Right to choose in oral auctions

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1 Economics Letters 95 (007) Right to choose in oral auctions Roberto Burguet Institute for Economic Analysis (CSIC), Campus UAB, Bellaterra, Barcelona, Spain Received 9 May 006; received in revised form 30 August 006; accepted 18 September 006 Available online February 007 Abstract Oral, right-to-choose auctions raise higher revenue when buyers are risk averse. When standard sequential and right-to-choose auctions allocate the objects in the same fashion, this means that sellers prefer the latter. 006 Elsevier B.V. All rights reserved. Keywords: Risk aversion; Sequential auctions JEL classification: D44 1. Introduction A right-to-choose (RTC) auction of several (similar) goods is a sequence of auction rounds where the winner of each chooses among the so far unsold goods. 1 This is, for instance, the usual way condominium units are auctioned (see Ashenfelter and Genesove (199)). In this note we show that this auction format allows the seller to cash on the buyers' risk aversion. As perhaps the simplest illustration, assume there are two potential buyers of two apartments, A and B. Each buyer is willing to pay 1 for one of the two apartments (and 0 for the other), but neither bidder knows I acknowledge support from the Spanish Ministry of Education through SEC C0-0 and from CREA. Tel.: ; fax: address: roberto.burguet@uab.es. 1 Menezes and Monteiro (1998) study closely related auctions called pooled auctions, where bidders submit a unique, sealed bid for K objects. The K highest bidders choose in order one of these objets. They characterize equilibrium of this auction when relative preferences over the objects are common to all bidders /$ - see front matter 006 Elsevier B.V. All rights reserved. doi: /j.econlet

2 168 R. Burguet / Economics Letters 95 (007) which apartment the other prefers. Ex-ante, either buyer prefers apartment A with a (independent) probability of 1/. If apartment A is auctioned first, a buyer who prefers apartment B will not bid. If the buyer prefers apartment A, she will be willing to match all rival bids up to 1. Something similar will happen in the second round, when apartment B is auctioned. Thus, with a probability of 1/, (both bidders like the same apartment and) bidders will raise their offers for their common preferred apartment until the price reaches 1, which will be the revenue that the seller will get (the other apartment will sell for 0). With a probability of 1/, buyers will be interested in different apartments and the seller will get zero revenues. Therefore, the expected revenue for the seller is 1/. If the seller chooses to auction the RTC, loosing the first round means that the rival will choose her preferred apartment (and leave the other available at a price of zero). Let the preferences be represented by the von Neumann Morgensten utility function u(θp) where P is the price paid, θ=1 if the buyer gets the apartment she prefers, and θ=0 otherwise. Then her optimal strategy today is to match all prices up to b, where uð1bþ ¼ 1 uð1þþ1 uð0þ: If buyers are risk neutral, b=1/ (which coincides with the seller's revenue) and both auction formats, the RTC and the standard sequential, raise the same expected revenue: 1/. However, if the buyers are risk averse, b N 1/, and then the RTC format raises more revenue in expected terms: buyers are willing to pay a premium in the less informative RTC auction. In this example, a bidder's behavior reveals information about her willingness to pay for her preferred apartment, but nothing about what this apartment may be. This makes the value of a subsequent auction round more uncertain for rival bidders. Thus, risk averse bidders are willing to pay a premium in order to secure their winning in the first round. What I do in this note is to extend this simple intuition to a model that includes a continuum of valuations and an arbitrary number of buyers. Gale and Hausch (1994) were pioneer in analyzing the RTC in auctions. In their paper, standard, sequential (sealed bid) auctions may be inferior to RTC auctions because what they call bottom-fishing. Bidders who prefer the object auction in second place still put very low bids in the first round to benefit in case competition turns out to be fierce for their preferred object and very low for the other, first auctioned object. The effect of bottom-fishing depends crucially on the authors' assumption that a bidder that obtains one object cannot attempt to acquire the other even if she prefers the latter. I do not make this assumption. However, the case I study is not more general, since I consider one-dimensional private information (about preferences). Recently, Goeree et al. (004) have conducted experiments based in this note. Their experimental data confirmed our predictions. For the single unit case, Maskin and Riley (1984) have shown that it is in the seller's interest to screen risk averse buyers by making their utility when winning and losing different. They also show that Eliaz et al. (004) also report experimental results on RTC auctions with risk neutral bidders, and argue that restricting the number of goods sold may increase revenues. Also related, in Burguet (005) I show that RTC auctions are efficient under risk neutrality, and characterize the optimal auction under this assumption.

3 R. Burguet / Economics Letters 95 (007) introducing uncertainty about the payment conditional on winning is no revenue enhancing. The RTC auction reduces the randomness of payments from an ex-ante point of view. RTC auctions pool together the randomness originated in the uncertainty about rivals' relative preferences over the objects. Payments conditioned on winning the preferred object depend on other buyers' willingness to pay for their respective preferred objects, but not on what these may be. In this sense, the results of this note are consistent with the general intuition given by Maskin and Riley.. The model There are N 3 potential buyers for two distinct objects, A and B. Buyer i's preferences, i=1,,, N are characterized by two parameters (t i, θ i ), where t i {A, B} the type, and θ i [0, 1], the valuation. For each i, t i takes the value A with a probability of 1/ and the value B with a probability of 1/. Also, θ i is the realization of a random variable with c.d.f. F and density f which takes positive values on [0, 1]. All random variables are independent and common knowledge. In addition, buyer i knows the realization of (t i, θ i ). The pay-off for buyer i with parameters (t i, θ i ) who obtains object t i at a price P is U½t i ; P; ðt i ; h i ÞŠ ¼ uðh i PÞ; ð1þ where u is a concave, von Neumann Morgensten utility function with u(0) = 0. Also, the pay-off for buyer i who pays P and gets either an object different from t i or no object is U½t; P; ðt i ; h i ÞŠ ¼ U½Ipt i ; P; ðt i ; h i ÞŠ ¼ uðpþ: ðþ For the moment, we assume that a buyer cannot buy both objects. Then, we do not need to define preferences for that case. 3. Standard sequential and right-to-choose auctions Assume object A is offered for sale in the first place. We model the English (oral ascending) auction as a clock auction (prices ascend continuously, bidders decide when to drop out, with no reentry). In principle, a strategy 3 for a buyer in the first auction (round) is a multidimensional mapping. It maps the set of parameters (t i, θ i ), the set of integers n {0, N }, and the set of vectors ( p 1,..., p n ) in R n, with p 1.. p n, into the set [ p n, ). 4 In fact, an (symmetric) equilibrium bidding strategy is quite straightforward in this case. Indeed, we should expect bidders to drop out immediately if their types do not coincide with the object being sold. Also, a buyer's 3 Given the symmetry of buyers, we will only consider anonymous bidding strategies, in which behavior does not depend on who took an action, but only on the nature of this action. 4 To avoid difficulties with what is the next moment in continuous time, we assume that whenever one buyer drops out, the clock stops and there is time for other buyers to drop out at that same price.

4 170 R. Burguet / Economics Letters 95 (007) best dropping-out policy is to stay until the clock reaches her willingness to pay in the round where the preferred object is auctioned. Proposition 1. In the standard sequential auction, an equilibrium (in weakly dominant strategies) is to drop out in each round at a price equal to the willingness to pay for the object being sold. Corollary 1. The object I is won by the buyer of type t=i with the highest valuation, if there is any. Corollary. Both the allocation and the prices are independent of the buyers' risk attitudes. In particular, the outcome of the auctions is the same as the one obtained when buyers are risk neutral. Now assume that the seller conducts an oral, ascending auction for the RTC. The rules of the auction are as in the standard sequential auction, but now the winner of the first round chooses whether to take object A or object B. Given all the symmetry that we have assumed, one would not expect the behavior to depend on the type. Moreover, one would expect that buyers with higher valuation would stay longer than buyers with lower valuation. Finally, in the second auction staying up to willingness to pay (whether 0 or θ) still is a dominant strategy. Thus, we propose a bidding strategy in the first round common for buyers of both types. This is a collection of N1 functions B n for n=0,1,, N, where B n maps the set of buyer's valuations and vectors of positive real numbers ( p 1,, p n )withp 1 p n, into the set [p n, ). We look for a symmetric, monotone equilibrium, that is, one in which B n (θ; p 1,, p n ) is non-decreasing in θ. If buyers behave in this fashion, the fact that one drops out at a given price in the first auction does not convey information with respect to her type. Also, even the first buyer to drop out expects to obtain her preferred object with positive probability in the second round. Indeed, since no information about the type is revealed, when a buyer drops out she still assigns positive probability to the event that all remaining buyers are of a type different to hers. We present equilibrium strategies with these properties in the Appendix. However, all that will be relevant is: Proposition. In equilibrium, bidders bid monotonically in the valuation and independently of their type in the first round. Proof. See Appendix. Corollary 3. The buyer with highest valuation obtains her preferred object. The remaining object is won by the buyer with highest willingness to pay for it among the remaining buyers. As opposed to what was the case in the sequential auction, here the attitude towards risk does play a role. In particular: Corollary 4. The seller expects higher revenue when the buyers are risk averse than when they are risk neutral. Proof. See Appendix. 4. Comparing the two mechanisms We could directly compare the revenues for the seller in both mechanisms. However, we will proceed in the tradition of revenue equivalence for risk neutral agents. Once this is done, the corollaries to Propositions 1 and will directly imply that risk aversion makes the RTC auction a more attractive one for a risk neutral seller.

5 R. Burguet / Economics Letters 95 (007) Thus, for any (anonymous, for economy of notation) mechanism (and one of its equilibria) for assigning both objects, let us specify the 6-tuple {x A, x B, P A, P B } where x I : Θ [0, 1], and P I : Θ R, I=A, B, where the value x I (θ) represents the probability that a buyer of type I obtains object I when her valuation is θ (and everybody behaves as the equilibrium predicts), and P I (θ) is the expected payment in such a case. Then, if buyers are risk neutral, incentive compatibility (a necessary condition for equilibrium behavior) requires h ¼ arg max za½0;1š x IðzÞhP I ðzþ; for all θ, for a buyer of type A. Under differentiability, the first order condition for this problem is the following differential equation dp I ðhþ dh ¼ dx IðhÞ dh h; which integrating (by parts) gives R I ðhþr I ð0þ ¼ Z h 0 x I ðzþdz; where R I (z)=x I (z) zp I (z) represents the equilibrium rents of a buyer of type I and valuation z. That means that the revenue equivalence holds in this situation, i.e., any two mechanisms that assign the objects in the same fashion and for which a buyer with valuation 0 obtains the same rents, for both type A and type B, give the seller the same expected revenue. We have seen that the allocation of the objects is the same in both mechanisms analyzed. Also, R I (0)=0. Then, the seller's expected revenue when buyers are risk neutral is also the same. On the other hand, the seller's revenue does not depend on risk attitudes, in the case of a sequential auction, but is higher in the RTC auction when buyers are risk averse. Thus: Proposition 3. Under risk neutrality, the seller expects the same revenue whether he uses a right-tochoose or a sequential auction. However, if buyers are risk averse, the former raises higher expected revenue than the latter. 5. Concluding remarks I have discussed the advantage of RTC auctions when buyers are risk averse. Sequential auctions reveal too much information to bidders, so that, if they loose in early rounds, they face less uncertain future competition. In contrast, auctioning the RTC can be considered as an instrument to reduce this flow of information in early rounds. By using this auction format, the seller is able to make buyers more willing to bid aggressively in these early rounds to avoid this risk. There are several ways in which the preferences considered in this paper can be extended with no change in the conclusions. Some of them are discussed in the working paper version of this note (Burguet, 1999). They include a positive willingness to pay for the less preferred object and multiple unit demands, under some restrictions. As in Burguet (005), the main assumption, however, is the restriction to one-dimensional private information.

6 17 R. Burguet / Economics Letters 95 (007) Appendix A Proof of Proposition. We propose the following dropping-out strategies, defined inductively. B 0 ðhþ ¼hu 1 1 N1 uðhþ! : Having defined B j (θ; p 1,, p j1 ), define B 1 j1 ( p j ; p 1,..., p j1 ) as the value of θ such that B j1 (θ; p 1,, p j1 )=p j. Then, for nn0 B n ðh; p 1 ; N ; p n Þ¼hu 1 1 N1 uðhþþ Xn! 1 Nk uðhb 1 k1 ð p k; p 1 ; N ; p k1 ÞÞ : We can check that all these functions are increasing in θ. This follows from the fact that 1 N1þ P n Nkb1 1 for nbn1, and the fact that u is decreasing. Thus, from (A) 1 N1 uvðþ¼uv h ðhb 0 ðþ h Þð1B 0 VðÞ h Þ; which shows both that B 0 (θ)n0 and that B 0 (θ)n1. Using the second fact inductively, we can show that indeed all the functions are monotonic. Also, for θ=b 1 n1 ( p n ; p 1,,p n1 ), B n (θ; p 1,, p n )=B n1 (θ; p 1,, p n1 ). That is, when a bidder drops out this does not induce another bidder to drop out with discrete probability. Continuity and monotonicity allow bidders to infer valuations of dropping-out bidders as B 1 j1 ( p j ; p 1,,p j1 ), without changing the ex-ante probability of their types. Then the dropping-out 1 N1uðhÞþ strategies P are standard equilibrium strategies for sequential English auctions: n NkuðhB 1 1 k1 ðp k; p 1 ; N ; p k1 ÞÞ is the expected profit if staying (infinitesimally) longer conditional on all remaining bidders having a valuation equal to the bidder's valuation. Proof of Corollary 4. The price in the second round is independent of risk attitudes. For any realization of (θ 1, θ, θ N ) with θ 1 θ θ N, B 1 k1 ( p k ; p 1,, p k1 ) for all k is also independent of risk attitudes (in both cases, bidding functions are monotonically). Thus, when u is strictly concave, 1 N1 uðh XN 1 Nk N1Þþ uðh N1B 1 k1 ð p k; p 1 ; N ; p k1 ÞÞ 1 N1! bu h N1 þ XN 1 Nk ðh N1B 1 k1 ð p k; p 1 ; N ; p k1 ÞÞ ; so that h N1 u 1 1 N1! uðh XN 1 Nk N1Þþ uðh N1B 1 k1 ð p k; p 1 ; N ; p k1 ÞÞ 1 N1! Nh N1 h N1 þ XN 1 Nk ðh N1B 1 k1 ð p k; p 1 ; N ; p k1 ÞÞ : The right hand side is the price in case u is linear (risk neutral bidders).

7 R. Burguet / Economics Letters 95 (007) References Ashenfelter, O., Genesove, D., 199. Testing for price anomalies in real-estate auctions. American Economic Review 8, Burguet, R., Right to Choose in Oral Auctions. Institute for Eco-nomic Analysis (CSIC). W.P Burguet, R., 005. The condominium problem; auctions for substitutes. Review of Economic Design 9, Eliaz, K., Offerman, T., Schotter, A., 004. Creating competition out of thin air: market thickening and right-to-choose auctions. CEPR Discussion Papers Gale, I., Hausch, D., Bottom-fishing and declining prices in sequential auctions. Games and Economic Behavior 7, Goeree, J.K., Wooders, J., Plott, C.R., 004. Bidders' choice auctions: raising revenues through the right to choose. Journal of the European Economic Association, Maskin, E., Riley, J., Optimal auctions with risk averse buyers. Econometrica 5 6, Menezes, F.M., Monteiro, P.K., Simultaneous pooled auctions. The Journal of Real Estate Finance and Economics 17 3, 19 3.