÂc. Thus, we have the standard result thatthere is rationing for the leisure type to lower the rent given away to the business type.

Under MPS, the rent R(kL ) is not a monotonically decreasing function of the refundkL to the leisure type. Instead, R(kL ) is zero at both ûq and û̄, and tends to be greater inthe middle of the support. In this case, subsidy as well as rationing can be used to reducethe rent to the business type. In order to obtain more insights about the nature of con-sumption distortion for the leisure type, we need to impose additional restrictions. Sup-pose that the rent function R( · ) is single-peaked at some z ∈ (ûq, û̄). This is satisfied if forexample GB and GL differ by a single mean-preserving-spread. For simplicity, we assumethat there is no ‘‘plateau’’ at z so that GB (û)HGL (û) for all ûFz and GB (û)FGL (û) for allûHz. An example of this is normal distributions with the same mean z and a greatervariance for type B.4

Under the single-peak assumption, the rent to the business type is the greatest whenthe refund for the leisure type equals the peak of the distributions, and it falls monoton-ically on either side of the peak. Whether it is optimal to subsidize (set kLFc) or ration(kLHc) the leisure type depends on how the loss of surplus due to distortions compareswith rent reduction. Note that when R( · ) is single-peaked at some z≠c, the optimal refundto the leisure type kL cannot lie between c and z. For example, if cFz, then any kL ∈ (c, z]cannot be optimal because by decreasing kL toward c the seller could increase the surplusS(kL ) and decrease the rent R(kL ). Furthermore, setting kLGc cannot be optimal, sinceby decreasing kL slightly below c, surplus from type L is not affected (because S ′(c)G0)but the rent to type B decreases. With the values between c and z excluded, one would

4. If U is a random variable with log-concave density function, and V has zero mean and is independentof U, then the distribution functions of UCV and U, as GB and GL respectively, have the above desired proper-ties. See Shaked and Shanthikumar (1994).


conjecture that it is optimal to subsidize consumption when the cost is low, and to rationwhen the cost is high. The intuition is that when the cost is below the peak of the rentfunction, rationing is too costly because it prevents many profitable trades, while whenthe cost is above the peak, subsidy means too many inefficient trades. The following tworesults give sufficient conditions under which such pattern of distortions is optimal. Thefirst one assumes symmetry of the density functions; the second one assumes that theproportion of business travellers is sufficiently small and�or the cost is sufficiently differentfrom the peak of the rent function.

Proposition 2.3. It is optimal to subsidize (ration) the low type when cFz (cHz) if atleast one of the following two conditions is satisfied: (i) gB and gL are symmetric around z;(ii) fBR(c)‰ fL (S(c)AS(z)).

Proof (i) Suppose that cFz. Since gB and gL are symmetric around z, the rent func-tion R( · ) is also symmetric around z. If kL ∈ (z, 2zAc], then an alternative menu with k̃LG2zAkL yields the same rent by symmetry and a greater surplus because c‰ k̃LFkL , acontradiction. Suppose kLH2zAc. Comparing the slope of S( · ) at any kH2zAc and atk̃G2zAk, we have

AS ′ (k)G(kAc)gL (k)G(kAc)gL (k̃)H(cAk̃)gL (k̃)GS ′ (kr).

Since S(k) reaches maximum at c, the above inequality implies that S(k̃)HS(k). Since R(k)is symmetric around z, an alternative menu with k̃LG2zAkL yields a greater profit, acontradiction. The proposition then follows from the fact that kL cannot lie between cand z. The argument is similar when cHz.

(ii) Suppose that cFz. Since kL cannot lie between c and z, either kLFc or kLHz.The profit of setting kLFc is at least fLS(c)AfBR(c), because choosing kL slightly belowc always reduces the rent without changing the surplus at the margin. On the other hand,the profit of setting kLHz is at most fLS(z), with maximum surplus and zero rent. IffBR(c)‰ fL (S(c)AS(z)), setting kLHz cannot be optimal. A similar argument holds whencHz. � �

According to Proposition 2.3, the pattern of consumption distortion is determinedby the comparison between the cost of the ticket and the peak of the rent function. Thecase in which leisure travellers are subsidized corresponds to the familiar ticketing patternthat leisure travellers buy discount tickets with lower or even zero refund while businesstravellers pay more for flexibility. Proposition 2.3 shows that greater uncertainty for busi-ness travellers, instead of greater likelihood to have high valuations, is the key to under-standing this pattern in terms of monopoly pricing.

To illustrate, let us reconsider the example presented in the introduction. By defi-nition, the surplus function is

S(k)G�12 if k ∈ [0, 1),

kA12k

2 if k ∈ [1, 2],

0 if k ∈ (2, 3],

and the rent function is

R(k)G�14k

2 if k ∈ [0, 1)14A

12(kA1)(kA2) if k ∈ [1, 2],

14 (kA3)2 if k ∈ (2, 3].


According to Proposition 2.2, the business type buys a contract with refund kBG1 andthe leisure type buys a contract with refund kLGargmaxk

13S(k)A2

3R(k). Figure 1 plots13S(k) and 2

3R(k). This example satisfies the first condition in Proposition 2.3 because thedensity functions are symmetric around zG1·5. Since the production cost equals 1 and isless than z, Proposition 2.3 predicts subsidy for the leisure type, and indeed from Figure1 the optimal refund for the leisure type is 0. Since the leisure type’s participation con-straint IRL binds, from kLG0 we have aLG1·5. Since kBG1, from ICB,L we have aBG

1·75. This verifies the optimality of the refund contracts given in the introduction.

FIGURE 1

A few remarks about the example are in order. First, in the optimal sequential con-tract, even the business type gets only a partial refund. Since a sequential contract withfull refund is formally identical to buying from the spot market at a price equal to therefund, this example shows that it is generally not optimal for a monopoly seller to offeronly an advance purchase discount and to induce the business type to buy from the spotmarket.5 Second, the business type gets zero surplus from the optimal sequential contract.This is a special result, because by Lemma 2.1, the business type should get a greatersurplus than the leisure type from any refund contract. What happens is that the leisuretype’s contract has a zero refund, and since the two types have the same mean, the businesstype gets the same zero expected surplus from the zero-refund contract of leisure type.6

Since ICB,L binds, the business type gets zero expected surplus from its partial-refundcontract. Finally, in this example, the first-best outcome is achieved by sequential screen-ing, but this result is also special. If the leisure type has a small probability of reachingvaluations on the interval [0, 1], then as long as such probability is small enough so that13S(k)A2

3R(k) still reaches maximum at kG0, a zero-refund contract for the leisure typecontinues to be optimal. But in this case, subsidy and over-production for the leisure typeoccurs with a positive probability, and efficiency of sequential screening disappears.

5. Dana (1998) shows that consumers with less demand uncertainty buy non-refundable advance tickets.His result is based on the use of advance purchase discount in choosing production capacity by competitivesellers. In his model, ticketing strategies are restricted to simple forms, and consumers facing greater demanduncertainty wait for the spot market instead of purchasing refund contracts as in our model.

6. Technically, in the proof of Lemma 2.1, the function max {kL , û} is not strictly convex.


3. SEQUENTIAL MECHANISM DESIGN: CONTINUOUS TYPE CASE

The analysis in the last section illustrates some general characteristics of sequential pricediscrimination. In this section, we show that these characteristics carry through in theabsence of the restriction to two ex ante types and the restriction to menus of refundcontracts. We discuss how type distribution and consumer learning affect the design ofsequential mechanisms, which can be done more satisfactorily in a model with continuoustypes. By solving for optimal sequential mechanisms for a number of simple and intuitiveparameterizations, we take a first step toward testing implications of the sequential screen-ing model.

In this section, we assume that types are continuously distributed over TG[tq , tr ], witha density function f (t) and distribution function F (t). Each type t is represented by adistribution of valuations over [ûq , û̄], with a differentiable density function g(û � t) andcumulative function G (û � t). As before, consumer type is privately learned before con-tracting. Note that we have assumed that the type space T is one-dimensional for sim-plicity, but this does not reduce the complexity of the type space, because each type is aprobability distribution and can vary in arbitrary ways. In the applications later in thissection, private type information will be about the expected valuation or the degree ofvaluation uncertainty.

As in the standard mechanism problem, the revelation principle (Myerson (1979),Harris and Townsend (1981)) allows us to simplify the problem of sequential mechanismdesign. We assume that the conditional distributions g(û � t) have the same support for allt ∈ T. This assumption makes it simpler to write down the incentive compatibility con-straints in the optimization problem.7 For each pair of reports t and û, let y(t, û) be theprobability of delivery and x(t, û) be the payment to the monopolist. The monopolistsolves the following mechanism design problem

maxx(t,û),y(t,û)

�tr

ûq�û̄

ûq

f (t)(x(t, û)Acy(t, û))g(û � t)dûdt,

subject to constraints

(IC2) ∀ t, ∀ û, û′, ûy(t, û)Ax(t, û)Âûy(t, û′ )Ax(t, û′ ),

(IC1) ∀ t, t ′, �û̄

ûq

(ûy(t, û)Ax(t, û))g(û � t)dûÂ�û̄

ûq

(ûy(t ′, û)Ax(t′, û))g(û � t)dû,

(IR) ∀ t, �û̄

ûq

(ûy(t, û)Ax(t, û))g(û �t )dûÂ0,

(R) ∀ t, ∀ û, 0‰y(t, û)‰1.

The first set of constraint (IC2) are the incentive compatibility constraints in period two.The second set (IC1) are the incentive compatibility constraints in period one. The third

7. The assumption of common support can be relaxed as long as supports of different types overlapsufficiently. More precisely, the condition is: for any type t and any two valuations û and û′, there is a type t ′(possibly t itself ) such that û and û′ are in the support of type t ′. If this condition holds, the optimizationproblem has incentive compatibility constraints for each type involving all valuations in the union of thesupports.


set (IR) are the individual reationality constraint in period one. The last set of constraint(R) requires the delivery rule to be feasible.8

Following the standard treatment of incentive compatibility constraints (Mirrlees(1971)), we can eliminate most of the period-two incentive constraints. Define u(t, û)Gûy(t, û)Ax(t, û) to be the consumer’s ex post surplus after he truthfully reports t and thenû. The following lemma shows that when the consumer draws a greater valuation, hereceives the good with a greater probability and has a greater consumer surplus. It alsoamounts to ‘‘localization’’ of IC2 constraints. In searching for the optimal sequentialmechanisms, we need only impose local constraints on the sequential mechanisms toensure that all IC2 constraints are satisfied. The proof the lemma is standard and thereforeskipped.9

Lemma 3.1. The period-two incentiûe compatibility constraints are satisfied if and onlyif (i) ∂u(t, û)�∂ûGy(t, û), and (ii) y(t, û) is non-decreasing in û for each t.

Our sequential mechanism design problem is related to static multi-dimensional pricediscrimination models (McAfee and McMillan (1988)). In these problems, the consumeris screened only once but he generally has more than one piece of private information(e.g., willingness to pay for two different goods), and the monopolist generally has morethan one instrument of price discrimination (e.g. quantities of the two goods sold to theconsumer). In our sequential mechanism design problem, the consumer is screened twice,but since the contract is signed in the first period, we can think of the sequential designproblem as a static problem in the first period, where the consumer chooses a contingentpackage of delivery probabilities and transfer payments. This static problem is multi-dimensional in a sense because, although the consumer has one piece of private infor-mation, the monopolist has many discrimination instruments in contingent packages ofdelivery probabilities and transfer payments. One difference between our problem and thestatic multi-dimensional problems is that in our problem the second-period screeningimposes IC2 constraints on the instruments that the monopolist can use, as stated inLemma 3.1, whereas in the static multi-dimensional problems, there is no such a prioriconstraint.

With the interpretation of our sequential mechanism design problem as a staticscreening problem, it becomes natural to ‘‘localize’’ period-one incentive compatibilityconstraints as in Lemma 3.1. Define U (t)G�û̄

ûqu(t, û)g(û � t)dû as the expected surplus of

consumer of type t and Y (t, û)G�û

ûqy(t, u)du as the cumulative delivery probability.10

8. Note that there is no period-two individual rationality constraint ûy(t, û)Ax(t, û)¤0 for all t and û.This corresponds to situations where up-front deposits are not fully refundable, or there are cancellation fees atthe consumption date. The absence of this constraint is important for our results. Within the class of determin-istic sequential mechanisms (menus of refund contracts), the ex post participation constraint implies that in eachrefund contract the advance payment does not exceed the refund. One can show that the monopolist cannot usethe combination of advance payment and refund to price discriminate, and therefore all types have the samecontract. The optimal menu of refund contract then coincides with the ex post monopolist pricing. Thisconclusion does not hold if the monopolist is not restricted to deterministic mechanisms, but the presence of theex post participation constraint clearly reduces the monopolist’s discriminatory power.

9. See, e.g. Stole (1996). We consider only sequential mechanisms with piece-wise differentiable deliveryrule y(t, û).

10. The reason for using the cumulative delivery probability is that optimal sequential mechanisms canhave piece-wise constant delivery rules, in which case the first-period second-order condition written in deriva-tives of the delivery probability does not capture the restrictions imposed by the incentive compatabilityconstraints.


Lemma 3.2. The period-one incentiûe compatibility constraints are satisfied only if (i)dU (t)�dtG−�û̄

ûqy(t, û)(∂G(û � t)�(∂t))dû; and (ii) �û̄

ûq(∂Y (t, û)�∂t)(∂g(û � t)�∂t)dûÂ0.

Proof. By the period-one incentive compatibility constraint,

U (t ′ )ÂU (t)C�û̄

ûq

(g(û � t′ )Ag(û � t))(ûy(t, û)Ax(t, û))dû.

Exchanging the roles of t and t ′, we have

U (t ′ )AU (t)‰�û̄

ûq

(g(û � t ′ )Ag(û � t))(ûy(t ′, û)Ax(t ′, û))dû.

To obtain (i), we combine the above two inequalities, divide them by t ′At (assumingt ′Ht), and let t ′ converge to t. Then.

dU (t)

dtG�

û̄

ûq

∂g(û � t)∂t

u(t, û)dûG−�û̄

ûq

∂G(û � t)∂t

y(t, û)dû,

where the last equality uses Lemma 3.1 and integration by parts. Condition (ii) can beobtained similarly by combining the two inequalities, dividing them by (tAt ′ )2 and lettingt ′ converge to t. ��

Lemma 3.2 parallels Lemma 3.1. The first condition is a local period-one first-ordercondition (FOC1), counterpart to the local period-two first-order condition (FOC2) inLemma 3.1 that ∂u(t, û)�∂ûGy(t, û); the second condition is a local period-one second-order condition (SOC1), counterpart to the local period-two second-order condition(SOC2) in Lemma 3.1 that y(t, û) is non-decreasing in û. However, the two lemmas differon an important point: the two local conditions in Lemma 3.2 are necessary but notsufficient for IC1 , whereas the two conditions in Lemma 3.1 are both necessary and suf-ficient for IC2 . Section 4 comments on this difference.

It is well-known that multi-product price discrimination problems are complex whenconsumers’ private information is multi-dimensional (Armstrong (1996), Rochet andChone (1998)). In our model, the consumer’s private information is a probability distri-bution on [ûq , û̄] that can vary quite arbitrarily. Little can be said about the properties ofthe optimal mechanism without making further assumptions. Since each type is a prob-ability distribution, FSD is a natural way to structure the type space T. In this case, wesay that type t is ‘‘higher’’ than t ′ if G(û � t)‰G(û � t ′ ) for all û, and that T is ordered byFSD if tHt ′ implies that t is higher than t ′ for any t, t ′ ∈ T. Another way to structure Tis through a particular kind of mean-preserving spread where all distributions G(û � t) crossat a single point z. In this case, we say that type t is ‘‘higher’’ than t ′ if G(û � t)ÂG(û � t ′ )for all ûFz and G(û � t)‰G(û � t ′ ) for all ûHz, and that T is ordered by MPS if tHt ′ impliesthat t is higher than t ′ for any t, t ′ ∈ T. As in the two-type case, the analyses of these twocases will be similar.

Following the standard practice of mechanism design, we obtain a ‘‘relaxed’’ problemby imposing the two local first-order conditions in Lemma 3.1 and Lemma 3.2 whileignoring the second-order conditions (and all but the lowest type IR constraint). ByLemma 3.2, we have

�t̄

tq

f (t)U (t)dtGU (tq)A�tr

tq�û̄

ûq

(1AF (t))y(t, û)∂G(û � t)

∂tdûdt.


Define

φ(t, û)G(ûAc)C(1AF (t))

f (t)

∂G(û �t)�∂tg(û � t)

.

The relaxed problem can be then written as maxy(t,û) �tr

tq �û̄

ûqφ(t, û)y(t, û)g(û � t) f (t)dûdt subject

to 0‰y(t, û)‰1. The solution to the relaxed problem is given by y(t, û)G1 for t and ûsuch that φ(t, û)H0 and 0 otherwise. There is no randomization.

If transfer payments x(t, û) can be found so that the solution y(t, û) to be relaxedproblem given above satisfies all IC1 and IC2 constraints, then the sequential mechanism{y(t, û), x(t, û)} is optimal. But since we have ignored the local second-order conditions inLemma 3.1 and Lemma 3.2, and since the two conditions in Lemma 3.2 are generallyinsufficient for IC1 , we need to impose some condition on y(t, û). The next result statesthat in the case of FSD, if the solution y(t, û) to the relaxed problem is monotone in botht and û, then transfer payments x(t, û) can be found such that the sequential mechanism{y(t, û), x(t, û)} solves the original problem.11

Lemma 3.3. Suppose that T is ordered by FSD. If a deliûery rule y(t, û) solûes therelaxed problem, and if y(t, û) is non-decreasing in t for all û and in û for all t, then thereexist transfer payments x(t, û) such that the sequential mechanism {y(t, û), x(t, û)} is optimal.

Proof. Since it solves the relaxed problem, y(t, û) is either 1 or 0 for any t and û. Byassumption, y(t, û) is non-decreasing in û for each t, so SOC2 implies that there exists k(t)for each t such that y(t, û)G0 if û‰k(t) and y(t, û)G1 if ûHk(t). By FOC2 , the transferpayments can be written as x(t, û)Gx0(t) if û‰k(t) and x(t, û)Gx1(t) if ûHk(t), with k(t)Gx1(t)Ax0(t). By Lemma 3.1, all IC2 constraints are satisfied.

The expected surplus of a type t consumer is

U (t)G−x0(t)C�û̄

k(t)

(1AG(û � t))dû.

Taking derivatives and using FOC1 , we obtain

Adx0(t)

dtA

dk(t)

dt(1AG(k(t) � t))G0.

The above condition gives a differential equation that can be used to find the functionx0(t), with the boundary condition that x0(tq) satisfies

U (tq)G−x0(tq)C�û̄

k(tq)

(1AG(û � tq))dûG0.

It remains to show that the sequential mechanism {k(t), x0(t), x1(t)} defined abovesatisfies all IC1 and IR constraints. The partial derivative of the expected utility U (t ′, t )of type t ′ when he claims to be type t is given by

∂U (t ′, t)∂t

G−dx0(t)

dtA

dk(t)

dt(1AG(k(t) � t ′ )).

11. Matthews and Moore (1987) make the same observation in a multi-dimensional screening problem,and call such mechanisms ‘‘attribute-ordered.’’


Since by assumption y(t, û) is non-decreasing in t for all û, dk(t)�dt‰0. Suppose t ′Ft.Then,

∂U (t ′, t )∂t

‰Adx0(t)

dtA

dk(t)

dt(1AG(k(t) � t))G0.

By integration we have U (t ′, t )‰U (t ′ ). The same reasoning applies if tFt ′. This showsthat the sequential mechanism {k(t), x0(t), x1(t)} satisfies all IC1 constraints. As in Lemma2.1, since all IC1 constraints are satisfied and since IR holds for the lowest type tq , all IRconstraints are satisfied. � �

In the case where T is ordered by MPS with all distribution functions passing througha single point z, depending on whether the cost c is low or high relative to z, the proof ofLemma 3.3 needs to be adapted. The statement of Lemma 3.3 holds for the case of MPSwith an additional restriction on the solution to the relaxed problem, namely no under-production if cFz and no over-production if cHz.

Lemma 3.4. Suppose that T is ordered by MPS with all distributions passing througha single point z. If cFz (resp. cHz) and y(t, û) solûes the relaxed problem with no under-production (oûer-production), and if y(t, û) is non-increasing (non-decreasing) in t for all ûand non-decreasing in û for all t, then there exists x(t, û) such that {y(t, û), x(t, û)} is optimal.

Proof. Define a sequential mechanism {k(t), x0(t), x1(t)} as in the proof of Lemma3.3. It suffices to show that all IC1 constraints are satisfied. Suppose cFz; the case of cHzis symmetric. By assumption y(t, û) is non-increasing in t for all û, so dk(t)�dtÂ0. Sincethere is no under-production, k(t)‰c for all t. Further, if t ′Ft, MPS impliesG(k(t) � t ′ )‰G(k(t) � t). Then, from the expression of ∂U (t ′, t )�∂t, we have

∂U (t ′, t )∂t

‰Adx0(t)

dtA

dk(t)

dt(1AG(k(t) � t ))G0.

By integration we have U (t ′, t )‰U (t ′ ). The same reasoning applies if tFt ′. � �

Before we present a few parameterizations where optimal sequential mechanisms canbe found by using Lemma 3.3 and Lemma 3.4, it is helpful to compare our model withthe standard one-dimensional nonlinear pricing problem. The coefficient φ(t, û) is anal-ogous to ‘‘virtual surplus’’ defined by Myerson (1981) in one-dimensional nonlinear pric-ing problems. As in nonlinear pricing problems, the first part of φ(t, û) corresponds tosocial surplus of type t with valuation û, and the second part represents the distortion. Ina nonlinear pricing problem, the second part contains only the ‘‘hazard rate’’ (1AF (t))�f (t), but in our sequential screening problem, it also contains an additional term (∂G(û � t )�∂t )�g(û � t ).12 The hazard rate measures the distortion due to eliciting truthful type infor-mation from t, for any valuation û. Distortions are larger with a greater hazard rate,because whatever surplus conceded to type t must also be given to all higher types. Theterm (∂G(û � t )�∂t )�g(û � t ) has a straightforward interpretation of an ‘‘informativeness mea-sure’’ (Baron and Besanko (1984)), as it represents how informative the consumer’s private

12. Whenever Lemma 3.3 and Lemma 3.4 apply, the sequential mechanism design problem is reduced tochoosing refund (cutoff valuation) as a function of type. Virtual surplus can be instead defined as the expectedtotal surplus for a given type and a given refund. This alternative definition of virtual surplus looks the sameas in a standard one-dimensional nonlinear pricing problem. We choose to define virtual surplus for a pair ofvaluation and type, because it applies even when the conditions of Lemma 3.3 and Lemma 3.4 do not hold.


type knowledge is about his valuation. It is zero if type and valuation are independentlydistributed, and is large if marginally different types have very different conditional distri-butions. Alternatively, holding G(û � t ) constant, we can think of û as a function of t, andthe informativeness measure is equal to −∂û�∂t. The measure then represents how mar-ginally differently types hit a fixed percentile G(û � t ) at different valuations. Distortionsare larger with a greater informativeness measure, because more rent must be concededin order for marginally different types not to claim to be type t with valuation û. Finally,in a nonnlinear pricing problem, the usual second-order condition (analogous to SOC2 inLemma 3.1) implies downward distortion—consumers of every type except for the highestone are rationed. Here, the direction of distortion is not necessarily downward becausethe ratio (∂G(û � t )�∂t)�g(û � t ) can be either positive or negative. We will discuss the case ofFSD and the case of MPS separately.

For the case of FSD, let us first consider the following ‘‘additive’’ structure of con-ditional distributions

ûGθtC(1Aθ)εt ,

where t is distributed over a positive range, θ∈ (0, 1), and εt is i.i.d. on the whole real linewith density h( · ) and distribution H ( · ).13 The distribution of û conditional on t is given

G(û � t )GH�ûAθt

1Aθ � .

Note that G(û � t ) satisfies FSD. The additive specification has some nice properties thatmake it an interesting benchmark case. Consumers face the same uncertainty regardingvaluation but have private information about their expected valuation for the good. Inthis linear case, the informativeness measure becomes a global one—it equals θ for alltypes and valuations. A greater θ means that more information is learned early ratherthan late and the consumer’s private type knowledge is more informative about valuationin that conditional distributions of valuations vary more with type.

With this additive specification, we have

φ(t, û)GûAcAθ(1AF (t))

f (t).

Under the standard monotone hazard rate assumption that the hazard rate (1AF (t))�f (t) is non-increasing in t (Fudenberg and Tirole (1991)), we have that φ(t, û)Â0 impliesφ(t ′, û′ )Â0 for any t ′Â t and û′Âû. The solution to the relaxed problem is monotone inû and t separately. By Lemma 3.3, it solves the original problem. The optimal deliveryrule is therefore given by

y(t, û)G�10if ûHcCθ(1AF (t))�f (t),

otherwise.

It is deterministic with a cutoff level for each type. Higher types have lower cutoffs. Thereis no production distortion for the highest type. We summarize the findings in the follow-ing proposition.

13. This formulation allows negative realized valuation. An example of negative valuation is a ticket-holder who is sick and must be paid to travel. Since the monopolist cannot force consumption (free disposal),in the profit maximization problem a distribution with a range of negative valuations is equivalent to one thathas an atom at zero valuation with all the probability weights of the negative valuations. This does not changethe results below. Note that in this and all other parameterizations of this section, we consider random variablesthat are distributed over the real line to guarantee that the conditional distributions have the same support.


Proposition 3.5. Suppose that the conditional distribution functions haûe an additiûestructure. Then, under the monotone hazard rate assumption, the optimal sequential mechan-ism is deterministic with larger under-production distortions for lower types and no under-production for the highest type.

The under-production distortion is larger when the consumer’s private knowledge ismore informative, because the monopolist prefers rationing the good to giving highertypes a large informational rent. In the polar case where θG0, type is completely unin-formative of valuation, and the monopolist achieves perfect discrimination with a sequen-tial mechanism. The monopolist sells the product in period one at the expected valuation,which is the same for all types, and allows the consumer to return the good for a refundequal to c. This refund policy guarantees social efficiency. In the other polar case whereθG1, the under-production distortion is the largest. Clearly, the optimal sequential mech-anism coincides with usual monopoly pricing after the consumer learns his valuation.

A characteristic of the optimal menu of refund contracts is that it is independent ofthe specification H ( · ) of the valuation shock ε t . For fixed θ, increasing the variance ofε t , for example, means that the consumer faces greater valuation uncertainty, yet this hasno effect on the optimal menu of refund contracts. What matters is not how much theconsumer knows about his valuation when he signs the contract, but how informative hisprivate type knowledge is about his valuation. The shock ε t may be interpreted as ademand shock. As long as it affects all consumer types independently so that it does notaffect the informativeness measure, the size of this demand shock has no impact on theoptimal sequential mechanism.

It is instructive at this point to compare the optimal sequential mechanism with expost monopoly pricing. Whereas the sequential mechanism is deterministic with lowercutoff levels of valuations for higher types, ex post monopoly pricing can be thought ofas a deterministic sequential mechanism with full refund and all types having the samecutoff level. The optimal sequential mechanism always yields greater profits than ex postoptimal monopoly pricing; the gains from sequential screening tend to be greater when θis close to 0. If sequential mechanisms involve greater implementation costs than ex postmonopolist pricing, perhaps due to the cost of registering consumers in advance, then oneis less likely to observe sequential screening in environments where conditional distri-butions of valuations vary substantially with type.

We can also extend Proposition 3.5 to a multiplicative specifications. Suppose thatûGtθε1Aθ

t where t is distributed over a positive range, θ∈ (0, 1), and ε t is i.i.d. on thewhole positive real line with density h( · ) and distribution H ( · ). This specification islog-linear and can be useful in constructing empirically testable implications.14 We have,

φ(t, û)GûAcAûθ(1AF (t))

tf (t).

As in the additive specification, the greater is θ, the more informative is the type as asignal of valuation, but here the informativeness measure is not uniform across types oracross valuations. Define λ (t)G(1AF (t))�(tf (t)) and suppose that λ (t)F1 for all t. Thisis satisfied as long as the range of t is sufficiently above zero, regardless of the value of θ.The solution to the relaxed problem is then

14. A specification with a special distribution that works out similarly is G(û � t)G1Aexp (−û�t).


y(t, û)G�10if ûHc�(1Aθλ (t)),

otherwise.

If λ (t) is non-increasing in t (a monotone hazard rate is sufficient for this but clearly notnecessary), the above solution has the monotone property required by Lemma 3.3 andtherefore solves the original problem.

For the case of MPS, perhaps the most natural class of distributions is given by thesame mean plus a multiplicative shock. Suppose that

ûGzCtε t ,

where ε t is i.i.d. on the whole real line with zero mean, density h( · ) and distribution H ( · ).Without loss of generality we assume that tqH0, so that greater t means greater dispersion.The distribution of û conditional on t is given by

G(û � t)GH�ûAz

t � .

It is easy to see that the distributions G(û � t) satisfy MPS and pass through the same pointat z, which is also the mean of the distributions. Consumers face the same expected valu-ation but have private information about the degree of valuation uncertainty. Consumersof higher types face greater valuation uncertainty. The informativeness measure is givenby (ûAz)�t. Higher types have less private information at the time of contracting.

With the above specification, we have

φ(t, û)GûAcA(ûAz)λ (t),

where λ (t) is as defined in the previous example, and assumed to be less than 1. Underthe standard monotone hazard rate assumption (sufficient but not necessary), λ (t) is non-increasing in t. Then, the solution to the relaxed problem is given by

y(t, û)G�10if ûH(cAλ (t)z)�(1Aλ (t)),

otherwise.

The assumption of λ (t)F1 guarantees that y(t, û) is non-decreasing in û for any t. It isstraightforward to show that the cutoff rule y(t, û) defined above has the propertiesrequired by Lemma 3.4: if cFz, then y(t, û) has no under-production and is non-increasingin t for all û; if cHz, then y(t, û) has no over-production and is non-decreasing in t for allû. The following proposition follows.

Proposition 3.6. Suppose that the ûaluation distributions are giûen by the same meanz plus a multiplicatiûe shock and (1AF (t))�(tf (t)) is non-decreasing in t and less than 1.Then, if cFz (resp. cHz), the optimal sequential mechanism is deterministic with greateroûer-production (under-production) distortions for types with smaller ûaluation uncertaintyand no distortion for the highest type.

Proposition 3.6 mirrors Proposition 2.3. When consumers have the same expectedvaluation but differ in the valuation uncertainty they face, the pattern of consumptiondistortion is determined by the comparison between the production cost and the expectedvaluation. In airplane ticket pricing, the cost of flying an additional passenger is typicallysmall compared to the average willingness to pay when plane capacity is not binding. In


this case, all travellers except for those with the greatest valuation uncertainty are subsid-ized and purchase advance tickets with a refund lower than the cost of the ticket. Travel-lers with greater uncertainty about their plans pay more in advance for greater flexibilityin terms of a higher refund.

Note that as in Proposition 3.5 the optimal menu of refund contracts does not dependon the specification H ( · ) of the valuation shock ε t . The variance of ε t , for example, canbe great or small, but it has no effect on the optimal menu of refund contracts because itis common to all types. Also, the multiplicative specification can be generalized by relaxingthe assumption that consumer type enters linearly with ε t . Finally, as in Proposition 3.5,consumption distortions (either downward or upward) are larger when the consumer’sprivate type knowledge is more informative about his valuation at the time of contracting.To see this, recall that the informativeness measure is given by (ûAz)�t. More privateinformation at the time of contracting can be represented by a leftward shift of the typedistribution. Since λ (t) is decreasing in t, this amounts to an increase in λ (t) for each t.The cutoff levels in the solution to the relaxed problem decrease if cFz and increase ifcHz. In either case, the consumption distortions are greater.

4. FURTHER COMMENTS ON SEQUENTIAL SCREENING

The optimal sequential mechanisms characterized in previous sections are all determin-istic. Deterministic sequential mechanisms are important because they are easy toimplement in practice, through refund contracts, option contracts or cancellation fees.Deterministic sequential mechanisms are also related to two-part tariffs in nonlinear pric-ing models, which are formally discussed in the mechanism design language by Laffontand Tirole (1986). A typical optimal non-linear pricing scheme is a direct mechanismrarely seen in practice; instead, two-part tariffs are often used. Although any concavenonlinear tariff can be implemented through a two-part tariff, the sequential feature ofthe consumer’s decision in two-part tariffs is purely artificial. One explanation for the useof two-part tariffs in practice, given by Laffont and Tirole, is that they are robust againstshocks. Our model of sequential screening, where the consumer self-selects twice, providesanother explanation. In our model, the shock mentioned by Laffont and Tirole can beviewed as the uncertainty faced by the consumer about his actual valuation at the time ofcontracting, which later becomes his additional private information.

Deterministic sequential mechanisms are also easy to characterize. Under either FSDor MPS, optimal deterministic mechanisms can be characterized with a local approach,because the design problem is reduced to a single-dimensional problem of choosing therefund as a function of type. Unfortunately, neither FSD nor MPS is sufficient to implythat optimal sequential mechanisms are deterministic. In our model of sequential screen-ing, as in the standard one-dimensional price discrimination literature, localization of theperiod-two incentive compatibility constraints is guaranteed due to the standard ‘‘singlecrossing’’ condition (Cooper (1984)) that a consumer with a greater realized valuation iswilling to pay more for an increase in delivery probability. This enables us to replace IC2

constraints by the two local conditions in Lemma 3.1. However, the two local period-oneincentive constraints in Lemma 3.2 are not sufficient to imply global period-one incentiveconstraints.15 This insufficiency results from the multi-dimensional nature of the sequential

15. In the multi-dimensional price discrimination literature, McAfee and McMillan (1988) have identifieda ‘‘generalized single crossing’’ condition that guarantees that local incentive compatibility constraints implyglobal constraints. But their condition requires that the dimension of consumer’s private information exceed thenumber of monopolist’ price discrimination instruments. This dimensionality condition is not satisfied in ourmodel: at the time of contracting, the consumer’s private information is one-dimensional, but the monopolisthas many instruments in contingent packages of delivery rule and payments.


mechanism design problem. To see this, note that from Lemma 3.3, under FSD a sufficientcondition for the IC1 constraints is that y(t, û) is non-decreasing in t for all û, but SOC1

states only that the delivery rule y(t, û) is non-decreasing in t ‘‘on average,’’ with theweights determined by local changes in the conditional distributions of valuations withrespect to type. Without further restrictions on type space T besides stochastic dominance,the weights can still change with type quite arbitrarily, and there is little hope that whatholds locally extends globally.16 When the delivery rule is not monotone increasing in typefor each valuation, ‘‘bunching’’ occurs both across valuations and across types. In thiscase, random delivery rules allow fine tuning by the monopolist.17 That the delivery ruleneed not be monotone in types is similar to the conclusion in the multi-product pricediscrimination literature that quantity or quality of each good need not be monotonewhen consumer demand characteristics are one-dimensional but the number of price dis-crimination instruments is greater than one. For example, Matthews and Moore (1987)show that if the monopolist in the model of Mussa and Rosen (1978) offers different levelsof warranty as well as quality, more eager consumers need not buy higher quality orreceive higher warranty.

To conclude the discussion of general sequential mechanisms, we use a two-typeexample of airplane ticket pricing to illustrate how random delivery rules can fine tunesequential screening in the absence of FSD or MPS. This example is closely related to theone presented in the introduction. As in Section 2, the two types are interpreted as businesstype and leisure type, with the business type facing greater valuation uncertainty in havingdistribution of valuations that second-order stochastically dominates that of the leisuretype. The difference here is that the business type has a smaller mean valuation than theleisure type. In particular, suppose that there are three possible valuations ûA2, û, andûC1, where û is a number greater than 2. The leisure type has a deterministic valuationof û. The business type is equally likely to draw valuation of ûA2 and ûC1, with a lowermean ûA1

2 . For simplicity we assume that the production cost is zero. We will see how atension arises under refund contracts between exploiting the high mean valuation of theleisure type and the fat tail of the business type, and how randomization can help finetune sequential discrimination.

First consider the following sequential mechanism with randomization. The selleroffers one contract with an advance price of û and zero refund, intended for the leisuretype, and another contract intended for the business type, with the same advance pricebut with an option of receiving a partial refund of 1

2û and then drawing a lottery withprobability 1

2 of getting the ticket. Note that if the business type purchases the partialrefund contract, it is strictly better off not claiming the refund if it draws the highervaluation of ûC1, whereas if the leisure type buys the partial refund contract, it is alwaysindifferent between claiming and not claiming the refund. The leisure type gets zero sur-plus from either contract (constraints IRL and ICL,B both hold with equality). The businesstype gets zero surplus from the partial refund contract (IRB binds) and negative surplusfrom the no-refund contract because it has a mean valuation smaller than the advanceprice of û (ICB,L does not bind). Since both participation constraints bind, the seller’s

16. It is possible to make stronger assumptions on the type space to reduce the dimension of the designproblem. For example, if in addition to satisfying FSD or MPS, the distribution functions G(û � t) are separablein t and û in that there exist functions α ( · ), β( · ) and γ ( · ) such that G(û � t)Gα (û)Cβ(t)γ (û) for all t and û, thenone can use a variation of the standard local approach in nonlinear pricing problems to characterize the optimalsequential mechanism.

17. Numerical examples of randomization in optimal sequential mechanism under FSD or MPS are avail-able from the authors.


expected profits are ΠGfLûCfB (14(ûA2)C1

2(ûC1)), where fL and fB are the fractions ofleisure type and business type respectively.

Next, consider any deterministic sequential mechanism. Since with the above randommechanism the seller extracts all the surplus from the leisure traveller and from the busi-ness traveller when he draws the higher valuation, for the profits from a deterministicmechanism to exceed Π, the business type must be induced to use the ticket with prob-ability one. But since the leisure type has a higher mean valuation, it will also use theticket with probability one. Since the participation constraint of the business type mustbe satisfied, the seller is essentially restricted to offering the ticket at an advance price ofûA1

2 with no refund, with expected profits of ûA12 . Therefore, if ûA1

2FΠ, or fLH(ûA2)�û, the optimal sequential mechanism must involve randomization.

The intuition behind this example can be understood as follows. It is straightforwardto argue that for a sequential mechanism to be optimal in this example, the seller mustsell to the leisure type with probability one, and the business type must be induced to usethe ticket with probability one when it draws the high valuation ûC1.18 The question iswhether to sell to the business type when it draws the low valuation. If a menu of refundcontracts is used, the seller faces an all-or-nothing choice: she must either sell to thebusiness type with probability one, or sell only when it draws the high valuation. In thefirst case, she is constrained to offering a single contract at an advance price of ûA1

2 andno refund. Since the leisure type has a higher mean valuation of û, a rent equal to 1

2 is leftto the leisure type. This rent is large if fL is large. In the second case, the seller can achieveseparation by offering one contract at an advance price of û with no refund to the leisuretype and another contract at an advance price of ûC1 with full refund to the businesstype. But this means that she has to give up expected surplus of 1

2 fB (ûA2) from thebusiness type when it draws the low valuation. Randomization provides an extra marginfor the seller to operate at. By providing a partial refund of 1

2û and a lottery to the businesstype when it draws the lower valuation, the seller leaves no rent to the leisure type andextracts half of the surplus from the business type when it draws the low valuation.

5. CONCLUDING REMARKS

The closest work to the present study is by Miravete (1996), who also considers the mon-opolist’s pricing problem when consumers face demand uncertainty. Miravete (1997) testssome empirical implications of his model. In contrast with our paper, he assumes continu-ous demand functions. This allows him to compare ex ante two-part tariffs (where con-sumers choose a tariff based on their expected demand) and ex post two-part tariffs (afterconsumers learn their actual demand). He shows that expected profits are higher underan ex post tariff if the variance of the type distribution is large enough. The generality ofhis results is compromised by the restriction to two-part tariffs, because ex ante two-parttariffs are generally not optimal. In the present paper where unit demand is assumed, theoptimal ex post tariff degenerates to standard monopolist pricing, and can be thought ofas a uniform sequential mechanism with full refund for all types. As a result, in out modelex post mechanisms are dominated by sequential mechanisms. Furthermore, our results(Proposition 3.5 and Proposition 3.6) indicate that profit gains from using a sequential

18. To prove the latter claim by contradiction, suppose that the probability of getting the ticket is pF1for some type after drawing its highest possible valuation. Then, the seller can increase this probability to oneand at the same time increase the corresponding payment from the type by û(1Ap). This will not affect theparticipation or the incentive constraint of any type, but will increase the profits by û(1Ap) when the buyerhappens to be this type and happens to draw the highest valuation.


mechanism depend not only on the type distribution, but on how informative consumers’initial private knowledge is about their valuations: if different types of consumers havevery different conditional distributions of valuations, then sequential mechanisms do notyield much greater expected profits than ex post monopolist pricing.

Our sequential mechanism design problem is related to the problems of dynamic pricediscrimination (Baron and Besanko (1984), Laffont and Tirole (1988, 1990)). An exampleof these problems is a monopolist facing a consumer making repeated purchases. Typi-cally, consumers have only one piece of private information and it does not change overtime. These problems focus on the implications of the monopolist’s ability to commit.With no change in consumers’ private information over time, the optimal dynamic mech-anism under commitment is static: it simply replicates the optimal static screening contractin every period. In contrast, our sequential screening problem is driven by the demanduncertainty, and consumption decisions are made based on new information.

Throughout the paper we assume that the monopolist can commit to a sequentialmechanism and examine how a sequential mechanism can be used by the monopolist toextract maximal surplus. If the monopolist cannot commit to a sequential mechanism,time-inconsistency problems mentioned by Coase (1972) arise. In our problem of sequen-tial screening, the monopolist may be tempted to renege both before and after consumerlearn their valuation. Lack of ability to commit to a sequential mechanism reduces themonopolist’s power to discriminate, which may explain why sometimes these types ofmechanisms are not observed in practice. Incorporating the commitment issue to thedesign of sequential mechanisms is an interesting topic for further research.

Although menus of refund contracts arise naturally from price discrimination underconsumer learning, they can be offered by producers for other reasons. One reason worthmentioning is that a menu of refund contract may allow a producer to learn about finalconsumer demand early on. This information can be valuable for production planningpurposes. Consider the airplane ticket pricing example in the introduction. The fractionof business travellers may be unknown to the ticket seller, but it is revealed by travellers’choices of refund contracts. This information is valuable to the seller if a productioncapacity decision must be made before the final demand is realized. Indeed, precisely dueto consumer demand uncertainty, capacity decisions are important in business such ashotels, airlines and car rental, as exemplified by the practice of ‘‘yield management,’’ or‘‘revenue management’’ (Dana (1999)). Pricing strategies such as advance purchase dis-counts and Saturday night stayover requirements in revenue management of the airlineindustry are generally understood to solve the peak-load problem and to implement pricediscrimination. Our present paper is the first to examine revenue management in the broadsense of sequential price discrimination. Combining sequential screening with peak-loadpricing seems to be a promising line of research in understanding industries where bothconsumer demand uncertainty and capacity decisions are important.

Acknowledgements. We received helpful comments from Michael Chwe, Hugo Hopenhayn, Jim Peck,Sherwin Rosen, Lars Stole, Wing Suen, and particularly Mark Armstrong, who sent us a note on the two-typecase under first-order stochastic dominance after we finished an earlier version of the paper. We are grateful toPatrick Bolton, the Editor, and the referees for suggesting that we investigate situations where first-order stochas-tic dominance does not hold.

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