Auction Design with Bidder Preferences and Resale

Transcription

1 Auction Design with Bidder Preferences and Resale Simon Loertscher Leslie M. Marx Preliminary May 28, 2014 Abstract We characterize equilibrium bid strategies in an auction in which preferred bidders receive a bid credit and bidders can engage in post-auction resale. We show that non-preferred bidders expected payoffs are maximized when there is no resale and that total expected surplus increases when resale is either prohibited or frictionless, relative to the case of intermediate resale frictions. Keywords: designated entities, spectrum JEL Classification: C72, D44, L13 We thank Martha Stancill for valuable comments. Department of Economics, Level 4, FBE Building, 111 Barry Street, University of Melbourne, Victoria 3010, Australia. simonl@unimelb.edu.au. The Fuqua School of Business, Duke University, 100 Fuqua Drive, Durham, NC 27708, USA: marx@duke.edu.

2 1 Introduction Auction designers sometimes value more than just the revenue from an auction. For example, an auction designer may place value on the efficiency of the allocation delivered by the auction and may have reasons to give preference to certain bidders over others. The U.S. Federal Communications Commission (FCC), which auctions spectrum licenses, was directed by lawmakers to be attentive to efficiency, revenue, and the success of certain types of bidders. 1 The FCC has addressed these objectives through ascending-bid auction designs that have reserve prices and that offer bid credits for preferred bidders. 2 In addition, the FCC has imposed restrictions on the ability of bidders receiving bid credits to lease or resell spectrum licenses they win at auction. 3 Restrictions on resale obviously limit the ability of later market transactions to correct inefficiencies in the auction outcome; however, they may be essential to the ability of the seller to achieve objectives associated with their bidder preferences. In order to analyze the effects of bid credits and the impact of restrictions on resale, one must understand equilibrium bidding behavior in an auction with bid credits. In this paper, we construct equilibrium bid strategies for an ascending-price auction with multiple preferred and nonpreferred bidders, bid credits for the preferred bidders, reserve prices, and post-auction resale. 1 (A) the development and rapid deployment of new technologies, products, and services for the benefit of the public, including those residing in rural areas, without administrative or judicial delays; (B) promoting economic opportunity and competition and ensuring that new and innovative technologies are readily accessible to the American people by avoiding excessive concentration of licenses and by disseminating licenses among a wide variety of applicants, including small businesses, rural telephone companies, and businesses owned by members of minority groups and women; (C) recovery for the public of a portion of the value of the public spectrum resource made available for commercial use and avoidance of unjust enrichment through the methods employed to award uses of that resource; and (D) efficient and intensive use of the electromagnetic spectrum. (Telecommunications Act of 1996, Pub. LA. No , 110 Stat. 56 (1996), Section 309(j)(3), available at 2 The FCC s program for designated entities, for example in Auction 73, offers bidding credits for small businesses and very small businesses of 15% and 25%, respectively. Bidders qualify for status as a designated entity based on their attributed average annual gross revenues for the preceding three years. (FCC Auction 73 Procedures Public Notice, p.22, available at 3 See Section (b)(3)(iv) of the FCC s rules. 1

3 Bid credits per se do not distort the incentives to bid truthfully in an ascendingprice auction once one accounts for the fact that given a bid credit the willingness to pay of a preferred bidder with value is (1 ) rather than, where0 1. The impact of bid credits is that in equilibrium the good is allocated inefficiently with positive probability. The possibility of ex post inefficiency, which is due to the use of bid credits, opens the scope for a resale market. However, anticipating the possible gains from the resale market now induces severely distorted incentives to bid in the auction, and potentially severe inefficiencies in the resulting equilibrium allocation. Non-preferred bidders have incentives to shade their bids while preferred bidders bid substantially above their bid-credit adjusted values. When resale is possible, non-preferred bidders with values greater than but sufficiently close to the reserve price exit the auction at the reserve price. Their chances of winning against a preferred bidder in the auction are slim because of their low values and because of the bid credit the preferred bidders are granted. Moreover, they have an additional incentive to drop out early, which is to conceal their type in order to receive more favorable offers in the resale market. Therefore, in equilibrium preferred bidders with low-tomoderate values expect positive surplus from participation in the resale market only. In contrast, preferred bidders in equilibrium correctly anticipate additional gains from the resale market and therefore remain active up to bids that exceed their bid-creditadjusted values, risking negative surplus from the purchase in exchange for the possibility of profits from resale. In the absence of resale, a seller that places weight on revenue, efficiency, and the surplus of a set of preferred bidders will optimally conduct an auction that involves reserve prices and a system of bid credits for the preferred bidders. Under certain conditions, a fixed percentage bid credit for preferred bidders is optimal. We take such an auction format as given and analyze the equilibrium response of bidders and the effects of resale on auction outcomes. 2

4 There is a related literature on auctions with resale. Haile (2003) considers an auction with symmetric bidders, but where each bidder only has a signal of its value at the time of the auction and learns its value afterwards, creating opportunities for resale. Zheng (2002) identifies conditions under which the outcomes of Myerson s (1981) optimal auction can be achieved when resale is permitted. Hafalir and Krishna (2008) consider an auction with two asymmetric bidders and resale. They examine how the existence of the resale market, which they model as a take-it-or-leave-it offer by the winner to the loser, affects equilibrium bidding at first and second price auctions. Gupta and Lebrun (1999) model resale differently, assuming that the bidders values are announced at the end of the auction. Garratt and Tröger (2006) assume that the value of one of the bidders is commonly known to be zero, but this bidder may participate in the auction in order to profit from resale opportunities. There is also a literature related to bidder preferences and bidder exclusion. Related to bidder exclusion, Bulow and Klemperer (1996) show that, when the seller s goal is revenue maximization, an auction with +1bidders produces greater expected revenue than any standard mechanism for selling to bidders. Brannman, Klein, and Weiss (1987) show that having more bidders increases expected revenue both in theory and in the data for a range of auction settings. Focusing on discrimination among bidders, Jehiel and Lamy (2013) examine a revenue maximizing seller has an incentive to discriminate among bidders, as in a Myersonian optimal auction, when entry is endogenous. They show that when there are an infinite number of potential bidders who must pay an entry fee to participate, then the optimal auction involves no discrimination. However, if there are also a finite number of bidders who pay no entry fee, then it is optimal for the seller to apply reserve prices to those bidders. Athey, Coey, and Levin (2013) examine set-asides for preferred bidders in U.S. Forest Service auctions, 4 estimating that set-asides reduced 4 In an auction with set-asides, a subset of the items being auctioned are reserved for targeted bidders. For example, in the United States, federal procurement contracts below a threshold dollar amount are typically reserved for small businesses. The governmentwide goals for promoting small businesses are set 3

5 revenue by 5% and reduced auction efficiency by 17%. Using data from a different time period, Brannman and Froeb (2000) estimate that eliminating the Forest Service setaside program would have increased auction revenues by 15%. Both Athey, Coey, and Levin (2013) and Brannman and Froeb (2000) conclude that a program of bid credits for preferred bidders is superior to set-asides. In Section 2, we present the model. In Section 3, we analyze the equilibrium. Section 4 discusses policy implications and conclude. 2 Model and Preliminaries 2.1 Setup Ascending Price Auction with Bid Credits We consider an auction for a single object with potential post-auction resale. Assume 1 preferred bidders and 1 non-preferred bidders. Each preferred bidder {1 } draws its value from, and each non-preferred bidder {1 } draws its value from where all distributions have common support [0 ] and positive densities on [0 ]. We consider an ascending-bid auction similar to the button auction of Milgrom and Weber (1982), where the price of the object rises continuously from zero and each bidder remains active until it exits. Once a bidder exits, it cannot reenter. When only one bidder remains active, the price stops if the price is greater than or equal to the reserve price 0, and otherwise the price continues to rise until the final bidder exits or the reserve price is reached and then stops. If the final price is greater than or equal to the reserve price, then the object is awarded to the final active bidder at the final price (either the price at which the second-to-last bidder exited or the reserve price), with preferred bidders out in 15 USC 644(g)(1), and the implementation as set-asides is described in the Federal Acquisitions Regulations, Section

6 receiving a bid credit so that they pay only 1 times the final price, where [0 1) is the bid credit. We assume that the seller and bidders observe the exit points of all bidders. We assume that bidders can exit immediately after one another, so that if one bidder is observed to exit at a bid of then other bidders observing that can also exit at abidof but with the the order of exits being observable. Resale Market We assume that with probability [0 1] resale is possible and that there is a negotiation over the resale of the object, with the possibility of resale being realized at the time the object is sold. If =0 then there is no resale. When resale is possible, the winning bidder can select one bidder with whom to negotiate. Negotiations take the form of randomized take-it-or-leave-it offers. In the case in which the negotiations involve one preferred and one non-preferred bidder, which will be the only relevant case, we assume that the preferred bidder makes the offer with probability and the non-preferred bidder makes the offer with probability 1. In what follows, we let and (1 ). Thus, if a preferred bidder wins, then with probability the preferred bidder makes an offer to the highest-bidding non-preferred bidder of a price at which it will sell the object, and with probability, the highest-bidding non-preferred bidder makes an offer to the winning preferred bidder of a price at which it would buy the object, and similarly if a non-preferred bidder wins (although, as we will see, in equilibrium there is no resale if a non-preferred bidder wins). Regularity Conditions In order to define the optimal resale offers, we introduce some notation. Let (; ) 1 () () be the virtual value and (; ) + () () be the virtual cost for a bidder drawing its value from distribution with non-zero density. Assume that for all and are regular in the sense of Myerson (1981) so that the virtual values and virtual costs are increasing when evaluated at those distributions. In addition, assume that for all [0 ], [0] and ; ( ) () 1 () is 5

7 increasing in, which holds if and only if 2( ()) 2 0 ()( () ()) (1) Observe that a sufficient condition for (1) to be satisfied is that the () exhibits an increasing hazard rate () (). 5 Preliminaries: Resale Offers Because bidders in our model will be updating beliefs about the values of other bidders from the observed bidding, it will be useful to define the optimal resale offers given updated beliefs. For example, if a bidder with value makes an offer to sell the object to bidder { 1 1 }, conditional on bidder s value being in [ ] where and [ ] [0 ] then the optimal resale offer is such that =. Thus, we can write the optimal resale offer as. ; ( ) () () () ; ( ) () () () Similarly, the optimal purchase offer by a bidder with value to bidder, conditional on bidder s value being in [ ] is ; ( ) (). Thus, it will be useful to define, for [0 ] with () () ˆ (; ) ; ( ) () () () if if = and if = and and ˆ (; ) ; ( ) () () () if if = and 0 if = and Defined this way, ˆ (; ) is the optimal resale offer by a bidder with value owning the object to bidder, where the selling bidder has beliefs that bidder s value 5 The hazard rate ()() is increasing if and only if () 2 () 0 (), which is a stronger requirement than (1). 6

8 is an element of [ ]. When the beliefs are degenerate at a point greater than the seller s value, then the seller sets a resale price of. Whenbeliefsaredegenerateata point less than or equal to the seller s value, then the seller s beliefs are that there are no positive gains from trade, so we assume a resale price of. Similarly, ˆ (; ) is the optimal purchase offer by a bidder with value to bidder that owns the object, where the offering bidder has beliefs that bidder s value is an element of [ ]. When beliefs are degenerate at a point less than the potential buyer s value, then the buyer offers to purchase at a price equal to. When beliefs are degenerate at a point greater than or equal to the potential buyer s value, then the buyer believes that there are not positive gains from trade, so we assume a purchase offer price of 0. Assumption (1) implies that for ˆ (;) Equilibrium 3.1 Equilibrium with zero reserve price In order to define the equilibrium for the case of multiple preferred and non-preferred bidders and positive reserve prices, it will be useful to begin by defining the equilibrium for the case of one preferred and one non-preferred bidder and a zero reserve price. For this case, we use the subscript to denote the single preferred bidder (instead of 1 )and the subscript to denote the single non-preferred bidder (instead of 1 ). Suppose that the preferred bidder bids according to bid function and the nonpreferred bidder bids according to bid function (where looks like p for preferred and looks like n for non-preferred). Specifically, the preferred bidder with value remains active if and only if the current auction price is less than (), and the non-preferred 6 To see this, note that for, ˆ (;) = + () () () and so = = ; ( ) () 1 () () () 2( ()) 2 ( () ()) 0 () which is positive by (1).,where (; ( ) () 1 () ) is s.t. 7

9 bidder with value remains active if and only if the current auction price is less than (). Suppose that is increasing on [0 (1 )] and that is initially zero and then increasing, i.e., there exists [0 ] such that for all [0 ]() =0 and () is increasing on ( ], with a discontinuity at. Further suppose that () (). and 1 We adopt the convention that the non-preferred bidder wins if both bidders exit the auction at the same price. Lemma 1 Assuming one preferred and one non-preferred bidder using bid functions and respectively, satisfying for all [0 ] () and () for profitable resale if the non-preferred bidder wins the auction.,thereisnoscope 1 Proof. Let and be the values of the non-preferred and preferred bidders, respectively. If the non-preferred bidder wins, then ( ) ( ) which implies that ( ) ( ) 1. Because thereisnoscopeforprofitable resale. Q.E.D. Later we confirm that our candidate equilibrium bidding strategies satisfy the conditions required for Lemma 1, but for now we proceed under the assumption that there is no resale if the non-preferred bidder wins the auction. The preferred bidder s expected payoff from continuing up to is its expected payoff from winning and possibly reselling the object. We characterize that expected payoff in the following lemma. To simplify notation, it will be useful to define the operator ˆ [ ] [ ]Pr(). Assuming =0conditional on one preferred bidder and one non-preferred bidder being active at the expected payoff to the preferred bidder with value from remaining active up to given that the non-preferred bidder remains active up to ( ) and 8

10 believes that the preferred bidder remains active up to ( ) is, for 0 (), 7 ( ) ˆ [( ( )(1 )) ( ) ] h + ˆ ˆ ( ; ) h + ˆ ˆ ; (( )) ˆ ˆ ( ; ) ( ) i ; (( )) ( ) i In the definition of,thefirst term corresponds to the payoff if the preferred bidder wins when the non-preferred bidder exits at ( ), in which case the non-preferred bidder reveals its value. The second term corresponds to the preferred bidder s payoff from the resale market when it wins and makes a resale offer of ˆ ( ; ). The third term corresponds to the preferred bidder s payoff from the resale market when it wins and finds it profitable to accept an offer from the non-preferred bidder to purchase the object at a price of ˆ ( ; (( )) ), which is the optimal offer by the non-preferred bidder in this case because the preferred bidder has revealed its value to be at least (( )). We can rewrite ( ) as, for 0 () ( )= + Z () Z () () max{inf{ ( ( )(1 )) ( ) + Z () ˆ ( ; (( )))} ()} 7 For =0 there is an additional term that does not depend on : h ˆ [ ]+ ˆ ˆ ( ;0 ) h + ˆ ˆ ( ;0) max{ ()}} ( ) ( ) ˆ ; (( )) ( ) i ˆ ( ;0 ) i ˆ ( ;0) This term gives the expected payoff to the preferred bidder from the case in which the non-preferred bidder has a value in [0 ] and exits immediately. In this case, the preferred bidder wins at a price of zero, the non-preferred bidder reveals itself to have a value in [0 ] and the non-preferred bidder learns nothing about the value of the preferred bidder. This term can be written as ( )+ R ˆ ( ;0 ) + R min{ inf{ ˆ ( ;0)}} ˆ ( ;0 ) ( ) ˆ ( ;0) ( ) 9

11 Assuming that for () the preferred bidder has off-equilibrium beliefs that the non-preferred bidder has a value of and remains active up to ˆ according to distribution with positive density on [() (1 )] then expected payoff to the preferred bidder with value from remaining active up to is, for (), h ˆ ( ) ˆ ˆ(1 i h i ) ˆ + ˆ ˆ h i + ˆ ˆ ; (ˆ) ˆ We can rewrite ˆ ( ) as, for (), ˆ ( )= Z ˆ(1 )+ ( )+ ˆ ; (ˆ) (ˆ)ˆ We now characterize the corresponding expected payoff for the non-preferred bidder. Assuming =0conditional on one preferred bidder and one non-preferred bidder being active at the expected payoff to the non-preferred bidder with value from remaining active up to given that the preferred bidder remains active up to ( ) 10

12 and believes that the non-preferred bidder remains active up to ( ),is,for0 (), 8 ( ) ˆ [ ( ) ( ) ] h i + ˆ ˆ ( ; () ()) ˆ ( ; () ()) ( ) h + ˆ ˆ ; () ˆ ; () ( )i In the definition of,thefirst term corresponds to the payoff if the non-preferred bidder wins when the preferred bidder exits at ( ). The second term corresponds to the non-preferred bidder s payoff from the resale market when it loses and receives a resale offer of ˆ ( ; () ()) basedonitsexitatabidof. The third term corresponds to the non-preferred bidder s payoff from the resale market when it loses and makes an offer to purchase the object at a price of thenon-preferredbidderinthiscase. ˆ ( ; () ), which is the optimal offer by We can rewrite ( ) as follows: 9 ( )= + Z ˆ () Z () () ( ; ()) Z ( ( )) ( ) + () max{0 ()} ( ) () ˆ ; () ( ) 8 For = =0 there is the following additional term that corresponds to the non-preferred bidder s expected payoff from resale if it exits the auction immediately at =0, in which case the preferred bidders wins at a price of zero and believes that the non-preferred bidder s value is in [0 ] and the non-preferred bidder learns nothing about the preferred bidder s value: h ˆ h + ˆ 0 R ˆ ( ;0) 0 ˆ ( ;0 ) ˆ ˆ ( ;0) This term can be written as: R ˆ ( ;0 ) + ˆ ˆ ( ;0 ) ˆ ( ;0 ) ( ;0 ) i i ˆ ( ;0) ( ) ( ) 9 Note that ˆ ( ; () ()) requires () in which case ( ; () ()) = (). 11

13 Assuming that for () the preferred bidder has off-equilibrium beliefs that the non-preferred bidder has a value of and remains active up to ˆ according to distribution with positive density on [() (1 )] then the expected payoff to the non-preferred bidder with value from remaining active up to is, for (), ˆ ( ) ˆ [ ( ) ( ) ] h + ˆ ˆ ; () ˆ ; () ( )i We can rewrite ˆ ( ) as, for (), ˆ ( ) = Z () () ( ( )) ( ) + Z ˆ () (; ()) ˆ ; () ( ) We now define the bid functions and and then in what follows we establish that these form an equilibrium of the game with no reserve price and only one preferred and one non-preferred bidder. Proposition 1 Assume either that and are the uniform distribution on [0 1] or that 0 is sufficiently small. There exist functions :[0 ] [0 (1 )] and :[0 ] [0 ] such that () is increasing on [0 (1 )] () there exists [0 ] such that for all [0 ]() = 0 and () is increasing on ( ] () for all [0 ] () by: 1. ˆ ( (()) ) and () for all [0 ] () and such functions are defined 1 =0and =() ˆ ( ) =0; =() 2. for all [0 (())) ((ˆ) ) ˆ n 3. for all [0 ) () =max 0 () s.t. 4. and for all [ (()) ] ˆ ((ˆ) ) ˆ =0and ˆ= ((ˆ) ) ˆ ˆ= =0; 12 ((ˆ) ) ˆ ˆ= =0 ˆ= =0 o ;

14 Proof. We first show in Lemma 2 in Appendix A that () and (()) are defined by condition 1. Second, we show in Lemma 3 in Appendix A that, given () and (()) conditions in 2 and 3 define ( ) for [0 (())) and ( ) for [0 ). Third, we show in Lemma 4 in Appendix A that, given (()) condition 4 defines ( ) for [ (()) ]. We complete the proof in the appendix with Lemma 5 showing that for and defined in this way, () () hold. Q.E.D. We can now show that the bid functions definedinproposition1constituteanequilibrium. Proposition 2 Assume either that and are the uniform distribution on [0 1] or that 0 is sufficiently small. If =0and there is one preferred bidder and one nonpreferred bidder, then there exists an equilibrium in which the preferred bidder remains active at bid if and only if ( ) and the non-preferred bidder remains active at bid if and only if ( ). Proof. Endow the preferred bidder with off-equilibrium beliefs conditional on observing bids greater than () that the non-preferred bidder has a value of and remains active up to ˆ according to distribution with positive density on [() (1 )]. For equilibrium, we need to establish that () arg max 0 ˆ ( ) (2) for [0 ) for [0 (())), arg max ˆ [0] ( (ˆ) ) (3) arg max ˆ [0] ( (ˆ) ), (4) 13

15 and for [ (()) ] arg max ˆ [0] ˆ ( (ˆ) ) (5) These conditions are satisfied by Proposition 1 as long as the second-order conditions hold, which we show in Lemmas 6 and 7 in Appendix B. Q.E.D. It is straightforward to solve for the equilibrium bid functions numerically, as shown in Figure Figure 1 shows and for different values of (0 1). As you can see from the figure, for small () approaches the line 1, but for large the preferred 1 bidder bids well above its value. For small () approaches the line but for large the non-preferred bidder exits immediately unless its value is sufficiently large. Figure 1: Bid functions assuming one preferred and one non-preferred bidder drawing values from [0 1] no reserve price, and a bid credit of =02 10 These calculations are numerical. The initial condition is (1) = 1 1 (1 2 ) 001 and the step 1 size is 1000.Because() is steep at values close to 1, we must use an initial condition slightly less than 1 1 (1 2 ), depending on the step size. 14

16 3.2 Equilibrium with a positive reserve price In order to introduce reserve prices, it will also be useful to define the non-preferred bidder s payoff from exit before and at the reserve price. If one preferred bidder and one non-preferred bidder are active at then the difference between the non-preferred bidder s expected payoff when it exits at and its expected payoff from bidding and then exiting (at the bid of ) is: h i ( ) ˆ ˆ ( ;0 ) () ˆ ( ;0 ) ( )Pr () h i ˆ ˆ ( ; ()) () ˆ ( ; ()) where the first term corresponds to the case where the non-preferred bidder exits at, causing the preferred bidder to infer that the non-preferred bidder s value is less than and purchases the object in the resale market based on the offer from the preferred bidder. The expected payoff associated with the non-preferred bidder getting to make the offer itself in the resale market is the same regardless of whether the non-preferred bidder exits at or so that term cancels. The second term corresponds to the non-preferred bidder winning the object at the reserve price. The third terms corresponds to the non-preferred bidder exiting at causing the preferred bidder to infer that the non-preferred bidder s value is greater than and purchases the object in the resale market based on the offer from the preferred bidder. Proposition 3 There exists an equilibrium in which preferred bidder with value remains active at bid if and only if: 1 some other preferred bidder is active and (1 ) or 2 (); and non-preferred bidder with value remains active at bid if and only if: 15

17 1 some other non-preferred bidder is active and or 2 () or 3 and ˆ where ˆ =sup{ ( ) 0}. Proof. To be completed. 4 Discussion Assuming no reserve price and one preferred and one non-preferred bidder, we can calculate auction outcomes numerically. These are shown in Table Table 1: Auction outcomes with one preferred and one non-preferred bidder drawing values from [0 1] reserve price =0 bid credit =02 revenue total 0 (theoretical) 8 25 = = = = (theoretical) = = = (theoretical) eff. resale = = = Numerically calculated values for closetoonemayhaveerror. 16

18 The data from Table 1 is displayed graphically in Figure 2. Figure 2: Auction outcomes as a function of for one preferred and one non-preferred bidder drawing values from [0 1] with no reserve price and bid credit =02. Resultsfrom =0and =1are theoretical, and the remainder are calculated numerically. As one might have anticipated, expected revenue decreases with resale and the expected surplus of the preferred bidder increases with resale (except possibly a small decrease close to =1). However, it is interesting that the expected surplus of the non-preferred bidder decreases and then increases with resale and that the total surplus decreases and then increases with resale. In particular, the non-preferred bidder prefers no resale to the case of frictionless matching in the resale market. Although, the absence of resale precludes the non-preferred bidder from buying the object on the secondary market, the absence of resale reduces the aggressiveness of bidding by the preferred bidder sufficiently that the non-preferred bidder is better off. These results imply that when there are moderate resale frictions, the non-preferred bidder s payoff and total payoff may be improved by a prohibition on resale. In fact, 17

19 total surplus is higher with no resale than with frictionless resale. If resale cannot be prohibited, then gains to total surplus may be possible if steps can be taken to reduce resale frictions. In conclusion, we examine the extent to which possibilities for post-auction resale affects auction outcomes when the seller uses reserve prices and bid credits for preferred bidders. This research highlights the need to consider the interaction between primary auction markets and secondary resale markets when designing the primary market mechanism. 18

20 A Appendix: Characterization of equilibrium Lemma 2 Condition 1 defines () and (()). ProofofLemma2: The condition ˆ ( (()) ) =0, together with (()) 0, =() implies that () = ˆ 1 1 (()) (()) + ( (())) (; (()) ) (()) 1 +(1 ) ˆ 1 (6) ; (()) wheretherangefollowsfrom [0 1]. For (()) 0 the condition ˆ ( ) 0 together with 0 (()) 0 implies that 0= (()) () + ˆ (; ) = (()) ˆ ; (()) ˆ (; (()) ) ˆ (; (()) ) =() = ˆ (; (()) ) + ( (())) which, together with the expression in (6) for () in terms of (()) defines (()). For (()) = the right side is nonnegative and for (()) 0 the right side is zero, so this with (()) 0 defines (()) [0 ]. Then, using this expression in (6), we have () 0. Q.E.D. Lemma 3 Conditions in 2 and 3 define ( ) for [0 (())) and ( ) for [0 ). 19

21 ProofofLemma3. For [0 (())) we can write ((ˆ) ) 0 = 0 (()) 0 () (()) ()(1 )+ ( (()) )+ ˆ ˆ ˆ= =0as (()); which implies that 0= ()(1 )+ ( (()) )+ ˆ (()); Evaluating this at = (( )) [0 (())) gives, for [0 (())) 0= (()) ()(1 )+ ( (())) + ˆ ; (()) (()) Differentiating ((ˆ) ) ˆ (7) =0with respect to we have for [0 ˆ= (())) 0 = 1 0 ()(1 )+ 0 (()) 0 ()+ ˆ ( (()); ) = 1 0 ()(1 )+ 0 (()) 0 () + 0 (()) 0 () ˆ (; ) + ˆ ( (()); ) = (()) = which, when evaluated at = (( )) gives 0 = 1 0 ( (( )))(1 )+ 0 (( )) 0 ( (( ))) + 0 (( )) 0 ( (( ))) ˆ (; (( )) ) + ˆ ( ; ) = = (( )) 20

22 which we can rewrite (using 0 (()) = 1 0 () and 0 ( (())) = 1 0 (()) )as 0 = ˆ 0 () 0 (()) 1 0 (()) (1 )+ 1 For [0 ] we can write ((ˆ)0) ˆ (; (()) ) 0 () 0 (()) + ˆ (; ) = ˆ= =0as = (()) (8) 0= 0 (()) 0 () (()) () ˆ ; (()) (9) () (()) + ˆ (; ((ˆ)) ) ˆ (; (()) ) ˆ (; (()) ). ˆ ˆ= ˆ (; (()) ) + ( (())) Letting 0 = (), 1 = (()) 2 = 0 (()) and 3 = 0 (), wecanrewrite (7) (9) as: 0=(1 ) 1 0 (1 )+ + ˆ (; 1 ) (10) 0 = 1 1 (1 )+ 1 (11) ˆ (; 1 ) + ˆ (; ) 2 3 = =1 and 0= 2 3 ( 1 ) 0 ˆ + (; ) 2 3 =1 ˆ (; 1 ) ˆ (; 1 ) ˆ (; 1 ) ( () ( 1 )) (12) ˆ (; 1 ) + ( 1 ) 21

23 Given and 0 equations (10) define 1 2 and 3 as functions of and 0. To see this,notethat(12)defines 2 3 as a function of 0 and 1. Substituting this into (11),wehaveanequationthatdefines 2 as a function of 0 and 1. Finally, (10) defines 1 as a function of and 0. Thus, given and () (7) (9) define (()) 0 (()) and 0 (). Starting from and () definedinlemma2,thisdefines () (()) 0 (()) and 0 () for all [0 ]. For [0 (())) we can define () { () =}. Q.E.D. Lemma 4 Condition 3 defines () for [ (()) ]. Proof. We can write ˆ ( (ˆ) ) ˆ ˆ= =0as 0 () ()(1 )+ ( )+ ˆ (; ) (()) = 0 which, together with 0 () 0 and (()) 0 implies that () = ( )+ ˆ (; ) (13) Q.E.D. Lemma 5 Bid functions and defined as above satisfy () () if and are the uniform distribution on [0 1] or if 0 is sufficiently small. [Proof assumes 0 sufficiently small. Proof for the uniform case is to be completed.] Proof. Conditions () and (): Toseethat is increasing for [0 (())) use (12) and let 0 to get for = ( ( 1 ) ()) ( 1 )( 0 ) 0. (14) 22

24 Using (11) and letting 0 we have 1 2 (1 ) = (15) which implies that 2 and 3 are positive. To see that is increasing for [ (()) ], note that by (13), Ã 0 () = Ã1 1 1!! ˆ (; ) 1 which is positive because ˆ (;) 0. Condition (): It is clear from (13) that for [ (()) ], (). To show 1 that for [0 (())), () we need to show that 1 ( ) or that 1 (1 )() (()) which is (1 ) 0 1 above. From (10), we know that 0 (1 ) =(1 ) ˆ (; 1 ) (16) which gives the result. Condition (): Wehaveshownthat(). From above, letting 0 3 = which implies that (). Q.E.D. 1 B Appendix: Second-order conditions Lemma 6 For 0 and 0 sufficiently close to zero, appropriate second-order conditions hold. Proof. Part (): Notethat ˆ ( ) lim 0 = 0 ()( ) () 23

25 and so ˆ ( ) =0implies 2ˆ ( ) lim = 0 () 0 2 () 0. In addition, lim 0 ˆ ( ) 0 so any optimum is interior. =0 Part (): Notethat ˆ ( ) = (1 )+ ( )+ ˆ ; () () and so ˆ ( ) =0implies that 2ˆ ( ) lim = (1 )() In addition, ˆ ( ) 0 so any optimum is interior. =0 Part (): Notethat ( ) lim = 0 ()( ) () 0 0 and so ( ) =0implies that 2 ( ) lim = 0 () () 0 In addition, lim 0 0 ( ) 0, sotheoptimumisinterior. =0 Part (): Notethat ( ) lim = 0 ()( (1 )) ()

26 and so ( ) =0implies that 2 ( ) lim = 0 ()(1 ) () 0. In addition, lim 0 0 () = =0 0 () ( ()) 0 for 0. Thus, for 0 the optimum is interior. Q.E.D. Lemma 7 For and uniform on [0 1], appropriate second-order conditions hold. Proof. Part (): Under the uniform distribution, ˆ ( ) µ µ = 0 () () 4 and at such that ˆ ( ) =0 we have 2ˆ ( ) 2 µ = 0 () 4 0 () 1 For () () is defined by ˆ i.e., using (13), () = (1 ) which implies that 0 () = 2ˆ ( ) and so 4 0 () 1 which implies that 0. Thisextendstoall given that () implies 0 () 0 (). Part (): Under the uniform distribution, ˆ ( ) = µ µ + (1 )+ ( )+ () () 2 25

27 At such that ˆ ( ) =0 we have 2ˆ µ ( ) = (1 ) () () For () () is defined by ˆ i.e., using (13), () = (1 ) which implies that 0 () = and so (1 ) () (=)0 iff (=)1 which implies that 2ˆ ( ) (=)0 if and only if (=)1. 2 Part (): Wewanttoshowthat 2 ((ˆ) ) 0. Fromthefirst-order condition, ˆ ˆ= 2 ((ˆ) ) =0 we have ˆ ˆ= ( ) =0. Because =() ((ˆ) ) =0holds for all ˆ ˆ= it is also the case that, differentiating with respect to ((ˆ) ) ˆ ˆ= =0 which µ implies ( ) =() 0 () =0 which using the first-order condition and 0 () 0 implies 2 ( ) + 2 ( ) =0 =() 2 =() Thus, we need only show that 2 ( ) 0. Under the uniform distribution, =() ( ) = 0 ()( (1 )) + 0 ()( () ) µ + 0 ()+ () () 2 Thus, 2 ( ) = 0 ()(1 ) which is positive if 1and zero if =1. Part (): Wewanttoshowthat 2 ((ˆ) ) 0. Fromthefirst-order condition, ˆ ˆ= 2 =0 we have ˆ= ( ) =0. Because =() ((ˆ) ) =0holds for all ˆ= ((ˆ) ) ˆ ˆ 26

28 it is also the case that, differentiating with respect to ((ˆ) ) ˆ ˆ= =0 which µ implies ( ) =() 0 () =0 which using the first-order condition and 0 () 0 implies 2 ( ) + 2 ( ) =0 =() 2 =() Thus, we need only show that 2 ( ) 0. Under the uniform distribution, =() ( ) = 0 () () () () 2 ()+ () Thus, 2 ( ) = 0 () () 0 [May be able to show that second-order conditions hold for general distributions, particularly if assume that 2 ˆ (;) condition for regularity may be sufficient.] Q.E.D. =0. May need a condition on the derivative of.the 27

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