revenue maximizing with return policy when buyers have uncertain valuations

10
Revenue maximizing with return policy when buyers have uncertain valuations Jun Zhang School of International Business Administration, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China abstract article info Article history: Received 7 February 2012 Received in revised form 1 August 2013 Accepted 5 August 2013 Available online 18 August 2013 JEL classication: C72 D44 D82 D83 L12 Keywords: Auctions Mechanism design Return policies Uncertain valuations This paper examines the optimal mechanism design problem when buyers have uncertain valuations. This uncer- tainty can only be resolved after the actual transactions take place and upon incurring signicant post-purchase cost. We focus on two different settings regarding how the seller values a returned object (salvage value). We rst study the case where the salvage value is exogenously determined. We nd that the revenue maximizing mechanism is deterministic and separable. We illustrate that the optimal revenue can be implemented by a mechanism with a no-questions-askedreturn policy. In addition, we show that linear return policiesare suboptimal when the hazard rates of initial estimates are monotone. We next examine the case where the salvage value is endogenously determined. We demonstrate that separabilityno longer holds and the recallof buyers is necessary in the optimal mechanism. © 2013 Elsevier B.V. All rights reserved. 1. Introduction In many auctions, buyers have uncertain valuations about the ob- jects. These uncertainties can only be resolved after the actual transac- tions take place and the buyers receive the objects. For example, in online auction sites such as eBay.com, buyers' valuations are subject to after-transaction shocks, adding to their initial estimates. These shocks can result from matching tastes or styles, complementarity with other products that the consumers have already owned, the conditions of the objects upon the arrivals of the shipments, etc. Similarly, in an estate auction, bidders may not know their ultimate values of a piece of furni- ture for sale until they have it in the house and see how well the color of the upholstery matches the carpet and the size ts relatively to existing pieces. When rms bid for the assets of a bankrupt company, they will not know their true values until they begin integrating them into their existing concern. 1 Foals are often auctioned before they are born in Japanese racehorse industry. Similarly, livestock breeders auction em- bryos in Australia, Canada, and the United States. Agricultural produce is auctioned off long before it is harvested. In the UK gas industry, the National Grid auctions off the transmission capacity rights long before the realization of demand. With these uncertainties, sellers can choose to auction off the objects in the traditional way, but more revenue may be raised by using a mech- anism with terms conditioning on the new information available later on. A simple way to implement this is to run an auction, and then allow the winner to return the object after the uncertainty resolves. Indeed, many sellers in various online auction sites, such as Amazon.com, eBay.com, Johareez.com and Yahoo.com, provide return services; buyers get their transaction prices refunded after paying some restocking fees and ship- ping fees when returning the objects. A recent search for antique auctions in eBay.com came out 161,729 items, and 108,150 (67%) of them came with certain return policies. The percentage of art auctions offering refunds is even higher, 131,944 out of 175,329 auctions had return poli- cies, representing 75% of the auctions. Other examples include the NHL (National Hockey League) online auctions, which provide a 7-Day, 100% Money-Back Guarantee, sellers in auctions for embryo, who guarantee International Journal of Industrial Organization 31 (2013) 452461 I am grateful to the editor Yossi Spiegel, two anonymous referees, James Amegashie, James Bergin, Li, Hao, Sumon Majumdar, Ruqu Wang, Jan Zabojnik and to the seminar par- ticipants at Monash University, Queen's University, the University of Queensland, Zhejiang University, the 2008 CEA conference, the 2008 World Congress of Game Theory, and the 2011 Shanghai University of Finance and Economics Micro Theory and Experimental Economics Conference for their valuable comments and suggestions. This paper was previously circulated under the title Auctions with Refund Policies as Optimal Selling Mechanism. All errors are the responsibility of the author. Tel.: +86 21 65907883. E-mail address: [email protected]. 1 We thank a referee for providing us with the above two motivating examples. 0167-7187/$ see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijindorg.2013.08.001 Contents lists available at ScienceDirect International Journal of Industrial Organization journal homepage: www.elsevier.com/locate/ijio

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International Journal of Industrial Organization 31 (2013) 452–461

Contents lists available at ScienceDirect

International Journal of Industrial Organization

j ourna l homepage: www.e lsev ie r .com/ locate / i j i o

Revenue maximizing with return policy when buyers haveuncertain valuations☆

Jun Zhang ⁎School of International Business Administration, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China

☆ I am grateful to the editor Yossi Spiegel, two anonymJames Bergin, Li, Hao, SumonMajumdar, RuquWang, Jan Zticipants atMonash University, Queen's University, the UnUniversity, the 2008 CEA conference, the 2008 World Con2011 Shanghai University of Finance and Economics MiEconomics Conference for their valuable comments anpreviously circulated under the title “Auctions with RefuMechanism”. All errors are the responsibility of the autho⁎ Tel.: +86 21 65907883.

E-mail address: [email protected].

0167-7187/$ – see front matter © 2013 Elsevier B.V. All rihttp://dx.doi.org/10.1016/j.ijindorg.2013.08.001

a b s t r a c t

a r t i c l e i n f o

Article history:Received 7 February 2012Received in revised form 1 August 2013Accepted 5 August 2013Available online 18 August 2013

JEL classification:C72D44D82D83L12

Keywords:AuctionsMechanism designReturn policiesUncertain valuations

This paper examines the optimalmechanism design problemwhen buyers have uncertain valuations. This uncer-tainty can only be resolved after the actual transactions take place and upon incurring significant post-purchasecost. We focus on two different settings regarding how the seller values a returned object (salvage value). Wefirst study the case where the salvage value is exogenously determined. We find that the revenue maximizingmechanism is deterministic and “separable”. We illustrate that the optimal revenue can be implemented by amechanism with a “no-questions-asked” return policy. In addition, we show that “linear return policies” aresuboptimalwhen the hazard rates of initial estimates aremonotone.We next examine the casewhere the salvagevalue is endogenously determined.We demonstrate that “separability” no longer holds and the “recall” of buyersis necessary in the optimal mechanism.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

In many auctions, buyers have uncertain valuations about the ob-jects. These uncertainties can only be resolved after the actual transac-tions take place and the buyers receive the objects. For example, inonline auction sites such as eBay.com, buyers' valuations are subject toafter-transaction shocks, adding to their initial estimates. These shockscan result from matching tastes or styles, complementarity with otherproducts that the consumers have already owned, the conditions ofthe objects upon the arrivals of the shipments, etc. Similarly, in an estateauction, bidders may not know their ultimate values of a piece of furni-ture for sale until they have it in the house and see howwell the color ofthe upholstery matches the carpet and the size fits relatively to existingpieces. When firms bid for the assets of a bankrupt company, they will

ous referees, James Amegashie,abojnik and to the seminar par-iversity of Queensland, Zhejianggress of Game Theory, and thecro Theory and Experimentald suggestions. This paper wasnd Policies as Optimal Sellingr.

ghts reserved.

not know their true values until they begin integrating them into theirexisting concern.1 Foals are often auctioned before they are born inJapanese racehorse industry. Similarly, livestock breeders auction em-bryos in Australia, Canada, and the United States. Agricultural produceis auctioned off long before it is harvested. In the UK gas industry, theNational Grid auctions off the transmission capacity rights long beforethe realization of demand.

With these uncertainties, sellers can choose to auction off the objectsin the traditional way, but more revenue may be raised by using a mech-anismwith terms conditioning on the new information available later on.A simple way to implement this is to run an auction, and then allow thewinner to return the object after the uncertainty resolves. Indeed, manysellers in various online auction sites, such as Amazon.com, eBay.com,Johareez.com and Yahoo.com, provide return services; buyers get theirtransaction prices refunded after paying some restocking fees and ship-ping feeswhen returning the objects. A recent search for antique auctionsin eBay.com came out 161,729 items, and 108,150 (67%) of them camewith certain return policies. The percentage of art auctions offeringrefunds is even higher, 131,944 out of 175,329 auctions had return poli-cies, representing 75% of the auctions. Other examples include the NHL(National Hockey League) online auctions, which provide a 7-Day, 100%Money-Back Guarantee, sellers in auctions for embryo, who guarantee

1 We thank a referee for providing us with the above two motivating examples.

453J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

pregnancy, and the National Grid, which promises to buy back capacityrights.

In this paper, we aim to examine the optimal auction design problemwhen buyers have uncertain valuations, and illustrate how auctionswith return policies can achieve optimality.2 With a more generousreturn policy, on one hand, the seller may end up with the returnedobject and pay back the buyer more often; on the other hand, buyerswill bid more aggressively. The optimal return policy balances thesetwo effects. We perform our analysis in an environment where a seller(she) sells one indivisible object to a few buyers. A buyer's (he) valua-tion depends on both his initial estimate and a shock. In the beginning,buyers observe their initial estimates privately. A buyer learns his shockonly after he wins, pays, and then receives the object for examination.Usually, the realization of thewinner's shock is privately observed. For ex-ample, whether the clothes fit the buyer or not is privately observableonly to the buyer himself. One of our results is that the seller can achievethe same revenue regardless this shock being publicly or privatelyobserved.

Since buyers learn more information after inspecting the object, theseller may have interest to reallocate the object after new informationflows in. As a result, one important issue is how the seller values thereturned object (salvage value). We first study the case where there ismuch fluctuation in the market place, and therefore, the salvage valueis exogenously determined. In the optimal mechanism, the seller firstallocates the object to the buyer with the highest “modified virtual ini-tial estimate”, provided that it is higher than the seller's reservationvalue. She then lets the winner return the object if his ex-post valuationis lower than a cutoff, which depends on and only on his initial estimate.The cutoff is lower than the socially efficient one, implying excess returnin the optimal mechanism. The optimal mechanism is proved to beseparable. Competition among buyers only affects the selection of thewinner. The return rule for when the winner should return the objectdepends only on the winner's initial estimate and his shock, not onthe number of buyers nor the losers' types. As a result, the seller shouldselect the return rules as if the winner were the only buyer. Therefore,the analysis of the optimal mechanism can be disaggregated into twoseparate issues: selecting the right winner and optimizing the returnpolicy.

We then illustrate that the optimal revenue can be implemented bya mechanism with “no-questions-asked” return policy, which does notrequire the seller to observe the winner's realized shock. In reality,return policies can take different forms, such as refund contracts, optioncontracts, and cancelation fees. Our results rationalize the wide adop-tion of return policies by sellers in worldwide. Furthermore, we findthat linear return policies widely observed in online auctions are oftensuboptimal. In a linear return policy, the seller charges a fixed fee plusa percentage fee (of the transaction price) for any return. Full refundpolicies, proportional restocking fees and flat cancelation fees are allexamples of linear return policies.

We second examine the casewhere the group of buyers is stable andready to interact again, and therefore, the salvage value is endogenouslydetermined. In a stylized symmetric two-player environment, the sellerfirst allocates the object to thewinner, who is the buyer with the higherinitial estimate. If his ex-post valuation is lower than a certain cutoff,which depends on the other buyer's (the loser's) initial estimate, hereturns the object, and then the seller allocates the object to the loser.In this case, if the loser's ex-post valuation is lower than another cutoff,the loser returns the object, and the seller allocates the object to thewinner again. Those features illustrate that the optimal mechanism isno longer separable and recalls are important in the mechanism.

2 Wang and Zhang (2010) consider a different type of uncertainty. They illustrate howreturn policies can beused tomitigate thewinner's curse in common value auctions. In an-other paper,Wang and Zhang (2011) consider return policies under an informedprincipalsetup.

The rest of the paper is organized as follows. In Section 2, we reviewthe literature. In Section 3, we describe the model. In Section 4, weexamine the optimal mechanism with exogenous salvage value. InSection 5, we characterize the optimal mechanism with endogenoussalvage value. And in Section 6, we conclude. All proofs are relegatedto an Appendix A.

2. Related literature

Our basic setup howbuyers gain information after buying is borrowedfrom a series of papers byHaile (2000, 2001, 2003)who uses this setup tomotivate auctions with resale. Whereas Haile assumes that buyers havethe option to re-auction the object, we consider the environment thatgives buyers an option to return it to the original seller who may thenresell among the other buyers, but do not allow buyers to resell on theirown.

The most closely related paper is Courty and Li (2000) on dynamicprice discrimination which is motivated by the sale of airline tickets.There, an airline sells a ticket with the option to return the ticket withsome costs prior to the flight. Airlines sell personalized tickets whichcannot be traded by the buyer. Therefore, it is assured that buyers donot have the option to resell on their own. The return option allowsthe airline to extract surplus that arises in the event when the customerhas downgraded his valuation. They provide a well executed generalmodel of dynamic price discrimination, beyond the motivating butvery fitting airline ticket pricing example. In their paper, airlines haveproduction functions with constant marginal costs and can serve theentire market. Therefore, there is no competition among consumers.Introducing competition among consumers to Courty and Li (2000)makes it difficult to analyze the direct mechanisms.

Our result that buyers cannot gain any informational rents for theirprivate information that is realized after the contract is signed has beenobtained in different environments. Our model setup is closest to Esoand Szentes (2007a) who consider the situation where a seller facesbuyers with initial estimates of the object and can control as well ascostlessly release additional private signals correlated to the buyers'valuations. Because the release of information is costless to the seller,she releases all information to all buyers. In our model, learning is costly,and it is usually not optimal to have all of the buyers learning theirshocks. We are interested in the optimal sequence of buyers learningtheir shocks and the role of return policy in the optimal mechanism.Crémer et al. (2009) construct an ingenious sequential selling mecha-nism, assuming buyers face a significant cost of drawing their valuations.However, in their paper potential buyers draw their valuations whenthey are offered to buy, whereas in our paper the buyer observes hisvaluation only after he has purchased. They find that the seller extractsall the surplus by inducing the efficient allocation since bidders do nothave private information prior to signing the contract. In contrast, ouroptimal mechanism generally does not induce an efficient allocation,since the buyers possess some private information before signing thecontracts. In Eso and Szentes (2007b), a consultant can reveal signalsthat affect her client's valuation and can make the payment rule condi-tional upon the client's actions. In their paper, it is the third party, theconsultant, who controls the signal revelation.

This paper is related to the literature on return policies and moneyback guarantees. Che (1996) considers consumers' risk aversions andobtains intuitive results. In his paper, a return policy provides insurancefor a consumer's ex-post loss. Since the buyer is risk averse and the selleris risk neutral, the seller earns the risk premium.However, to induce theseller to provide return policies, consumers must be highly risk averse.Furthermore, providing a return policy can never be optimal whenbuyers are risk neutral. In our paper, the seller is better off by providingreturn policies, even if she is facing risk neutral buyers. Davis et al.(1995) consider risk neutral buyers and find that when the salvagevalue of a product is relatively large, the seller gains from return policies.This is consistent with our findings. Both Che (1996) and Davis et al.

454 J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

(1995) only compare selling with the full return policy and no returnpolicy and compute the conditions under which the full return policyis better than the no return policy. In addition, they assume that thereis only one representative consumer. In contrast, we characterize theoptimal return policy under competition among consumers.

3. The model

A revenuemaximizing seller (she) has a single indivisible object, anddesigns optimal mechanisms to allocate the object to n privately in-formed buyers, indexed by i ∈ {1,⋯,n}. All parties are risk neutral. Thereservation value of the seller is publicly known as v0, which is thevalue the seller gets if she retains the object. Buyer i's (he) valuation ofthe objectVi is composed of two parts, Vi = Wi + Φi, whereWi is his ini-tial estimate andΦi is his shock. Let vi,wi andϕi denote their realizations,respectively. In the beginning, each buyer i privately observes his initialestimatewi; no one can directly observe any of the shocks and all partiesonly know the joint distribution of the shocks. However, if the sellersends the object to a buyer, he can learn of his shock after inspection.

Whenever the seller sends the object to a buyer, the buyer incurs acost Cw. For example, in online auctions, the buyer has to pay for theshipping cost and this cost could be significant for items like TVs, com-puters and furniture. There is also a cost Cr incurred by the buyer whenhe sends the object back to the seller after inspection. This cost includesthe shipping cost for sending the object back to the seller and thedisutilityassociatedwith the time and effort of doing so. For the costs Cw and Cr, wecan also assume that they are incurred by the seller or shared between theseller and the buyer. This does not change the structure of the optimalmechanism but it changes the seller's revenue by a constant. We canalso allow Cr to be negative to capture the situation where buyers maygain some utility from using the object even though it will be returnedlater on.

An important issue is how the seller values the returned object,i.e., the salvage value. Wewill focus on two different settings dependingon the specific applications. In Section 4, we assume that the seller re-ceives an exogenous salvage value of the object S. This applies to the sit-uationwhere the buyers change frequently in themarket place such as inonline auctions, so that one could reasonably assume that the value of thereturn object depends on expectations of future demand, which impliesthat the salvage value is exogenous.3 With exogenous salvage value,the seller can choose only one of the buyers to inspect the item and tolearn of his shock. As a result, in the seller's mechanism, she specifieswho can win the chance for inspection (the winner), whether the objectshould be returned after inspection, andmonetary transfers frombuyers.In Section 5, we examine endogenous salvage value. This is applicable ifthe group of bidders is stable and ready to interact again. As a result,the salvage value is a function of the current demand, and therefore, isendogenously determined. In this case, the seller's mechanism needs tospecify the sequence of buyers' inspections, whether a buyer shouldreturn the object after inspection, and monetary transfers.

Buyers' initial estimates and their shocks are independent and initialestimates are independent across buyers. However, shocks may be cor-related across buyers. The distribution of buyer i's initial estimatewi is Fi,with associated density function fi and support Wi ¼ wi;wi½ �. Let F de-note the joint distribution of the initial estimates. The joint distributionof buyers' shocks is G and the marginal distribution of buyer i's shock is

Gi, with density function gi and support Φi ¼ ϕi;ϕi

h i. We assume that

the support of the shock covers the whole real space to avoid cumber-some notations. We also assume that the hazard rate function of anybuyer i's initial estimate f i wið Þ

1−Fi wið Þ is increasing.

3 The assumption of exogenous salvage valuemay be applicabledue to other factors. Forexample, if an object is in a sealed original package, the seller needs to send the returnedobject to themanufacturer, or sell it as an open box or refurbished item. Therefore, the sal-vage value represents the reduced form valuation of a returned object in these situations.

4. Exogenous salvage value

When the salvage value is exogenously determined, the sellerchooses one of the buyers (thewinner) and sends him the object for in-spection. After the winner learns his shock, the seller then decideswhether the object should be returned. If the object is not returned,the winner consumes the object and the game ends; if the object isreturned, the seller obtains an exogenous salvage value and the gameends. The revelation principle enables us to restrict our attention todirect mechanisms when searching for the optimal mechanism.However, even direct mechanisms are difficult to characterize in oursituation. After all buyers announce their initial estimates in the firstreporting stage, the seller has to control for the type of informationshe would like to reveal regarding those reports. This informationrevelation within stages affects the winner's incentive compatibilityconstraint in the second reporting stage, if either the return rules orthe monetary transfers depend on the losers' reports. However, itis complicated to formulate all possible information revelation ruleswithin stages in the determination of the optimal mechanism.

To overcome this difficulty, we first examine a subset of the feasiblemechanisms in the (original) problem and find the optimal mechanismwithin this subset of mechanisms.We then show that this optimalmech-anism generates the same revenue as that from the optimal mechanismin a relaxed problem where the realization of the winner's shock is pub-licly observed. When the realization of the winner's shock is publicly ob-served, the seller can always ignore the information. As a result, theoptimal revenue in the original problem can never be higher than thatin this relaxedproblem, andwe can conclude that the optimalmechanismin the above subset ofmechanisms is also optimal in the original problem.

The subset that we examine is a class of two-stage direct mecha-nismswith full information revelation within stages. This directmecha-nism works as follows:

1. Knowing their initial estimates privately, buyers make participationdecisions. If a buyer, say i, participates, he reports his initial estimatesewi to the seller confidentially. Let ew ¼ ewi; ⋯; ewnð Þ denote the vectorof reports.

2. Buyer i wins the object with probability xi ewð Þ.3. All buyers' reports about their initial estimates are made public.4. If buyer i is thewinner, his realized shockϕi is privately observed and

he sends report eϕi to the seller.5. The winning buyer i keeps the object with probability yi ew; eϕi

� �and

returns the object to the seller with probability 1−yi ew; eϕi

� �.

6. Contingent monetary transfers t j ew; eϕi

� �are made from buyer j to

the seller.

Let x ¼ x1 ewð Þ; ⋯; xn ewð Þð Þ be the vector capturing the winningprobabilities of the buyers. Let y ¼ y1 ew;ϕ1ð Þ; ⋯; yn ew;ϕnð Þð Þ be thevector capturing the keeping probabilities of the buyers. Finallylet t ¼ t1 ew;ϕ1ð Þ; ⋯; t1 ew;ϕnð Þ; ⋯; tn ew;ϕnð Þð Þ be the vector capturingthe monetary transfers of the buyers. We call xi ewð Þyi ew;ϕið Þ buyeri's consuming probability conditional on his shock.

Let N represent the set of buyers, i.e., N = {1,⋯,n}. Let W and Φdenote respectively the sets of all possible combinations of the buyers'initial estimates and the shocks, i.e., W = W1 × ⋯ × Wn and Φ =Φ1 × ⋯ × Φn. LetW−i andΦ−i denote respectively the sets of all possi-ble combinations of the initial estimates and the shocks which mightbe held by buyers other than i, i.e., W−i = × j ∈ N,j ≠ iWj and Φ−i =× j ∈ N,j ≠ iΦj.

The seller maximizes her revenue by choosing (x, y, t). This mech-anism must satisfy the incentive compatibility constraints in bothreporting stages and the participation constraints. Furthermore, sincethere is only one unit of the object to be allocated, the following addi-tional conditions must be satisfied:

Xni¼1

xi wð Þ≤1and xi wð Þ≥0; ∀i∈ N;∀w∈ W; ð1Þ

455J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

0≤ yi w;ϕið Þ≤1; ∀i∈ N; ∀w∈ W; ∀ϕi ∈ Φi: ð2Þ

As there are two reporting stages in the model, we examine theincentive compatibility and participation constraints starting in thesecond reporting stage of the mechanism.

4.1. Incentive compatibility constraints in the second reporting stage (IC2)

At the beginning of the second reporting stage, all reports about ini-tial estimates are made public. Without loss of generality, let buyer i bethe winner. Since only the winner has the chance to learn his shock,only he needs to make a report about his shock. Given that all theother buyers have truthfully reported their initial estimates in the first

reporting stage, we let eUi eϕi;ϕi; ewi;wi;w−i

� �denote buyer i's payoff in

the second reporting stage. Here, he reports his initial estimate as ewi

in the first reporting stage, and reports his shock as eϕi in the secondreporting stage. Formally,

eUi eϕi;ϕi; ewi;wi;w−i

� �¼ wi þ ϕið Þyi ewi;w−i;

eϕi

� �−Cr 1−yi ewi;w−i;

eϕi

� �h i−ti ewi;w−i;

eϕi

� �¼ wi þ ϕi þ Crð Þyi ewi;w−i;

eϕi

� �−Cr−ti ewi;w−i;

eϕi

� �:

ð3ÞIf he keeps the object, he obtains the utility from consuming the ob-

ject; if he returns the object, he has to pay the cost Cr; and the monetarytransfer is deducted from his payoff.

His incentive compatibility constraint in the second reporting stagerequires that it is optimal for him to report his realized shock truthfully,if he has truthfully reported his initial estimate in the first reportingstage, i.e.,

eUiϕi;ϕi;wi;wi;w−ið Þ≥ eUi eϕi;ϕi;wi;wi;w−i

� �;

∀i∈ N; ∀w∈ W; ∀ϕi;eϕi ∈ Φi IC2ð Þ:

ð4Þ

No participation constraint is needed in the second reporting stage,since buyers cannot quit as long as they have chosen to participate inthe beginning. The following lemma simplifies (IC2).

Lemma 1. The incentive compatibility constraints in the secondreporting stage, as defined in (4), is satisfied if and only if the followingconditions hold: ∀i∈ N; ∀w∈ W; ∀ϕi;

eϕi ∈ Φi

if ϕi≤eϕi; thenyi w;ϕið Þ≤yi w; eϕi

� �; ð5Þ

eUiϕi;ϕi;wi;wi;w−ið Þ ¼ eUi

ϕi;ϕ

i;wi;wi;w−i

� �þ ∫ϕi

ϕi

yi w; ξð Þdξ: ð6Þ

The proof is quite standard and is omitted here (seeMyerson, 1981).

In order to characterize the incentive compatibility constraints inthe first reporting stage, we need to examine another situation in thesecond reporting stage. We need to know if the winner has lied abouthis initial estimate in the first reporting stage, how he should manipu-late the report of his shock in the second reporting stage. We have thefollowing lemma.

Lemma 2. Suppose that thewinner hasmisreported his initial estimatein thefirst reporting stage as ewi. In the second reporting stage, it is optimalfor the winner to report his shock as eϕ�

i such that

eϕ�i ¼ wi þ ϕi−ewi: ð7Þ

Furthermore, his payoff eUi eϕ�i ;ϕi; ewi;wi;w−i

� �satisfies:

eUi eϕ�i ;ϕi; ewi;wi;w−i

� �¼ eUi eϕ�

i ;eϕ�i ; ewi; ewi;w−i

� �: ð8Þ

If the winner has misreported his initial estimate, he will correct hislie in the second reporting stage, i.e., he will choose to report his shocksuch that the sum of the reports of his initial estimate and shock isequal to his true ex-post valuation. Furthermore, his payoff is equal tothat of an honest buyer with an initial estimate ewi and a realizedshock eϕ�

i . The intuition behind the result is as follows. Given that thewinner has reported his initial estimate as ewi, his problem of how to re-port his shock is the same as if he had initial estimate ewi and realizedshock eϕ�

i . (IC2) then ensures that truthfully reporting one's shock isoptimal, if he has truthfully reported his initial estimate. Therefore, itis optimal for the winner to report his shock as eϕ�

i .

4.2. Incentive compatibility constraints and participation constraints in thefirst reporting stage (IC1 and PC)

From Lemma 2, we know how thewinner shouldmanipulate the re-port of his shock if he deviates in the first reporting stage. Therefore, wecan formulate a representative buyer's problem in the first reportingstage. Assuming that all other buyers truthfully report their initial esti-mates, buyer i's first stage payoff, if he reports ewi, is given by:

Ui ewi;wið Þ ¼ ∫W−i

∫Φixi ewi;w−ið Þ eUi eϕ�

i ;ϕi; ewi;wi;w−i

� �−Cw

h idGi ϕið ÞdF−i w−ið Þ

−Xj≠i

∫W−i

∫Φ jx j ewi;w−ið Þti ewi;w−i;ϕ j

� �dGj ϕ j

� �dF−i w−ið Þ:

ð9Þ

If he wins, he obtains the continuation value of the second reportingstage eUi eϕ�

i ;ϕi; ewi;wi;w−i

� �, less the cost of learning his shock Cw; if he

loses, he still has to pay when another buyer wins. In the first reportingstage, he does not knowother buyers' initial estimates.More importantly,he does not know his shock, but knows how to report his shock contin-gent upon its realization.

The incentive compatibility constraints and participation constraintsin the first reporting stage require:

Ui wi;wið Þ≥Ui ewi;wið Þ; ∀i∈ N; ∀ewi;wi ∈ Wi IC1ð Þ; ð10Þ

Ui wi;wið Þ≥0; ∀i∈ N; ∀wi ∈ Wi PCð Þ: ð11Þ

Define

Qi wi;ϕið Þ ¼ ∫W−i

xi wi;w−ið Þyi wi;w−i;ϕið ÞdF−i w−ið Þ; ð12Þ

Xi wið Þ ¼ ∫W−i

xi wi;w−ið ÞdF−i w−ið Þ: ð13Þ

Here, Qi(wi,ϕi) is buyer i's expected consuming probability condi-tional on his shock, and Xi(wi) is his expected winning probability. Amechanism is feasible if it satisfies the constraints (1), (2), (4), (10)and (11). The following lemma simplifies the characterization of thefeasible mechanism. The proof is standard and can be found in onlineAppendix A.

Lemma 3. In the direct two-stage mechanisms with full informationrevelation within stages, a mechanism (x, y, t) is feasible if and only ifthe following conditions hold: ∀ i ∈ N, ∀ w ∈ W, ∀ ϕi ∈ Φi,

∫wiewi∫ΦiQ i z;ϕið ÞdGi ϕið Þdz≥∫wiewi

∫ΦiQ i ewi; zþ ϕi−ewið ÞdGi ϕið Þdz; ð14Þ

456 J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

Ui wi;wið Þ ¼ Ui wi;wið Þ þ ∫wi

wi∫ΦiQ i ξ;ϕið ÞdGi ϕið Þdξ; ; ð15Þ

Ui wi;wið Þ≥0; ð16Þ

(1), (2), (5) and (6).

4.2.1. The seller's problemNow, the seller's problem is to maximize her expected revenue by

choosing (x, y, t), subject to feasibility constraints. Formally,

maxx;y;t

Xni¼1

Xnj¼1

∫W∫Φixi wð Þti w;ϕ j

� �dGj ϕ j

� �dF wð Þ

þ v0∫W1−

Xni¼1

xi wð Þ" #

dF wð Þ

þ SXni¼1

∫W∫Φi

xi wð Þ 1−yi w;ϕið Þ½ �f gdGi ϕið ÞdF wð Þ;

ð17Þ

subject to: (x, y, t) is feasible as defined in Lemma 3.The seller's revenue is composed of three parts: monetary transfers

from all buyers, the reservation value if she retains the object, and thesalvage value if the winner returns the object. Define the followingtwo functions: Ji wið Þ ¼ wi−1−Fi wið Þ

f i wið Þ and

J i wið Þ ¼ ∫ϕi

− Ji wið ÞþS−Crϕi þ Ji wið Þ−Sþ Cr½ �dGi ϕið Þ−Cw−Cr þ S;

where Ji(wi) is the virtual initial estimate, and J i wið Þ is the modified vir-tual initial estimate, whichmeasures themarginal revenue by taking intoconsideration the possible shock realization, which will be explained indetail later on.

The following proposition characterizes the optimal two-stage directmechanism with full information revelation within stages, and demon-strates that it is also optimal in the original problem.

Proposition 1. A mechanism (x, y, t) satisfying the following condi-tions, is an optimal two-stage direct mechanism with full informationrevelation within stages.

Allocation rules:

xi wð Þ ¼ 1 if J i wið Þ ¼ argmax J j wj

� �n on

j¼1; v0

� �0 otherwise

;

8<: ð18Þ

yi w;ϕið Þ ¼ 1 if wi þ ϕi ≥1−Fi wið Þf i wið Þ þ S−Cr

0 otherwise;

8<: ð19Þ

Monetary transfers:

ti w;ϕ j

� �¼

0 if i ≠ j

ti wið Þ if i ¼ j and wi þ ϕi≥1−Fi wið Þf i wið Þ þ S−Cr ;

ti wið Þ−1−Fi wið Þf i wið Þ −S otherwise

8>>>><>>>>:ð20Þ

where ti(wi) is defined as

ti wið Þ ¼ 1Xi wið Þ

n∫Φi

wi þ ϕi þ Crð ÞQi wi;ϕið ÞdGi ϕið Þ− Cw þ Crð ÞXi wið Þ

−∫wi

wi∫ΦiQ i ξ;ϕið ÞdGi ϕið Þdξ

oþ 1þ Fi wið Þ

f i wið Þ þ S� �

G − Ji wið Þ þ S−Crð Þ:

ð21Þ

Seller's revenue:

Revenue ¼ ∫W

max J i wið Þf gni¼1; v0�

dF wð Þ: ð22Þ

This revenue is equal to the optimal revenue when the winner’sshock is publicly observed. As a result, the constructed mechanism isoptimal in the original problem.

The above proposition illustrates that the seller can achieve the samerevenue as if she can observe the winner's realized shock. The intuitionis as follows. In the beginning, buyers have no information advantagesover the seller regarding their shocks. Since the seller is proposing thecontract and has all the bargaining power, she extracts all the benefitsfrom the shocks.

In the optimalmechanism, the allocation rules (x,y) are uniquely de-termined. However, there is a lot of freedom in choosing the monetarytransfers since buyers are risk neutral. In fact, a generalized revenueequivalence theorem holds. As long as two incentive compatible mech-anisms induce the same allocation rules, and the informational rents forthe lowest initial estimates are the same, they will generate the sameexpected revenue for the seller. In the constructed optimal mechanism,only the winner needs to pay. The winner's payment only depends onhis own initial estimate and shock. Furthermore, conditional on returnor not, the winner's payment is independent of the shock.

Since the optimal mechanism can be fully characterized by the allo-cation rules, as defined in Eqs. (18) and (19), it is helpful to discuss themseparately. In the standardMyerson's setup, when the seller assigns theobject to a buyer (who is also the final owner of the object), she obtainsthe virtual valuation of that buyer. The virtual valuation can be inter-preted as the total surplus minus the buyer's informational rent, mea-sured by the hazard rate. If the seller retains the object, there is noinformational rent. Therefore, given a vector of reports v, the seller allo-cates the object to the one with the highest virtual valuation (includingherself). In our model, shocks do not generate informational rent to thebuyers as they are observed after the mechanism is proposed and theseller has all the bargaining power. Therefore, the informational rentis only generated by the private initial estimate.

First consider the choice of yi(⋅) in Stage 5. If the seller asks thewinnerto keep the object, thewinnerwill become the finalwinner. The total sur-plus is thenwi + ϕi and the informational rent is 1−F wið Þ

f wið Þ , and the seller ob-tains Ji(wi) + ϕi. If the seller asks the buyer to return the object, the sellerwill become thefinalwinner. Then the total surplus is S − Cr and the rentis zero, and the seller obtains S − Cr. Therefore, it is optimal for the sellerto ask the winner to return the object if and only if Ji(wi) + ϕi ≤ S − Cr.Now consider the choice of xi(⋅) in Stage 2. If the seller keeps the object,then she obtains v0. If she allocates the object to a buyer, say i, howmuch the seller can obtain depends on who will be the final owner,which is determined by the rule yi(⋅) in Stage 5. She obtainsJi(wi) + ϕi − Cw if ϕi turns out to be small so that the final owner isbuyer i; she obtains S − Cw − Cr if ϕi turns out to be big so that thefinal owner is the seller. Therefore, after simple algebra, the seller obtainsan expected payoff equal to J i wið Þ. Hence, the seller should allocate theobject to the one with the highest J i wið Þ, provided that it is higher thanv0. Similar arguments will be used to determine the optimal mechanismin Section 5 as well.

Note that the return rule yi(w,ϕi) only depends on thewinner's initialestimate and his shock, not on the number of buyers nor on the losers'types. This implies:

Corollary 1. The optimalmechanism is separable: thewinner selectionand the return rules can be determined independently.

Competition among buyers only affects the selection of the winner.The seller should select the return rules as if the winner were the onlybuyer. As a result, the analysis of the optimal mechanism can be dis-aggregated into two issues.

457J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

The first issue is how to select the winner. This rule is deterministic.According to Eq. (18), the winner is the one with the highest modifiedvirtual initial estimate, provided that it is higher than the seller's reser-vation value. In Myerson (1981), the marginal revenue for selling tobuyer i is given by the virtual valuation Ji(wi). Since buyers willlearn more information after they win and may incur other extracosts, J i wið Þ represents the marginal revenue for selling to buyer iin our model. When buyers' shocks are symmetric, i.e., Gi = Gj, ∀ i,j ∈ N, Myerson's optimal auction should be held to select the winner,since the buyer with the highest modified virtual initial estimate isthe one with the highest virtual initial estimate. When buyers areex-ante symmetric, i.e., Gi = Gj and Fi = Fj, ∀ i, j ∈ N, a first-priceor second-price auction will select the “best” buyer.

The second issue is how to select the return rules. According toEq. (19), they are cutoff rules and deterministic. If the realization ofthe winner's shock is higher than the cutoff ϕi

∗ = − Ji(wi) + S − Cr,the winner keeps the object; otherwise, he returns the object. A sociallyefficient return rule should have the winner keeping the object if andonly if his ex-post valuation is higher than S − Cr. By examiningEq. (19), we have the following corollary.

Corollary 2. In the optimal return rules, there is no distortion at the top(at v ¼ v), and there is downward distortion everywhere else. This dis-tortion is smaller, and thus there is less return, if thewinner has a higherinitial estimate.

This is a result of the trade-off between efficiency and rent extrac-tion. It provides an explanation for the “excessive returns puzzle” inMatthews and Persico (2005). The distortion is only introduced by thefact that initial estimates are buyers' private information. Hazard ratefunctions of initial estimates measure the distortion introduced byeliciting truthful information. Sincewe assume that the virtual valuationfunctions of initial estimates are increasing, the distortion is smaller fora higher initial estimate. This implies that the winner with a higher ini-tial estimate should return the object less often.

According to Eq. (19), return happenswith positive probability in theoptimal mechanism as we have assumed that the shock covers thewhole real space. Furthermore, either decreasing the salvage value S orincreasing the cost of making a return Cr can lead to less return. This issummarized in the following corollary.

Corollary 3. It is strictly revenue enhancing to allow for return. Thereshould be less return if the salvage value is lower or the cost of makinga return is higher.

Indeed, in online auctions, sellers for jewelry and clothes,whose costof return is lower, are more likely to provide better return policies thanthose for TV, furniture and computers.

4 A complete proof is available upon request.5 Since players are risk neutral, only expected monetary transfers matter.

4.3. Implementation

The above optimal mechanism requires the seller to observe thewinner's realized shock. In this subsection, we will demonstrate thatthe optimal revenue can always be implemented by a mechanismwhich is independent of the winner's realized shock.

Proposition 2. The optimal revenue can be induced by the followingmechanism, which is independent of the winner's realized shock.Buyers report their initial estimates ewi; ⋯; ewnð Þ . Buyer i wins if he hasthe highest modified virtual initial estimate (including v0), and paysan amount equal to ti ewið Þ defined in (21). After the winner learns ofhis shock, he has the option to return the object for a refund equal to1−Fi ewi

�f i ewi

� þ S.

In the return stage, if the winner has truthfully reported his initialestimate, he returns the object if and only if wi þ ϕi≤ 1−Fi wið Þ

f i wið Þ þ S−Cr .

The monetary transfers are constructed such that truthfully reportingis optimal, thus the one with the highest modified virtual initial esti-mate wins if it is higher than v0. As a result, the induced allocationrules in the abovemechanism coincide with that in the optimal mecha-nism. The revenue equivalency theorem then implies that it yields theoptimal revenue.

In the above optimalmechanism, return policies donot depend on thewinner's shock. This explains why “no-questions-asked” return policiesare frequently adopted in reality; there is no need for sellers to knowbuyers' shocks. Thus, we have

Corollary 4. The optimal revenue can be achievedwith a “no-questions-asked” return policy.

In reality, most of the adopted return policies are linear, where con-sumers have to pay for fixed fee plus a percentage fee (of the transac-tion price) between zero and one for refund. However, in the aboveoptimal mechanism, the amount of refund the winner can obtain isequal to 1−Fi wið Þ

f i wið Þ þ S. Since we assume that hazard rate functions of initialestimates are increasing (and thus the inverse hazard rate functions aredecreasing), we obtain the following corollary.

Corollary 5. The amount of refund is decreasing in the winner's initialestimate, implying the sub-optimality of linear return policies.

The property of increasing hazard rate is satisfied bymany common-ly used distributions, including uniform, normal, logistic, chi-squared,exponential and Laplace distributions. This suggests that the commonlyobserved linear return policies in standard auctions are not optimal ingeneral. With linear return policies, in first-price auctions, the amountof refund is an increasing function of the winner's bid, and therefore,is increasing in the winner's initial estimate. Similar properties holdfor second-price auctions. According to our analysis, these return poli-cies are not optimal.

5. Endogenous salvage value

In this section,we consider the case of endogenous salvage value.Weassume that when the object is returned to the seller, she can send theobject to other buyers. For simplicity, we focus on a special case of thegeneral setup. We assume that there are two ex-ante symmetric buyerswith independent shocks. As demonstrated in the previous section,whether the realized shock is privately or publicly observable does notinfluence the seller's optimal revenue. This also holds when the salvagevalue is endogenously determined.4 As a result, we focus on publicly ob-servable shocks.

Again, we can limit our search of the optimal mechanism to directmechanisms. In contrast to the exogenous salvage value case, the sellerneeds to further specify whether to send the object to the loser for in-spection, and the final allocation of the object after the loser also learnshis shock. If the two buyers report ew ¼ ew1; ew2ð Þ, buyers win the objectwith probabilityx1 ewð Þ andx2 ewð Þ. If buyer i ∈ {1,2} is thewinner and hisrealized shock is ϕi, the object is reassigned to the buyerswith probabil-ity y1 ew;ϕið Þ and y2 ew;ϕið Þ. With probability 1−y1 ew;ϕið Þ−y2 ew;ϕið Þ, theseller takes the object. If the seller or the winner, buyer i, obtains theobject, the game ends; if the loser, buyer j, obtains the object and his re-alized shock is ϕj, then the seller determines the final allocation of the

object: assigning the object to buyers with probabilities z1 ew;ϕi;ϕ j

� �and z2 ew;ϕi;ϕ j

� �. Of course, expected monetary transfers t1 ew1ð Þ; t2ew2ð Þ are made from buyers to the seller.5

458 J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

Whenever the seller sends the object to a buyer, the cost Cw isdeducted from the buyer's payoff. Whenever a buyer returns the ob-ject, the cost Cr is deducted from the buyer's payoff. Although thegame involves a lot of stages, buyers only need to act once, i.e., to re-port their initial estimates. All the rules are controlled by the sellercontingent on the publicly observable realizations of the shocks.Let x ¼ x1 ewð Þ; x2 ewð Þð Þ , y ¼ y1 ew;ϕ1ð Þ; y2 ew;ϕ2ð Þð Þ , z ¼ z1 ew;ϕ1;ϕ2ð Þ;ðz2 ew;ϕ1;ϕ2ð Þ; z1 ew;ϕ2;ϕ1ð Þ; z2 ew;ϕ2;ϕ1ð ÞÞ be the vectors capturingall the probabilities. Finally let t ¼ t1 ew1ð Þ; ⋯; tn ewnð Þð Þ be the vectorcapturing all the expected monetary transfers. Then (x,y,z,t) definea direct mechanism.

A standard approach similar to the earlier sections can be applied tosolve for the optimal mechanism. However, for concreteness, we willadopt a more straightforward approach which is similar to the discus-sion after Proposition 1. We know that in this environment, in the opti-mal mechanism the seller obtains the total surplus minus the finalwinning buyer's informational rent. We assume that the seller's reserva-tion value is low enough so that the seller never wants to retain the ob-ject, and thus we can focus on the allocation between the two buyers.First consider the choice of z. In this stage, the object is in the hand ofbuyer j. If the seller allocates the object to buyer i, then she obtainsJ(wi) + ϕi − Cr − Cw, where Cr, Cw are deducted since buyer j mustfirst return the object to the seller who then can send it to buyer iagain; if she allocates the object to buyer j, she obtains J(wj) + ϕj.6 Defineϕj∗(wi,wj,ϕi) = J(wi) + ϕi − Cr − Cw − J(wj). The allocation rule is as

follows:

zi w;ϕi;ϕ j

� �¼ 1 if ϕ j ≤ ϕ�

j wi;wj;ϕi

� �0 if otherwise

(ð23Þ

z j w;ϕi;ϕ j

� �¼ 0 if ϕ j ≤ ϕ�

j wi;wj;ϕi

� �1 if otherwise

(ð24Þ

Now consider the choice of y, given the decision for z. In this stage,the object is in the hand of buyer i. If the seller allocates the object to i,then the seller obtains J(wi) + ϕi since the game ends there and buyeri becomes the final owner; if the seller allocates the object to j, thenhow much the seller obtains depends on the allocation rule for z. If theshock ϕj turns out to be lower than ϕj

∗(wi,wj,ϕi), buyer i will be thefinal winner. The total surplus is then wi + ϕi minus all the transactioncosts involved. Here, buyer i first needs to return the object to the seller,who then sends it to buyer j, who then returns it back to the seller, whofinally sends the object back to buyer i again. The informational rent is1−F wið Þf wið Þ . If the shock ϕj turns out to be greater than ϕj

∗(wi,wj,ϕi), buyer jwill be the final winner, and how much the seller obtains can be calcu-lated in a similarway. Hence, the seller's expected payoff to sell to buyerj is:

eJ wi;wj;ϕi

� �¼ ∫ϕ�

j wi ;wj ;ϕið Þ−∞ J wið Þ þ ϕi−Cr−Cwð ÞdG ϕ j

� �þ∫∞

ϕ�j wi ;wj ;ϕið Þ J wj

� �þ ϕ j

� �dG ϕ j

� �−Cr−Cw

ð25Þ

Denote the solution of ϕi to

eJ wi;wj;ϕi

� �− J wið Þ−ϕi ¼ 0 ð26Þ

as ϕi∗(wi,wj). It is easy to verify that the left-hand-side of Eq. (25) is

strictly decreasing in ϕi almost everywhere, and the maximum is

6 Here, we have not deducted the costs involved before the current stage; these costsare the same for any decision the seller can make.

positive and the minimum is negative. Therefore ϕi∗(wi,wj) is uniquely

determined. As a result, the allocation rule is

yi w;ϕið Þ ¼ 1 if ϕi ≥ ϕ�i wi;wj

� �0 if otherwise

(ð27Þ

yj w;ϕið Þ 0 if ϕi ≥ ϕ�i wi;wj

� �1 if otherwise

(ð28Þ

Now consider the choice of x, knowing the decisions for y and z. If theseller allocates the object to i, then the seller obtains

bJ wi;wj

� �¼ ∫∞

ϕ�i wi ;wjð Þ J wið Þ þ ϕið ÞdG ϕið Þ

þ ∫ϕ�i wi ;wjð Þ

−∞eJ wi;wj;ϕi

� �dG ϕið Þ−Cw

ð29Þ

Since buyers are symmetric, switching i and j reveals that the func-tion forms are symmetric, and both are increasing in the initial esti-mate.7 Thus, the allocation rule is

xi wð Þ ¼ 1 if wi ≥ wj0 if otherwise

�ð30Þ

xj wð Þ ¼ 0 if wi ≥ wj1 if otherwise

�ð31Þ

The following proposition summarizes the optimal mechanism.

Proposition 3. With endogenous salvage value, the seller first allocatesthe object to the buyerwith higher initial estimate, say buyer i. Then shelets him keep the object if his shock is higher than ϕi

∗(wi,wj); and askshim to return the object otherwise. If the object is returned, the sellersends the object to the other buyer j. If buyer j′s shock is higher thanϕj∗(wi,wj,ϕi), he keeps the object; otherwise, he returns the object,

which is then sent to buyer i again.

We know that ϕi∗(wi,wj) depends on the buyer j′s initial estimate.

This implies that the separability of the optimal mechanism no longerholds. This is intuitive since what happens in later stages depends onthe loser's initial estimate. This proposition also reveals that recalls areimportant for optimality as the seller needs to send the object to thewinner again if the loser's shock is too low.

Here, we assume that the shocks are independent. Suppose there is apositive correlation between them. If thewinner's realized shock is low,it is quite likely that the loser's shock will be low as well. For instance,the seller may find it less attractive to take the object back and havethe loser try it out, i.e., the winner keeps the object more often. In addi-tion, if the object is indeed sent to the loser, the winner will be the finalwinner more often as the loser is also likely to have a low shock. Notethat if ϕ1 and ϕ2 are perfectly correlated, then we are back to the exog-enous salvage value case as the sellerwill notwant to resell the object tothe loser and the salvage value is v0.

6. Conclusions and discussions

In this paper, we investigate the optimal auction design problemwhen buyers have uncertain valuations. With exogenous salvage valuefor the seller, the optimal mechanism is deterministic and separable.

7 When the two players are asymmetric, the object is allocated to the one with a highermarginal revenue defined above. When there are more than two players, similar processcan be used to compute the optimal mechanism, although it involves much more compli-cated notation.

459J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

The seller first allocates the object to the buyer with the highestmodified virtual initial estimate, provided that it is higher than theseller's reservation value. She then lets the winner return theobject, if his ex-post valuation is lower than a cutoff. Auctionswith return policies and reserve prices can implement this optimalmechanism. With endogenous salvage value for the seller, the opti-mal mechanism is no longer separable, and involves sequential real-ization of the shocks ranked by the buyers' initial estimates. The returndecisions are deterministic and of cutoff rules. Recalls are important foroptimality.

The structure of buyers' valuations is less restrictive than it ap-pears. First, the assumption that a buyer's initial estimate and hisshock are independent is not losing too much generality. It is nowwell-known that if they are correlated with each other, ProbabilityIntegral Transform Theorem can be used to rewrite buyers' true val-uations as a function of two independent components with furtherrestrictions on the valuation function (see, for example, Eso andSzentes (2007a) and Pavan et al. (2008). Second, we assume an addi-tive structure of buyers' true valuation. It is straightforward to extendthe analysis to a general functional form, i.e., a buyer's true valuationis a function of his initial estimate and the shock. In this case, we needfurther restrictions on the function in the analysis. However, none ofthe main results shall be affected. We believe that the current settingis more transparent and more intuitive.

In some cases, awinner can resell the object instead of returning it tothe seller. In online auctions, when thewinnerwants to resell the object,he needs to open a new auction, and it is usually hard for him to find thelosers who were in the previous auction to participate. As a result, heoften faces a new set of buyers. Therefore, such a resale opportunity pro-vides him with an exogenously determined reduced value. If we rede-fine buyers' valuations by incorporating this resale value, our analysiscan apply. Nevertheless, when the winner can resell the object to thelosers who were in the previous auction, the mechanism design prob-lem becomes much more complicated. The winner needs to updatehis belief about the losers' private information based onwhat happenedin the initial auction, which is determined by the initial seller's mecha-nism. Note that the same problem arises in most of the existing litera-ture. For example, in Myerson (1981) the optimal mechanism favorsthe weak buyers and the revenue maximizing allocation is not efficientamong buyers. As a result, when the seller cannot prevent inter-bidderresale, it would actually happen afterward, and it is not clear whetherthe same revenue could be achieved. Therefore, Myerson (1981) re-quires the assumption of no resale for the theory to work, which isalsowhat we need. Nevertheless, the issue of resale has attracted signif-icant attentions in the recent literature. Zheng (2002) provides the firstcomprehensive analysis, and illustrates that under certain conditions,Myerson's revenue is still achievable even if resales cannot beprohibited. In this sense, the optimal mechanism in our paper providesan upper bound revenue and the same revenue may be achieved whenresale cannot be prevented.8

Online auctions differ from the traditional ones as there are muchmore uncertainty involved. The current paper examines one type of un-certainty. Nevertheless, uncertainty can also result from the quality ofthe object, known as the “informed-principal” problem since the sellershave better knowledge about their own products than the buyers. How-ever, most of the online auction sites have credible rating system andconsumer's protection program, which may partially resolve the in-formed principal problem. For example, eBay provides the “top-ratedseller” title to excellent sellers, as well as the “eBay buyer protection”program which protects buyers if the items do not arrive or do not fitthe descriptions.

8 Calzolari and Pavan (2006) and Zhang and Wang (forthcoming) characterize the op-timal mechanism when Myerson's revenue cannot be achieved.

Under such systems, the informed principal problem is minimized.In addition, sellers usually have different return policies for differentreasons. If the problem is about the quality of the product, then the sell-ermay providewarranties or very generous return policies; if the returnis due to issues on the side of the consumers, such as a consumerchanges his mind, then the seller usually charges some handling andrestocking fees for return. We treat these two different types of uncer-tainty separately, andwe focus on the latter. Furthermore, on eBay.com,identical goods are usually being sold bydifferent sellers. Itmaybe of in-terest to investigate how competitions among sellers would affect thereturn policies.

Appendix A. Proofs

Proof for Lemma 2.

eUi eϕi;ϕi; ewi;wi;w−i

� �¼ wi þ ϕi þ Crð Þyi ewi;w−i;ϕið Þ−Cr−ti ewi;w−i;

eϕi

� �¼ ewi þ wi þ ϕi−ewið Þ þ Cr½ �yi ewi;w−i;ϕið Þ−Cr−ti ewi;w−i;

eϕi

� �≤ ewi þ wi þ ϕi−ewið Þ þ Cr½ �yi ewi;w−i;wi þ ϕi−ewið Þ−Cr−ti ewi;w−i;wi þ ϕi−ewið Þ¼ eUi

wi þ ϕi−ewi;ϕi; ewi;wi;w−ið Þ:ð32Þ

The inequality follows from the fact that it is optimal for a winnerwith the initial estimate ewi and shock wi þ ϕi−ewi to truthfully reporthis shock, if he has already truthfully reported his initial estimate inthe first reporting stage. From the above formula, it is optimal for thewinner to report his shock as eϕ∗

i ¼ wi þ ϕi−ewi , if he has reported hisinitial estimate as ewi.

The payoff becomes

eUi eϕ�i ;ϕi; ewi;wi;w−i

� �¼ wi þ ϕi þ Crð Þyi ewi;w−i;

eϕ�i

� �−Cr−ti ewi;w−i;

eϕ�i

� �¼ ewi þ eϕ�

i þ Cr

� �yi ewi;w−i;

eϕ�i

� �−Cr−ti ewi;w−i;

eϕ�i

� �¼ eUi eϕ�

i ;eϕ�i ; ewi; ewi;w−i

� �:

ð33Þ

This means that his payoff is equal to the payoff of an honest buyerwith an initial estimate ewi and a realized shock eϕ∗

i . Q.E.D.

Proof for Proposition 1. Function Ui(wi,wi) can be formulated asfollows:

Ui wi;wið Þ¼

ZΦi

wi þ ϕi þ Crð ÞQi wi;ϕið ÞdGi ϕið Þ− Cw þ Crð ÞXi wið Þ

−Xnj¼1

ZW−i

ZΦ j

x j wð Þti w;ϕ j

� �dGj ϕ j

� �dF−i w−ið Þ:

ð34Þ

Rearranging terms:

Xnj¼1

ZW−i

ZΦ j

x j wð Þti w;ϕ j

� �dGj ϕ j

� �dF−i w−ið Þ

¼Z

Φi

wi þ ϕi þ Crð ÞQi wi;ϕið ÞdGi ϕið Þ− Cw þ Crð ÞXi wið Þ

−Ui wi;wið Þ−Z wi

wi

ZΦi

Q i ξ;ϕið ÞdGi ϕið Þdξ:

ð35Þ

The equality follows from Eq. (15).

460 J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

Now we can calculate buyer i's expected payment from the prospec-tive of the seller:

Xnj¼1

ZW

ZΦ j

x j wð Þti w;ϕ j

� �dGj ϕ j

� �dF wð Þ

¼Z

W

ZΦi

wi þ ϕi þ Crð Þxi wð Þyi w;ϕið ÞdGi ϕið ÞdF wð Þ−Z

WCw þ Crð Þxi wð ÞdF wð Þ

−Z

Wi

Z wi

wi

ZΦi

ZW−i

xi ξ;w−ið Þyi ξ;w−i;ϕið ÞdF−i w−ið ÞdGi ϕið ÞdξdFi wið Þ−Ui wi;wið Þ

¼Z

W

ZΦi

wi þ ϕi þ Cr−1−Fi wið Þf i wið Þ

� �xi wð Þyi w;ϕið ÞdGi ϕið ÞdF wð Þ

−Z

WCw þ Crð Þxi wð ÞdF wð Þ−Ui wi;wið Þ;

ð36Þ

The second step made used of integration by parts. The seller'srevenue becomes:

Xni¼1

ZW

ZΦi

wi þ ϕi þ Cr−1−Fi wið Þf i wið Þ

� xi wð Þyi w;ϕið ÞdGi ϕið ÞdF wð Þ

−Xni¼1

ZW

Cw þ Crð Þxi wð ÞdF wð Þ−Xni¼1

Ui wi;wið Þ

þ v0

ZW

1−Xni¼1

xi wð Þ" #

dF wð Þ

þ SXni¼1

ZW

ZΦi

xi wð Þ 1−yi w;ϕið Þ½ �f gdGi ϕið ÞdF wð Þ;

¼Z

W

Xni¼1

ZΦi

wi þ ϕi þ Cr−S−1−Fi wið Þf i wið Þ

� �yi w;ϕið ÞdGi ϕið Þ−Cr−Cw−v0 þ S

( )xi wð ÞdF wð Þ

þ v0−Xni¼1

Ui wi;wið Þ

ð37Þ

The remaining constraints are (5), (6), (14), (16), (1) and (2).We canfirst ignore (5), (6) and (14), and thenprove that the optimalmechanismsatisfies these constraints. First, it is obvious that it is optimal to setUi wi;wið Þ ¼ 0; ∀i∈N.

Second, it is optimal to choose yi(w,ϕi) as follows:

yi w;ϕið Þ ¼ 1 if ϕi≥− Ji wið Þ þ S−Cr0 otherwise

;

�ð38Þ

where Ji wið Þ ¼ wi−1− Fi wið Þf i wið Þ is the virtual initial estimate. Since the hazard

rate functions of the initial estimates are increasing, the virtual initialestimate function Ji(wi) is also increasing.

Now the objective function becomes:

ZW

Xni¼1

Z ϕi

− Ji wið Þ−CrþSϕi þ Ji wið Þ þ Cr−S½ �dGi ϕið Þ−Cr−Cw−v0 þ S

( )xi wð ÞdF wð Þ þ v0

¼Z

W

Xni¼1

J i wið Þ−v0f gxi wð ÞdF wð Þ þ v0:;

ð39Þ

where J j wj

� �¼ ∫ϕ j

− J j w jð Þ−CrþSϕ j þ J j wj

� �þ Cr−S

h idGj ϕ j

� �−Cw−

Cr þ S is the modified virtual initial estimate. Therefore, it is clear thatthe choice of xi(w) should be as follows:

xi wð Þ ¼ 1 if J i wið Þ ¼ argmax J j wj

� �n on

j¼1; v0

� �0 otherwise

:

8<: ð40Þ

It is helpful to note that the modified virtual initial estimate is alsoincreasing:

J ′j wj

� �¼

Z ϕ j

− J j w jð Þ−CrþSJ′j wj

� �dGj ϕ j

� �¼ J′j wj

� �1−G − J j wj−Cr þ S

� �� �h i≥0:

ð41Þ

Thus, the seller's revenue becomes:ZWmax J i wið Þ; v0f gni¼1dF wð Þ: ð42Þ

Now, we only need to show that the allocation rules (18) and (19)satisfy the monotonicity constraints (5) and (14).

It is obvious that (5) is satisfied given (19). It is much more compli-cated to verity (14). Let us consider the case ewi≤wi.

For z≥ ewi,

xi z;w−ið Þ ¼ 1 if J i zð Þ ¼ argmax J j wj

� �n oj≠i

; J i zð Þ; v0� �

0 otherwise;

8<: ð43Þ

xi ewi;w−ið Þ ¼ 1 if J i ewið Þ ¼ argmax J j wj

� �n oj≠i

; J i ewið Þ; v0� �

0 otherwise:

8<:ð44Þ

Since J i �ð Þ is increasing, we obtain

xi z;w−ið Þ≥xi ewi;w−ið Þ; ð45Þ

In addition,

yi z;w−i;ϕið Þ ¼ 1 if ϕi≥−zþ 1−Fi zð Þf i zð Þ þ S−Cr

0 otherwise;

8<: ð46Þ

yi ewi;w−i; zþ ϕi−ewið Þ ¼(1 if zþ ϕi−ewi≥−ewi þ

1−Fi ewið Þf i ewið Þ þ S−Cr

0 otherwise

¼ 1 if ϕi≥−zþ 1−Fi ewið Þf i ewið Þ þ S−Cr

0 otherwise:

8<:ð47Þ

Since the hazard rate functions of initial estimates are increasing, we

obtain 1− Fi ewi

�f i wið Þ ≥ 1−Fi zð Þ

f i zð Þ . Therefore, by comparing (46) and (47), we obtain:

yi z;w−i;ϕið Þ≥yi ewi;w−i; zþ ϕi−ewið Þ: ð48Þ

Combining (45) and (48) provides us with:

xi z;w−ið Þyi z;w−i;ϕið Þ≥xi ewi;w−ið Þyi ewi;w−i; zþ ϕi−ewið Þ ð49Þ

⇒Z

Φi

ZW−i

xi z;w−ið Þyi z;w−i;ϕið ÞdF−i w−ið ÞdGi ϕið Þ

≥Z

Φi

ZW−i

xi ewi;w−ið Þyi ewi;w−i; zþ ϕi−ewið ÞdF−i w−ið ÞdGi ϕið Þð50Þ

⇒Z

Φi

Q i z;ϕið ÞdGi ϕið Þ≥Z

Φi

Q i ewi; zþ ϕi−ewið ÞdGi ϕið Þ ð51Þ

⇒Z wiewi

ZΦi

Q i z;ϕið ÞdGi ϕið Þdz≥Z wiewi

ZΦi

Q i ewi; zþ ϕi−ewið ÞdGi ϕið Þdz:

ð52Þ(50) follows because (49) is valid for any ϕi and w−i; (52) follows be-cause (51) is valid for any z≥ ewi . The same argument can be used forthe case wi≤ ewi.

461J. Zhang / International Journal of Industrial Organization 31 (2013) 452–461

Eq. (6) is equivalent to

ti w;ϕið Þ ¼ wi þ ϕi þ Crð Þyi w;ϕið Þ−Z ϕi

ϕi

yi w; ξð Þdξ−Cr−eUi ϕi;ϕ

i;wi;wi;w−i

� �:

ð53Þ

Given the return rules yi(w,ϕi), we can pin down the winner's mone-tary transfer. From Eqs. (53) and (19), when ϕi ≤ −Ji(wi) + S − Cr,

ti w;ϕið Þ ¼ −Cr−eUiϕ

i;ϕ

i;wi;wi;w−i

� �; ð54Þ

when ϕi ≥ −Ji(wi) + S − Cr,

ti w;ϕið Þ¼ wi þ ϕi þ Crð Þ−

Z ϕi

− Ji wið ÞþS−Cr

dξ−Cr−eUiϕ

i;ϕ

i;wi;wi;w−i

� �¼ wi þ ϕi þ Crð Þ− ϕi þwi−

1−Fi wið Þf i wið Þ −Sþ Cr

� �−Cr−eUi

ϕi;ϕ

i;wi;wi;w−i

� �¼ 1−Fi wið Þ

f i wið Þ þ S−Cr−eUiϕ

i;ϕ

i;wi;wi;w−i

� �:

ð55Þ

We thus yield the monetary function in the proposition by

letting only the winner pay and ti wið Þ ¼ 1− Fi wið Þf i wið Þ þ S−Cr−eUi

ϕi;ϕ

i;wi;wi;w−i

� �. The restriction on the ti(wi) is then pinned

down by (35).When the winner's shock is publicly observable, the revelation

principal applies and the seller's problem is a small variation ofMyerson (1981). It can be shown that the optimal mechanism yieldsthe same revenue. Since the seller cannot generate more revenuewhen the winner's shock is privately observable, the constructedmechanism is optimal in the original problem.

Q.E.D.

Proof for Proposition 2. In the return stage, the winner, buyer i, keepsthe object if and only if consuming the object is better than returning it,

i.e., wi þ ϕi≥1− Fi ewi

�f i ewi

� þ S−Cr .

In the auction stage, given that all the other buyers truthfully reporttheir initial estimates, we can formulate buyer i's payoff in the auctionstage. If ewi≤ J i v0ð Þ, then u ewi;wið Þ ¼ 0; if otherwise

u ewi;wið Þ¼ ∏

j≠iFi J−1

j J i ewið Þð Þ� �

�(Z 1− F ewi

�f ewi

� þS−Cr−wi

ϕi

1−F ewið Þf ewið Þ þ S−Cr

� �dGi ϕið Þ

þZ ϕi

1− F ewi

�f ewi

� þS−Cr−wi

wi þ ϕi½ �dGi ϕið Þ−ti ewið Þ) ð56Þ

To determine the equilibrium,we use the following lemmadevelopedin McAfee and Vincent (1993).

Lemma 4. Suppose that v(r,x) is twice continuously differentiable, andv12(r,x) ≥ 0 ∀ x, r. Then

v r; xð Þ≤v x; xð Þ∀ r; x⇔v1 x; xð Þ ¼ 0 ∀ r; x:

The assumption on the cross partial derivative ensures that theobjective function is pseudoconcave. As a result, the second order con-dition for the optimization problem is satisfied, and the first order con-dition is necessary and sufficient for optimality. In our model, we have:

∂u ewi;wið Þ∂wi

¼ ∏j≠i

Fi J−1j J i ewið Þð Þ

� �� 1−Gi

1−F ewið Þf ewið Þ þ S−Cr−wi

� � �; ð57Þ

∂2u ewi;wið Þ∂wi∂ewi

¼d∏ j≠i Fi J−1

j J i ewið Þð Þ� �dewi

1−Gi1−F ewið Þf ewið Þ þ S−Cr−wi

� � �

−∏j≠i

Fi J−1j J i ewið Þð Þ

� �gi

1−F ewið Þf ewið Þ þ S−Cr−wi

� d 1−F ewi

�f ewi

�� �dewi|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}b0

N0:

ð58Þ

The last step follows from the assumption that the hazard rate func-tions of initial estimates are increasing. Therefore, from Lemma 4, FOC isnecessary and sufficient for the maximization problem:

∂u ewi;wið Þ∂ewi

jewi¼wi¼ 0

It can be checked that truthfully reporting is optimal since the pay-ment function ti ewið Þ is constructed from the ICs in Proposition 1. Q.E.D.

Appendix B. Supplementary data

Supplementary data to this article can be found online at http://dx.doi.org/10.1016/j.ijindorg.2013.08.001.

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