Common Value Auctions with Return Policies

Transcription

1 Common Value Auctions with Return Policies Ruqu Wang Jun Zhang October 11, 2011 Abstract This paper eamines the role of return policies in common value auctions. In second-price auctions with continuous signals and discrete common values, we characterize the unique symmetric equilibrium for any given (linear) return policy. We find that, surprisingly, a more generous return policy actually hurts the buyers. We then eamine the optimal linear return policy that maimizes the seller s revenue, and find that it must be in the form of a fied fee, and no percentage fee should be charged for returns. This resembles many real life refund policies with which the sellers refund the full purchasing prices minus some shipping and handling charges. In general, the seller s revenue is the difference between the social welfare and the consumer surplus. With a more generous return policy, the latter is always lower and the former can be lower or higher. We characterize the optimal linear return policy in second-price auctions and then etend the analysis to first-price auctions with a simpler setup. Given the same return policy, second-price auctions generate more revenue than first-price auctions. Keywords: auctions, return policies, common value JEL classification: D44, D72, D82 We thank Bram Cadsby, Rene Kirkegaard, and seminar participants at University of Guelph, Boston University, Tsinghua University, Yonsei University, the 2009 CEA conference in Toronto, and the 2009 Canadian Economic Theory Conference in Montreal for helpful comments. Wang s research is supported by the Social Sciences and Humanities Research Council of Canada. Economics Department, Queen s University, Kingston, Ontario, K7L 3N6, Canada. Phone: +1(613) , wangr@queensu.ca. School of International Business Administration, Shanghai University of Finance and Economics. Phone: +1(613) , zhangjunqueens@gmail.com. 1

2 1 Introduction Traditional auctions have a history of thousands of years and could date back to the fifth century B.C. However, refunds are rarely allowed. Recently, the rapid growth of internet commerce makes online auctions etremely popular. These auctions create a problem for both the buyers and the sellers. As a buyer is unable to personally eamine the good before bidding, bidders are usually facing a lot of uncertainty. Nevertheless, consumers could learn more information after receiving and inspecting the object upon winning. Such situations make online auctions significantly different from traditional ones, and bring an active role for return policies. A casual survey on ebay.com and Amazon.com shows that more than half of the sellers provide a refund policy for returns. How would a return policy affect buyers behavior? Would the sellers (and the buyers) benefit from such a return policy? How should a revenue maimizing seller select the proper return policy? These are some of the issues we will investigate in this paper. In a private-value auction, return policies do not affect a buyer s bidding strategy, since he never bids more than his valuation. In contrast, with interdependent or common values, which captures many real life applications, return policies induce bidders to bid more aggressively. Returns could happen with positive probability after the winning bidder receives the good and learns more information about its true value. In this paper, we focus on the common-value model in Wilson [13], since it is the simplest model accounting for interdependent and correlated values. 1 This model is widely used to analyze oil, gas and mineral rights auctions. In this paper, buyers receive independent signals conditional upon the true value of the object in our common-value auction models with return policies. To make the analysis as simple as possible, this common value is assumed to take discrete values, while the signals are continuously distributed. We mainly analyze the behavior of bidders in second-price auctions, even though we also consider first-price auctions in a simpler setup. We focus on linear refund policies where the seller could charge a percentage fee in addition to a fied fee. The popularity of linear return policies is because they, just like linear pricing, are easy to implement in practice. Results from the literature on return policies in retailer stores, such as Che [1], predict that consumers are better off with a more generous return policy if there is no competition among buyers. It turns out that, surprisingly, a more generous return policy actually hurts consumers in auctions. This counter intuitive result is from the fact that a more generous return policy not only protects consumers from bad shocks, but also induce buyers to bid more aggressively in the competition, which lowers the consumer surplus. 1 Resale can introduce common value components to a good of private value in nature. (See Haile [3], for eample.) Milgrom and Weber [11] have characterized the equilibria in a very general model of correlated values, but with return policies, it is difficult to characterize the equilibria. Nevertheless, the qualitative results should remain valid. 2

3 We also eamine how return policies affect the seller s revenue. The phenomenon, known as the winner s curse, is well recognized in the auction literature. Winning could mean that the winner has overestimated the object value, since his bid is higher than those from other bidders. As the number of bidders increases, the winner s curse becomes more severe and bidders bid even more cautiously. However, if a return policy is in place, buyers will bid more aggressively, since the winner can get a refund by returning the object. A return policy acts as an insurance against overestimation and overcomes some of the winner s curse. In fact, a return policy can do more than mitigating the winner s curse. When the return policy is generous enough, bidders may bid more than the unconditional estimates of the object value. For eample, if the seller implements the full refund policy, then it is obvious that bidders will bid very high in the auction. Of course, returns could negatively impact the seller s revenue as well as the efficiency of trading, as the seller usually has a lower value for keeping the object. By selecting a proper return policy, the seller can achieve a higher revenue by balancing the trade off between higher bids and efficiency losses. We find that the optimal linear return policy must be in the form of a fied fee, and no percentage fee should be charges. This eplains the widely adopted return policies in reality: the seller refunds the full purchasing price minus some fied shipping and handling fees. In general, the seller s revenue is the difference between the social surplus and the consumer surplus. As we mentioned above, with a more generous return policy, the consumer surplus is always lower. However, the social surplus could go either way. If the social surplus is higher, which happens if and only if the lowest common value of the object is low, the seller s revenue is unambiguously higher. In this case, a full refund policy is optimal. If the social surplus is lower, then there is no clear cut solution. In this case, we characterize the optimal return policy. We then etend our model to first-price auctions in a simpler setup, where the social surplus remains constant regardless of the return policies. We find that a more generous return policy induces a higher revenue and a lower consumer surplus as in second-price auctions. In addition, second-price auctions dominates first-price auctions in terms of the seller s revenue. There is a huge traditional literature on auctions. However, few papers consider return policies. Zhang [15] considers private values which are subject to shocks after transaction, and illustrates how return policies can be part of the optimal mechanism. Hafalir and Yektas [?] consider second-price auctions under a special case of the information structure in Zhang [15], and compare the revenues among spot auctions, forward auctions, and forward auctions with full return policy. Huang, Qiu, and Matsubara [4] recently consider an algorithm for multi-unit auctions with partial refund for bid withdrawals that are caused by eogenous reasons. The paper provides an analysis from the perspectives of artificial intelligence, and bidders strategic behaviors are not the focus. In theory, there eist optimal mechanisms for sellers to maimize revenue. 2 However, 2 The optimal auction with independent values has been established by Myerson [12]. Matthews [9] and Maskin and Riley [8] characterize the optimal mechanism with risk averse buyers and independent values. 3

4 those optimal mechanisms are not commonly observed in reality, partly because too much detail regarding the underlining environment is required for the seller to design an optimal mechanism. The discrepancy between theory and common practice prompts the claim that a set of simplicity and robustness criteria should be imposed on the trading mechanisms. 3 Our auction model with return policies satisfies those simplicity criteria, and the return policies do not depend on much of the detail of the environment. As we shall show in this paper, return policies, while being simple instruments, are effective in revenue improving under certain circumstances. The rest of this paper is organized as follows. In Section 2, we set up the model. In Section 3, we characterize the bidders equilibrium strategies in second-price auctions and perform some preliminary analysis. In Section 4, we characterize the optimal linear return policy in second-price auctions. In Section 5, we characterize the bidders equilibrium strategies in first-price auctions and establish the revenue ranking among different auction formats. In Section 6, we conclude. All proofs are relegated to an appendi. 2 The model Suppose that there are two bidders, bidders 1 and 2, bidding for one object. The object value is the same to both bidders. Let V denote this common value. Assume that V = v H with probability µ H, and V = v L with probability µ L = 1 µ H, where v H > v L. The distribution for V is common knowledge. Before the bidding starts, bidder i receives a private signal i, i = 1, 2. This signal is correlated to V. However, it is independently distributed across the bidders conditional on V. If V = v H, then i follows a distribution with p.d.f. f H ( ) and c.d.f. F H ( ). If V = v L, then i follows a distribution with p.d.f. f L ( ) and c.d.f. F L ( ). Assume that F H ( ) and F L ( ) have a common support, [, ]. Assume that ρ() = f H() is increasing in, i.e., F f L () H dominates F L in likelihood ratio. This will ensure that a higher signal implies a higher probability of V = v H. The lemma below lists a few properties implied by this assumption; its proof is standard and is thus omitted. Lemma 1 Suppose that F H dominates F L in likelihood ratio, i.e., ρ() = f H() f L () in. Then is increasing 1. F H dominates F L in hazard rate, i.e. f H () 1 F H () f L() 1 F H (),. With correlated values, (almost) full surplus etraction can be achieved using the mechanism in Cremer and Mclean [2] and McAfee and Reny [10]. 3 Hurwicz [5] illustrates the need for mechanisms that are independent of the parameters of the model. Wilson [14] points out that a desirable property of a trading rule is that it does not rely on features of the agents. Lopomo [6] [7] restricts to mechanisms with simplicity and robustness. 4

5 2. F H dominates F L in reversed hazard rate, i.e. 3. F H() F L () is increasing,. f H () f L(),. F H () F L () Now we describe the return policy in our model. Let p be the transaction price (i.e., the price the winning bidder pays) in the auction. If the winning bidder returns the object, the seller charges a fee c = a + γp. Here, we restrict our analysis to linear fees since, besides simplicity of calculations, we are not aware of any return policies beyond linear structure in reality. Note that a is the fied fee, such as handling charges or service charges, and γ is the percentage fee, such as restocking fees. In addition, we assume that there is a transaction cost (such as shipping cost) for the winning bidder to return the object, and it is denoted by c B. We will put some restrictions on the return policy to ensure an interesting equilibrium. We assume that γ 0 and a c B. If γ < 0, then it is a weakly dominant strategy for a buyer to bid an infinitely large amount to win and then return the object for refund. In this case, the seller will lose an infinite amount, and it cannot occur in equilibrium. If γ = 0 and a < c B, then a buyer can still win and return the object for refund, ending up with a positive surplus. These restrictions are in place to make sure that a buyer cannot make money by using the win-and-return strategy. 4 The game proceeds in three stages: 1. Nature selects V = v H or V = v L. Conditional on V, each bidder receives an independent signal. 2. A second-price auction (or a first-price auction) with return policy c = a + γp is held. The winner pays accordingly and obtains the object. 3. The winner learns the true V and decides whether or not to return the object to the seller for a refund. Here, we assume that the winner learns costlessly the true value of V after he obtains the object. (The analysis is similar if he learns more but imperfect information about V.) This assumption is motivated by the fact that in online auctions, after a buyer receives the object, he will usually learn more about its value. In auctions for oil, gas and mineral rights, for eample, the winners usually learn more information by doing more tests and analysis after winning. 4 When γ = 0 and a = c B, there are multiple equilibria, including those where every bidder bids an amount greater than or equal to v H and then the winner always returns the object. In these equilibria, the buyers have zero surplus and the seller makes zero profit. As we shall show later, when γ converges to zero from above and a converges to c B from above, the equilibrium converges to the equilibrium where each bidder bids v H and the object is returned only when V = v L. 5

6 In the following analysis, we will focus on the symmetric perfect Bayesian equilibrium with strictly increasing bidding function in the auction. We will start by analyzing the last stage of the game, where the winner makes the return decision. In the following section, we will eamine the second-price auctions. We will then eamine the first-price auctions in a later section. 3 Second-price auctions Suppose that a second-price auction with a return policy is held in Stage 2 of the game. The transaction price is then equal to the second highest bid in the auction. Suppose that in the symmetric perfect Bayesian equilibrium, both bidders adopt B( ), a strictly increasing bidding function, in the auction stage. We restrict our attention to bidding functions taking values in [v L, v H ]. We can do so because a buyer should not bid more than v H ; bidding more than v H sometimes gives him a negative surplus and it is dominated by bidding v H ; if the lowest type bidder bids less than v L, then by increasing his bid to v L he wins with a positive probability and thus receives a positive epected surplus. 3.1 Equilibrium analysis In this subsection, we shall characterize the buyers equilibrium bidding function. We first consider Stage 3, the return stage. Suppose that buyer 1 receives signal, bids B( ), and wins the object. Since it is a second-price auction, he pays B( 2 ), the bid of buyer 2. If he learns that V = v H, he will not return the object since his payment is less than v H. If he learns that V = v L, he returns the object for refund when v L < B( 2 ) [a + γb( 2 )] c B, i.e., when B( 2 ) > v L+a+c B. Otherwise, he keeps the object. As a result, there can only be three different situations in the return stage. Case 1: B() v L+a+c B, and the winner keeps the object all the time. Case 2: B() v L+a+c B, and the winner returns the object whenever V = v L. Case 3: B() < v L+a+c B < B(), and the winner returns the object based on a cutoff rule when V = v L : return when B( 2 ) is high and not return otherwise. Given the winner s strategies in Stage 3 (the return stage), we net eamine Stage 2 (the auction stage). We focus on symmetric strictly increasing bidding functions. In a two-bidder, second-price, common-value auction with no return policies, from Milgrom and Weber [11], a bidder with signal bids E(V, ), the epected object value conditional on the other bidder having the same signal, where E(V, ) = µ Hv H ρ() 2 + µ L v L µ H ρ() 2 + µ L. 6

7 Define Γ() E(V, ). In later analysis, the values of Γ() and Γ() play an important role in characterizing the equilibrium bidding function. The properties of the equilibrium we will characterize depend partially on the value of v L +a+c B. As is shown in the proof of Proposition 1 in the appendi, if v L+a+c B is lower than Γ(), the winning bidder would always return the object when V = v L. If v L+a+c B is higher than Γ(), the winning bidder would never return the object when V = v L. If v L +a+c B is intermediate, the winning bidder would sometimes return the object when V = v L. Furthermore, the intervals for v L+a+c B in the above three cases do not overlap with each other and they cover the entire positive real space. Thus we can conclude that a unique symmetric perfect Bayesian Nash equilibrium eists for any value of v L+a+c B. Define as the solution to v L+a+c B = Γ( ) if v L+a+c B [Γ(), Γ()]. Since Γ() is strictly increasing, is unique and in [, ]. These results are characterized in the following proposition. Proposition 1 In the second-price common value auction with return policy a + γp, the unique symmetric perfect Bayesian equilibrium is characterized in three cases as follows: Γ(), each bidder adopts the follow- Case 1 (No return equilibrium): If v L+a+c B ing strictly increasing bidding function: B() = B 1 () Γ() µ Hv H ρ() 2 + µ L v L µ H ρ() 2 + µ L. Γ(), each bidder adopts the fol- The winning bidder never returns the object. Case 2 (Always return equilibrium): If v L+a+c B lowing strictly increasing bidding function: B() = B 2 () µ Hv H ρ() 2 (a + c B )µ L µ H ρ() 2 + γµ L. The winning bidder always returns the object whenever V = v L. Case 3 (Partial return equilibrium): If Γ() < v L+a+c B the following strictly increasing bidding function: B 1 (), if ; B() = B 2 (), if. < Γ(), each bidder adopts The winning bidder returns the object when V = v L if he pays more than v L+a+c B. 7

8 In Case 1 of the above proposition, since the return policy is never eecuted, the bidding function coincides with the one with no return policy. Obviously, a very strict return policy is equivalent to a no return policy. In Case 2, the bidding function B 2 () is in fact the equilibrium bidding function for a game where the winner is forced to return the object if the realized value of V is v L. It is equal to the price that the bidder will break even if he pays that price (i.e., when the other bidder also has signal ) and receives v H when V = v H, and pays γ percent of that price plus a + c B and gets 0 when V = v L. That is, B 2 () is the solution to (v H B)µ H f H () 2 + ( γb a c B )µ L f L () 2 = 0. In Case 3, whether the winning bidder returns the object or not depends on how much he pays. It is easy to show that B 1 () B 2 () for and B 1 () B 2 () for, with equality at the cutoff. Therefore, the bidding function is the maimum of the two functions in Cases 1 and 2. However, as we shall show in a later section, this pattern is not valid for first-price auctions. Given any return policy a + γp, there eists a unique symmetric equilibrium. When γ 0 and a c B, we have B() v H. We know that when γ = 0 and a = c B, there eist many equilibria. First, bidding v H and the winner always returns the object when V = v L and keeps the object when V = v H is an equilibrium. Second, bidding any amount more than v H and the winner returns the object all the time is also an equilibrium. However, bidding more than v H is weakly dominated by bidding v H. Thus, for simplicity and continuity, we select the equilibrium with B() = v H and the winner always returns the object whenever V = v L and keeps the object whenever V = v H as the equilibrium for γ = 0 and a = c B. This is captured in Case 2 with γ = 0 and a = c B. We call the return policy with γ = 0 and a = c B the full refund with full cost reimbursement policy, as the seller gives out full refund of the transaction price plus reimbursing the winning buyer s cost of shipping back the object. 3.2 The effects of return policies In this section, we shall study how return policies affect buyers epected surplus (i.e., consumer surplus) and the epected gain from trade (i.e., social welfare). We shall defer our analysis on the effects of return policies on the seller s profit (i.e., producer surplus) to the net subsection, where we characterize the optimal return policy for the seller. Denote the consumer surplus as CS(a, γ), and the total surplus as W (a, γ), respectively. Let v 0 be the seller s reservation value for the object, which is also the salvage value for the seller when the winner returns the object. 5 We assume that v 0 < v H ; otherwise, the seller will have no incentive to sell the object. Meanwhile, we do not impose restrictions on the relationship between v 0 and v L. In the following analysis, we treat the three different cases separately. 5 We can also assume that the salvage value is different from the reservation value, but it would not affect the analysis. 8

9 3.2.1 Case 1: No return equilibrium In the no return equilibrium, the object always goes to the buyer with the higher signal, and he never returns the object. The consumer surplus CS(a, γ) can be calculated using the second order statistics of the two signals. When V = v H, its c.d.f is given by 2F H ( ) F H ( ) 2 ; when V = v L, it is given by 2F L ( ) F L ( ) 2. One buyer s epected surplus (which is half of the consumer surplus) is 1 CS(a, γ) 2 = µ H [v H B 1 ( 2 )](1 F H ( 2 ))f H ( 2 )d 2 +µ L [v L B 1 ( 2 )](1 F L ( 2 ))f L ( 2 )d 2 = µ H v H + µ L v L Meanwhile, the social welfare is Γ( 2 )[µ H (1 F H ( 2 ))f H ( 2 ) + µ L (1 F L ( 2 ))f L ( 2 )]d 2. (1) W (a, γ) = µ H v H + µ L v L v 0 In this case, within the range of this no return equilibrium, the consumer surplus and the social welfare do not depend on the return policy, since the object is never returned and the buyers strategies are not affected by the return policy. We have the following lemma. Lemma 2 In the range of no return equilibrium, the consumer surplus and the social welfare do not depend on a and γ Case 2: Always return equilibrium Within the range of the always return equilibrium, the winning bidder (i.e., the buyer with the higher signal) keeps the object if V = v H and always returns it if V = v L. Therefore, a buyer s epected surplus is 1 CS(a, γ) 2 = µ H [v H B 2 ( 2 )][1 F H ( 2 )]f H ( 2 )d 2 +µ L [ a γb 2 ( 2 ) c B ][1 F L ( 2 )]f L ( 2 )d 2 (2) Note that c B is the winner s cost of shipping back the object. 9

10 We first consider how the fied fee a affects the consumer surplus. 1 CS(a, γ) 2 a B 2 ( 2 ) = µ H [1 F H ( 2 )]f H ( 2 )d 2 a [a + γb 2 ( 2 )] µ L [1 F L ( 2 )]f L ( 2 )d 2 a { µ L µ H = [1 F µ H ρ( 2 ) 2 H ( 2 )]f H ( 2 ) + γµ L γµ L µ L [1 + = ][1 F µ H ρ( 2 ) 2 L ( 2 )]f L ( 2 ) } d 2 + γµ L µ L µ H f H ( 2 ) 2 { 1 F L ( 2 ) µ H ρ( 2 ) 2 + γµ L f L ( 2 ) 1 F H( 2 )} d2 0 (3) f H ( 2 ) The inequality at the end follows from the first part of Lemma 1. Therefore, we can conclude that the consumer surplus is increasing in a. Now we eamine how the percentage fee affects the consumer surplus. 1 R(a, γ) 2 γ B 2 ( 2 ) = µ H [1 F H ( 2 )]f H ( 2 )d 2 γ [γb 2 ( 2 )] µ L [1 F L ( 2 )]f L ( 2 )d 2 = = = γ { µ H v H ρ( 2 ) 2 µ L [µ H ρ( 2 ) 2 + γµ L ] µ H[1 F 2 H ( 2 )]f H ( 2 ) + µ Hv H ρ( 2 ) 2 µ H ρ( 2 ) 2 µ [µ H ρ( 2 ) 2 + γµ L ] 2 L [1 F L ( 2 )]f L ( 2 ) } d 2 µ H v H ρ( 2 ) 2 µ H µ { L ρ(2 ) 2 [1 F [µ H ρ( 2 ) 2 + γµ L ] 2 L ( 2 )]f L ( 2 ) [1 F H ( 2 )]f H ( 2 ) } d 2 µ 2 Hµ L v H ρ( 2 ) 2 f H ( 2 ) 2 { 1 F L ( 2 ) 1 F H( 2 ) [µ H ρ( 2 ) 2 + γµ L ] 2 f L ( 2 ) f H ( 2 ) Again, the consumer surplus is increasing in γ. The social welfare in this equilibrium is given by W (a, γ) = µ H (v H v 0 ) µ L c B, } d2 0 (4) since the winning bidder returns the object to the seller when V = v L and no gain is generated in this case. Because this event is not affected by the return policy in this equilibrium, the 10

11 social welfare does not change with γ or a. We have the following lemma. Lemma 3 In the range of always return equilibrium, a more generous return policy (lower a or lower γ) induces lower consumer surplus, while the social welfare is unaffected by the return policy. This result is more or less counter intuitive. Usually, as in the case of return policies in retailer store, when a more generous return policy is provided, it protects the consumers better when bad shock happens and they should be better off. However, in an auction, buyers are competing with each other. A more generous return policy induces buyers to bid more aggressively, and thus lowers consumer surplus. In our model, the first effect is always less than the second one. This is because bidders always have a higher estimate of the probability of returns in their equilibrium strategy calculation than what actually happens. In their equilibrium calculation, because it is a second-price auction, a bidder assumes (correctly) that the other bidder has the same signal when calculating his breakeven bid. But this bid is paid to the seller only when the other buyer has a higher signal and wins. This higher signal reduces the probability of having V = v L (in which case the winner will return). Note that the return policy does not affect the social welfare simply because it does not alter the final allocation of the object Case 3: Partial return equilibrium In a partial return equilibrium, after learning that V = v L, the winning bidder returns the object when the price (i.e., the bid of the other bidder) he pays is high, and keeps the object otherwise. In this case, a buyer s epected surplus is given by: 1 CS(a, γ) 2 { = µ H [v H B 1 ( 2 )][1 F H ( 2 )]f H ( 2 )d 2 + [v H B 2 ( 2 )][1 F H ( 2 )]f H ( 2 )d 2 { +µ L [v L B 1 ( 2 )][1 F L ( 2 )]f L ( 2 )d 2 + [ a γb 2 ( 2 ) c B ][1 F L ( 2 )]f L ( 2 )d 2 Note that is a function of a and γ. We first consider the effect of the fied fee a. 1 CS(a, γ) 2 a } }. (5) 11

12 = µ H {[v H B 1 ( )] [v H B 2 ( )]}[1 F H ( )]f H ( ) d }{{} da =0 B 2 ( 2 ) µ H [1 F H ( 2 )]f H ( 2 )d 2 a [a + γb 2 ( 2 )] µ L [1 F L ( 2 )]f L ( 2 )d 2 a +µ L [[v L B 1 ( )] + a + γb 2 ( ) + c B ][1 F L ( )]f L ( ) d }{{} da =0 µ L µ H f H ( 2 ) 2 = { 1 F L ( 2 ) µ H ρ( 2 ) 2 + γµ L f L ( 2 ) 1 F H( 2 )} d2 0 (6) f H ( 2 ) Therefore, the consumer surplus is increasing in a. Now consider the effect of the percentage fee γ. 1 CS(a, γ) 2 γ = µ H {[v H B 1 ( )] [v H B 2 ( )]}[1 F H ( )]f H ( ) d }{{} dγ =0 B 2 ( 2 ) µ H [1 F H ( 2 )]f H ( 2 )d 2 γ [γb 2 ( 2 )] µ L [1 F L ( 2 )]f L ( 2 )d 2 γ +µ L [[v L B 1 ( )] + a + γb 2 ( ) + c B ][1 F L ( )]f L ( ) d }{{} dγ µ 2 = Hµ L v H ρ( 2 ) 2 f H ( 2 ) 2 { 1 F L ( 2 ) [µ H ρ( 2 ) 2 + γµ L ] 2 f L ( 2 ) Therefore, the consumer surplus is increasing in γ. =0 In this equilibrium, the social welfare is: Therefore, W (a, γ) = µ H (v H v 0 ) +µ L [ ( c B )d[2f L ( 2 ) F L ( 2 ) 2 ] + 1 F H( 2 )} d2 0 (7) f H ( 2 ) (v L v 0 )d[2f L ( 2 ) F L ( 2 ) 2 ] ]. (8) W (a, γ) a = µ L (v L + c B v 0 ) a 2f L( )[1 F L ( )], (9) 12

13 W (a, γ) γ = µ L (v L + c B v 0 ) γ 2f L( )[1 F L ( )]. (10) As a result, the social welfare is increasing in a and γ if v L + c B v 0, and decreasing in a and γ if v L + c B v 0. These properties are summarized in the following lemma. Lemma 4 Within the range of the partial return equilibria, with a more generous return policy (i.e., a lower a or γ) the consumer surplus is lower, and the social welfare is higher if v L + c B v 0 and is lower if v L + c B v 0. The intuition behind this lemma is similar to Lemma 3. The etra effect a more generous return policy has in this lemma on the consumer surplus is that the winner will return the object more often. However, this does not affect the consumer surplus at the margin. In contrast, it does affect the social welfare. Combining Lemmas 2-4, we conclude that the following properties are valid for any of the three types of equilibria in the game. Proposition 2 With a more generous return policy (lower a or γ), buyers surplus is lower, and the social welfare is higher if v L + c B v 0 and is lower if v L + c B v 0. 4 The optimal return policy In this section, we will eamine the effect of the return policy on the seller s revenue and characterize the optimal linear return policy for the seller. For epositional purpose, in what follows, define the revenue maimizing return policies within the three different cases as the optimal return policy for the no return equilibrium, the optimal return policy for the full return equilibrium, and the optimal return policy for the partial return equilibrium, respectively. We define the best return policy among these three policies as the overall optimal return policy. Denote the seller s revenue as R(a, γ). It is obvious that R(a, γ) = W (a, γ) CS(a, γ). According to Proposition 2, if v L + c B v 0, then a more generous return policy reduces the consumer surplus and at the same time increases the social welfare. Therefore, the most generous return policy, i.e., the full return with full cost reimbursement policy will maimize the seller s revenue. Proposition 3 Suppose that v L + c B v 0. Then the seller s optimal linear return policy is the full refund with full cost reimbursement policy. In the rest of the analysis in this section, we will focus on the case where v L +c B > v 0. In this case, both the social welfare and the consumer surplus are lower when the return policy 13

14 becomes more generous. This means that the seller s revenue may not change monotonically with the return policy. Depending on the parameter values in a return policy, from the earlier section, we know that there are three cases of equilibria. In what follows, we will eamine the return policies inducing each case of equilibrium separately and characterize the optimal policy for each case of equilibrium. Then the overall optimal return policy must be the best policy among the three case-specific optimal policies: the optimal policy for the no-return equilibrium, the optimal policy for the full-return equilibrium, and the optimal policy for the partial-return equilibrium. According to Lemma 2 and Lemma 4, any policy inducing the no-return equilibrium is an optimal policy for the no-return equilibrium. The optimal policy to induce full return is a = c B, γ = 0, i.e., the full refund with full cost reimbursement policy. Below, we first make a comparison between these two optimal policies. In the optimal policy inducing full return in equilibrium, both bidders bid v H, and the seller s revenue is µ H v H + µ L (v 0 c B ). Comparing this with the seller s revenue from an optimal policy inducing no return in equilibrium, we have the following proposition. Proposition 4 The optimal policy inducing no return in equilibrium is better than the optimal policy inducing full return in equilibrium if and only if 2 E(V 2, 2 )[µ H (1 F H ( 2 ))f H ( 2 ) + µ L (1 F L ( 2 ))f L ( 2 )]d 2 µ H v H + µ L (v 0 c B ). Given this proposition, the remaining analysis is to characterize the optimal policy that induces partial return in equilibrium. How a partial return policy affects the revenue depends on the parameter values. Since R(a, γ) = W (a, γ) CS(a, γ), by eamining Equations (6), (7), (9), and (10), there is no clear conclusion about how a and γ would affect the revenue. We proceed as follows using an indirect method. In a policy inducing partial return, the seller can choose a and γ, which then uniquely determine the cutoff through Γ( ) = v L +a+c B. Alternatively, if we allow the seller to choose γ and directly, it is equivalent to allowing the seller to choose γ and a indirectly, where a = (1 γ)γ( ) v L c B. Therefore, we can rewrite the seller s revenue as a function of γ and : 1 2 R(γ, ) { = µ H B 1 ( 2 )[1 F H ( 2 )]f H ( 2 )d 2 + { +µ L B 1 ( 2 )[1 F L ( 2 )]f L ( 2 )d 2 + +µ L B 2 ( 2 )[1 F H ( 2 )]f H ( 2 )d 2 [a + γb 2 ( 2 )][1 F L ( 2 )]f L ( 2 )d 2 v 0 [1 F L ( 2 )]f L ( 2 )d 2. (11) } } 14

15 We first eamine how γ affects the revenue. 1 R(γ, ) 2 γ [ B 2 ( 2 ) = µ H + B2 ( 2 ) a γ { a γ + [γb2 ( 2 )] γ ] a [1 F H ( 2 )]f H ( 2 )d 2 γ } + γ B2 ( 2 ) a [1 F L ( 2 )]f L ( 2 )d 2 a γ +µ L [ B 2 ( 2 ) [γb 2 ] ( 2 )] = µ H [1 F H ( 2 )]f H ( 2 ) + µ L [1 F L ( 2 )]f L ( 2 ) d 2 γ γ { B 2 [ ] } ( 2 ) + µ H [1 F H ( 2 )]f H ( 2 ) + µ L 1 + γ B2 ( 2 ) a [1 F L ( 2 )]f L ( 2 ) a a γ d 2 [ µ L [µ H v H ρ() 2 µ L (a + c B )] = µ H [1 F [µ H ρ() 2 + γµ L ] 2 H ( 2 )]f H ( 2 ) µ H ρ() 2 [µ H v H ρ() 2 ] µ L (a + c B )] +µ L [1 F [µ H ρ() 2 + γµ L ] 2 L ( 2 )]f L ( 2 ) d 2 { µ L + µ H [1 F µ H ρ() 2 H ( 2 )]f H ( 2 ) + γµ L µ H ρ() 2 } a +µ L [1 F µ H ρ() 2 L ( 2 )]f L ( 2 ) + γµ L γ d 2 µ L µ H [µ H v H ρ() 2 µ L (a + c B )]f H ( 2 ) 2 = { 1 F L ( 2 ) 1 F H( 2 )} d2 [µ H ρ( 2 ) 2 + γµ L ] 2 f L ( 2 ) f H ( 2 ) µ H µ L f H ( 2 ) 2 { 1 F L ( 2 ) 1 F H( 2 )} Γ( )d [µ H ρ( 2 ) γµ L ] f L ( 2 ) f H ( 2 ) µ L µ H {µ H v H ρ() 2 µ L [(1 γ)γ( ) v L ] [µ H ρ( 2 ) 2 + γµ L ]Γ( )} f H ( 2 ) 2 = [µ H ρ( 2 ) 2 + γµ L ] 2 { 1 F L ( 2 ) 1 F H( 2 )} d2 f L ( 2 ) f H ( 2 ) µ L µ H f H ( 2 ) 2 { = µh ρ() 2 [v [µ H ρ( 2 ) 2 + γµ L ] 2 H Γ( )] µ L [Γ( ) v L ] } { 1 F L ( 2 ) 1 F H( 2 )} d2 f L ( 2 ) f H ( 2 ) µ L µ H f H ( 2 ) 2 [µ L µ H (v H v L )][ρ() 2 ρ( ) 2 ] = { 1 F L ( 2 ) 1 F H( 2 )} d2 [µ H ρ( 2 ) 2 + γµ L ] 2 µ H ρ( ) 2 + µ L f L ( 2 ) f H ( 2 ) 0 (12) This means the seller s revenue is decreasing in γ if is fied. This implies the following proposition. Proposition 5 Given a fied, among all policies inducing partial return, the seller s 15

16 revenue is decreasing in γ, implying that the optimal policy inducing partial return must have γ = 0 (i.e., no percentage fee). The intuition behind this proposition is as follows. Given cutoff, the seller can choose a combination of a fied fee and a percentage fee to fit this cutoff. However, using a percentage fee diminishes the seller revenue since it distorts the bids downward (as higher winning bids are punished more in the case of returning the object). In contrast, a fied fee is a lump sum transfer and does not have this distortion. Therefore, to maimize the seller s revenue, a percentage fee is inferior. Now we eamine the optimal cutoff level of. 1 R(γ, ) 2 = µ H [B 1 ( ) B 2 ( )][1 F H ( )]f H ( ) d }{{} dγ =0 B 2 ( 2 ) a +µ H a [1 F H( 2 )]f H ( 2 )d 2 [ ] +µ L 1 + γ B2 ( 2 ) a a [1 F L( 2 )]f L ( 2 )d 2 +µ L [B 1 ( ) a γb 2 ( )][1 F L ( )]f L ( ) µ L v 0 [1 F L ( )]f L ( ) }{{} =v L +c B = µ L (v L + c B v 0 )[1 F L ( )]f L ( ) }{{} social welfare effect 0 µ H µ L f H ( 2 ) 2 { 1 F H ( 2 ) 1 F L( 2 ) } a [µ H ρ( 2 ) 2 + γµ L ] f H ( 2 ) f L ( 2 ) d 2 }{{} consumer surplus effect 0 (13) From earlier analysis, we conclude that γ = 0 in the optimal policy inducing partial return. At γ = 0, we have 1 R(γ, ) 2 = µ L (v L + c B v 0 )[1 F L ( )]f L ( ) }{{} social welfare effect 0 2(v H v L )µ H µ L ρ( )ρ S ) µ (µ H ρ( ) 2 + µ L ) 2 L f L ( 2 ) { 2 1 F H ( 2 ) 1 F L( 2 )} d2 f H ( 2 ) f L ( 2 ) }{{} consumer surplus effect 0 (14) In the above epression, either the consumer surplus effect or the social welfare effect could dominate. One observation is that if v H v L is very small, then the overall sign is positive 16

17 and the optimal policy inducing partial return is the no return policy (as a limiting case of policies inducing partial return). This is summarized in the following lemma. Lemma 5 When v 0 < v L + c B and v H v L is small enough, the optimal policy inducing partial return is a = Γ() v L c B and γ = 0, such that the cutoff =. This result is true because when v H v L is very small, providing a more generous return policy cannot increase the bids much and the consumer surplus effect is actually quite small. As a result, the social welfare effect dominates. Therefore, the optimal policy should be the least generous, inducing the least return, i.e., no return. Meanwhile, Proposition 4 implies that the optimal policy inducing no return is better than the optimal policy inducing full return; in the case of v L converging to v H, the revenue from a policy inducing no return converges to v H and the revenue from a policy inducing full return converges to µ H v H. We have the following proposition: Proposition 6 When v 0 < v L + c B and v H v L is small enough, the overall optimal return policy is a policy that induces no return. Propositions 3 and 6 cover the situation where the overall optimal return policy is a corner solution. If the overall optimal return policy is an interior solution, then it must be a policy inducing partial return and satisfy the first order condition implied by (14). We have the following proposition. Proposition 7 If the overall optimal return policy is an interior solution, then it must be a policy inducing partial return and satisfy the following condition: 2(v H v L )µ H µ L ρ( )ρ S ) (µ H ρ( ) 2 + µ L ) 2 µ L f L ( 2 ) { 1 F L ( 2 ) 1 F H( 2 )} d2 f L ( 2 ) f H ( 2 ) +µ L (v L + c B v 0 )[1 F L ( )]f L ( ) = 0. (15) The proposition below characterizes some of the properties of an interior overall optimal return policy. Proposition 8 The optimal cutoff is increasing in v H v L, v 0 and decreasing in v L, c B. The intuition behind this proposition is that v H v L changes only the consumer surplus. A higher v H v L means a stronger consumer surplus effect and implies a more generous return policy. v 0, v L, c B only changes the social welfare effect (when keeping v H v L constant). Eample: Suppose that v 0 = 0, v H = v L, v L to be specified later, and µ H = µ L = 0.5, c B = 0. We set γ = 0 as this is always optimal for the seller, and eamine how the seller s 17

18 revenue is affected by the return policy by changing, which then uniquely determines the value of a. For [0, 1], F H () = 2, F L () = 2 2, f H () = 2, f L () = 2 2. Then ρ() = f H() = f L. Note that ρ() is indeed strictly increasing as previously assumed. () 1 Note also that with the above specifications, the case of always return equilibrium vanishes. Furthermore, no return equilibrium is revenue equivalent to the partial return equilibrium with = 1. Thus, the optimal policy inducing partial return will be the overall optimal return policy. We will vary the value of v L and let it take the values of 0.01, 0.15, 0.25, and 0.4, respectively. The results are shown in Figure 4. When v L = 0.01, the seller s revenue is decreasing in ; the overall optimal return policy is = 0, i.e., the full refund with full cost reimbursement policy a = c B = 0. When v L = 0.15, the seller s revenue first increases, then decreases, and then increases in ; the overall optimal return policy is a partial refund policy with = When v L = 0.25, the seller s revenue first increases, then decreases, and then increases in ; the overall optimal return policy is the no refund policy. When v L = 0.4, the seller s revenue is increasing in ; the no refund policy is optimal again. This eample also illustrates the difficulties in determining the condition for an interior overall optimal return policy as the revenue function is not well behaved. However, as predicted, as v L increases, the overall optimal return policy becomes less generous. 5 First-price auctions In this section, we eamine return policies in a first-price auction. Since it is a first-price auction, the transaction price is the winning bid. Again, we focus on an equilibrium where every bidder adopts the same strictly increasing bidding function B F ( ). Similarly to the second-price auctions, we can establish that the range of the bidding function is a subset of [v L, v H ]. We first eamine the winning bidder s return decision. Suppose that a bidder has signal but bids B F ( ). If he wins, he pays B F ( ) for the object. When the realization of the object value is V = v H, he will keep the object, since he pays less than v H. When V = v L, he will return the object if v L < ()B F ( ) a c B, i.e., > (B F ) 1 ( v L+a+c B only be three different situations in the return stage. Case 1: (B F ) 1 ( v L+a+c B this case the winner always keeps the object. Case 2: (B F ) 1 ( v L+a+c B ). As a result, there can ), and in ), and in this case ) <, and the winner always returns the object when V = v L. Case 3: < (B F ) 1 ( v L+a+c B in this case the winner returns the object when V = v L if and only if > (B F ) 1 ( v L+a+c B We first define two functions: L 1 (α ) = e α L 2 (α ; γ) = e α µ H f H (s) 2 +µ L f L (s) 2 µ H f H (s)f H (s)+µ L f L (s)f L (s) ds, µ H f H (s) 2 +γµ L f L (s) 2 µ H f H (s)f H (s)+γµ L f L (s)f L (s) ds. ). 18

19 Lemma 6 L 1 (α ) and L 2 (α ; γ) are both proper c.d.f. s of α with support [, ]. In a two-bidder first-price common-value auction without any return policy, according to Milgrom and Weber [11], all bidders would bid according to the same strictly increasing function: Γ F () = E(V α, α)dl 1 (α ) = µ H v H ρ(α) 2 + µ L v L µ H ρ(α) 2 + µ L dl 1 (α ). Note that Γ F () = Γ(). As in the second-price auctions, there are two cutoffs for γ that are important in the characterization of the bidders equilibrium bidding function: Γ F () and Γ F (). As in the second-price auctions, the value of v L+a+c B plays an important role in the return decision. Define F as the solution to v L+a+c B = Γ F ( F ) for v L+a+c B (Γ F (), Γ F ()). Since Γ F () is strictly increasing, F is unique and belongs to (, ). Also define A = v L + a + c B 1 γ F µ H v H ρ(α) 2 µ L V L µ H ρ(α) 2 + γµ L dl 2 (α F ; γ). Similarly to the second-price auctions, we have the following proposition. Proposition 9 In a first-price common-value auction with return policy c = a + γp, the unique symmetric equilibrium can be characterized as follows in three cases. Case 1: When γ Γ F (), each bidder adopts the following strictly increasing bidding function: B F () = B F 1 () = Γ F µ H v H ρ(α) 2 + µ L v L () = dl µ H ρ(α) 2 1 (α ). + µ L The winning bidder never returns the object. Case 2: When γ Γ F (), each bidder adopts the following strictly increasing bidding function: B F () = B F 2 v H µ H ρ(α) 2 µ L (a + c B ) () = dl µ H ρ(α) 2 2 (α ; γ). + γµ L The winning bidder always returns the object when V = v L. Case 3: When Γ F () < γ < Γ F (), each bidder adopts the following strictly increasing bidding function: B F 1 (), if F, B F () = B F 2 () + AL 2 ( F ; γ), if F. 19

20 The winning bidder returns the object if he pays more than B F ( F ) when V = v L. In Case 1 of the above proposition, since the return policy is never eecuted, the bidding function coincides with the one with no return policy. A very strict return policy is equivalent to not allowing returns. The bidding function B F 1 () corresponds to the one in Milgrom and Weber [11]. In Case 2, the function B 2 () is in fact the equilibrium bidding function for the auction when the winner is forced to return the object if the realized value is v L. In Case 3, however, the bidding function is no long the maimum of the two individual bidding functions in Cases 1 and 2 as in the second-price auctions. For the same reason as in the second-price auctions, when γ = 0 and a = c B, there are multiple equilibria and we choose the equilibrium where B F () = v H and where the winner always returns the object whenever V = v L and keeps the object whenever V = v H. This is captured by Case 2 using γ = 0 and a = c B in the above proposition. Due to the compleity of the equilibrium bidding function in a first-price auction, we will eamine only the special case where v L = 0, a = c B below. In this case, the object is either in perfect condition (high common value), or totally useless (zero common value). The winner always keeps the object when the realized common value is high, and always returns it when the realized common value is zero, regardless of value of γ. Because the return policy does not alter the allocation of the object, the social welfare is the same among different return policies. Therefore, the effect of return policies on revenue is the opposite of consumer surplus effect. Here we keep the value of a unchanged and vary the value of γ, as this is the only way to keep the social welfare unchanged. In what follows, we can make use of a generalized version of the linkage principle to show how the return policy affects the revenue, and thus simplify the analysis greatly. 6 Let M(, ) be the epected payment from a bidder who has signal but reports. We have the following proposition. Proposition 10 Let A and B be two auctions with return policies. In both auctions, the bidder with the highest bid wins. Furthermore, the winner always keeps the object if V = v H, and always returns it if V = v L. Suppose that in each auction, there is a symmetric and strictly increasing equilibrium bidding function with the properties that (i) for all, M A 2 (, ) M B 2 (, ), where subscripts denote the derivatives; (ii) M A (, ) = M B (, ) = 0. Then the seller s epected revenue from A is at least as large as the epected revenue from B. Given any return policy and auction format, the winner with the highest bid wins. The winner always returns the object when V = v L and always keeps the object when V = v H. Thus, both first-price and second-price auctions with any return policy can be regarded as 6 Unfortunately, if v L 0 or a c B, the allocation of the object is affected by the return policy and thus the linkage principle does not apply to this case. 20

21 a proper mechanism in the above proposition. Note that the equilibrium strategy for a first-price auction is characterized by Case 2 in Proposition 9 if v L = 0, a = c B. Similarly to the second-price auctions, ranking in the first-price auctions with different percentage fees can also be obtained. We have the following proposition. Proposition 11 Suppose that v L = 0 and a = c B. A more generous return policy (with a lower γ) generates more revenue in the first-price auctions. The intuition here is the same as in second-price auctions. The following proposition compares the revenues generated from the first-price and the second-price auctions given the same return policy. Proposition 12 Suppose that v L = 0 and a = c B. Given the same return policy, a second-price auction generates at least as much revenue as a first-price auction. This proposition shows that the result in Milgrom and Weber [11] that second-price auctions generates weakly more revenue than first-price auctions can be generalized to auctions with return policies. In Milgrom and Weber [11], the result can be derived directly from the linkage principle. In contrast, the linkage principle cannot be applied directly to auctions with return policies. This is because for the linkage principle to work, the difference between the two epected payment functions must be increasing in a bidder s reported type. This property is satisfied among second-price auctions with different γs, as well as among first-price auctions with different γs. But when we compare a first-price auction with a second-price auction with the same γ, the property is no longer valid. (See the proof of Proposition 12 for details.) Nevertheless, revenue ranking is still possible here. This is because in a second-price auction, the seller receives the bid of a bidder only when the other bidder has a higher signal, and this higher signal makes V = v H more likely to happen than the first bidder originally thought. That is, the object gets returned less often in the seller s revenue calculation than in a bidder s surplus calculation. However, this effect is absent in the first-price auction. Therefore, the total epected revenue for the seller is higher in the second-price auction. 6 Conclusion This paper investigates how return policies affect buyers bidding strategies in auctions and the respective seller s revenue. We mainly focus on second-price auctions and analyze firstprice auctions in a simpler setup. Providing a return policy undoubtedly induces buyers to bid more aggressively, and thus, hurts the consumer surplus. Nevertheless, a more generous return policy improves the social welfare if and only if the low common value is high. Since the seller s revenue is difference between the social welfare and the consumer surplus, designing 21

22 the optimal return policy is a more delicate issue. When the low common value is high, the more generous a return policy is, the more epected revenue the auction generates for the seller. Another surprising result is that the overall optimal refund policy should have no percentage fee. Furthermore, the standard results in Milgrom and Weber [11] that secondprice auction generate more revenue than first-price auctions in common value auctions can be etended to the case of return policies. Auctions with return policies are more complicated to analyze than standard auctions, as the winning bidder may return the object when he obtains more information regarding the object value. Therefore, a higher bid induced by a more generous return policy may not be beneficial to the seller. This paper shows that when the efficiency losses from the returns are not significant, a more generous return policy helps the seller. Since a seller can also use return policies to signal the quality of the object, we should epect to see return policies in many auctions as we have witnessed in online auctions, where buyers have less confidence in the quality of the objects. 7 Appendi Proof for Proposition 1 Case 1: Never return We first characterize the symmetric equilibrium bidding function in the case where the winning bidder never returns the object after winning. Let B 1 ( ) denote the bidding function in this case. Consider buyer 1. Suppose that buyer 1 s signal is and he pretends to have signal and bids B 1 ( ). Given that when the realization of the value is v L, bidder 1 will keep the object if he wins, his epected surplus in the auction is given by: where Π 1 (, ) = P r(v = v H 1 = )E{[V B 1 ( 2 )]I{ 2 < } 1 =, V = v H } +P r(v = v L 1 = )E{[V B 1 ( 2 )]I{ 2 < } 1 =, V = v L } = µ H () [v H B 1 ( 2 )]df H ( 2 ) + µ L () [v L B 1 ( 2 )]df L ( 2 ), (16) µ H () Pr(V = v H 1 = ) = Pr( 1 = V = v H ) Pr(V = v H ) Pr( 1 = V = v H ) Pr(V = v H ) + Pr( 1 = V = v L ) Pr(V = v L ) = f H ()µ H, f H ()µ H + f L ()µ L (17) and where µ L () = Pr(V = v L 1 = ) = 1 µ H (). It is important to note that µ H () is 22